Optimal. Leaf size=62 \[ -\frac{(b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac{b (b c-a d) \log \left (c+d x^2\right )}{d^3}+\frac{b^2 x^2}{2 d^2} \]
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Rubi [A] time = 0.058084, antiderivative size = 62, 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} \[ -\frac{(b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac{b (b c-a d) \log \left (c+d x^2\right )}{d^3}+\frac{b^2 x^2}{2 d^2} \]
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 )^2}{\left (c+d x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{(c+d x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{d^2}+\frac{(-b c+a d)^2}{d^2 (c+d x)^2}-\frac{2 b (b c-a d)}{d^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac{b^2 x^2}{2 d^2}-\frac{(b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac{b (b c-a d) \log \left (c+d x^2\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0447732, size = 56, normalized size = 0.9 \[ \frac{-\frac{(b c-a d)^2}{c+d x^2}+2 b (a d-b c) \log \left (c+d x^2\right )+b^2 d x^2}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 97, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2\,{d}^{2}}}+{\frac{b\ln \left ( d{x}^{2}+c \right ) a}{{d}^{2}}}-{\frac{{b}^{2}\ln \left ( d{x}^{2}+c \right ) c}{{d}^{3}}}-{\frac{{a}^{2}}{2\,d \left ( d{x}^{2}+c \right ) }}+{\frac{abc}{{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987961, size = 100, normalized size = 1.61 \begin{align*} \frac{b^{2} x^{2}}{2 \, d^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \,{\left (d^{4} x^{2} + c d^{3}\right )}} - \frac{{\left (b^{2} c - a b d\right )} \log \left (d x^{2} + c\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41026, size = 200, normalized size = 3.23 \begin{align*} \frac{b^{2} d^{2} x^{4} + b^{2} c d x^{2} - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \,{\left (b^{2} c^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (d^{4} x^{2} + c d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.911734, size = 68, normalized size = 1.1 \begin{align*} \frac{b^{2} x^{2}}{2 d^{2}} + \frac{b \left (a d - b c\right ) \log{\left (c + d x^{2} \right )}}{d^{3}} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 c d^{3} + 2 d^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16857, size = 149, normalized size = 2.4 \begin{align*} \frac{{\left (d x^{2} + c\right )} b^{2}}{2 \, d^{3}} + \frac{{\left (b^{2} c - a b d\right )} \log \left (\frac{{\left | d x^{2} + c \right |}}{{\left (d x^{2} + c\right )}^{2}{\left | d \right |}}\right )}{d^{3}} - \frac{\frac{b^{2} c^{2} d}{d x^{2} + c} - \frac{2 \, a b c d^{2}}{d x^{2} + c} + \frac{a^{2} d^{3}}{d x^{2} + c}}{2 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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