3.1833 \(\int \left (a+\frac{b}{x^2}\right )^3 x \, dx\)

Optimal. Leaf size=40 \[ \frac{a^3 x^2}{2}+3 a^2 b \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0626124, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{a^3 x^2}{2}+3 a^2 b \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3*x,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00912048, size = 40, normalized size = 1. \[ \frac{a^3 x^2}{2}+3 a^2 b \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3*x,x]

[Out]

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

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Maple [A]  time = 0.008, size = 35, normalized size = 0.9 \[ -{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{3\,a{b}^{2}}{2\,{x}^{2}}}+{\frac{{x}^{2}{a}^{3}}{2}}+3\,{a}^{2}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3*x,x)

[Out]

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

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Maxima [A]  time = 1.44224, size = 50, normalized size = 1.25 \[ \frac{1}{2} \, a^{3} x^{2} + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac{6 \, a b^{2} x^{2} + b^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219988, size = 53, normalized size = 1.32 \[ \frac{2 \, a^{3} x^{6} + 12 \, a^{2} b x^{4} \log \left (x\right ) - 6 \, a b^{2} x^{2} - b^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.35052, size = 36, normalized size = 0.9 \[ \frac{a^{3} x^{2}}{2} + 3 a^{2} b \log{\left (x \right )} - \frac{6 a b^{2} x^{2} + b^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3*x,x)

[Out]

a**3*x**2/2 + 3*a**2*b*log(x) - (6*a*b**2*x**2 + b**3)/(4*x**4)

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GIAC/XCAS [A]  time = 0.235653, size = 49, normalized size = 1.22 \[ \frac{1}{2} \, a^{3} x^{2} + 3 \, a^{2} b{\rm ln}\left ({\left | x \right |}\right ) - \frac{6 \, a b^{2} x^{2} + b^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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