Optimal. Leaf size=91 \[ -\frac{a^2}{b (a+b x) (b c-a d)^2}-\frac{c^2}{d (c+d x) (b c-a d)^2}-\frac{2 a c \log (a+b x)}{(b c-a d)^3}+\frac{2 a c \log (c+d x)}{(b c-a d)^3} \]
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Rubi [A] time = 0.0648745, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ -\frac{a^2}{b (a+b x) (b c-a d)^2}-\frac{c^2}{d (c+d x) (b c-a d)^2}-\frac{2 a c \log (a+b x)}{(b c-a d)^3}+\frac{2 a c \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{x^2}{(a+b x)^2 (c+d x)^2} \, dx &=\int \left (\frac{a^2}{(b c-a d)^2 (a+b x)^2}-\frac{2 a b c}{(b c-a d)^3 (a+b x)}+\frac{c^2}{(b c-a d)^2 (c+d x)^2}-\frac{2 a c d}{(-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac{a^2}{b (b c-a d)^2 (a+b x)}-\frac{c^2}{d (b c-a d)^2 (c+d x)}-\frac{2 a c \log (a+b x)}{(b c-a d)^3}+\frac{2 a c \log (c+d x)}{(b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.148262, size = 71, normalized size = 0.78 \[ \frac{-(b c-a d) \left (\frac{a^2}{b (a+b x)}+\frac{c^2}{d (c+d x)}\right )-2 a c \log (a+b x)+2 a c \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 92, normalized size = 1. \begin{align*} -{\frac{{c}^{2}}{ \left ( ad-bc \right ) ^{2}d \left ( dx+c \right ) }}-2\,{\frac{ac\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}-{\frac{{a}^{2}}{ \left ( ad-bc \right ) ^{2}b \left ( bx+a \right ) }}+2\,{\frac{ac\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.25837, size = 327, normalized size = 3.59 \begin{align*} -\frac{2 \, a c \log \left (b x + 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{2 \, a c \log \left (d x + 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{a b c^{2} + a^{2} c d +{\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} +{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} +{\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.407, size = 597, normalized size = 6.56 \begin{align*} -\frac{a b^{2} c^{3} - a^{3} c d^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x + 2 \,{\left (a b^{2} c d^{2} x^{2} + a^{2} b c^{2} d +{\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (a b^{2} c d^{2} x^{2} + a^{2} b c^{2} d +{\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4} +{\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} +{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.80327, size = 437, normalized size = 4.8 \begin{align*} - \frac{2 a c \log{\left (x + \frac{- \frac{2 a^{5} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{8 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{12 a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + \frac{8 a^{2} b^{3} c^{4} d}{\left (a d - b c\right )^{3}} + 2 a^{2} c d - \frac{2 a b^{4} c^{5}}{\left (a d - b c\right )^{3}} + 2 a b c^{2}}{4 a b c d} \right )}}{\left (a d - b c\right )^{3}} + \frac{2 a c \log{\left (x + \frac{\frac{2 a^{5} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{8 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{12 a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{3}} - \frac{8 a^{2} b^{3} c^{4} d}{\left (a d - b c\right )^{3}} + 2 a^{2} c d + \frac{2 a b^{4} c^{5}}{\left (a d - b c\right )^{3}} + 2 a b c^{2}}{4 a b c d} \right )}}{\left (a d - b c\right )^{3}} - \frac{a^{2} c d + a b c^{2} + x \left (a^{2} d^{2} + b^{2} c^{2}\right )}{a^{3} b c d^{3} - 2 a^{2} b^{2} c^{2} d^{2} + a b^{3} c^{3} d + x^{2} \left (a^{2} b^{2} d^{4} - 2 a b^{3} c d^{3} + b^{4} c^{2} d^{2}\right ) + x \left (a^{3} b d^{4} - a^{2} b^{2} c d^{3} - a b^{3} c^{2} d^{2} + b^{4} c^{3} d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16687, size = 207, normalized size = 2.27 \begin{align*} \frac{2 \, a b c \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + 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{a^{2} b}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (b x + a\right )}} + \frac{b c^{2}}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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