Optimal. Leaf size=61 \[ -\frac{(b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac{d (b c-a d) \log \left (a+b x^2\right )}{b^3}+\frac{d^2 x^2}{2 b^2} \]
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Rubi [A] time = 0.0580752, antiderivative size = 61, 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 = {444, 43} \[ -\frac{(b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac{d (b c-a d) \log \left (a+b x^2\right )}{b^3}+\frac{d^2 x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(c+d x)^2}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{d^2}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)^2}+\frac{2 d (b c-a d)}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{d^2 x^2}{2 b^2}-\frac{(b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac{d (b c-a d) \log \left (a+b x^2\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.046709, size = 56, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{a+b x^2}+2 d (b c-a d) \log \left (a+b x^2\right )+b d^2 x^2}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 97, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{2}}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ){d}^{2}a}{{b}^{3}}}+{\frac{\ln \left ( b{x}^{2}+a \right ) dc}{{b}^{2}}}-{\frac{{a}^{2}{d}^{2}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{acd}{{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{c}^{2}}{2\,b \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 = 1.06481, size = 99, normalized size = 1.62 \begin{align*} \frac{d^{2} x^{2}}{2 \, b^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} + \frac{{\left (b c d - a d^{2}\right )} \log \left (b x^{2} + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43955, size = 200, normalized size = 3.28 \begin{align*} \frac{b^{2} d^{2} x^{4} + a b d^{2} x^{2} - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} + 2 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.906413, size = 68, normalized size = 1.11 \begin{align*} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 a b^{3} + 2 b^{4} x^{2}} + \frac{d^{2} x^{2}}{2 b^{2}} - \frac{d \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13733, size = 150, normalized size = 2.46 \begin{align*} \frac{{\left (b x^{2} + a\right )} d^{2}}{2 \, b^{3}} - \frac{{\left (b c d - a d^{2}\right )} \log \left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} - \frac{\frac{b^{3} c^{2}}{b x^{2} + a} - \frac{2 \, a b^{2} c d}{b x^{2} + a} + \frac{a^{2} b d^{2}}{b x^{2} + a}}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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