3.2223 \(\int \frac{x^5}{(a+b \sqrt{x})^8} \, dx\)

Optimal. Leaf size=203 \[ \frac{2 a^{11}}{7 b^{12} \left (a+b \sqrt{x}\right )^7}-\frac{11 a^{10}}{3 b^{12} \left (a+b \sqrt{x}\right )^6}+\frac{22 a^9}{b^{12} \left (a+b \sqrt{x}\right )^5}-\frac{165 a^8}{2 b^{12} \left (a+b \sqrt{x}\right )^4}+\frac{220 a^7}{b^{12} \left (a+b \sqrt{x}\right )^3}-\frac{462 a^6}{b^{12} \left (a+b \sqrt{x}\right )^2}+\frac{924 a^5}{b^{12} \left (a+b \sqrt{x}\right )}-\frac{240 a^3 \sqrt{x}}{b^{11}}+\frac{36 a^2 x}{b^{10}}+\frac{660 a^4 \log \left (a+b \sqrt{x}\right )}{b^{12}}-\frac{16 a x^{3/2}}{3 b^9}+\frac{x^2}{2 b^8} \]

[Out]

(2*a^11)/(7*b^12*(a + b*Sqrt[x])^7) - (11*a^10)/(3*b^12*(a + b*Sqrt[x])^6) + (22*a^9)/(b^12*(a + b*Sqrt[x])^5)
- (165*a^8)/(2*b^12*(a + b*Sqrt[x])^4) + (220*a^7)/(b^12*(a + b*Sqrt[x])^3) - (462*a^6)/(b^12*(a + b*Sqrt[x])
^2) + (924*a^5)/(b^12*(a + b*Sqrt[x])) - (240*a^3*Sqrt[x])/b^11 + (36*a^2*x)/b^10 - (16*a*x^(3/2))/(3*b^9) + x
^2/(2*b^8) + (660*a^4*Log[a + b*Sqrt[x]])/b^12

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Rubi [A]  time = 0.198654, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2 a^{11}}{7 b^{12} \left (a+b \sqrt{x}\right )^7}-\frac{11 a^{10}}{3 b^{12} \left (a+b \sqrt{x}\right )^6}+\frac{22 a^9}{b^{12} \left (a+b \sqrt{x}\right )^5}-\frac{165 a^8}{2 b^{12} \left (a+b \sqrt{x}\right )^4}+\frac{220 a^7}{b^{12} \left (a+b \sqrt{x}\right )^3}-\frac{462 a^6}{b^{12} \left (a+b \sqrt{x}\right )^2}+\frac{924 a^5}{b^{12} \left (a+b \sqrt{x}\right )}-\frac{240 a^3 \sqrt{x}}{b^{11}}+\frac{36 a^2 x}{b^{10}}+\frac{660 a^4 \log \left (a+b \sqrt{x}\right )}{b^{12}}-\frac{16 a x^{3/2}}{3 b^9}+\frac{x^2}{2 b^8} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*Sqrt[x])^8,x]

[Out]

(2*a^11)/(7*b^12*(a + b*Sqrt[x])^7) - (11*a^10)/(3*b^12*(a + b*Sqrt[x])^6) + (22*a^9)/(b^12*(a + b*Sqrt[x])^5)
- (165*a^8)/(2*b^12*(a + b*Sqrt[x])^4) + (220*a^7)/(b^12*(a + b*Sqrt[x])^3) - (462*a^6)/(b^12*(a + b*Sqrt[x])
^2) + (924*a^5)/(b^12*(a + b*Sqrt[x])) - (240*a^3*Sqrt[x])/b^11 + (36*a^2*x)/b^10 - (16*a*x^(3/2))/(3*b^9) + x
^2/(2*b^8) + (660*a^4*Log[a + b*Sqrt[x]])/b^12

