3.1620 \(\int \frac{x^2}{\left (a+\frac{b}{x}\right )^2} \, dx\)

Optimal. Leaf size=58 \[ -\frac{b^4}{a^5 (a x+b)}-\frac{4 b^3 \log (a x+b)}{a^5}+\frac{3 b^2 x}{a^4}-\frac{b x^2}{a^3}+\frac{x^3}{3 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0926245, antiderivative size = 58, 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 \[ -\frac{b^4}{a^5 (a x+b)}-\frac{4 b^3 \log (a x+b)}{a^5}+\frac{3 b^2 x}{a^4}-\frac{b x^2}{a^3}+\frac{x^3}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0373014, size = 54, normalized size = 0.93 \[ \frac{a^3 x^3-3 a^2 b x^2-\frac{3 b^4}{a x+b}-12 b^3 \log (a x+b)+9 a b^2 x}{3 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b/x)^2,x]

[Out]

(9*a*b^2*x - 3*a^2*b*x^2 + a^3*x^3 - (3*b^4)/(b + a*x) - 12*b^3*Log[b + a*x])/(3
*a^5)

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Maple [A]  time = 0.01, size = 57, normalized size = 1. \[ 3\,{\frac{{b}^{2}x}{{a}^{4}}}-{\frac{b{x}^{2}}{{a}^{3}}}+{\frac{{x}^{3}}{3\,{a}^{2}}}-{\frac{{b}^{4}}{{a}^{5} \left ( ax+b \right ) }}-4\,{\frac{{b}^{3}\ln \left ( ax+b \right ) }{{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x)^2,x)

[Out]

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

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Maxima [A]  time = 1.44351, size = 80, normalized size = 1.38 \[ -\frac{b^{4}}{a^{6} x + a^{5} b} - \frac{4 \, b^{3} \log \left (a x + b\right )}{a^{5}} + \frac{a^{2} x^{3} - 3 \, a b x^{2} + 9 \, b^{2} x}{3 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218982, size = 99, normalized size = 1.71 \[ \frac{a^{4} x^{4} - 2 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a b^{3} x - 3 \, b^{4} - 12 \,{\left (a b^{3} x + b^{4}\right )} \log \left (a x + b\right )}{3 \,{\left (a^{6} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a^4*x^4 - 2*a^3*b*x^3 + 6*a^2*b^2*x^2 + 9*a*b^3*x - 3*b^4 - 12*(a*b^3*x + b
^4)*log(a*x + b))/(a^6*x + a^5*b)

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Sympy [A]  time = 1.41563, size = 54, normalized size = 0.93 \[ - \frac{b^{4}}{a^{6} x + a^{5} b} + \frac{x^{3}}{3 a^{2}} - \frac{b x^{2}}{a^{3}} + \frac{3 b^{2} x}{a^{4}} - \frac{4 b^{3} \log{\left (a x + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b/x)**2,x)

[Out]

-b**4/(a**6*x + a**5*b) + x**3/(3*a**2) - b*x**2/a**3 + 3*b**2*x/a**4 - 4*b**3*l
og(a*x + b)/a**5

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GIAC/XCAS [A]  time = 0.222332, size = 84, normalized size = 1.45 \[ -\frac{4 \, b^{3}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{5}} - \frac{b^{4}}{{\left (a x + b\right )} a^{5}} + \frac{a^{4} x^{3} - 3 \, a^{3} b x^{2} + 9 \, a^{2} b^{2} x}{3 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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