\(\int (a+b \sqrt {x})^{15} x^4 \, dx\) [60]

Optimal result

Integrand size = 15, antiderivative size = 202 \[ \int \left (a+b \sqrt {x}\right )^{15} x^4 \, dx=-\frac {a^9 \left (a+b \sqrt {x}\right )^{16}}{8 b^{10}}+\frac {18 a^8 \left (a+b \sqrt {x}\right )^{17}}{17 b^{10}}-\frac {4 a^7 \left (a+b \sqrt {x}\right )^{18}}{b^{10}}+\frac {168 a^6 \left (a+b \sqrt {x}\right )^{19}}{19 b^{10}}-\frac {63 a^5 \left (a+b \sqrt {x}\right )^{20}}{5 b^{10}}+\frac {12 a^4 \left (a+b \sqrt {x}\right )^{21}}{b^{10}}-\frac {84 a^3 \left (a+b \sqrt {x}\right )^{22}}{11 b^{10}}+\frac {72 a^2 \left (a+b \sqrt {x}\right )^{23}}{23 b^{10}}-\frac {3 a \left (a+b \sqrt {x}\right )^{24}}{4 b^{10}}+\frac {2 \left (a+b \sqrt {x}\right )^{25}}{25 b^{10}} \] Output:

-1/8*a^9*(a+b*x^(1/2))^16/b^10+18/17*a^8*(a+b*x^(1/2))^17/b^10-4*a^7*(a+b* 
x^(1/2))^18/b^10+168/19*a^6*(a+b*x^(1/2))^19/b^10-63/5*a^5*(a+b*x^(1/2))^2 
0/b^10+12*a^4*(a+b*x^(1/2))^21/b^10-84/11*a^3*(a+b*x^(1/2))^22/b^10+72/23* 
a^2*(a+b*x^(1/2))^23/b^10-3/4*a*(a+b*x^(1/2))^24/b^10+2/25*(a+b*x^(1/2))^2 
5/b^10
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.93 \[ \int \left (a+b \sqrt {x}\right )^{15} x^4 \, dx=\frac {3268760 a^{15} x^5+44574000 a^{14} b x^{11/2}[phone] a^{13} b^2 x^6+1144066000 a^{12} b^3 x^{13/2}[phone] a^{11} b^4 x^7+6544057520 a^{10} b^5 x^{15/2}[phone] a^9 b^6 x^8+12373218000 a^8 b^7 x^{17/2}[phone] a^7 b^8 x^9+8610602000 a^6 b^9 x^{19/2}[phone] a^5 b^{10} x^{10}[phone] a^4 b^{11} x^{21/2}[phone] a^3 b^{12} x^{11}[phone] a^2 b^{13} x^{23/2}[phone] a b^{14} x^{12}+1307504 b^{15} x^{25/2}}{16343800} \] Input:

Integrate[(a + b*Sqrt[x])^15*x^4,x]
 

Output:

(3268760*a^15*x^5 + 44574000*a^14*b*x^(11/2) + 286016500*a^13*b^2*x^6 + 11 
44066000*a^12*b^3*x^(13/2) + 3187041000*a^11*b^4*x^7 + 6544057520*a^10*b^5 
*x^(15/2) + 10225089875*a^9*b^6*x^8 + 12373218000*a^8*b^7*x^(17/2) + 11685 
817000*a^7*b^8*x^9 + 8610602000*a^6*b^9*x^(19/2) + 4908043140*a^5*b^10*x^1 
0 + 2124694000*a^4*b^11*x^(21/2) + 676039000*a^3*b^12*x^11 + 149226000*a^2 
*b^13*x^(23/2) + 20429750*a*b^14*x^12 + 1307504*b^15*x^(25/2))/16343800
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 204, 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.

