\(\int (a+b \sqrt {x})^{15} x^5 \, dx\) [59]

Optimal result

Integrand size = 15, antiderivative size = 242 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=-\frac {a^{11} \left (a+b \sqrt {x}\right )^{16}}{8 b^{12}}+\frac {22 a^{10} \left (a+b \sqrt {x}\right )^{17}}{17 b^{12}}-\frac {55 a^9 \left (a+b \sqrt {x}\right )^{18}}{9 b^{12}}+\frac {330 a^8 \left (a+b \sqrt {x}\right )^{19}}{19 b^{12}}-\frac {33 a^7 \left (a+b \sqrt {x}\right )^{20}}{b^{12}}+\frac {44 a^6 \left (a+b \sqrt {x}\right )^{21}}{b^{12}}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{22}}{b^{12}}+\frac {660 a^4 \left (a+b \sqrt {x}\right )^{23}}{23 b^{12}}-\frac {55 a^3 \left (a+b \sqrt {x}\right )^{24}}{4 b^{12}}+\frac {22 a^2 \left (a+b \sqrt {x}\right )^{25}}{5 b^{12}}-\frac {11 a \left (a+b \sqrt {x}\right )^{26}}{13 b^{12}}+\frac {2 \left (a+b \sqrt {x}\right )^{27}}{27 b^{12}} \] Output:

-1/8*a^11*(a+b*x^(1/2))^16/b^12+22/17*a^10*(a+b*x^(1/2))^17/b^12-55/9*a^9* 
(a+b*x^(1/2))^18/b^12+330/19*a^8*(a+b*x^(1/2))^19/b^12-33*a^7*(a+b*x^(1/2) 
)^20/b^12+44*a^6*(a+b*x^(1/2))^21/b^12-42*a^5*(a+b*x^(1/2))^22/b^12+660/23 
*a^4*(a+b*x^(1/2))^23/b^12-55/4*a^3*(a+b*x^(1/2))^24/b^12+22/5*a^2*(a+b*x^ 
(1/2))^25/b^12-11/13*a*(a+b*x^(1/2))^26/b^12+2/27*(a+b*x^(1/2))^27/b^12
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.77 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {17383860 a^{15} x^6+240699600 a^{14} b x^{13/2}[phone] a^{13} b^2 x^7+6327725040 a^{12} b^3 x^{15/2}[phone] a^{11} b^4 x^8+36849692880 a^{10} b^5 x^{17/2}[phone] a^9 b^6 x^9+70651666800 a^8 b^7 x^{19/2}[phone] a^7 b^8 x^{10}[phone] a^6 b^9 x^{21/2}[phone] a^5 b^{10} x^{11}[phone] a^4 b^{11} x^{23/2}[phone] a^3 b^{12} x^{12}[phone] a^2 b^{13} x^{25/2}[phone] a b^{14} x^{13}+7726160 b^{15} x^{27/2}}{104303160} \] Input:

Integrate[(a + b*Sqrt[x])^15*x^5,x]
 

Output:

(17383860*a^15*x^6 + 240699600*a^14*b*x^(13/2) + 1564547400*a^13*b^2*x^7 + 
 6327725040*a^12*b^3*x^(15/2) + 17796726675*a^11*b^4*x^8 + 36849692880*a^1 
0*b^5*x^(17/2) + 58004146200*a^9*b^6*x^9 + 70651666800*a^8*b^7*x^(19/2) + 
67119083460*a^7*b^8*x^10 + 49717839600*a^6*b^9*x^(21/2) + 28474762680*a^5* 
b^10*x^11 + 12380331600*a^4*b^11*x^(23/2) + 3954828150*a^3*b^12*x^12 + 876 
146544*a^2*b^13*x^(25/2) + 120349800*a*b^14*x^13 + 7726160*b^15*x^(27/2))/ 
104303160
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02, 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.

