Optimal. Leaf size=107 \[ \frac{a}{2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{c}{2 \left (c+d x^2\right ) (b c-a d)^2}+\frac{(a d+b c) \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{(a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
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Rubi [A] time = 0.107765, antiderivative size = 107, 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} \[ \frac{a}{2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{c}{2 \left (c+d x^2\right ) (b c-a d)^2}+\frac{(a d+b c) \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{(a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
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{a b}{(b c-a d)^2 (a+b x)^2}+\frac{b (b c+a d)}{(b c-a d)^3 (a+b x)}-\frac{c d}{(b c-a d)^2 (c+d x)^2}-\frac{d (b c+a d)}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{2 (b c-a d)^2 \left (a+b x^2\right )}+\frac{c}{2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{(b c+a d) \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{(b c+a d) \log \left (c+d x^2\right )}{2 (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.0667347, size = 86, normalized size = 0.8 \[ \frac{\frac{a (b c-a d)}{a+b x^2}+\frac{c (b c-a d)}{c+d x^2}+(a d+b c) \log \left (a+b x^2\right )-(a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 188, normalized size = 1.8 \begin{align*}{\frac{d\ln \left ( d{x}^{2}+c \right ) a}{2\, \left ( ad-bc \right ) ^{3}}}+{\frac{\ln \left ( d{x}^{2}+c \right ) bc}{2\, \left ( ad-bc \right ) ^{3}}}+{\frac{acd}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{b{c}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{\ln \left ( b{x}^{2}+a \right ) ad}{2\, \left ( ad-bc \right ) ^{3}}}-{\frac{b\ln \left ( b{x}^{2}+a \right ) c}{2\, \left ( ad-bc \right ) ^{3}}}+{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{abc}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.987347, size = 308, normalized size = 2.88 \begin{align*} \frac{{\left (b c + a d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac{{\left (b c + a d\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac{{\left (b c + a d\right )} x^{2} + 2 \, a c}{2 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60208, size = 595, normalized size = 5.56 \begin{align*} \frac{2 \, a b c^{2} - 2 \, a^{2} c d +{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2} +{\left ({\left (b^{2} c d + a b d^{2}\right )} x^{4} + a b c^{2} + a^{2} c d +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) -{\left ({\left (b^{2} c d + a b d^{2}\right )} x^{4} + a b c^{2} + a^{2} c d +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.39294, size = 507, normalized size = 4.74 \begin{align*} \frac{2 a c + x^{2} \left (a d + b c\right )}{2 a^{3} c d^{2} - 4 a^{2} b c^{2} d + 2 a b^{2} c^{3} + x^{4} \left (2 a^{2} b d^{3} - 4 a b^{2} c d^{2} + 2 b^{3} c^{2} d\right ) + x^{2} \left (2 a^{3} d^{3} - 2 a^{2} b c d^{2} - 2 a b^{2} c^{2} d + 2 b^{3} c^{3}\right )} + \frac{\left (a d + b c\right ) \log{\left (x^{2} + \frac{- \frac{a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} + \frac{4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d - \frac{b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac{\left (a d + b c\right ) \log{\left (x^{2} + \frac{\frac{a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} - \frac{4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d + \frac{b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{2 \left (a d - b c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22421, size = 240, normalized size = 2.24 \begin{align*} \frac{\frac{a b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (b x^{2} + a\right )}} - \frac{{\left (b^{3} c + a b^{2} d\right )} \log \left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{b^{2} c d}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d\right )}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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