Optimal. Leaf size=85 \[ -\frac{a}{(c+d x) (b c-a d)^2}-\frac{c}{2 d (c+d x)^2 (b c-a d)}-\frac{a b \log (a+b x)}{(b c-a d)^3}+\frac{a b \log (c+d x)}{(b c-a d)^3} \]
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Rubi [A] time = 0.0523724, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ -\frac{a}{(c+d x) (b c-a d)^2}-\frac{c}{2 d (c+d x)^2 (b c-a d)}-\frac{a b \log (a+b x)}{(b c-a d)^3}+\frac{a b \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x}{(a+b x) (c+d x)^3} \, dx &=\int \left (-\frac{a b^2}{(b c-a d)^3 (a+b x)}+\frac{c}{(b c-a d) (c+d x)^3}+\frac{a d}{(-b c+a d)^2 (c+d x)^2}-\frac{a b d}{(-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac{c}{2 d (b c-a d) (c+d x)^2}-\frac{a}{(b c-a d)^2 (c+d x)}-\frac{a b \log (a+b x)}{(b c-a d)^3}+\frac{a b \log (c+d x)}{(b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.045354, size = 85, normalized size = 1. \[ -\frac{a}{(c+d x) (b c-a d)^2}+\frac{c}{2 d (c+d x)^2 (a d-b c)}-\frac{a b \log (a+b x)}{(b c-a d)^3}+\frac{a b \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 84, normalized size = 1. \begin{align*} -{\frac{a}{ \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+{\frac{c}{ \left ( 2\,ad-2\,bc \right ) d \left ( dx+c \right ) ^{2}}}-{\frac{ab\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}+{\frac{ab\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 = 1.19763, size = 281, normalized size = 3.31 \begin{align*} -\frac{a b \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{a b \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{2 \, a d^{2} x + b c^{2} + a c d}{2 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} +{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.42468, size = 497, normalized size = 5.85 \begin{align*} -\frac{b^{2} c^{3} - a^{2} c d^{2} + 2 \,{\left (a b c d^{2} - a^{2} d^{3}\right )} x + 2 \,{\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (b x + a\right ) - 2 \,{\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (d x + c\right )}{2 \,{\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4} +{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{2} + 2 \,{\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.72268, size = 400, normalized size = 4.71 \begin{align*} - \frac{a b \log{\left (x + \frac{- \frac{a^{5} b d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} - \frac{6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac{4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d - \frac{a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} + \frac{a b \log{\left (x + \frac{\frac{a^{5} b d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} + \frac{6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac{4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d + \frac{a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} - \frac{a c d + 2 a d^{2} x + b c^{2}}{2 a^{2} c^{2} d^{3} - 4 a b c^{3} d^{2} + 2 b^{2} c^{4} d + x^{2} \left (2 a^{2} d^{5} - 4 a b c d^{4} + 2 b^{2} c^{2} d^{3}\right ) + x \left (4 a^{2} c d^{4} - 8 a b c^{2} d^{3} + 4 b^{2} c^{3} d^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16943, size = 223, normalized size = 2.62 \begin{align*} -\frac{a b^{2} \log \left ({\left | b x + a \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 b d \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac{b^{2} c^{3} - a^{2} c d^{2} + 2 \,{\left (a b c d^{2} - a^{2} d^{3}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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