Optimal. Leaf size=41 \[ \frac{B \log \left (a+b x^2\right )}{2 b^2}-\frac{A b-a B}{2 b^2 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0363941, antiderivative size = 41, 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} \[ \frac{B \log \left (a+b x^2\right )}{2 b^2}-\frac{A b-a B}{2 b^2 \left (a+b x^2\right )} \]
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 )}{\left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b (a+b x)^2}+\frac{B}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{A b-a B}{2 b^2 \left (a+b x^2\right )}+\frac{B \log \left (a+b x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0122215, size = 41, normalized size = 1. \[ \frac{a B-A b}{2 b^2 \left (a+b x^2\right )}+\frac{B \log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 47, normalized size = 1.2 \begin{align*}{\frac{B\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}-{\frac{A}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{Ba}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03535, size = 54, normalized size = 1.32 \begin{align*} \frac{B a - A b}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} + \frac{B \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 = 1.22481, size = 92, normalized size = 2.24 \begin{align*} \frac{B a - A b +{\left (B b x^{2} + B a\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.507564, size = 36, normalized size = 0.88 \begin{align*} \frac{B \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{- A b + B a}{2 a b^{2} + 2 b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23471, size = 88, normalized size = 2.15 \begin{align*} -\frac{B{\left (\frac{\log \left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{2} + a\right )} b}\right )}}{2 \, b} - \frac{A}{2 \,{\left (b x^{2} + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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