\(\displaystyle \int x^4 \left (a+b \sqrt {x}\right )^{15} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{15} x^{9/2}d\sqrt {x}\)

\(\Big \downarrow \) 49

\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{24}}{b^9}-\frac {9 a \left (a+b \sqrt {x}\right )^{23}}{b^9}+\frac {36 a^2 \left (a+b \sqrt {x}\right )^{22}}{b^9}-\frac {84 a^3 \left (a+b \sqrt {x}\right )^{21}}{b^9}+\frac {126 a^4 \left (a+b \sqrt {x}\right )^{20}}{b^9}-\frac {126 a^5 \left (a+b \sqrt {x}\right )^{19}}{b^9}+\frac {84 a^6 \left (a+b \sqrt {x}\right )^{18}}{b^9}-\frac {36 a^7 \left (a+b \sqrt {x}\right )^{17}}{b^9}+\frac {9 a^8 \left (a+b \sqrt {x}\right )^{16}}{b^9}-\frac {a^9 \left (a+b \sqrt {x}\right )^{15}}{b^9}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {a^9 \left (a+b \sqrt {x}\right )^{16}}{16 b^{10}}+\frac {9 a^8 \left (a+b \sqrt {x}\right )^{17}}{17 b^{10}}-\frac {2 a^7 \left (a+b \sqrt {x}\right )^{18}}{b^{10}}+\frac {84 a^6 \left (a+b \sqrt {x}\right )^{19}}{19 b^{10}}-\frac {63 a^5 \left (a+b \sqrt {x}\right )^{20}}{10 b^{10}}+\frac {6 a^4 \left (a+b \sqrt {x}\right )^{21}}{b^{10}}-\frac {42 a^3 \left (a+b \sqrt {x}\right )^{22}}{11 b^{10}}+\frac {36 a^2 \left (a+b \sqrt {x}\right )^{23}}{23 b^{10}}+\frac {\left (a+b \sqrt {x}\right )^{25}}{25 b^{10}}-\frac {3 a \left (a+b \sqrt {x}\right )^{24}}{8 b^{10}}\right )\)

Input:

Int[(a + b*Sqrt[x])^15*x^4,x]
 

Output:

2*(-1/16*(a^9*(a + b*Sqrt[x])^16)/b^10 + (9*a^8*(a + b*Sqrt[x])^17)/(17*b^ 
10) - (2*a^7*(a + b*Sqrt[x])^18)/b^10 + (84*a^6*(a + b*Sqrt[x])^19)/(19*b^ 
10) - (63*a^5*(a + b*Sqrt[x])^20)/(10*b^10) + (6*a^4*(a + b*Sqrt[x])^21)/b 
^10 - (42*a^3*(a + b*Sqrt[x])^22)/(11*b^10) + (36*a^2*(a + b*Sqrt[x])^23)/ 
(23*b^10) - (3*a*(a + b*Sqrt[x])^24)/(8*b^10) + (a + b*Sqrt[x])^25/(25*b^1 
0))
 