\(\displaystyle \int x^5 \left (a+b \sqrt {x}\right )^{15} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{15} x^{11/2}d\sqrt {x}\)

\(\Big \downarrow \) 49

\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{26}}{b^{11}}-\frac {11 a \left (a+b \sqrt {x}\right )^{25}}{b^{11}}+\frac {55 a^2 \left (a+b \sqrt {x}\right )^{24}}{b^{11}}-\frac {165 a^3 \left (a+b \sqrt {x}\right )^{23}}{b^{11}}+\frac {330 a^4 \left (a+b \sqrt {x}\right )^{22}}{b^{11}}-\frac {462 a^5 \left (a+b \sqrt {x}\right )^{21}}{b^{11}}+\frac {462 a^6 \left (a+b \sqrt {x}\right )^{20}}{b^{11}}-\frac {330 a^7 \left (a+b \sqrt {x}\right )^{19}}{b^{11}}+\frac {165 a^8 \left (a+b \sqrt {x}\right )^{18}}{b^{11}}-\frac {55 a^9 \left (a+b \sqrt {x}\right )^{17}}{b^{11}}+\frac {11 a^{10} \left (a+b \sqrt {x}\right )^{16}}{b^{11}}-\frac {a^{11} \left (a+b \sqrt {x}\right )^{15}}{b^{11}}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {a^{11} \left (a+b \sqrt {x}\right )^{16}}{16 b^{12}}+\frac {11 a^{10} \left (a+b \sqrt {x}\right )^{17}}{17 b^{12}}-\frac {55 a^9 \left (a+b \sqrt {x}\right )^{18}}{18 b^{12}}+\frac {165 a^8 \left (a+b \sqrt {x}\right )^{19}}{19 b^{12}}-\frac {33 a^7 \left (a+b \sqrt {x}\right )^{20}}{2 b^{12}}+\frac {22 a^6 \left (a+b \sqrt {x}\right )^{21}}{b^{12}}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^{22}}{b^{12}}+\frac {330 a^4 \left (a+b \sqrt {x}\right )^{23}}{23 b^{12}}-\frac {55 a^3 \left (a+b \sqrt {x}\right )^{24}}{8 b^{12}}+\frac {11 a^2 \left (a+b \sqrt {x}\right )^{25}}{5 b^{12}}+\frac {\left (a+b \sqrt {x}\right )^{27}}{27 b^{12}}-\frac {11 a \left (a+b \sqrt {x}\right )^{26}}{26 b^{12}}\right )\)

Input:

Int[(a + b*Sqrt[x])^15*x^5,x]
 

Output:

2*(-1/16*(a^11*(a + b*Sqrt[x])^16)/b^12 + (11*a^10*(a + b*Sqrt[x])^17)/(17 
*b^12) - (55*a^9*(a + b*Sqrt[x])^18)/(18*b^12) + (165*a^8*(a + b*Sqrt[x])^ 
19)/(19*b^12) - (33*a^7*(a + b*Sqrt[x])^20)/(2*b^12) + (22*a^6*(a + b*Sqrt 
[x])^21)/b^12 - (21*a^5*(a + b*Sqrt[x])^22)/b^12 + (330*a^4*(a + b*Sqrt[x] 
)^23)/(23*b^12) - (55*a^3*(a + b*Sqrt[x])^24)/(8*b^12) + (11*a^2*(a + b*Sq 
rt[x])^25)/(5*b^12) - (11*a*(a + b*Sqrt[x])^26)/(26*b^12) + (a + b*Sqrt[x] 
)^27/(27*b^12))
 

