Optimal. Leaf size=92 \[ -\frac{b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac{b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]
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Rubi [A] time = 0.0792166, antiderivative size = 92, 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, 44} \[ -\frac{b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac{b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 444
Rule 44
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{1}{(a+b x)^2 (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{(b c-a d)^2 (a+b x)^2}-\frac{2 b^2 d}{(b c-a d)^3 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)^2}+\frac{2 b d^2}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b}{2 (b c-a d)^2 \left (a+b x^2\right )}-\frac{d}{2 (b c-a d)^2 \left (c+d x^2\right )}-\frac{b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac{b d \log \left (c+d x^2\right )}{(b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.0679679, size = 77, normalized size = 0.84 \[ \frac{\frac{b (a d-b c)}{a+b x^2}+\frac{d (a d-b c)}{c+d x^2}-2 b d \log \left (a+b x^2\right )+2 b d \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 143, normalized size = 1.6 \begin{align*} -{\frac{bd\ln \left ( d{x}^{2}+c \right ) }{ \left ( ad-bc \right ) ^{3}}}-{\frac{a{d}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{bdc}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{b\ln \left ( b{x}^{2}+a \right ) d}{ \left ( ad-bc \right ) ^{3}}}-{\frac{abd}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}c}{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 = 0.988063, size = 290, normalized size = 3.15 \begin{align*} -\frac{b d \log \left (b x^{2} + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{b d \log \left (d x^{2} + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{2 \, b d x^{2} + b c + a d}{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 = 1.6321, size = 508, normalized size = 5.52 \begin{align*} -\frac{b^{2} c^{2} - a^{2} d^{2} + 2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} d^{2} x^{4} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (b^{2} d^{2} x^{4} + 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 (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 = 3.84365, size = 408, normalized size = 4.43 \begin{align*} - \frac{b d \log{\left (x^{2} + \frac{- \frac{a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} - \frac{b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac{b d \log{\left (x^{2} + \frac{\frac{a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} + \frac{b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} - \frac{a d + b c + 2 b d x^{2}}{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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15689, size = 220, normalized size = 2.39 \begin{align*} \frac{b^{2} d \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^{3}}{2 \,{\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{b d^{2}}{2 \,{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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