Optimal. Leaf size=98 \[ \frac{b^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{b^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3}+\frac{b}{2 \left (c+d x^2\right ) (b c-a d)^2}+\frac{1}{4 \left (c+d x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.0749927, antiderivative size = 98, 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} \[ \frac{b^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{b^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3}+\frac{b}{2 \left (c+d x^2\right ) (b c-a d)^2}+\frac{1}{4 \left (c+d x^2\right )^2 (b c-a d)} \]
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 ) \left (c+d x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(a+b x) (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac{b}{2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{b^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{b^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.0468274, size = 98, normalized size = 1. \[ \frac{b^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac{b^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3}+\frac{b}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{1}{4 \left (c+d x^2\right )^2 (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 176, normalized size = 1.8 \begin{align*}{\frac{\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\, \left ( ad-bc \right ) ^{3}}}-{\frac{{a}^{2}{d}^{2}}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{cabd}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{bda}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}c}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03754, size = 285, normalized size = 2.91 \begin{align*} \frac{b^{2} \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{b^{2} \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{2 \, b d x^{2} + 3 \, b c - a d}{4 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c 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 = 1.56677, size = 506, normalized size = 5.16 \begin{align*} \frac{3 \, b^{2} c^{2} - 4 \, a b c d + 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} + 2 \, b^{2} c d x^{2} + b^{2} c^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (b^{2} d^{2} x^{4} + 2 \, b^{2} c d x^{2} + b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} +{\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{4} + 2 \,{\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c 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 = 4.26234, size = 391, normalized size = 3.99 \begin{align*} \frac{b^{2} \log{\left (x^{2} + \frac{- \frac{a^{4} b^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b^{3} c d^{3}}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{4} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{5} c^{3} d}{\left (a d - b c\right )^{3}} + a b^{2} d - \frac{b^{6} c^{4}}{\left (a d - b c\right )^{3}} + b^{3} c}{2 b^{3} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac{b^{2} \log{\left (x^{2} + \frac{\frac{a^{4} b^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{3} c d^{3}}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{4} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{5} c^{3} d}{\left (a d - b c\right )^{3}} + a b^{2} d + \frac{b^{6} c^{4}}{\left (a d - b c\right )^{3}} + b^{3} c}{2 b^{3} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac{- a d + 3 b c + 2 b d x^{2}}{4 a^{2} c^{2} d^{2} - 8 a b c^{3} d + 4 b^{2} c^{4} + x^{4} \left (4 a^{2} d^{4} - 8 a b c d^{3} + 4 b^{2} c^{2} d^{2}\right ) + x^{2} \left (8 a^{2} c d^{3} - 16 a b c^{2} d^{2} + 8 b^{2} c^{3} d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15733, size = 235, normalized size = 2.4 \begin{align*} \frac{b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac{b^{2} d \log \left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )}} + \frac{3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2} + 2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2}}{4 \,{\left (d x^{2} + c\right )}^{2}{\left (b c - a d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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