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 = 23.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\frac {2 b^{15} x^{\frac {27}{2}}}{27}+\frac {15 a \,b^{14} x^{13}}{13}+\frac {42 a^{2} b^{13} x^{\frac {25}{2}}}{5}+\frac {455 a^{3} b^{12} x^{12}}{12}+\frac {2730 a^{4} b^{11} x^{\frac {23}{2}}}{23}+273 a^{5} b^{10} x^{11}+\frac {1430 a^{6} b^{9} x^{\frac {21}{2}}}{3}+\frac {1287 a^{7} b^{8} x^{10}}{2}+\frac {12870 a^{8} b^{7} x^{\frac {19}{2}}}{19}+\frac {5005 a^{9} b^{6} x^{9}}{9}+\frac {6006 a^{10} b^{5} x^{\frac {17}{2}}}{17}+\frac {1365 a^{11} b^{4} x^{8}}{8}+\frac {182 a^{12} b^{3} x^{\frac {15}{2}}}{3}+15 a^{13} b^{2} x^{7}+\frac {30 a^{14} b \,x^{\frac {13}{2}}}{13}+\frac {a^{15} x^{6}}{6}\)	\(168\)
default	\(\frac {2 b^{15} x^{\frac {27}{2}}}{27}+\frac {15 a \,b^{14} x^{13}}{13}+\frac {42 a^{2} b^{13} x^{\frac {25}{2}}}{5}+\frac {455 a^{3} b^{12} x^{12}}{12}+\frac {2730 a^{4} b^{11} x^{\frac {23}{2}}}{23}+273 a^{5} b^{10} x^{11}+\frac {1430 a^{6} b^{9} x^{\frac {21}{2}}}{3}+\frac {1287 a^{7} b^{8} x^{10}}{2}+\frac {12870 a^{8} b^{7} x^{\frac {19}{2}}}{19}+\frac {5005 a^{9} b^{6} x^{9}}{9}+\frac {6006 a^{10} b^{5} x^{\frac {17}{2}}}{17}+\frac {1365 a^{11} b^{4} x^{8}}{8}+\frac {182 a^{12} b^{3} x^{\frac {15}{2}}}{3}+15 a^{13} b^{2} x^{7}+\frac {30 a^{14} b \,x^{\frac {13}{2}}}{13}+\frac {a^{15} x^{6}}{6}\)	\(168\)
orering	\(\frac {\left (45465480 b^{40} x^{20}-647229338 a^{2} b^{38} x^{19}[phone] a^{4} b^{36} x^{18}-17496401598 a^{6} b^{34} x^{17}[phone] a^{8} b^{32} x^{16}-101038796859 a^{10} b^{30} x^{15}[phone] a^{12} b^{28} x^{14}-184459485463 a^{14} b^{26} x^{13}[phone] a^{16} b^{24} x^{12}-117102180195 a^{18} b^{22} x^{11}[phone] a^{20} b^{20} x^{10}-24022355175 a^{22} b^{18} x^{9}[phone] a^{24} b^{16} x^{8}[phone] a^{28} x^{6} b^{12}[phone] a^{30} x^{5} b^{10}[phone] a^{32} x^{4} b^{8}[phone] a^{34} x^{3} b^{6}[phone] a^{36} x^{2} b^{4}[phone] a^{38} x \,b^{2}[phone] a^{40}\right ) \left (a +b \sqrt {x}\right )^{15}}{312909480 b^{12} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (4457400 b^{40} x^{20}-66043810 a^{2} b^{38} x^{19}[phone] a^{4} b^{36} x^{18}-1944044622 a^{6} b^{34} x^{17}[phone] a^{8} b^{32} x^{16}-12321804495 a^{10} b^{30} x^{15}[phone] a^{12} b^{28} x^{14}-24926957495 a^{14} b^{26} x^{13}[phone] a^{16} b^{24} x^{12}-17742754575 a^{18} b^{22} x^{11}[phone] a^{20} b^{20} x^{10}-4141785375 