3.1820 \(\int \left (a+\frac{b}{x^2}\right )^2 x \, dx\)

Optimal. Leaf size=27 \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0515083, antiderivative size = 27, 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 \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2*x,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00895152, size = 27, normalized size = 1. \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2*x,x]

[Out]

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

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Maple [A]  time = 0.008, size = 24, normalized size = 0.9 \[ -{\frac{{b}^{2}}{2\,{x}^{2}}}+{\frac{{a}^{2}{x}^{2}}{2}}+2\,ab\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2*x,x)

[Out]

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

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Maxima [A]  time = 1.43881, size = 32, normalized size = 1.19 \[ \frac{1}{2} \, a^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{b^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222102, size = 36, normalized size = 1.33 \[ \frac{a^{2} x^{4} + 4 \, a b x^{2} \log \left (x\right ) - b^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.16373, size = 24, normalized size = 0.89 \[ \frac{a^{2} x^{2}}{2} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2*x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229232, size = 32, normalized size = 1.19 \[ \frac{1}{2} \, a^{2} x^{2} + 2 \, a b{\rm ln}\left ({\left | x \right |}\right ) - \frac{b^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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