Optimal. Leaf size=35 \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 b^2}+\frac{B x^2}{2 b} \]
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Rubi [A] time = 0.0299063, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {444, 43} \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 b^2}+\frac{B x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{B}{b}+\frac{A b-a B}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{B x^2}{2 b}+\frac{(A b-a B) \log \left (a+b x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0108656, size = 31, normalized size = 0.89 \[ \frac{(A b-a B) \log \left (a+b x^2\right )+b B x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 40, normalized size = 1.1 \begin{align*}{\frac{B{x}^{2}}{2\,b}}+{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ) Ba}{2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991694, size = 42, normalized size = 1.2 \begin{align*} \frac{B x^{2}}{2 \, b} - \frac{{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32872, size = 65, normalized size = 1.86 \begin{align*} \frac{B b x^{2} -{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.375242, size = 27, normalized size = 0.77 \begin{align*} \frac{B x^{2}}{2 b} - \frac{\left (- A b + B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16417, size = 43, normalized size = 1.23 \begin{align*} \frac{B x^{2}}{2 \, b} - \frac{{\left (B a - A b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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