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+b \sqrt{x}\right )^8} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^{11}}{(a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{120 a^3}{b^{11}}+\frac{36 a^2 x}{b^{10}}-\frac{8 a x^2}{b^9}+\frac{x^3}{b^8}-\frac{a^{11}}{b^{11} (a+b x)^8}+\frac{11 a^{10}}{b^{11} (a+b x)^7}-\frac{55 a^9}{b^{11} (a+b x)^6}+\frac{165 a^8}{b^{11} (a+b x)^5}-\frac{330 a^7}{b^{11} (a+b x)^4}+\frac{462 a^6}{b^{11} (a+b x)^3}-\frac{462 a^5}{b^{11} (a+b x)^2}+\frac{330 a^4}{b^{11} (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^{11}}{7 b^{12} \left (a+b \sqrt{x}\right )^7}-\frac{11 a^{10}}{3 b^{12} \left (a+b \sqrt{x}\right )^6}+\frac{22 a^9}{b^{12} \left (a+b \sqrt{x}\right )^5}-\frac{165 a^8}{2 b^{12} \left (a+b \sqrt{x}\right )^4}+\frac{220 a^7}{b^{12} \left (a+b \sqrt{x}\right )^3}-\frac{462 a^6}{b^{12} \left (a+b \sqrt{x}\right )^2}+\frac{924 a^5}{b^{12} \left (a+b \sqrt{x}\right )}-\frac{240 a^3 \sqrt{x}}{b^{11}}+\frac{36 a^2 x}{b^{10}}-\frac{16 a x^{3/2}}{3 b^9}+\frac{x^2}{2 b^8}+\frac{660 a^4 \log \left (a+b \sqrt{x}\right )}{b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.182618, size = 174, normalized size = 0.86 \[ \frac{414295 a^8 b^3 x^{3/2}+171745 a^7 b^4 x^2-90993 a^6 b^5 x^{5/2}-127351 a^5 b^6 x^3-45913 a^4 b^7 x^{7/2}-3465 a^3 b^8 x^4+385 a^2 b^9 x^{9/2}+365001 a^9 b^2 x+154007 a^{10} b \sqrt{x}+27720 a^4 \left (a+b \sqrt{x}\right )^7 \log \left (a+b \sqrt{x}\right )+25961 a^{11}-77 a b^{10} x^5+21 b^{11} x^{11/2}}{42 b^{12} \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*Sqrt[x])^8,x]

[Out]

(25961*a^11 + 154007*a^10*b*Sqrt[x] + 365001*a^9*b^2*x + 414295*a^8*b^3*x^(3/2) + 171745*a^7*b^4*x^2 - 90993*a
^6*b^5*x^(5/2) - 127351*a^5*b^6*x^3 - 45913*a^4*b^7*x^(7/2) - 3465*a^3*b^8*x^4 + 385*a^2*b^9*x^(9/2) - 77*a*b^
10*x^5 + 21*b^11*x^(11/2) + 27720*a^4*(a + b*Sqrt[x])^7*Log[a + b*Sqrt[x]])/(42*b^12*(a + b*Sqrt[x])^7)

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Maple [A]  time = 0.011, size = 174, normalized size = 0.9 \begin{align*} 36\,{\frac{{a}^{2}x}{{b}^{10}}}-{\frac{16\,a}{3\,{b}^{9}}{x}^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{2\,{b}^{8}}}+660\,{\frac{{a}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{12}}}-240\,{\frac{{a}^{3}\sqrt{x}}{{b}^{11}}}+{\frac{2\,{a}^{11}}{7\,{b}^{12}} \left ( a+b\sqrt{x} \right ) ^{-7}}-{\frac{11\,{a}^{10}}{3\,{b}^{12}} \left ( a+b\sqrt{x} \right ) ^{-6}}+22\,{\frac{{a}^{9}}{{b}^{12} \left ( a+b\sqrt{x} \right ) ^{5}}}-{\frac{165\,{a}^{8}}{2\,{b}^{12}} \left ( a+b\sqrt{x} \right ) ^{-4}}+220\,{\frac{{a}^{7}}{{b}^{12} \left ( a+b\sqrt{x} \right ) ^{3}}}-462\,{\frac{{a}^{6}}{{b}^{12} \left ( a+b\sqrt{x} \right ) ^{2}}}+924\,{\frac{{a}^{5}}{{b}^{12} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b*x^(1/2))^8,x)

[Out]

36*a^2*x/b^10-16/3*a*x^(3/2)/b^9+1/2*x^2/b^8+660*a^4*ln(a+b*x^(1/2))/b^12-240*a^3*x^(1/2)/b^11+2/7*a^11/b^12/(
a+b*x^(1/2))^7-11/3*a^10/b^12/(a+b*x^(1/2))^6+22*a^9/b^12/(a+b*x^(1/2))^5-165/2*a^8/b^12/(a+b*x^(1/2))^4+220*a
^7/b^12/(a+b*x^(1/2))^3-462*a^6/b^12/(a+b*x^(1/2))^2+924*a^5/b^12/(a+b*x^(1/2))

