\(\int (a+b \sqrt {x})^{15} x^3 \, dx\) [61]

Optimal result

Integrand size = 15, antiderivative size = 162 \[ \int \left (a+b \sqrt {x}\right )^{15} x^3 \, dx=-\frac {a^7 \left (a+b \sqrt {x}\right )^{16}}{8 b^8}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{17}}{17 b^8}-\frac {7 a^5 \left (a+b \sqrt {x}\right )^{18}}{3 b^8}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{19}}{19 b^8}-\frac {7 a^3 \left (a+b \sqrt {x}\right )^{20}}{2 b^8}+\frac {2 a^2 \left (a+b \sqrt {x}\right )^{21}}{b^8}-\frac {7 a \left (a+b \sqrt {x}\right )^{22}}{11 b^8}+\frac {2 \left (a+b \sqrt {x}\right )^{23}}{23 b^8} \] Output:

-1/8*a^7*(a+b*x^(1/2))^16/b^8+14/17*a^6*(a+b*x^(1/2))^17/b^8-7/3*a^5*(a+b* 
x^(1/2))^18/b^8+70/19*a^4*(a+b*x^(1/2))^19/b^8-7/2*a^3*(a+b*x^(1/2))^20/b^ 
8+2*a^2*(a+b*x^(1/2))^21/b^8-7/11*a*(a+b*x^(1/2))^22/b^8+2/23*(a+b*x^(1/2) 
)^23/b^8
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15 \[ \int \left (a+b \sqrt {x}\right )^{15} x^3 \, dx=\frac {490314 a^{15} x^4+6537520 a^{14} b x^{9/2}[phone] a^{13} b^2 x^5+162249360 a^{12} b^3 x^{11/2}[phone] a^{11} b^4 x^6+906100272 a^{10} b^5 x^{13/2}[phone] a^9 b^6 x^7+1682757648 a^8 b^7 x^{15/2}[phone] a^7 b^8 x^8+1154833680 a^6 b^9 x^{17/2}[phone] a^5 b^{10} x^9+281801520 a^4 b^{11} x^{19/2}[phone] a^3 b^{12} x^{10}[phone] a^2 b^{13} x^{21/2}+2674440 a b^{14} x^{11}+170544 b^{15} x^{23/2}}{1961256} \] Input:

Integrate[(a + b*Sqrt[x])^15*x^3,x]
 

Output:

(490314*a^15*x^4 + 6537520*a^14*b*x^(9/2) + 41186376*a^13*b^2*x^5 + 162249 
360*a^12*b^3*x^(11/2) + 446185740*a^11*b^4*x^6 + 906100272*a^10*b^5*x^(13/ 
2) + 1402298040*a^9*b^6*x^7 + 1682757648*a^8*b^7*x^(15/2) + 1577585295*a^7 
*b^8*x^8 + 1154833680*a^6*b^9*x^(17/2) + 654405752*a^5*b^10*x^9 + 28180152 
0*a^4*b^11*x^(19/2) + 89237148*a^3*b^12*x^10 + 19612560*a^2*b^13*x^(21/2) 
+ 2674440*a*b^14*x^11 + 170544*b^15*x^(23/2))/1961256
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 49, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \left (a+b \sqrt {x}\right )^{15} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{15} x^{7/2}d\sqrt {x}\)

\(\Big \downarrow \) 49

\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{22}}{b^7}-\frac {7 a \left (a+b \sqrt {x}\right )^{21}}{b^7}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^{20}}{b^7}-\frac {35 a^3 \left (a+b \sqrt {x}\right )^{19}}{b^7}+\frac {35 a^4 \left (a+b \sqrt {x}\right )^{18}}{b^7}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^{17}}{b^7}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{16}}{b^7}-\frac {a^7 \left (a+b \sqrt {x}\right )^{15}}{b^7}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {a^7 \left (a+b \sqrt {x}\right )^{16}}{16 b^8}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{17}}{17 b^8}-\frac {7 a^5 \left (a+b \sqrt {x}\right )^{18}}{6 b^8}+\frac {35 a^4 \left (a+b \sqrt {x}\right )^{19}}{19 b^8}-\frac {7 a^3 \left (a+b \sqrt {x}\right )^{20}}{4 b^8}+\frac {a^2 \left (a+b \sqrt {x}\right )^{21}}{b^8}+\frac {\left (a+b \sqrt {x}\right )^{23}}{23 b^8}-\frac {7 a \left (a+b \sqrt {x}\right )^{22}}{22 b^8}\right )\)