a^{22} b^{18} x^{9}[phone] a^{24} b^{16} x^{8}[phone] a^{28} x^{6} b^{12}[phone] a^{30} x^{5} b^{10}[phone] a^{32} x^{4} b^{8}[phone] a^{34} x^{3} b^{6}[phone] a^{36} x^{2} b^{4}[phone] a^{38} x \,b^{2}[phone] a^{40}\right ) \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} x^{\frac {9}{2}} b}{2}+5 \left (a +b \sqrt {x}\right )^{15} x^{4}\right )}{782273700 b^{12} \left (-b^{2} x +a^{2}\right )^{14} x^{4}}\)	\(498\)
trager	\(\frac {a \left (1080 b^{14} x^{12}+35490 a^{2} b^{12} x^{11}+1080 b^{14} x^{11}+255528 a^{4} b^{10} x^{10}+35490 x^{10} b^{12} a^{2}+1080 b^{14} x^{10}+602316 a^{6} b^{8} x^{9}+255528 b^{10} x^{9} a^{4}+35490 x^{9} b^{12} a^{2}+1080 b^{14} x^{9}+520520 a^{8} b^{6} x^{8}+602316 a^{6} b^{8} x^{8}+255528 b^{10} x^{8} a^{4}+35490 b^{12} x^{8} a^{2}+1080 b^{14} x^{8}+159705 a^{10} b^{4} x^{7}+520520 b^{6} x^{7} a^{8}+602316 x^{7} b^{8} a^{6}+255528 b^{10} x^{7} a^{4}+35490 a^{2} b^{12} x^{7}+1080 x^{7} b^{14}+14040 a^{12} b^{2} x^{6}+159705 b^{4} x^{6} a^{10}+520520 b^{6} x^{6} a^{8}+602316 b^{8} x^{6} a^{6}+255528 a^{4} b^{10} x^{6}+35490 a^{2} b^{12} x^{6}+1080 b^{14} x^{6}+156 a^{14} x^{5}+14040 b^{2} x^{5} a^{12}+159705 b^{4} x^{5} a^{10}+520520 b^{6} x^{5} a^{8}+602316 a^{6} b^{8} x^{5}+255528 a^{4} b^{10} x^{5}+35490 a^{2} b^{12} x^{5}+1080 b^{14} x^{5}+156 a^{14} x^{4}+14040 a^{12} b^{2} x^{4}+159705 b^{4} x^{4} a^{10}+520520 x^{4} b^{6} a^{8}+602316 a^{6} b^{8} x^{4}+255528 a^{4} b^{10} x^{4}+35490 b^{12} x^{4} a^{2}+1080 b^{14} x^{4}+156 a^{14} x^{3}+14040 a^{12} b^{2} x^{3}+159705 a^{10} b^{4} x^{3}+520520 a^{8} b^{6} x^{3}+602316 a^{6} b^{8} x^{3}+255528 b^{10} x^{3} a^{4}+35490 b^{12} x^{3} a^{2}+1080 b^{14} x^{3}+156 a^{14} x^{2}+14040 a^{12} b^{2} x^{2}+159705 a^{10} b^{4} x^{2}+520520 a^{8} b^{6} x^{2}+602316 b^{8} x^{2} a^{6}+255528 b^{10} x^{2} a^{4}+35490 b^{12} x^{2} a^{2}+1080 b^{14} x^{2}+156 a^{14} x +14040 a^{12} b^{2} x +159705 a^{10} b^{4} x +520520 a^{8} b^{6} x +602316 a^{6} b^{8} x +255528 b^{10} x \,a^{4}+35490 b^{12} x \,a^{2}+1080 b^{14} x +156 a^{14}+14040 a^{12} b^{2}+159705 a^{10} b^{4}+520520 a^{8} b^{6}+602316 a^{6} b^{8}+255528 a^{4} b^{10}+35490 a^{2} b^{12}+1080 b^{14}\right ) \left (-1+x \right )}{936}+\frac {2 b \,x^{\frac {13}{2}} \left (482885 x^{7} b^{14}[phone] 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 [phone] a^{14}\right )}{13037895}\)	\(832\)

