3.1831 \(\int \left (a+\frac{b}{x^2}\right )^3 x^3 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0692082, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a^3 x^4}{4}+\frac{3}{2} a^2 b x^2+3 a b^2 \log (x)-\frac{b^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3*x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.012776, size = 40, normalized size = 1. \[ \frac{a^3 x^4}{4}+\frac{3}{2} a^2 b x^2+3 a b^2 \log (x)-\frac{b^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3*x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3*x^3,x)

[Out]

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

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Maxima [A]  time = 1.44562, size = 49, normalized size = 1.22 \[ \frac{1}{4} \, a^{3} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac{b^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219785, size = 51, normalized size = 1.27 \[ \frac{a^{3} x^{6} + 6 \, a^{2} b x^{4} + 12 \, a b^{2} x^{2} \log \left (x\right ) - 2 \, b^{3}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.24156, size = 37, normalized size = 0.92 \[ \frac{a^{3} x^{4}}{4} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} \log{\left (x \right )} - \frac{b^{3}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3*x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22813, size = 47, normalized size = 1.18 \[ \frac{1}{4} \, a^{3} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{b^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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