Optimal. Leaf size=126 \[ \frac{b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac{\log (x) (2 a d+b c)}{a^2 c^3}+\frac{d^2}{2 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2}-\frac{1}{2 a c^2 x^2} \]
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Rubi [A] time = 0.144465, antiderivative size = 126, 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} \[ \frac{b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac{\log (x) (2 a d+b c)}{a^2 c^3}+\frac{d^2}{2 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2}-\frac{1}{2 a c^2 x^2} \]
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 ) \left (c+d x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x) (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a c^2 x^2}+\frac{-b c-2 a d}{a^2 c^3 x}+\frac{b^4}{a^2 (-b c+a d)^2 (a+b x)}-\frac{d^3}{c^2 (b c-a d) (c+d x)^2}-\frac{d^3 (3 b c-2 a d)}{c^3 (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a c^2 x^2}+\frac{d^2}{2 c^2 (b c-a d) \left (c+d x^2\right )}-\frac{(b c+2 a d) \log (x)}{a^2 c^3}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^2}-\frac{d^2 (3 b c-2 a d) \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.236739, size = 117, normalized size = 0.93 \[ \frac{1}{2} \left (\frac{b^3 \log \left (a+b x^2\right )}{a^2 (b c-a d)^2}-\frac{2 \log (x) (2 a d+b c)}{a^2 c^3}+\frac{\frac{c d^2}{\left (c+d x^2\right ) (b c-a d)}+\frac{d^2 (2 a d-3 b c) \log \left (c+d x^2\right )}{(b c-a d)^2}-\frac{c}{a x^2}}{c^3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 170, normalized size = 1.4 \begin{align*}{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) a}{{c}^{3} \left ( ad-bc \right ) ^{2}}}-{\frac{3\,{d}^{2}\ln \left ( d{x}^{2}+c \right ) b}{2\,{c}^{2} \left ( ad-bc \right ) ^{2}}}-{\frac{{d}^{3}a}{2\,{c}^{2} \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{b{d}^{2}}{2\,c \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{1}{2\,a{c}^{2}{x}^{2}}}-2\,{\frac{\ln \left ( x \right ) d}{a{c}^{3}}}-{\frac{\ln \left ( x \right ) b}{{a}^{2}{c}^{2}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2} \left ( ad-bc \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17474, size = 254, normalized size = 2.02 \begin{align*} \frac{b^{3} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} - \frac{{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac{b c^{2} - a c d +{\left (b c d - 2 \, a d^{2}\right )} x^{2}}{2 \,{\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{4} +{\left (a b c^{4} - a^{2} c^{3} d\right )} x^{2}\right )}} - \frac{{\left (b c + 2 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 14.2487, size = 601, normalized size = 4.77 \begin{align*} -\frac{a b^{2} c^{4} - 2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} +{\left (a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x^{2} -{\left (b^{3} c^{3} d x^{4} + b^{3} c^{4} x^{2}\right )} \log \left (b x^{2} + a\right ) +{\left ({\left (3 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4}\right )} x^{4} +{\left (3 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \,{\left ({\left (b^{3} c^{3} d - 3 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4}\right )} x^{4} +{\left (b^{3} c^{4} - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{2} b^{2} c^{5} d - 2 \, a^{3} b c^{4} d^{2} + a^{4} c^{3} d^{3}\right )} x^{4} +{\left (a^{2} b^{2} c^{6} - 2 \, a^{3} b c^{5} d + a^{4} c^{4} d^{2}\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 [B] time = 1.15844, size = 347, normalized size = 2.75 \begin{align*} \frac{b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )}} - \frac{{\left (3 \, b c d^{3} - 2 \, a d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )}} + \frac{b^{3} c^{2} d x^{4} + b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - 4 \, a^{3} d^{3} x^{2} - 2 \, a b^{2} c^{3} + 4 \, a^{2} b c^{2} d - 2 \, a^{3} c d^{2}}{4 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )}{\left (d x^{4} + c x^{2}\right )}} - \frac{{\left (b c + 2 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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