Input:

int((a+b*x^(1/2))^15*x^5,x,method=_RETURNVERBOSE)
 

Output:

2/27*b^15*x^(27/2)+15/13*a*b^14*x^13+42/5*a^2*b^13*x^(25/2)+455/12*a^3*b^1 
2*x^12+2730/23*a^4*b^11*x^(23/2)+273*a^5*b^10*x^11+1430/3*a^6*b^9*x^(21/2) 
+1287/2*a^7*b^8*x^10+12870/19*a^8*b^7*x^(19/2)+5005/9*a^9*b^6*x^9+6006/17* 
a^10*b^5*x^(17/2)+1365/8*a^11*b^4*x^8+182/3*a^12*b^3*x^(15/2)+15*a^13*b^2* 
x^7+30/13*a^14*b*x^(13/2)+1/6*a^15*x^6
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.71 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {15}{13} \, a b^{14} x^{13} + \frac {455}{12} \, a^{3} b^{12} x^{12} + 273 \, a^{5} b^{10} x^{11} + \frac {1287}{2} \, a^{7} b^{8} x^{10} + \frac {5005}{9} \, a^{9} b^{6} x^{9} + \frac {1365}{8} \, a^{11} b^{4} x^{8} + 15 \, a^{13} b^{2} x^{7} + \frac {1}{6} \, a^{15} x^{6} + \frac {2}{13037895} \, {\left (482885 \, b^{15} x^{13} + 54759159 \, a^{2} b^{13} x^{12} + 773770725 \, a^{4} b^{11} x^{11} + 3107364975 \, a^{6} b^{9} x^{10} + 4415729175 \, a^{8} b^{7} x^{9} + 2303105805 \, a^{10} b^{5} x^{8} + 395482815 \, a^{12} b^{3} x^{7} + 15043725 \, a^{14} b x^{6}\right )} \sqrt {x} \] Input:

integrate((a+b*x^(1/2))^15*x^5,x, algorithm="fricas")
 

Output:

15/13*a*b^14*x^13 + 455/12*a^3*b^12*x^12 + 273*a^5*b^10*x^11 + 1287/2*a^7* 
b^8*x^10 + 5005/9*a^9*b^6*x^9 + 1365/8*a^11*b^4*x^8 + 15*a^13*b^2*x^7 + 1/ 
6*a^15*x^6 + 2/13037895*(482885*b^15*x^13 + 54759159*a^2*b^13*x^12 + 77377 
0725*a^4*b^11*x^11 + 3107364975*a^6*b^9*x^10 + 4415729175*a^8*b^7*x^9 + 23 
03105805*a^10*b^5*x^8 + 395482815*a^12*b^3*x^7 + 15043725*a^14*b*x^6)*sqrt 
(x)
 

Sympy [A] (verification not implemented)

Time = 1.77 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.88 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {a^{15} x^{6}}{6} + \frac {30 a^{14} b x^{\frac {13}{2}}}{13} + 15 a^{13} b^{2} x^{7} + \frac {182 a^{12} b^{3} x^{\frac {15}{2}}}{3} + \frac {1365 a^{11} b^{4} x^{8}}{8} + \frac {6006 a^{10} b^{5} x^{\frac {17}{2}}}{17} + \frac {5005 a^{9} b^{6} x^{9}}{9} + \frac {12870 a^{8} b^{7} x^{\frac {19}{2}}}{19} + \frac {1287 a^{7} b^{8} x^{10}}{2} + \frac {1430 a^{6} b^{9} x^{\frac {21}{2}}}{3} + 273 a^{5} b^{10} x^{11} + \frac {2730 a^{4} b^{11} x^{\frac {23}{2}}}{23} + \frac {455 a^{3} b^{12} x^{12}}{12} + \frac {42 a^{2} b^{13} x^{\frac {25}{2}}}{5} + \frac {15 a b^{14} x^{13}}{13} + \frac {2 b^{15} x^{\frac {27}{2}}}{27} \] Input:

integrate((a+b*x**(1/2))**15*x**5,x)
 

Output:

