Optimal. Leaf size=50 \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{2 a x^2} \]
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Rubi [A] time = 0.0476906, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^3 \left (a+b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A}{a x^2}+\frac{-A b+a B}{a^2 x}-\frac{b (-A b+a B)}{a^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{A}{2 a x^2}-\frac{(A b-a B) \log (x)}{a^2}+\frac{(A b-a B) \log \left (a+b x^2\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0203936, size = 49, normalized size = 0.98 \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 a^2}+\frac{\log (x) (a B-A b)}{a^2}-\frac{A}{2 a x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 56, normalized size = 1.1 \begin{align*} -{\frac{A}{2\,a{x}^{2}}}-{\frac{A\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{\ln \left ( x \right ) B}{a}}+{\frac{\ln \left ( b{x}^{2}+a \right ) Ab}{2\,{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00942, size = 65, normalized size = 1.3 \begin{align*} -\frac{{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{{\left (B a - A b\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{A}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18832, size = 111, normalized size = 2.22 \begin{align*} -\frac{{\left (B a - A b\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (B a - A b\right )} x^{2} \log \left (x\right ) + A a}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.790597, size = 41, normalized size = 0.82 \begin{align*} - \frac{A}{2 a x^{2}} + \frac{\left (- A b + B a\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (- A b + B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14143, size = 96, normalized size = 1.92 \begin{align*} \frac{{\left (B a - A b\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{{\left (B a b - A b^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} - \frac{B a x^{2} - A b x^{2} + A a}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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