3.2221 \(\int \frac{x^5}{\left (a+b \sqrt{x}\right )^8} \, dx\)

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

[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^1
2*(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 = 0.432792, 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 \[ \frac{2 a^{11}}{7 b^{12} \left (a+b \sqrt{x}\right )^7}-\frac{11 a^{10}}{3 b^{12} \left (a+b \sqrt{x}\right )^6}+\frac{22 a^9}{b^{12} \left (a+b \sqrt{x}\right )^5}-\frac{165 a^8}{2 b^{12} \left (a+b \sqrt{x}\right )^4}+\frac{220 a^7}{b^{12} \left (a+b \sqrt{x}\right )^3}-\frac{462 a^6}{b^{12} \left (a+b \sqrt{x}\right )^2}+\frac{924 a^5}{b^{12} \left (a+b \sqrt{x}\right )}+\frac{660 a^4 \log \left (a+b \sqrt{x}\right )}{b^{12}}-\frac{240 a^3 \sqrt{x}}{b^{11}}+\frac{36 a^2 x}{b^{10}}-\frac{16 a x^{3/2}}{3 b^9}+\frac{x^2}{2 b^8} \]

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^1
2*(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 in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a^{11}}{7 b^{12} \left (a + b \sqrt{x}\right )^{7}} - \frac{11 a^{10}}{3 b^{12} \left (a + b \sqrt{x}\right )^{6}} + \frac{22 a^{9}}{b^{12} \left (a + b \sqrt{x}\right )^{5}} - \frac{165 a^{8}}{2 b^{12} \left (a + b \sqrt{x}\right )^{4}} + \frac{220 a^{7}}{b^{12} \left (a + b \sqrt{x}\right )^{3}} - \frac{462 a^{6}}{b^{12} \left (a + b \sqrt{x}\right )^{2}} + \frac{924 a^{5}}{b^{12} \left (a + b \sqrt{x}\right )} + \frac{660 a^{4} \log{\left (a + b \sqrt{x} \right )}}{b^{12}} - \frac{240 a^{3} \sqrt{x}}{b^{11}} + \frac{72 a^{2} \int ^{\sqrt{x}} x\, dx}{b^{10}} - \frac{16 a x^{\frac{3}{2}}}{3 b^{9}} + \frac{x^{2}}{2 b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(a+b*x**(1/2))**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) + 2
2*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))) + 660*a**4*log(a + b*sqrt(x))/b**12 - 240*a**3*sqrt(x)/
b**11 + 72*a**2*Integral(x, (x, sqrt(x)))/b**10 - 16*a*x**(3/2)/(3*b**9) + x**2/
(2*b**8)

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Mathematica [A]  time = 0.075844, size = 174, normalized size = 0.86 \[ \frac{25961 a^{11}+154007 a^{10} b \sqrt{x}+365001 a^9 b^2 x+414295 a^8 b^3 x^{3/2}+171745 a^7 b^4 x^2-90993 a^6 b^5 x^{5/2}-127351 a^5 b^6 x^3-45913 a^4 b^7 x^{7/2}+27720 a^4 \left (a+b \sqrt{x}\right )^7 \log \left (a+b \sqrt{x}\right )-3465 a^3 b^8 x^4+385 a^2 b^9 x^{9/2}-77 a b^{10} x^5+21 b^{11} x^{11/2}}{42 b^{12} \left (a+b \sqrt{x}\right )^7} \]

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 = 0.018, size = 174, normalized size = 0.9 \[ 36\,{\frac{x{a}^{2}}{{b}^{10}}}-{\frac{16\,a}{3\,{b}^{9}}{x}^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{2\,{b}^{8}}}+660\,{\frac{{a}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{12}}}-240\,{\frac{{a}^{3}\sqrt{x}}{{b}^{11}}}+{\frac{2\,{a}^{11}}{7\,{b}^{12}} \left ( a+b\sqrt{x} \right ) ^{-7}}-{\frac{11\,{a}^{10}}{3\,{b}^{12}} \left ( a+b\sqrt{x} \right ) ^{-6}}+22\,{\frac{{a}^{9}}{{b}^{12} \left ( a+b\sqrt{x} \right ) ^{5}}}-{\frac{165\,{a}^{8}}{2\,{b}^{12}} \left ( a+b\sqrt{x} \right ) ^{-4}}+220\,{\frac{{a}^{7}}{{b}^{12} \left ( a+b\sqrt{x} \right ) ^{3}}}-462\,{\frac{{a}^{6}}{{b}^{12} \left ( a+b\sqrt{x} \right ) ^{2}}}+924\,{\frac{{a}^{5}}{{b}^{12} \left ( a+b\sqrt{x} \right ) }} \]