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Maxima [A]  time = 0.97311, size = 266, normalized size = 1.31 \begin{align*} \frac{660 \, a^{4} \log \left (b \sqrt{x} + a\right )}{b^{12}} + \frac{{\left (b \sqrt{x} + a\right )}^{4}}{2 \, b^{12}} - \frac{22 \,{\left (b \sqrt{x} + a\right )}^{3} a}{3 \, b^{12}} + \frac{55 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2}}{b^{12}} - \frac{330 \,{\left (b \sqrt{x} + a\right )} a^{3}}{b^{12}} + \frac{924 \, a^{5}}{{\left (b \sqrt{x} + a\right )} b^{12}} - \frac{462 \, a^{6}}{{\left (b \sqrt{x} + a\right )}^{2} b^{12}} + \frac{220 \, a^{7}}{{\left (b \sqrt{x} + a\right )}^{3} b^{12}} - \frac{165 \, a^{8}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{12}} + \frac{22 \, a^{9}}{{\left (b \sqrt{x} + a\right )}^{5} b^{12}} - \frac{11 \, a^{10}}{3 \,{\left (b \sqrt{x} + a\right )}^{6} b^{12}} + \frac{2 \, a^{11}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*x^(1/2))^8,x, algorithm="maxima")

[Out]

660*a^4*log(b*sqrt(x) + a)/b^12 + 1/2*(b*sqrt(x) + a)^4/b^12 - 22/3*(b*sqrt(x) + a)^3*a/b^12 + 55*(b*sqrt(x) +
a)^2*a^2/b^12 - 330*(b*sqrt(x) + a)*a^3/b^12 + 924*a^5/((b*sqrt(x) + a)*b^12) - 462*a^6/((b*sqrt(x) + a)^2*b^
12) + 220*a^7/((b*sqrt(x) + a)^3*b^12) - 165/2*a^8/((b*sqrt(x) + a)^4*b^12) + 22*a^9/((b*sqrt(x) + a)^5*b^12)
- 11/3*a^10/((b*sqrt(x) + a)^6*b^12) + 2/7*a^11/((b*sqrt(x) + a)^7*b^12)

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Fricas [B]  time = 1.35887, size = 906, normalized size = 4.46 \begin{align*} \frac{21 \, b^{18} x^{9} + 1365 \, a^{2} b^{16} x^{8} - 10143 \, a^{4} b^{14} x^{7} - 27195 \, a^{6} b^{12} x^{6} + 227094 \, a^{8} b^{10} x^{5} - 540190 \, a^{10} b^{8} x^{4} + 661465 \, a^{12} b^{6} x^{3} - 455091 \, a^{14} b^{4} x^{2} + 167867 \, a^{16} b^{2} x - 25961 \, a^{18} + 27720 \,{\left (a^{4} b^{14} x^{7} - 7 \, a^{6} b^{12} x^{6} + 21 \, a^{8} b^{10} x^{5} - 35 \, a^{10} b^{8} x^{4} + 35 \, a^{12} b^{6} x^{3} - 21 \, a^{14} b^{4} x^{2} + 7 \, a^{16} b^{2} x - a^{18}\right )} \log \left (b \sqrt{x} + a\right ) - 8 \,{\left (28 \, a b^{17} x^{8} + 1064 \, a^{3} b^{15} x^{7} - 13083 \, a^{5} b^{13} x^{6} + 48580 \, a^{7} b^{11} x^{5} - 92323 \, a^{9} b^{9} x^{4} + 101376 \, a^{11} b^{7} x^{3} - 65373 \, a^{13} b^{5} x^{2} + 23100 \, a^{15} b^{3} x - 3465 \, a^{17} b\right )} \sqrt{x}}{42 \,{\left (b^{26} x^{7} - 7 \, a^{2} b^{24} x^{6} + 21 \, a^{4} b^{22} x^{5} - 35 \, a^{6} b^{20} x^{4} + 35 \, a^{8} b^{18} x^{3} - 21 \, a^{10} b^{16} x^{2} + 7 \, a^{12} b^{14} x - a^{14} b^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*x^(1/2))^8,x, algorithm="fricas")

[Out]

1/42*(21*b^18*x^9 + 1365*a^2*b^16*x^8 - 10143*a^4*b^14*x^7 - 27195*a^6*b^12*x^6 + 227094*a^8*b^10*x^5 - 540190
*a^10*b^8*x^4 + 661465*a^12*b^6*x^3 - 455091*a^14*b^4*x^2 + 167867*a^16*b^2*x - 25961*a^18 + 27720*(a^4*b^14*x
^7 - 7*a^6*b^12*x^6 + 21*a^8*b^10*x^5 - 35*a^10*b^8*x^4 + 35*a^12*b^6*x^3 - 21*a^14*b^4*x^2 + 7*a^16*b^2*x - a
^18)*log(b*sqrt(x) + a) - 8*(28*a*b^17*x^8 + 1064*a^3*b^15*x^7 - 13083*a^5*b^13*x^6 + 48580*a^7*b^11*x^5 - 923
23*a^9*b^9*x^4 + 101376*a^11*b^7*x^3 - 65373*a^13*b^5*x^2 + 23100*a^15*b^3*x - 3465*a^17*b)*sqrt(x))/(b^26*x^7
- 7*a^2*b^24*x^6 + 21*a^4*b^22*x^5 - 35*a^6*b^20*x^4 + 35*a^8*b^18*x^3 - 21*a^10*b^16*x^2 + 7*a^12*b^14*x - a
^14*b^12)