Input:

Int[(a + b*Sqrt[x])^15*x^3,x]
 

Output:

2*(-1/16*(a^7*(a + b*Sqrt[x])^16)/b^8 + (7*a^6*(a + b*Sqrt[x])^17)/(17*b^8 
) - (7*a^5*(a + b*Sqrt[x])^18)/(6*b^8) + (35*a^4*(a + b*Sqrt[x])^19)/(19*b 
^8) - (7*a^3*(a + b*Sqrt[x])^20)/(4*b^8) + (a^2*(a + b*Sqrt[x])^21)/b^8 - 
(7*a*(a + b*Sqrt[x])^22)/(22*b^8) + (a + b*Sqrt[x])^23/(23*b^8))
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 23.85 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {2 b^{15} x^{\frac {23}{2}}}{23}+\frac {15 b^{14} x^{11} a}{11}+10 a^{2} b^{13} x^{\frac {21}{2}}+\frac {91 a^{3} b^{12} x^{10}}{2}+\frac {2730 a^{4} b^{11} x^{\frac {19}{2}}}{19}+\frac {1001 b^{10} x^{9} a^{5}}{3}+\frac {10010 a^{6} b^{9} x^{\frac {17}{2}}}{17}+\frac {6435 b^{8} x^{8} a^{7}}{8}+858 a^{8} b^{7} x^{\frac {15}{2}}+715 a^{9} b^{6} x^{7}+462 a^{10} b^{5} x^{\frac {13}{2}}+\frac {455 a^{11} b^{4} x^{6}}{2}+\frac {910 a^{12} b^{3} x^{\frac {11}{2}}}{11}+21 a^{13} b^{2} x^{5}+\frac {10 a^{14} b \,x^{\frac {9}{2}}}{3}+\frac {a^{15} x^{4}}{4}\)	\(168\)
default	\(\frac {2 b^{15} x^{\frac {23}{2}}}{23}+\frac {15 b^{14} x^{11} a}{11}+10 a^{2} b^{13} x^{\frac {21}{2}}+\frac {91 a^{3} b^{12} x^{10}}{2}+\frac {2730 a^{4} b^{11} x^{\frac {19}{2}}}{19}+\frac {1001 b^{10} x^{9} a^{5}}{3}+\frac {10010 a^{6} b^{9} x^{\frac {17}{2}}}{17}+\frac {6435 b^{8} x^{8} a^{7}}{8}+858 a^{8} b^{7} x^{\frac {15}{2}}+715 a^{9} b^{6} x^{7}+462 a^{10} b^{5} x^{\frac {13}{2}}+\frac {455 a^{11} b^{4} x^{6}}{2}+\frac {910 a^{12} b^{3} x^{\frac {11}{2}}}{11}+21 a^{13} b^{2} x^{5}+\frac {10 a^{14} b \,x^{\frac {9}{2}}}{3}+\frac {a^{15} x^{4}}{4}\)	\(168\)
orering	\(\frac {\left (1666680 b^{36} x^{18}-23678484 a^{2} b^{34} x^{17}[phone] a^{4} b^{32} x^{16}-637591955 a^{6} b^{30} x^{15}[phone] a^{8} b^{28} x^{14}-3669868895 a^{10} b^{26} x^{13}[phone] a^{12} b^{24} x^{12}-6690157175 a^{14} b^{22} x^{11}[phone] a^{16} b^{20} x^{10}-4262132875 a^{18} b^{18} x^{9}[phone] a^{20} b^{16} x^{8}[phone] a^{24} b^{12} x^{6}[phone] a^{26} b^{10} x^{5}[phone] a^{28} x^{4} b^{8}[phone] a^{30} x^{3} b^{6}[phone] a^{32} x^{2} b^{4}[phone] a^{34} x \,b^{2}[phone] a^{36}\right ) \left (a +b \sqrt {x}\right )^{15}}{9806280 b^{8} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (116280 b^{36} x^{18}-1732572 