3.1633 \(\int \frac{x^2}{\left (a+\frac{b}{x}\right )^3} \, dx\)

Optimal. Leaf size=77 \[ \frac{b^5}{2 a^6 (a x+b)^2}-\frac{5 b^4}{a^6 (a x+b)}-\frac{10 b^3 \log (a x+b)}{a^6}+\frac{6 b^2 x}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^3}{3 a^3} \]

[Out]

(6*b^2*x)/a^5 - (3*b*x^2)/(2*a^4) + x^3/(3*a^3) + b^5/(2*a^6*(b + a*x)^2) - (5*b
^4)/(a^6*(b + a*x)) - (10*b^3*Log[b + a*x])/a^6

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Rubi [A]  time = 0.12168, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{b^5}{2 a^6 (a x+b)^2}-\frac{5 b^4}{a^6 (a x+b)}-\frac{10 b^3 \log (a x+b)}{a^6}+\frac{6 b^2 x}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^3}{3 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x)^3,x]

[Out]

(6*b^2*x)/a^5 - (3*b*x^2)/(2*a^4) + x^3/(3*a^3) + b^5/(2*a^6*(b + a*x)^2) - (5*b
^4)/(a^6*(b + a*x)) - (10*b^3*Log[b + a*x])/a^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3 a^{3}} - \frac{3 b \int x\, dx}{a^{4}} + \frac{6 b^{2} x}{a^{5}} + \frac{b^{5}}{2 a^{6} \left (a x + b\right )^{2}} - \frac{5 b^{4}}{a^{6} \left (a x + b\right )} - \frac{10 b^{3} \log{\left (a x + b \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a+b/x)**3,x)

[Out]

x**3/(3*a**3) - 3*b*Integral(x, x)/a**4 + 6*b**2*x/a**5 + b**5/(2*a**6*(a*x + b)
**2) - 5*b**4/(a**6*(a*x + b)) - 10*b**3*log(a*x + b)/a**6

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Mathematica [A]  time = 0.0803045, size = 63, normalized size = 0.82 \[ \frac{2 a^3 x^3-9 a^2 b x^2-\frac{3 b^4 (10 a x+9 b)}{(a x+b)^2}-60 b^3 \log (a x+b)+36 a b^2 x}{6 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b/x)^3,x]

[Out]

(36*a*b^2*x - 9*a^2*b*x^2 + 2*a^3*x^3 - (3*b^4*(9*b + 10*a*x))/(b + a*x)^2 - 60*
b^3*Log[b + a*x])/(6*a^6)

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Maple [A]  time = 0.01, size = 72, normalized size = 0.9 \[ 6\,{\frac{{b}^{2}x}{{a}^{5}}}-{\frac{3\,b{x}^{2}}{2\,{a}^{4}}}+{\frac{{x}^{3}}{3\,{a}^{3}}}+{\frac{{b}^{5}}{2\,{a}^{6} \left ( ax+b \right ) ^{2}}}-5\,{\frac{{b}^{4}}{{a}^{6} \left ( ax+b \right ) }}-10\,{\frac{{b}^{3}\ln \left ( ax+b \right ) }{{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x)^3,x)

[Out]

6*b^2*x/a^5-3/2*b*x^2/a^4+1/3*x^3/a^3+1/2*b^5/a^6/(a*x+b)^2-5*b^4/a^6/(a*x+b)-10
*b^3*ln(a*x+b)/a^6

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Maxima [A]  time = 1.45262, size = 109, normalized size = 1.42 \[ -\frac{10 \, a b^{4} x + 9 \, b^{5}}{2 \,{\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}} - \frac{10 \, b^{3} \log \left (a x + b\right )}{a^{6}} + \frac{2 \, a^{2} x^{3} - 9 \, a b x^{2} + 36 \, b^{2} x}{6 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x)^3,x, algorithm="maxima")

[Out]

-1/2*(10*a*b^4*x + 9*b^5)/(a^8*x^2 + 2*a^7*b*x + a^6*b^2) - 10*b^3*log(a*x + b)/
a^6 + 1/6*(2*a^2*x^3 - 9*a*b*x^2 + 36*b^2*x)/a^5

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Fricas [A]  time = 0.218979, size = 144, normalized size = 1.87 \[ \frac{2 \, a^{5} x^{5} - 5 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} + 63 \, a^{2} b^{3} x^{2} + 6 \, a b^{4} x - 27 \, b^{5} - 60 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \log \left (a x + b\right )}{6 \,{\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x)^3,x, algorithm="fricas")

[Out]

1/6*(2*a^5*x^5 - 5*a^4*b*x^4 + 20*a^3*b^2*x^3 + 63*a^2*b^3*x^2 + 6*a*b^4*x - 27*
b^5 - 60*(a^2*b^3*x^2 + 2*a*b^4*x + b^5)*log(a*x + b))/(a^8*x^2 + 2*a^7*b*x + a^
6*b^2)

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Sympy [A]  time = 1.74563, size = 83, normalized size = 1.08 \[ - \frac{10 a b^{4} x + 9 b^{5}}{2 a^{8} x^{2} + 4 a^{7} b x + 2 a^{6} b^{2}} + \frac{x^{3}}{3 a^{3}} - \frac{3 b x^{2}}{2 a^{4}} + \frac{6 b^{2} x}{a^{5}} - \frac{10 b^{3} \log{\left (a x + b \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b/x)**3,x)

[Out]

-(10*a*b**4*x + 9*b**5)/(2*a**8*x**2 + 4*a**7*b*x + 2*a**6*b**2) + x**3/(3*a**3)
- 3*b*x**2/(2*a**4) + 6*b**2*x/a**5 - 10*b**3*log(a*x + b)/a**6

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GIAC/XCAS [A]  time = 0.229542, size = 99, normalized size = 1.29 \[ -\frac{10 \, b^{3}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{6}} - \frac{10 \, a b^{4} x + 9 \, b^{5}}{2 \,{\left (a x + b\right )}^{2} a^{6}} + \frac{2 \, a^{6} x^{3} - 9 \, a^{5} b x^{2} + 36 \, a^{4} b^{2} x}{6 \, a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x)^3,x, algorithm="giac")

[Out]

-10*b^3*ln(abs(a*x + b))/a^6 - 1/2*(10*a*b^4*x + 9*b^5)/((a*x + b)^2*a^6) + 1/6*
(2*a^6*x^3 - 9*a^5*b*x^2 + 36*a^4*b^2*x)/a^9