Kate Hudson Over the Moon for Daughter Rani Rose

Kate Hudson is over the moon for her new daughter, Rani Rose. The 39-year-old mom of three "can't stop smiling" about the latest addition to her family, a source told PEOPLE on Monday.

"She's always wanted a daughter so this is really a dream come true," the source said, adding that Rani's older brothers are excited by the news as well.

"And her big brothers are so excited — they'll be so protective and sweet to her always," the source continued.

As previously reported, Hudson and boyfriend Danny Fujikawa welcomed their daughter on Oct. 2. Hudson is also mom to 7-year-old son Bingham Bellamy from her relationship with ex-fiancé Matt Bellamy, and 14-year-old son Ryder Russell, from her marriage to ex-husband Chris Robinson.

Back in April, she enlisted her sons for help with the gender reveal, sharing an Instagram video of the boys helping Hudson and her family pop four black balloons filled with pink confetti, signaling that Hudson would be having a girl.

Hudson shared the first photo of Rani via social media on Saturday, just four days after she arrived in the world.

In the photo, Rani was fast asleep, swaddled in a striped newborn blanket and wearing an adorable pink hat with a bow. "Our little rosebud," Hudson wrote in the caption.

In the initial birth announcement, Hudson explained the meeting behind the name, Rani Rose Hudson Fujikawa.

"We have decided to name our daughter Rani (pronounced Ronnie) after her grandfather, Ron Fujikawa. Ron was the most special man who we all miss dearly. To name her after him is an honor," Hudson explained on Instagram. Fujikawa's father, Ron, died in October 2012.

"Everyone is doing well and happy as can be," she wrote at the time of the birth. "Our family thanks you for all the love and blessings that have been sent our way and we send our right back."

Hudson might not be finished expanding her family just yet. She said in an interview with SiriusXM in 2016 that she's open to having three more kids.

"I really want more kids. You know, anywhere between four and six," she said. "I just love them [kids]. They're a pain in my a—, and they're the best pain in my a— ever. I just love every bit of it. I love the insanity. I love the chaos. I love laughing with them. I love when I gotta raise my voice at them. I love gathering them all together. I love when I have all the kids."