a**15*x**6/6 + 30*a**14*b*x**(13/2)/13 + 15*a**13*b**2*x**7 + 182*a**12*b* 
*3*x**(15/2)/3 + 1365*a**11*b**4*x**8/8 + 6006*a**10*b**5*x**(17/2)/17 + 5 
005*a**9*b**6*x**9/9 + 12870*a**8*b**7*x**(19/2)/19 + 1287*a**7*b**8*x**10 
/2 + 1430*a**6*b**9*x**(21/2)/3 + 273*a**5*b**10*x**11 + 2730*a**4*b**11*x 
**(23/2)/23 + 455*a**3*b**12*x**12/12 + 42*a**2*b**13*x**(25/2)/5 + 15*a*b 
**14*x**13/13 + 2*b**15*x**(27/2)/27
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.83 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{27}}{27 \, b^{12}} - \frac {11 \, {\left (b \sqrt {x} + a\right )}^{26} a}{13 \, b^{12}} + \frac {22 \, {\left (b \sqrt {x} + a\right )}^{25} a^{2}}{5 \, b^{12}} - \frac {55 \, {\left (b \sqrt {x} + a\right )}^{24} a^{3}}{4 \, b^{12}} + \frac {660 \, {\left (b \sqrt {x} + a\right )}^{23} a^{4}}{23 \, b^{12}} - \frac {42 \, {\left (b \sqrt {x} + a\right )}^{22} a^{5}}{b^{12}} + \frac {44 \, {\left (b \sqrt {x} + a\right )}^{21} a^{6}}{b^{12}} - \frac {33 \, {\left (b \sqrt {x} + a\right )}^{20} a^{7}}{b^{12}} + \frac {330 \, {\left (b \sqrt {x} + a\right )}^{19} a^{8}}{19 \, b^{12}} - \frac {55 \, {\left (b \sqrt {x} + a\right )}^{18} a^{9}}{9 \, b^{12}} + \frac {22 \, {\left (b \sqrt {x} + a\right )}^{17} a^{10}}{17 \, b^{12}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{11}}{8 \, b^{12}} \] Input:

integrate((a+b*x^(1/2))^15*x^5,x, algorithm="maxima")
 

Output:

2/27*(b*sqrt(x) + a)^27/b^12 - 11/13*(b*sqrt(x) + a)^26*a/b^12 + 22/5*(b*s 
qrt(x) + a)^25*a^2/b^12 - 55/4*(b*sqrt(x) + a)^24*a^3/b^12 + 660/23*(b*sqr 
t(x) + a)^23*a^4/b^12 - 42*(b*sqrt(x) + a)^22*a^5/b^12 + 44*(b*sqrt(x) + a 
)^21*a^6/b^12 - 33*(b*sqrt(x) + a)^20*a^7/b^12 + 330/19*(b*sqrt(x) + a)^19 
*a^8/b^12 - 55/9*(b*sqrt(x) + a)^18*a^9/b^12 + 22/17*(b*sqrt(x) + a)^17*a^ 
10/b^12 - 1/8*(b*sqrt(x) + a)^16*a^11/b^12
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {2}{27} \, b^{15} x^{\frac {27}{2}} + \frac {15}{13} \, a b^{14} x^{13} + \frac {42}{5} \, a^{2} b^{13} x^{\frac {25}{2}} + \frac {455}{12} \, a^{3} b^{12} x^{12} + \frac {2730}{23} \, a^{4} b^{11} x^{\frac {23}{2}} + 273 \, a^{5} b^{10} x^{11} + \frac {1430}{3} \, a^{6} b^{9} x^{\frac {21}{2}} + \frac {1287}{2} \, a^{7} b^{8} x^{10} + \frac {12870}{19} \, a^{8} b^{7} x^{\frac {19}{2}} + \frac {5005}{9} \, a^{9} b^{6} x^{9} + \frac {6006}{17} \, a^{10} b^{5} x^{\frac {17}{2}} + \frac {1365}{8} \, a^{11} b^{4} x^{8} + \frac {182}{3} \, a^{12} b^{3} x^{\frac {15}{2}} + 15 \, a^{13} b^{2} x^{7} + \frac {30}{13} \, a^{14} b x^{\frac {13}{2}} + \frac {1}{6} \, a^{15} x^{6} \] Input:

integrate((a+b*x^(1/2))^15*x^5,x, algorithm="giac")
 

Output:

2/27*b^15*x^(27/2) + 15/13*a*b^14*x^13 + 42/5*a^2*b^13*x^(25/2) + 455/12*a 
^3*b^12*x^12 + 2730/23*a^4*b^11*x^(23/2) + 273*a^5*b^10*x^11 + 1430/3*a^6* 
b^9*x^(21/2) + 1287/2*a^7*b^8*x^10 + 12870/19*a^8*b^7*x^(19/2) + 5005/9*a^ 
9*b^6*x^9 + 6006/17*a^10*b^5*x^(17/2) + 1365/8*a^11*b^4*x^8 + 182/3*a^12*b 
^3*x^(15/2) + 15*a^13*b^2*x^7 + 30/13*a^14*b*x^(13/2) + 1/6*a^15*x^6
 