a^{2} b^{34} x^{17}[phone] a^{4} b^{32} x^{16}-51696645 a^{6} b^{30} x^{15}[phone] a^{8} b^{28} x^{14}-333624445 a^{10} b^{26} x^{13}[phone] a^{12} b^{24} x^{12}-692085225 a^{14} b^{22} x^{11}[phone] a^{16} b^{20} x^{10}-511455945 a^{18} b^{18} x^{9}[phone] a^{20} b^{16} x^{8}[phone] a^{24} b^{12} x^{6}[phone] a^{26} b^{10} x^{5}[phone] a^{28} x^{4} b^{8}[phone] a^{30} x^{3} b^{6}[phone] a^{32} x^{2} b^{4}[phone] a^{34} x \,b^{2}[phone] a^{36}\right ) \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} x^{\frac {5}{2}} b}{2}+3 \left (a +b \sqrt {x}\right )^{15} x^{2}\right )}{14709420 b^{8} \left (-b^{2} x +a^{2}\right )^{14} x^{2}}\)	\(454\)
trager	\(\frac {a \left (360 b^{14} x^{10}+12012 x^{9} b^{12} a^{2}+360 b^{14} x^{9}+88088 b^{10} x^{8} a^{4}+12012 b^{12} x^{8} a^{2}+360 b^{14} x^{8}+212355 x^{7} b^{8} a^{6}+88088 b^{10} x^{7} a^{4}+12012 a^{2} b^{12} x^{7}+360 x^{7} b^{14}+188760 b^{6} x^{6} a^{8}+212355 b^{8} x^{6} a^{6}+88088 a^{4} b^{10} x^{6}+12012 a^{2} b^{12} x^{6}+360 b^{14} x^{6}+60060 b^{4} x^{5} a^{10}+188760 b^{6} x^{5} a^{8}+212355 a^{6} b^{8} x^{5}+88088 a^{4} b^{10} x^{5}+12012 a^{2} b^{12} x^{5}+360 b^{14} x^{5}+5544 a^{12} b^{2} x^{4}+60060 b^{4} x^{4} a^{10}+188760 x^{4} b^{6} a^{8}+212355 a^{6} b^{8} x^{4}+88088 a^{4} b^{10} x^{4}+12012 b^{12} x^{4} a^{2}+360 b^{14} x^{4}+66 a^{14} x^{3}+5544 a^{12} b^{2} x^{3}+60060 a^{10} b^{4} x^{3}+188760 a^{8} b^{6} x^{3}+212355 a^{6} b^{8} x^{3}+88088 b^{10} x^{3} a^{4}+12012 b^{12} x^{3} a^{2}+360 b^{14} x^{3}+66 a^{14} x^{2}+5544 a^{12} b^{2} x^{2}+60060 a^{10} b^{4} x^{2}+188760 a^{8} b^{6} x^{2}+212355 b^{8} x^{2} a^{6}+88088 b^{10} x^{2} a^{4}+12012 b^{12} x^{2} a^{2}+360 b^{14} x^{2}+66 a^{14} x +5544 a^{12} b^{2} x +60060 a^{10} b^{4} x +188760 a^{8} b^{6} x +212355 a^{6} b^{8} x +88088 b^{10} x \,a^{4}+12012 b^{12} x \,a^{2}+360 b^{14} x +66 a^{14}+5544 a^{12} b^{2}+60060 a^{10} b^{4}+188760 a^{8} b^{6}+212355 a^{6} b^{8}+88088 a^{4} b^{10}+12012 a^{2} b^{12}+360 b^{14}\right ) \left (-1+x \right )}{264}+\frac {2 b \,x^{\frac {9}{2}} \left (10659 x^{7} b^{14}+1225785 a^{2} b^{12} x^{6}[phone] a^{4} b^{10} x^{5}[phone] a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}[phone] a^{10} b^{4} x^{2}[phone] a^{12} b^{2} x +408595 a^{14}\right )}{245157}\)	\(668\)

