3.1846 \(\int \frac{x}{a+\frac{b}{x^2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0575099, 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 \[ \frac{x^2}{2 a}-\frac{b \log \left (a x^2+b\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{2}} \frac{1}{a}\, dx}{2} - \frac{b \log{\left (a x^{2} + b \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b/x**2),x)

[Out]

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

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Mathematica [A]  time = 0.00865874, size = 27, normalized size = 1. \[ \frac{x^2}{2 a}-\frac{b \log \left (a x^2+b\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x^2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 24, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,a}}-{\frac{b\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^2),x)

[Out]

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

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Maxima [A]  time = 1.45208, size = 31, normalized size = 1.15 \[ \frac{x^{2}}{2 \, a} - \frac{b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/a - 1/2*b*log(a*x^2 + b)/a^2

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Fricas [A]  time = 0.223204, size = 30, normalized size = 1.11 \[ \frac{a x^{2} - b \log \left (a x^{2} + b\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(a*x^2 - b*log(a*x^2 + b))/a^2

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Sympy [A]  time = 1.16343, size = 20, normalized size = 0.74 \[ \frac{x^{2}}{2 a} - \frac{b \log{\left (a x^{2} + b \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226527, size = 32, normalized size = 1.19 \[ \frac{x^{2}}{2 \, a} - \frac{b{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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