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {a^{15}\,x^6}{6}+\frac {2\,b^{15}\,x^{27/2}}{27}+\frac {15\,a\,b^{14}\,x^{13}}{13}+\frac {30\,a^{14}\,b\,x^{13/2}}{13}+15\,a^{13}\,b^2\,x^7+\frac {1365\,a^{11}\,b^4\,x^8}{8}+\frac {5005\,a^9\,b^6\,x^9}{9}+\frac {1287\,a^7\,b^8\,x^{10}}{2}+273\,a^5\,b^{10}\,x^{11}+\frac {455\,a^3\,b^{12}\,x^{12}}{12}+\frac {182\,a^{12}\,b^3\,x^{15/2}}{3}+\frac {6006\,a^{10}\,b^5\,x^{17/2}}{17}+\frac {12870\,a^8\,b^7\,x^{19/2}}{19}+\frac {1430\,a^6\,b^9\,x^{21/2}}{3}+\frac {2730\,a^4\,b^{11}\,x^{23/2}}{23}+\frac {42\,a^2\,b^{13}\,x^{25/2}}{5} \] Input:

int(x^5*(a + b*x^(1/2))^15,x)
 

Output:

(a^15*x^6)/6 + (2*b^15*x^(27/2))/27 + (15*a*b^14*x^13)/13 + (30*a^14*b*x^( 
13/2))/13 + 15*a^13*b^2*x^7 + (1365*a^11*b^4*x^8)/8 + (5005*a^9*b^6*x^9)/9 
 + (1287*a^7*b^8*x^10)/2 + 273*a^5*b^10*x^11 + (455*a^3*b^12*x^12)/12 + (1 
82*a^12*b^3*x^(15/2))/3 + (6006*a^10*b^5*x^(17/2))/17 + (12870*a^8*b^7*x^( 
19/2))/19 + (1430*a^6*b^9*x^(21/2))/3 + (2730*a^4*b^11*x^(23/2))/23 + (42* 
a^2*b^13*x^(25/2))/5
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.74 \[ \int \left (a+b \sqrt {x}\right )^{15} x^5 \, dx=\frac {x^{6} \left (240699600 \sqrt {x}\, a^{14} b [phone] \sqrt {x}\, a^{12} b^{3} x [phone] \sqrt {x}\, a^{10} b^{5} x^{2}[phone] \sqrt {x}\, a^{8} b^{7} x^{3}[phone] \sqrt {x}\, a^{6} b^{9} x^{4}[phone] \sqrt {x}\, a^{4} b^{11} x^{5}[phone] \sqrt {x}\, a^{2} b^{13} x^{6}+7726160 \sqrt {x}\, b^{15} x^{7}[phone] a^{15}[phone] a^{13} b^{2} x [phone] a^{11} b^{4} x^{2}[phone] a^{9} b^{6} x^{3}[phone] a^{7} b^{8} x^{4}[phone] a^{5} b^{10} x^{5}[phone] a^{3} b^{12} x^{6}[phone] a \,b^{14} x^{7}\right )}{104303160} \] Input:

int((a+b*x^(1/2))^15*x^5,x)
 

Output:

(x**6*(240699600*sqrt(x)*a**14*b + 6327725040*sqrt(x)*a**12*b**3*x + 36849 
692880*sqrt(x)*a**10*b**5*x**2 + 70651666800*sqrt(x)*a**8*b**7*x**3 + 4971 
7839600*sqrt(x)*a**6*b**9*x**4 + 12380331600*sqrt(x)*a**4*b**11*x**5 + 876 
146544*sqrt(x)*a**2*b**13*x**6 + 7726160*sqrt(x)*b**15*x**7 + 17383860*a** 
15 + 1564547400*a**13*b**2*x + 17796726675*a**11*b**4*x**2 + 58004146200*a 
**9*b**6*x**3 + 67119083460*a**7*b**8*x**4 + 28474762680*a**5*b**10*x**5 + 
 3954828150*a**3*b**12*x**6 + 120349800*a*b**14*x**7))/104303160