Optimal. Leaf size=90 \[ -\frac{b x^2 (b c-a d)}{d^3}+\frac{c (b c-a d)^2}{2 d^4 \left (c+d x^2\right )}+\frac{(b c-a d) (3 b c-a d) \log \left (c+d x^2\right )}{2 d^4}+\frac{b^2 x^4}{4 d^2} \]
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Rubi [A] time = 0.0975002, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ -\frac{b x^2 (b c-a d)}{d^3}+\frac{c (b c-a d)^2}{2 d^4 \left (c+d x^2\right )}+\frac{(b c-a d) (3 b c-a d) \log \left (c+d x^2\right )}{2 d^4}+\frac{b^2 x^4}{4 d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (a+b x)^2}{(c+d x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{2 b (b c-a d)}{d^3}+\frac{b^2 x}{d^2}-\frac{c (b c-a d)^2}{d^3 (c+d x)^2}+\frac{(b c-a d) (3 b c-a d)}{d^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b (b c-a d) x^2}{d^3}+\frac{b^2 x^4}{4 d^2}+\frac{c (b c-a d)^2}{2 d^4 \left (c+d x^2\right )}+\frac{(b c-a d) (3 b c-a d) \log \left (c+d x^2\right )}{2 d^4}\\ \end{align*}
Mathematica [A] time = 0.0633547, size = 87, normalized size = 0.97 \[ \frac{2 \left (a^2 d^2-4 a b c d+3 b^2 c^2\right ) \log \left (c+d x^2\right )+4 b d x^2 (a d-b c)+\frac{2 c (b c-a d)^2}{c+d x^2}+b^2 d^2 x^4}{4 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 142, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}{x}^{4}}{4\,{d}^{2}}}+{\frac{ab{x}^{2}}{{d}^{2}}}-{\frac{{b}^{2}c{x}^{2}}{{d}^{3}}}+{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{d}^{2}}}-2\,{\frac{\ln \left ( d{x}^{2}+c \right ) abc}{{d}^{3}}}+{\frac{3\,\ln \left ( d{x}^{2}+c \right ){b}^{2}{c}^{2}}{2\,{d}^{4}}}+{\frac{{a}^{2}c}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{ab{c}^{2}}{{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}{c}^{3}}{2\,{d}^{4} \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 = 0.993973, size = 144, normalized size = 1.6 \begin{align*} \frac{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}}{2 \,{\left (d^{5} x^{2} + c d^{4}\right )}} + \frac{b^{2} d x^{4} - 4 \,{\left (b^{2} c - a b d\right )} x^{2}}{4 \, d^{3}} + \frac{{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37787, size = 327, normalized size = 3.63 \begin{align*} \frac{b^{2} d^{3} x^{6} + 2 \, b^{2} c^{3} - 4 \, a b c^{2} d + 2 \, a^{2} c d^{2} -{\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )} x^{4} - 4 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x^{2} + 2 \,{\left (3 \, b^{2} c^{3} - 4 \, a b c^{2} d + a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (d^{5} x^{2} + c d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.10964, size = 97, normalized size = 1.08 \begin{align*} \frac{b^{2} x^{4}}{4 d^{2}} + \frac{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}}{2 c d^{4} + 2 d^{5} x^{2}} + \frac{x^{2} \left (a b d - b^{2} c\right )}{d^{3}} + \frac{\left (a d - 3 b c\right ) \left (a d - b c\right ) \log{\left (c + d x^{2} \right )}}{2 d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15326, size = 220, normalized size = 2.44 \begin{align*} \frac{\frac{{\left (d x^{2} + c\right )}^{2}{\left (b^{2} - \frac{2 \,{\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )}}{{\left (d x^{2} + c\right )} d}\right )}}{d^{3}} - \frac{2 \,{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{{\left | d x^{2} + c \right |}}{{\left (d x^{2} + c\right )}^{2}{\left | d \right |}}\right )}{d^{3}} + \frac{2 \,{\left (\frac{b^{2} c^{3} d^{2}}{d x^{2} + c} - \frac{2 \, a b c^{2} d^{3}}{d x^{2} + c} + \frac{a^{2} c d^{4}}{d x^{2} + c}\right )}}{d^{5}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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