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 = 1.46113, size = 266, normalized size = 1.31 \[ \frac{660 \, a^{4} \log \left (b \sqrt{x} + a\right )}{b^{12}} + \frac{{\left (b \sqrt{x} + a\right )}^{4}}{2 \, b^{12}} - \frac{22 \,{\left (b \sqrt{x} + a\right )}^{3} a}{3 \, b^{12}} + \frac{55 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2}}{b^{12}} - \frac{330 \,{\left (b \sqrt{x} + a\right )} a^{3}}{b^{12}} + \frac{924 \, a^{5}}{{\left (b \sqrt{x} + a\right )} b^{12}} - \frac{462 \, a^{6}}{{\left (b \sqrt{x} + a\right )}^{2} b^{12}} + \frac{220 \, a^{7}}{{\left (b \sqrt{x} + a\right )}^{3} b^{12}} - \frac{165 \, a^{8}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{12}} + \frac{22 \, a^{9}}{{\left (b \sqrt{x} + a\right )}^{5} b^{12}} - \frac{11 \, a^{10}}{3 \,{\left (b \sqrt{x} + a\right )}^{6} b^{12}} + \frac{2 \, a^{11}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*sqrt(x) + a)^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 + 92
4*a^5/((b*sqrt(x) + a)*b^12) - 462*a^6/((b*sqrt(x) + a)^2*b^12) + 220*a^7/((b*sq
rt(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^1
2)

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Fricas [A]  time = 0.241692, size = 386, normalized size = 1.9 \[ -\frac{77 \, a b^{10} x^{5} + 3465 \, a^{3} b^{8} x^{4} + 127351 \, a^{5} b^{6} x^{3} - 171745 \, a^{7} b^{4} x^{2} - 365001 \, a^{9} b^{2} x - 25961 \, a^{11} - 27720 \,{\left (7 \, a^{5} b^{6} x^{3} + 35 \, a^{7} b^{4} x^{2} + 21 \, a^{9} b^{2} x + a^{11} +{\left (a^{4} b^{7} x^{3} + 21 \, a^{6} b^{5} x^{2} + 35 \, a^{8} b^{3} x + 7 \, a^{10} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) - 7 \,{\left (3 \, b^{11} x^{5} + 55 \, a^{2} b^{9} x^{4} - 6559 \, a^{4} b^{7} x^{3} - 12999 \, a^{6} b^{5} x^{2} + 59185 \, a^{8} b^{3} x + 22001 \, a^{10} b\right )} \sqrt{x}}{42 \,{\left (7 \, a b^{18} x^{3} + 35 \, a^{3} b^{16} x^{2} + 21 \, a^{5} b^{14} x + a^{7} b^{12} +{\left (b^{19} x^{3} + 21 \, a^{2} b^{17} x^{2} + 35 \, a^{4} b^{15} x + 7 \, a^{6} b^{13}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*sqrt(x) + a)^8,x, algorithm="fricas")

[Out]

-1/42*(77*a*b^10*x^5 + 3465*a^3*b^8*x^4 + 127351*a^5*b^6*x^3 - 171745*a^7*b^4*x^
2 - 365001*a^9*b^2*x - 25961*a^11 - 27720*(7*a^5*b^6*x^3 + 35*a^7*b^4*x^2 + 21*a
^9*b^2*x + a^11 + (a^4*b^7*x^3 + 21*a^6*b^5*x^2 + 35*a^8*b^3*x + 7*a^10*b)*sqrt(
x))*log(b*sqrt(x) + a) - 7*(3*b^11*x^5 + 55*a^2*b^9*x^4 - 6559*a^4*b^7*x^3 - 129
99*a^6*b^5*x^2 + 59185*a^8*b^3*x + 22001*a^10*b)*sqrt(x))/(7*a*b^18*x^3 + 35*a^3
*b^16*x^2 + 21*a^5*b^14*x + a^7*b^12 + (b^19*x^3 + 21*a^2*b^17*x^2 + 35*a^4*b^15
*x + 7*a^6*b^13)*sqrt(x))

