Optimal. Leaf size=156 \[ -\frac{b^3}{2 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}-\frac{2 \log (x) (a d+b c)}{a^3 c^3}-\frac{1}{2 a^2 c^2 x^2}-\frac{d^3}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2}+\frac{d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3} \]
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Rubi [A] time = 0.20896, antiderivative size = 156, 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, 88} \[ -\frac{b^3}{2 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}-\frac{2 \log (x) (a d+b c)}{a^3 c^3}-\frac{1}{2 a^2 c^2 x^2}-\frac{d^3}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2}+\frac{d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rubi steps
\begin{align*} \int \frac{1}{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{1}{x^2 (a+b x)^2 (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 c^2 x^2}-\frac{2 (b c+a d)}{a^3 c^3 x}+\frac{b^4}{a^2 (-b c+a d)^2 (a+b x)^2}+\frac{2 b^4 (-b c+2 a d)}{a^3 (-b c+a d)^3 (a+b x)}+\frac{d^4}{c^2 (b c-a d)^2 (c+d x)^2}+\frac{2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a^2 c^2 x^2}-\frac{b^3}{2 a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac{d^3}{2 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac{2 (b c+a d) \log (x)}{a^3 c^3}+\frac{b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}+\frac{d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.200367, size = 157, normalized size = 1.01 \[ \frac{1}{2} \left (-\frac{b^3}{a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac{2 b^3 (2 a d-b c) \log \left (a+b x^2\right )}{a^3 (a d-b c)^3}-\frac{4 \log (x) (a d+b c)}{a^3 c^3}-\frac{1}{a^2 c^2 x^2}-\frac{d^3}{c^2 \left (c+d x^2\right ) (b c-a d)^2}+\frac{2 d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 254, normalized size = 1.6 \begin{align*}{\frac{{d}^{4}\ln \left ( d{x}^{2}+c \right ) a}{{c}^{3} \left ( ad-bc \right ) ^{3}}}-2\,{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) b}{{c}^{2} \left ( ad-bc \right ) ^{3}}}-{\frac{{d}^{4}a}{2\,{c}^{2} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{3}b}{2\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{1}{2\,{a}^{2}{c}^{2}{x}^{2}}}-2\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{3}}}-2\,{\frac{\ln \left ( x \right ) b}{{a}^{3}{c}^{2}}}+2\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) d}{{a}^{2} \left ( ad-bc \right ) ^{3}}}-{\frac{{b}^{4}\ln \left ( b{x}^{2}+a \right ) c}{{a}^{3} \left ( ad-bc \right ) ^{3}}}-{\frac{{b}^{3}d}{2\,a \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}c}{2\,{a}^{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 = 1.06165, size = 514, normalized size = 3.29 \begin{align*} \frac{{\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac{{\left (2 \, b c d^{3} - a d^{4}\right )} \log \left (d x^{2} + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + 2 \,{\left (b^{3} c^{2} d - a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x^{2}}{2 \,{\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} +{\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{4} +{\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2}\right )}} - \frac{{\left (b c + a d\right )} \log \left (x^{2}\right )}{a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 53.814, size = 1287, normalized size = 8.25 \begin{align*} -\frac{a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + 2 \,{\left (a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} +{\left (2 \, a b^{4} c^{5} - 3 \, a^{2} b^{3} c^{4} d + 3 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}\right )} x^{2} - 2 \,{\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{6} +{\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x^{4} +{\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left ({\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} +{\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} +{\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 4 \,{\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} +{\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} +{\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{3} b^{4} c^{6} d - 3 \, a^{4} b^{3} c^{5} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{3} - a^{6} b c^{3} d^{4}\right )} x^{6} +{\left (a^{3} b^{4} c^{7} - 2 \, a^{4} b^{3} c^{6} d + 2 \, a^{6} b c^{4} d^{3} - a^{7} c^{3} d^{4}\right )} x^{4} +{\left (a^{4} b^{3} c^{7} - 3 \, a^{5} b^{2} c^{6} d + 3 \, a^{6} b c^{5} d^{2} - a^{7} c^{4} d^{3}\right )} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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