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 = 0.83

method	result	size
derivativedivides	\(\frac {2 b^{15} x^{\frac {25}{2}}}{25}+\frac {5 a \,b^{14} x^{12}}{4}+\frac {210 a^{2} b^{13} x^{\frac {23}{2}}}{23}+\frac {455 a^{3} b^{12} x^{11}}{11}+130 a^{4} b^{11} x^{\frac {21}{2}}+\frac {3003 a^{5} b^{10} x^{10}}{10}+\frac {10010 a^{6} b^{9} x^{\frac {19}{2}}}{19}+715 a^{7} b^{8} x^{9}+\frac {12870 a^{8} b^{7} x^{\frac {17}{2}}}{17}+\frac {5005 a^{9} b^{6} x^{8}}{8}+\frac {2002 a^{10} b^{5} x^{\frac {15}{2}}}{5}+195 a^{11} b^{4} x^{7}+70 a^{12} b^{3} x^{\frac {13}{2}}+\frac {35 a^{13} b^{2} x^{6}}{2}+\frac {30 a^{14} b \,x^{\frac {11}{2}}}{11}+\frac {a^{15} x^{5}}{5}\)	\(168\)
default	\(\frac {2 b^{15} x^{\frac {25}{2}}}{25}+\frac {5 a \,b^{14} x^{12}}{4}+\frac {210 a^{2} b^{13} x^{\frac {23}{2}}}{23}+\frac {455 a^{3} b^{12} x^{11}}{11}+130 a^{4} b^{11} x^{\frac {21}{2}}+\frac {3003 a^{5} b^{10} x^{10}}{10}+\frac {10010 a^{6} b^{9} x^{\frac {19}{2}}}{19}+715 a^{7} b^{8} x^{9}+\frac {12870 a^{8} b^{7} x^{\frac {17}{2}}}{17}+\frac {5005 a^{9} b^{6} x^{8}}{8}+\frac {2002 a^{10} b^{5} x^{\frac {15}{2}}}{5}+195 a^{11} b^{4} x^{7}+70 a^{12} b^{3} x^{\frac {13}{2}}+\frac {35 a^{13} b^{2} x^{6}}{2}+\frac {30 a^{14} b \,x^{\frac {11}{2}}}{11}+\frac {a^{15} x^{5}}{5}\)	\(168\)
orering	\(-\frac {\left (-38407930 b^{38} x^{19}[phone] a^{2} b^{36} x^{18}-3615112254 a^{4} b^{34} x^{17}[phone] a^{6} b^{32} x^{16}-41486940495 a^{8} b^{30} x^{15}[phone] a^{10} b^{28} x^{14}-131474402675 a^{12} b^{26} x^{13}[phone] a^{14} b^{24} x^{12}-141562505175 a^{16} b^{22} x^{11}[phone] a^{18} b^{20} x^{10}-52449271875 a^{20} b^{18} x^{9}[phone] a^{22} b^{16} x^{8}[phone] a^{26} b^{12} x^{6}[phone] a^{28} x^{5} b^{10}[phone] a^{30} x^{4} b^{8}[phone] a^{32} x^{3} b^{6}[phone] a^{34} x^{2} b^{4}[phone] a^{36} x \,b^{2}[phone] a^{38}\right ) \left (a +b \sqrt {x}\right )^{15}}{245157000 b^{10} \left (-b^{2} x +a^{2}\right )^{14}}+\frac {\left (-817190 b^{38} x^{19}[phone] a^{2} b^{36} x^{18}-84072378 a^{4} b^{34} x^{17}[phone] a^{6} b^{32} x^{16}-1063767705 a^{8} b^{30} x^{15}[phone] a^{10} b^{28} x^{14}-3756411505 a^{12} b^{26} x^{13}[phone] a^{14} b^{24} x^{12}-4566532425 a^{16} b^{22} x^{11}[phone] a^{18} b^{20} x^{10}-1942565625 a^{20} b^{18} x^{9}[phone] a^{22} b^{16} x^{8}[phone] a^{26} b^{12} x^{6}[phone] a^{28} x^{5} b^{10}[phone] a^{30} x^{4} b^{8}[phone] a^{32} x^{3} b^{6}[phone] a^{34} x^{2} b^{4}[phone] a^{36} x \,b^{2}[phone] a^{38}\right ) \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} x^{\frac {7}{2}} b}{2}+4 \left (a +b \sqrt {x}\right )^{15} x^{3}\right )}{122578500 b^{10} \left (-b^{2} x +a^{2}\right )^{14} x^{3}}\)	\(476\)
trager	\(\frac {a \left (550 b^{14} x^{11}+18200 x^{10} b^{12} a^{2}+550 b^{14} x^{10}+132132 b^{10} x^{9} a^{4}+18200 x^{9} b^{12} a^{2}+550 b^{14} x^{9}+314600 a^{6} b^{8} x^{8}+132132 b^{10} x^{8} a^{4}+18200 b^{12} x^{8} a^{2}+550 b^{14} x^{8}+275275 b^{6} x^{7} a^{8}+314600 x^{7} b^{8} a^{6}+132132 b^{10} x^{7} a^{4}+18200 a^{2} b^{12} x^{7}+550 x^{7} b^{14}+85800 b^{4} x^{6} a^{10}+275275 b^{6} x^{6} a^{8}+314600 b^{8} x^{6} a^{6}+132132 a^{4} b^{10} x^{6}+18200 a^{2} b^{12} x^{6}+550 b^{14} x^{6}+7700 b^{2} x^{5} a^{12}+85800 b^{4} x^{5} a^{10}+275275 b^{6} x^{5} a^{8}+314600 a^{6} b^{8} x^{5}+132132 a^{4} b^{10} x^{5}+18200 a^{2} b^{12} x^{5}+550 b^{14} x^{5}+88 a^{14} x^{4}+7700 a^{12} b^{2} x^{4}+85800 b^{4} x^{4} a^{10}+275275 x^{4} b^{6} a^{8}+314600 a^{6} b^{8} x^{4}+132132 a^{4} b^{10} x^{4}+18200 b^{12} x^{4} a^{2}+550 b^{14} x^{4}+88 a^{14} x^{3}+7700 a^{12} b^{2} x^{3}+85800 a^{10} b^{4} x^{3}+275275 a^{8} b^{6} x^{3}+314600 a^{6} b^{8} x^{3}+132132 b^{10} x^{3} a^{4}+18200 b^{12} x^{3} a^{2}+550 b^{14} x^{3}+88 a^{14} x^{2}+7700 a^{12} b^{2} x^{2}+85800 a^{10} b^{4} x^{2}+275275 a^{8} b^{6} x^{2}+314600 b^{8} x^{2} a^{6}+132132 b^{10} x^{2} a^{4}+18200 b^{12} x^{2} a^{2}+550 b^{14} x^{2}+88 a^{14} x +7700 a^{12} b^{2} x +85800 a^{10} b^{4} x +275275 a^{8} b^{6} x +314600 a^{6} b^{8} x +132132 b^{10} x \,a^{4}+18200 b^{12} x \,a^{2}+550 b^{14} x +88 a^{14}+7700 a^{12} b^{2}+85800 a^{10} b^{4}+275275 a^{8} b^{6}+314600 a^{6} b^{8}+132132 a^{4} b^{10}+18200 a^{2} b^{12}+550 b^{14}\right ) \left (-1+x \right )}{440}+\frac {2 b \,x^{\frac {11}{2}} \left (81719 x^{7} b^{14}+9326625 a^{2} b^{12} x^{6}[phone] a^{4} b^{10} x^{5}[phone] a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}[phone] a^{10} b^{4} x^{2}[phone] a^{12} b^{2} x +2785875 a^{14}\right )}{2042975}\)	\(750\)