Input:

int((a+b*x^(1/2))^15*x^3,x,method=_RETURNVERBOSE)
 

Output:

2/23*b^15*x^(23/2)+15/11*b^14*x^11*a+10*a^2*b^13*x^(21/2)+91/2*a^3*b^12*x^ 
10+2730/19*a^4*b^11*x^(19/2)+1001/3*b^10*x^9*a^5+10010/17*a^6*b^9*x^(17/2) 
+6435/8*b^8*x^8*a^7+858*a^8*b^7*x^(15/2)+715*a^9*b^6*x^7+462*a^10*b^5*x^(1 
3/2)+455/2*a^11*b^4*x^6+910/11*a^12*b^3*x^(11/2)+21*a^13*b^2*x^5+10/3*a^14 
*b*x^(9/2)+1/4*a^15*x^4
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.07 \[ \int \left (a+b \sqrt {x}\right )^{15} x^3 \, dx=\frac {15}{11} \, a b^{14} x^{11} + \frac {91}{2} \, a^{3} b^{12} x^{10} + \frac {1001}{3} \, a^{5} b^{10} x^{9} + \frac {6435}{8} \, a^{7} b^{8} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac {455}{2} \, a^{11} b^{4} x^{6} + 21 \, a^{13} b^{2} x^{5} + \frac {1}{4} \, a^{15} x^{4} + \frac {2}{245157} \, {\left (10659 \, b^{15} x^{11} + 1225785 \, a^{2} b^{13} x^{10} + 17612595 \, a^{4} b^{11} x^{9} + 72177105 \, a^{6} b^{9} x^{8} + 105172353 \, a^{8} b^{7} x^{7} + 56631267 \, a^{10} b^{5} x^{6} + 10140585 \, a^{12} b^{3} x^{5} + 408595 \, a^{14} b x^{4}\right )} \sqrt {x} \] Input:

integrate((a+b*x^(1/2))^15*x^3,x, algorithm="fricas")
 

Output:

15/11*a*b^14*x^11 + 91/2*a^3*b^12*x^10 + 1001/3*a^5*b^10*x^9 + 6435/8*a^7* 
b^8*x^8 + 715*a^9*b^6*x^7 + 455/2*a^11*b^4*x^6 + 21*a^13*b^2*x^5 + 1/4*a^1 
5*x^4 + 2/245157*(10659*b^15*x^11 + 1225785*a^2*b^13*x^10 + 17612595*a^4*b 
^11*x^9 + 72177105*a^6*b^9*x^8 + 105172353*a^8*b^7*x^7 + 56631267*a^10*b^5 
*x^6 + 10140585*a^12*b^3*x^5 + 408595*a^14*b*x^4)*sqrt(x)
 

Sympy [A] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.29 \[ \int \left (a+b \sqrt {x}\right )^{15} x^3 \, dx=\frac {a^{15} x^{4}}{4} + \frac {10 a^{14} b x^{\frac {9}{2}}}{3} + 21 a^{13} b^{2} x^{5} + \frac {910 a^{12} b^{3} x^{\frac {11}{2}}}{11} + \frac {455 a^{11} b^{4} x^{6}}{2} + 462 a^{10} b^{5} x^{\frac {13}{2}} + 715 a^{9} b^{6} x^{7} + 858 a^{8} b^{7} x^{\frac {15}{2}} + \frac {6435 a^{7} b^{8} x^{8}}{8} + \frac {10010 a^{6} b^{9} x^{\frac {17}{2}}}{17} + \frac {1001 a^{5} b^{10} x^{9}}{3} + \frac {2730 a^{4} b^{11} x^{\frac {19}{2}}}{19} + \frac {91 a^{3} b^{12} x^{10}}{2} + 10 a^{2} b^{13} x^{\frac {21}{2}} + \frac {15 a b^{14} x^{11}}{11} + \frac {2 b^{15} x^{\frac {23}{2}}}{23} \] Input:

integrate((a+b*x**(1/2))**15*x**3,x)
 

