Optimal. Leaf size=33 \[ \frac {b}{2 a^2 \left (a x^2+b\right )}+\frac {\log \left (a x^2+b\right )}{2 a^2} \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac {b}{2 a^2 \left (a x^2+b\right )}+\frac {\log \left (a x^2+b\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x} \, dx &=\int \frac {x^3}{\left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b}{a (b+a x)^2}+\frac {1}{a (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b}{2 a^2 \left (b+a x^2\right )}+\frac {\log \left (b+a x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.82 \[ \frac {\frac {b}{a x^2+b}+\log \left (a x^2+b\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 35, normalized size = 1.06 \[ \frac {{\left (a x^{2} + b\right )} \log \left (a x^{2} + b\right ) + b}{2 \, {\left (a^{3} x^{2} + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 32, normalized size = 0.97 \[ -\frac {x^{2}}{2 \, {\left (a x^{2} + b\right )} a} + \frac {\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.91 \[ \frac {b}{2 \left (a \,x^{2}+b \right ) a^{2}}+\frac {\ln \left (a \,x^{2}+b \right )}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 32, normalized size = 0.97 \[ \frac {b}{2 \, {\left (a^{3} x^{2} + a^{2} b\right )}} + \frac {\log \left (a x^{2} + b\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 29, normalized size = 0.88 \[ \frac {\ln \left (a\,x^2+b\right )}{2\,a^2}+\frac {b}{2\,a^2\,\left (a\,x^2+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 29, normalized size = 0.88 \[ \frac {b}{2 a^{3} x^{2} + 2 a^{2} b} + \frac {\log {\left (a x^{2} + b \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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