Input:

int((a+b*x^(1/2))^15*x^4,x,method=_RETURNVERBOSE)
 

Output:

2/25*b^15*x^(25/2)+5/4*a*b^14*x^12+210/23*a^2*b^13*x^(23/2)+455/11*a^3*b^1 
2*x^11+130*a^4*b^11*x^(21/2)+3003/10*a^5*b^10*x^10+10010/19*a^6*b^9*x^(19/ 
2)+715*a^7*b^8*x^9+12870/17*a^8*b^7*x^(17/2)+5005/8*a^9*b^6*x^8+2002/5*a^1 
0*b^5*x^(15/2)+195*a^11*b^4*x^7+70*a^12*b^3*x^(13/2)+35/2*a^13*b^2*x^6+30/ 
11*a^14*b*x^(11/2)+1/5*a^15*x^5
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.86 \[ \int \left (a+b \sqrt {x}\right )^{15} x^4 \, dx=\frac {5}{4} \, a b^{14} x^{12} + \frac {455}{11} \, a^{3} b^{12} x^{11} + \frac {3003}{10} \, a^{5} b^{10} x^{10} + 715 \, a^{7} b^{8} x^{9} + \frac {5005}{8} \, a^{9} b^{6} x^{8} + 195 \, a^{11} b^{4} x^{7} + \frac {35}{2} \, a^{13} b^{2} x^{6} + \frac {1}{5} \, a^{15} x^{5} + \frac {2}{2042975} \, {\left (81719 \, b^{15} x^{12} + 9326625 \, a^{2} b^{13} x^{11} + 132793375 \, a^{4} b^{11} x^{10} + 538162625 \, a^{6} b^{9} x^{9} + 773326125 \, a^{8} b^{7} x^{8} + 409003595 \, a^{10} b^{5} x^{7} + 71504125 \, a^{12} b^{3} x^{6} + 2785875 \, a^{14} b x^{5}\right )} \sqrt {x} \] Input:

integrate((a+b*x^(1/2))^15*x^4,x, algorithm="fricas")
 

Output:

5/4*a*b^14*x^12 + 455/11*a^3*b^12*x^11 + 3003/10*a^5*b^10*x^10 + 715*a^7*b 
^8*x^9 + 5005/8*a^9*b^6*x^8 + 195*a^11*b^4*x^7 + 35/2*a^13*b^2*x^6 + 1/5*a 
^15*x^5 + 2/2042975*(81719*b^15*x^12 + 9326625*a^2*b^13*x^11 + 132793375*a 
^4*b^11*x^10 + 538162625*a^6*b^9*x^9 + 773326125*a^8*b^7*x^8 + 409003595*a 
^10*b^5*x^7 + 71504125*a^12*b^3*x^6 + 2785875*a^14*b*x^5)*sqrt(x)
 

Sympy [A] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.04 \[ \int \left (a+b \sqrt {x}\right )^{15} x^4 \, dx=\frac {a^{15} x^{5}}{5} + \frac {30 a^{14} b x^{\frac {11}{2}}}{11} + \frac {35 a^{13} b^{2} x^{6}}{2} + 70 a^{12} b^{3} x^{\frac {13}{2}} + 195 a^{11} b^{4} x^{7} + \frac {2002 a^{10} b^{5} x^{\frac {15}{2}}}{5} + \frac {5005 a^{9} b^{6} x^{8}}{8} + \frac {12870 a^{8} b^{7} x^{\frac {17}{2}}}{17} + 715 a^{7} b^{8} x^{9} + \frac {10010 a^{6} b^{9} x^{\frac {19}{2}}}{19} + \frac {3003 a^{5} b^{10} x^{10}}{10} + 130 a^{4} b^{11} x^{\frac {21}{2}} + \frac {455 a^{3} b^{12} x^{11}}{11} + \frac {210 a^{2} b^{13} x^{\frac {23}{2}}}{23} + \frac {5 a b^{14} x^{12}}{4} + \frac {2 b^{15} x^{\frac {25}{2}}}{25} \] Input:

integrate((a+b*x**(1/2))**15*x**4,x)
 

Output:

a**15*x**5/5 + 30*a**14*b*x**(11/2)/11 + 35*a**13*b**2*x**6/2 + 70*a**12*b 
**3*x**(13/2) + 195*a**11*b**4*x**7 + 2002*a**10*b**5*x**(15/2)/5 + 5005*a 
**9*b**6*x**8/8 + 12870*a**8*b**7*x**(17/2)/17 + 715*a**7*b**8*x**9 + 1001 
0*a**6*b**9*x**(19/2)/19 + 3003*a**5*b**10*x**10/10 + 130*a**4*b**11*x**(2 
1/2) + 455*a**3*b**12*x**11/11 + 210*a**2*b**13*x**(23/2)/23 + 5*a*b**14*x 
**12/4 + 2*b**15*x**(25/2)/25
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.82 \[ \int \left (a+b \sqrt {x}\right )^{15} x^4 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{25}}{25 \, b^{10}} - \frac {3 \, {\left (b \sqrt {x} + a\right )}^{24} a}{4 \, b^{10}} + \frac {72 \, {\left (b \sqrt {x} + a\right )}^{23} a^{2}}{23 \, b^{10}} - \frac {84 \, {\left (b \sqrt {x} + a\right )}^{22} a^{3}}{11 \, b^{10}} + \frac {12 \, {\left (b \sqrt {x} + a\right )}^{21} a^{4}}{b^{10}} - \frac {63 \, {\left (b \sqrt {x} + a\right )}^{20} a^{5}}{5 \, b^{10}} + \frac {168 \, {\left (b \sqrt {x} + a\right )}^{19} a^{6}}{19 \, b^{10}} - \frac {4 \, {\left (b \sqrt {x} + a\right )}^{18} a^{7}}{b^{10}} + \frac {18 \, {\left (b \sqrt {x} + a\right )}^{17} a^{8}}{17 \, b^{10}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{9}}{8 \, b^{10}} \] Input:

integrate((a+b*x^(1/2))^15*x^4,x, algorithm="maxima")
 

Output:

2/25*(b*sqrt(x) + a)^25/b^10 - 3/4*(b*sqrt(x) + a)^24*a/b^10 + 72/23*(b*sq 
rt(x) + a)^23*a^2/b^10 - 84/11*(b*sqrt(x) + a)^22*a^3/b^10 + 12*(b*sqrt(x) 
 + a)^21*a^4/b^10 - 63/5*(b*sqrt(x) + a)^20*a^5/b^10 + 168/19*(b*sqrt(x) + 
 a)^19*a^6/b^10 - 4*(b*sqrt(x) + a)^18*a^7/b^10 + 18/17*(b*sqrt(x) + a)^17 
*a^8/b^10 - 1/8*(b*sqrt(x) + a)^16*a^9/b^10
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.83 \[ \int \left (a+b \sqrt {x}\right )^{15} x^4 \, dx=\frac {2}{25} \, b^{15} x^{\frac {25}{2}} + \frac {5}{4} \, a b^{14} x^{12} + \frac {210}{23} \, a^{2} b^{13} x^{\frac {23}{2}} + \frac {455}{11} \, a^{3} b^{12} x^{11} + 130 \, a^{4} b^{11} x^{\frac {21}{2}} + \frac {3003}{10} \, a^{5} b^{10} x^{10} + \frac {10010}{19} \, a^{6} b^{9} x^{\frac {19}{2}} + 715 \, a^{7} b^{8} x^{9} + \frac {12870}{17} \, a^{8} b^{7} x^{\frac {17}{2}} + \frac {5005}{8} \, a^{9} b^{6} x^{8} + \frac {2002}{5} \, a^{10} b^{5} x^{\frac {15}{2}} + 195 \, a^{11} b^{4} x^{7} + 70 \, a^{12} b^{3} x^{\frac {13}{2}} + \frac {35}{2} \, a^{13} b^{2} x^{6} + \frac {30}{11} \, a^{14} b x^{\frac {11}{2}} + \frac {1}{5} \, a^{15} x^{5} \] Input:

integrate((a+b*x^(1/2))^15*x^4,x, algorithm="giac")
 

Output:

2/25*b^15*x^(25/2) + 5/4*a*b^14*x^12 + 210/23*a^2*b^13*x^(23/2) + 455/11*a 
^3*b^12*x^11 + 130*a^4*b^11*x^(21/2) + 3003/10*a^5*b^10*x^10 + 10010/19*a^ 
6*b^9*x^(19/2) + 715*a^7*b^8*x^9 + 12870/17*a^8*b^7*x^(17/2) + 5005/8*a^9* 
b^6*x^8 + 2002/5*a^10*b^5*x^(15/2) + 195*a^11*b^4*x^7 + 70*a^12*b^3*x^(13/ 
2) + 35/2*a^13*b^2*x^6 + 30/11*a^14*b*x^(11/2) + 1/5*a^15*x^5
 