Output:

a**15*x**4/4 + 10*a**14*b*x**(9/2)/3 + 21*a**13*b**2*x**5 + 910*a**12*b**3 
*x**(11/2)/11 + 455*a**11*b**4*x**6/2 + 462*a**10*b**5*x**(13/2) + 715*a** 
9*b**6*x**7 + 858*a**8*b**7*x**(15/2) + 6435*a**7*b**8*x**8/8 + 10010*a**6 
*b**9*x**(17/2)/17 + 1001*a**5*b**10*x**9/3 + 2730*a**4*b**11*x**(19/2)/19 
 + 91*a**3*b**12*x**10/2 + 10*a**2*b**13*x**(21/2) + 15*a*b**14*x**11/11 + 
 2*b**15*x**(23/2)/23
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int \left (a+b \sqrt {x}\right )^{15} x^3 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{23}}{23 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{22} a}{11 \, b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{21} a^{2}}{b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{20} a^{3}}{2 \, b^{8}} + \frac {70 \, {\left (b \sqrt {x} + a\right )}^{19} a^{4}}{19 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{18} a^{5}}{3 \, b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )}^{17} a^{6}}{17 \, b^{8}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{7}}{8 \, b^{8}} \] Input:

integrate((a+b*x^(1/2))^15*x^3,x, algorithm="maxima")
 

Output:

2/23*(b*sqrt(x) + a)^23/b^8 - 7/11*(b*sqrt(x) + a)^22*a/b^8 + 2*(b*sqrt(x) 
 + a)^21*a^2/b^8 - 7/2*(b*sqrt(x) + a)^20*a^3/b^8 + 70/19*(b*sqrt(x) + a)^ 
19*a^4/b^8 - 7/3*(b*sqrt(x) + a)^18*a^5/b^8 + 14/17*(b*sqrt(x) + a)^17*a^6 
/b^8 - 1/8*(b*sqrt(x) + a)^16*a^7/b^8
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.03 \[ \int \left (a+b \sqrt {x}\right )^{15} x^3 \, dx=\frac {2}{23} \, b^{15} x^{\frac {23}{2}} + \frac {15}{11} \, a b^{14} x^{11} + 10 \, a^{2} b^{13} x^{\frac {21}{2}} + \frac {91}{2} \, a^{3} b^{12} x^{10} + \frac {2730}{19} \, a^{4} b^{11} x^{\frac {19}{2}} + \frac {1001}{3} \, a^{5} b^{10} x^{9} + \frac {10010}{17} \, a^{6} b^{9} x^{\frac {17}{2}} + \frac {6435}{8} \, a^{7} b^{8} x^{8} + 858 \, a^{8} b^{7} x^{\frac {15}{2}} + 715 \, a^{9} b^{6} x^{7} + 462 \, a^{10} b^{5} x^{\frac {13}{2}} + \frac {455}{2} \, a^{11} b^{4} x^{6} + \frac {910}{11} \, a^{12} b^{3} x^{\frac {11}{2}} + 21 \, a^{13} b^{2} x^{5} + \frac {10}{3} \, a^{14} b x^{\frac {9}{2}} + \frac {1}{4} \, a^{15} x^{4} \] Input:

integrate((a+b*x^(1/2))^15*x^3,x, algorithm="giac")
 

Output:

2/23*b^15*x^(23/2) + 15/11*a*b^14*x^11 + 10*a^2*b^13*x^(21/2) + 91/2*a^3*b 
^12*x^10 + 2730/19*a^4*b^11*x^(19/2) + 1001/3*a^5*b^10*x^9 + 10010/17*a^6* 
b^9*x^(17/2) + 6435/8*a^7*b^8*x^8 + 858*a^8*b^7*x^(15/2) + 715*a^9*b^6*x^7 
 + 462*a^10*b^5*x^(13/2) + 455/2*a^11*b^4*x^6 + 910/11*a^12*b^3*x^(11/2) + 
 21*a^13*b^2*x^5 + 10/3*a^14*b*x^(9/2) + 1/4*a^15*x^4
 