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Sympy [A]  time = 19.5863, size = 2048, normalized size = 10.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b*x**(1/2))**8,x)

[Out]

Piecewise((27720*a**11*log(a/b + sqrt(x))/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a*
*4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) + 1
4124*a**11/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b
**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) + 194040*a**10*b*sqrt(x)*log(a/b +
sqrt(x))/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b*
*16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) + 71148*a**10*b*sqrt(x)/(42*a**7*b*
*12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b
**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) + 582120*a**9*b**2*x*log(a/b + sqrt(x))/(42*a**7*b**12 +
294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*
x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) + 116424*a**9*b**2*x/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x)
+ 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x
**3 + 42*b**19*x**(7/2)) + 970200*a**8*b**3*x**(3/2)*log(a/b + sqrt(x))/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x
) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18
*x**3 + 42*b**19*x**(7/2)) + 970200*a**7*b**4*x**2*log(a/b + sqrt(x))/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x)
+ 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x
**3 + 42*b**19*x**(7/2)) - 242550*a**7*b**4*x**2/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x +
1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/
2)) + 582120*a**6*b**5*x**(5/2)*log(a/b + sqrt(x))/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x
+ 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(
7/2)) - 339570*a**6*b**5*x**(5/2)/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15
*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) + 194040*a*
*5*b**6*x**3*log(a/b + sqrt(x))/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x
**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) - 210210*a**5
*b**6*x**3/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b
**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) + 27720*a**4*b**7*x**(7/2)*log(a/b
+ sqrt(x))/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*
b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) - 57750*a**4*b**7*x**(7/2)/(42*a*
*7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a
**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) - 3465*a**3*b**8*x**4/(42*a**7*b**12 + 294*a**6*b**
13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 29
4*a*b**18*x**3 + 42*b**19*x**(7/2)) + 385*a**2*b**9*x**(9/2)/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**
5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*
b**19*x**(7/2)) - 77*a*b**10*x**5/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15
*x**(3/2) + 1470*a**3*b**16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)) + 21*b**11*
x**(11/2)/(42*a**7*b**12 + 294*a**6*b**13*sqrt(x) + 882*a**5*b**14*x + 1470*a**4*b**15*x**(3/2) + 1470*a**3*b*
*16*x**2 + 882*a**2*b**17*x**(5/2) + 294*a*b**18*x**3 + 42*b**19*x**(7/2)), Ne(b, 0)), (x**6/(6*a**8), True))

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Giac [A]  time = 1.09967, size = 193, normalized size = 0.95 \begin{align*} \frac{660 \, a^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{12}} + \frac{38808 \, a^{5} b^{6} x^{3} + 213444 \, a^{6} b^{5} x^{\frac{5}{2}} + 494340 \, a^{7} b^{4} x^{2} + 615615 \, a^{8} b^{3} x^{\frac{3}{2}} + 434049 \, a^{9} b^{2} x + 164087 \, a^{10} b \sqrt{x} + 25961 \, a^{11}}{42 \,{\left (b \sqrt{x} + a\right )}^{7} b^{12}} + \frac{3 \, b^{24} x^{2} - 32 \, a b^{23} x^{\frac{3}{2}} + 216 \, a^{2} b^{22} x - 1440 \, a^{3} b^{21} \sqrt{x}}{6 \, b^{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*x^(1/2))^8,x, algorithm="giac")

[Out]

660*a^4*log(abs(b*sqrt(x) + a))/b^12 + 1/42*(38808*a^5*b^6*x^3 + 213444*a^6*b^5*x^(5/2) + 494340*a^7*b^4*x^2 +
615615*a^8*b^3*x^(3/2) + 434049*a^9*b^2*x + 164087*a^10*b*sqrt(x) + 25961*a^11)/((b*sqrt(x) + a)^7*b^12) + 1/
6*(3*b^24*x^2 - 32*a*b^23*x^(3/2) + 216*a^2*b^22*x - 1440*a^3*b^21*sqrt(x))/b^32