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.83 \[ \int \left (a+b \sqrt {x}\right )^{15} x^4 \, dx=\frac {a^{15}\,x^5}{5}+\frac {2\,b^{15}\,x^{25/2}}{25}+\frac {5\,a\,b^{14}\,x^{12}}{4}+\frac {30\,a^{14}\,b\,x^{11/2}}{11}+\frac {35\,a^{13}\,b^2\,x^6}{2}+195\,a^{11}\,b^4\,x^7+\frac {5005\,a^9\,b^6\,x^8}{8}+715\,a^7\,b^8\,x^9+\frac {3003\,a^5\,b^{10}\,x^{10}}{10}+\frac {455\,a^3\,b^{12}\,x^{11}}{11}+70\,a^{12}\,b^3\,x^{13/2}+\frac {2002\,a^{10}\,b^5\,x^{15/2}}{5}+\frac {12870\,a^8\,b^7\,x^{17/2}}{17}+\frac {10010\,a^6\,b^9\,x^{19/2}}{19}+130\,a^4\,b^{11}\,x^{21/2}+\frac {210\,a^2\,b^{13}\,x^{23/2}}{23} \] Input:

int(x^4*(a + b*x^(1/2))^15,x)
 

Output:

(a^15*x^5)/5 + (2*b^15*x^(25/2))/25 + (5*a*b^14*x^12)/4 + (30*a^14*b*x^(11 
/2))/11 + (35*a^13*b^2*x^6)/2 + 195*a^11*b^4*x^7 + (5005*a^9*b^6*x^8)/8 + 
715*a^7*b^8*x^9 + (3003*a^5*b^10*x^10)/10 + (455*a^3*b^12*x^11)/11 + 70*a^ 
12*b^3*x^(13/2) + (2002*a^10*b^5*x^(15/2))/5 + (12870*a^8*b^7*x^(17/2))/17 
 + (10010*a^6*b^9*x^(19/2))/19 + 130*a^4*b^11*x^(21/2) + (210*a^2*b^13*x^( 
23/2))/23
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.88 \[ \int \left (a+b \sqrt {x}\right )^{15} x^4 \, dx=\frac {x^{5} \left (44574000 \sqrt {x}\, a^{14} b [phone] \sqrt {x}\, a^{12} b^{3} x [phone] \sqrt {x}\, a^{10} b^{5} x^{2}[phone] \sqrt {x}\, a^{8} b^{7} x^{3}[phone] \sqrt {x}\, a^{6} b^{9} x^{4}[phone] \sqrt {x}\, a^{4} b^{11} x^{5}[phone] \sqrt {x}\, a^{2} b^{13} x^{6}+1307504 \sqrt {x}\, b^{15} x^{7}+3268760 a^{15}[phone] a^{13} b^{2} x [phone] a^{11} b^{4} x^{2}[phone] a^{9} b^{6} x^{3}[phone] a^{7} b^{8} x^{4}[phone] a^{5} b^{10} x^{5}[phone] a^{3} b^{12} x^{6}[phone] a \,b^{14} x^{7}\right )}{16343800} \] Input:

int((a+b*x^(1/2))^15*x^4,x)
 

Output:

(x**5*(44574000*sqrt(x)*a**14*b + 1144066000*sqrt(x)*a**12*b**3*x + 654405 
7520*sqrt(x)*a**10*b**5*x**2 + 12373218000*sqrt(x)*a**8*b**7*x**3 + 861060 
2000*sqrt(x)*a**6*b**9*x**4 + 2124694000*sqrt(x)*a**4*b**11*x**5 + 1492260 
00*sqrt(x)*a**2*b**13*x**6 + 1307504*sqrt(x)*b**15*x**7 + 3268760*a**15 + 
286016500*a**13*b**2*x + 3187041000*a**11*b**4*x**2 + 10225089875*a**9*b** 
6*x**3 + 11685817000*a**7*b**8*x**4 + 4908043140*a**5*b**10*x**5 + 6760390 
00*a**3*b**12*x**6 + 20429750*a*b**14*x**7))/16343800