3.1807 \(\int \left (a+\frac{b}{x^2}\right ) x \, dx\)

Optimal. Leaf size=13 \[ \frac{a x^2}{2}+b \log (x) \]

[Out]

(a*x^2)/2 + b*Log[x]

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Rubi [A]  time = 0.0135458, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{a x^2}{2}+b \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*x,x]

[Out]

(a*x^2)/2 + b*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*x,x)

[Out]

a*Integral(x, x) + b*log(x)

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Mathematica [A]  time = 0.00526564, size = 13, normalized size = 1. \[ \frac{a x^2}{2}+b \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*x,x]

[Out]

(a*x^2)/2 + b*Log[x]

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Maple [A]  time = 0.003, size = 12, normalized size = 0.9 \[{\frac{a{x}^{2}}{2}}+b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*x,x)

[Out]

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

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Maxima [A]  time = 1.42977, size = 19, normalized size = 1.46 \[ \frac{1}{2} \, a x^{2} + \frac{1}{2} \, b \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*x^2 + 1/2*b*log(x^2)

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Fricas [A]  time = 0.234629, size = 15, normalized size = 1.15 \[ \frac{1}{2} \, a x^{2} + b \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*x^2 + b*log(x)

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Sympy [A]  time = 0.143045, size = 10, normalized size = 0.77 \[ \frac{a x^{2}}{2} + b \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*x,x)

[Out]

a*x**2/2 + b*log(x)

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GIAC/XCAS [A]  time = 0.228145, size = 16, normalized size = 1.23 \[ \frac{1}{2} \, a x^{2} + b{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*x^2 + b*ln(abs(x))