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.03 \[ \int \left (a+b \sqrt {x}\right )^{15} x^3 \, dx=\frac {a^{15}\,x^4}{4}+\frac {2\,b^{15}\,x^{23/2}}{23}+\frac {15\,a\,b^{14}\,x^{11}}{11}+\frac {10\,a^{14}\,b\,x^{9/2}}{3}+21\,a^{13}\,b^2\,x^5+\frac {455\,a^{11}\,b^4\,x^6}{2}+715\,a^9\,b^6\,x^7+\frac {6435\,a^7\,b^8\,x^8}{8}+\frac {1001\,a^5\,b^{10}\,x^9}{3}+\frac {91\,a^3\,b^{12}\,x^{10}}{2}+\frac {910\,a^{12}\,b^3\,x^{11/2}}{11}+462\,a^{10}\,b^5\,x^{13/2}+858\,a^8\,b^7\,x^{15/2}+\frac {10010\,a^6\,b^9\,x^{17/2}}{17}+\frac {2730\,a^4\,b^{11}\,x^{19/2}}{19}+10\,a^2\,b^{13}\,x^{21/2} \] Input:

int(x^3*(a + b*x^(1/2))^15,x)
 

Output:

(a^15*x^4)/4 + (2*b^15*x^(23/2))/23 + (15*a*b^14*x^11)/11 + (10*a^14*b*x^( 
9/2))/3 + 21*a^13*b^2*x^5 + (455*a^11*b^4*x^6)/2 + 715*a^9*b^6*x^7 + (6435 
*a^7*b^8*x^8)/8 + (1001*a^5*b^10*x^9)/3 + (91*a^3*b^12*x^10)/2 + (910*a^12 
*b^3*x^(11/2))/11 + 462*a^10*b^5*x^(13/2) + 858*a^8*b^7*x^(15/2) + (10010* 
a^6*b^9*x^(17/2))/17 + (2730*a^4*b^11*x^(19/2))/19 + 10*a^2*b^13*x^(21/2)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.10 \[ \int \left (a+b \sqrt {x}\right )^{15} x^3 \, dx=\frac {x^{4} \left (6537520 \sqrt {x}\, a^{14} b [phone] \sqrt {x}\, a^{12} b^{3} x [phone] \sqrt {x}\, a^{10} b^{5} x^{2}[phone] \sqrt {x}\, a^{8} b^{7} x^{3}[phone] \sqrt {x}\, a^{6} b^{9} x^{4}[phone] \sqrt {x}\, a^{4} b^{11} x^{5}[phone] \sqrt {x}\, a^{2} b^{13} x^{6}+170544 \sqrt {x}\, b^{15} x^{7}+490314 a^{15}[phone] a^{13} b^{2} x [phone] a^{11} b^{4} x^{2}[phone] a^{9} b^{6} x^{3}[phone] a^{7} b^{8} x^{4}[phone] a^{5} b^{10} x^{5}[phone] a^{3} b^{12} x^{6}+2674440 a \,b^{14} x^{7}\right )}{1961256} \] Input:

int((a+b*x^(1/2))^15*x^3,x)
 

Output:

(x**4*(6537520*sqrt(x)*a**14*b + 162249360*sqrt(x)*a**12*b**3*x + 90610027 
2*sqrt(x)*a**10*b**5*x**2 + 1682757648*sqrt(x)*a**8*b**7*x**3 + 1154833680 
*sqrt(x)*a**6*b**9*x**4 + 281801520*sqrt(x)*a**4*b**11*x**5 + 19612560*sqr 
t(x)*a**2*b**13*x**6 + 170544*sqrt(x)*b**15*x**7 + 490314*a**15 + 41186376 
*a**13*b**2*x + 446185740*a**11*b**4*x**2 + 1402298040*a**9*b**6*x**3 + 15 
77585295*a**7*b**8*x**4 + 654405752*a**5*b**10*x**5 + 89237148*a**3*b**12* 
x**6 + 2674440*a*b**14*x**7))/1961256