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Sympy [A]  time = 156.991, size = 2154, normalized size = 10.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((803880*a**11*log(a/b + sqrt(x))/(1218*a**7*b**12 + 8526*a**6*b**13*sq
rt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 +
25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)) + 774576*a
**11/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**
4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b*
*18*x**3 + 1218*b**19*x**(7/2)) + 5627160*a**10*b*sqrt(x)*log(a/b + sqrt(x))/(12
18*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*
x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3
+ 1218*b**19*x**(7/2)) + 4618152*a**10*b*sqrt(x)/(1218*a**7*b**12 + 8526*a**6*b
**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**16
*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)) + 1
6881480*a**9*b**2*x*log(a/b + sqrt(x))/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x
) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 255
78*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)) + 11040876*a**
9*b**2*x/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630
*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*
a*b**18*x**3 + 1218*b**19*x**(7/2)) + 28135800*a**8*b**3*x**(3/2)*log(a/b + sqrt
(x))/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**
4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b*
*18*x**3 + 1218*b**19*x**(7/2)) + 12774300*a**8*b**3*x**(3/2)/(1218*a**7*b**12 +
8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 4263
0*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x
**(7/2)) + 28135800*a**7*b**4*x**2*log(a/b + sqrt(x))/(1218*a**7*b**12 + 8526*a*
*6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b
**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2))
+ 5740350*a**7*b**4*x**2/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**
5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17
*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)) + 16881480*a**6*b**5*x**(5/
2)*log(a/b + sqrt(x))/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b*
*14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**
(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)) - 2182950*a**6*b**5*x**(5/2)/(1
218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15
*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**
3 + 1218*b**19*x**(7/2)) + 5627160*a**5*b**6*x**3*log(a/b + sqrt(x))/(1218*a**7*
b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2)
+ 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*
b**19*x**(7/2)) - 3541230*a**5*b**6*x**3/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt
(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 2
5578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)) + 803880*a**
4*b**7*x**(7/2)*log(a/b + sqrt(x))/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) +
25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a
**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)) - 1309770*a**4*b**
7*x**(7/2)/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 426
30*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 852
6*a*b**18*x**3 + 1218*b**19*x**(7/2)) - 100485*a**3*b**8*x**4/(1218*a**7*b**12 +
8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 4263
0*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x
**(7/2)) + 11165*a**2*b**9*x**(9/2)/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) +
25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*
a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)) - 2233*a*b**10*x*
*5/(1218*a**7*b**12 + 8526*a**6*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*
b**15*x**(3/2) + 42630*a**3*b**16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**1
8*x**3 + 1218*b**19*x**(7/2)) + 609*b**11*x**(11/2)/(1218*a**7*b**12 + 8526*a**6
*b**13*sqrt(x) + 25578*a**5*b**14*x + 42630*a**4*b**15*x**(3/2) + 42630*a**3*b**
16*x**2 + 25578*a**2*b**17*x**(5/2) + 8526*a*b**18*x**3 + 1218*b**19*x**(7/2)),
Ne(b, 0)), (x**6/(6*a**8), True))

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GIAC/XCAS [A]  time = 0.235223, size = 193, normalized size = 0.95 \[ \frac{660 \, a^{4}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{12}} + \frac{38808 \, a^{5} b^{6} x^{3} + 213444 \, a^{6} b^{5} x^{\frac{5}{2}} + 494340 \, a^{7} b^{4} x^{2} + 615615 \, a^{8} b^{3} x^{\frac{3}{2}} + 434049 \, a^{9} b^{2} x + 164087 \, a^{10} b \sqrt{x} + 25961 \, a^{11}}{42 \,{\left (b \sqrt{x} + a\right )}^{7} b^{12}} + \frac{3 \, b^{24} x^{2} - 32 \, a b^{23} x^{\frac{3}{2}} + 216 \, a^{2} b^{22} x - 1440 \, a^{3} b^{21} \sqrt{x}}{6 \, b^{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*sqrt(x) + a)^8,x, algorithm="giac")

[Out]

660*a^4*ln(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