Optimal. Leaf size=61 \[ -\frac{c}{d (c+d x) (b c-a d)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]
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Rubi [A] time = 0.0355882, antiderivative size = 61, 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} \[ -\frac{c}{d (c+d x) (b c-a d)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x}{(a+b x) (c+d x)^2} \, dx &=\int \left (-\frac{a b}{(b c-a d)^2 (a+b x)}+\frac{c}{(b c-a d) (c+d x)^2}+\frac{a d}{(-b c+a d)^2 (c+d x)}\right ) \, dx\\ &=-\frac{c}{d (b c-a d) (c+d x)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.0295518, size = 60, normalized size = 0.98 \[ \frac{c}{d (c+d x) (a d-b c)}-\frac{a \log (a+b x)}{(b c-a d)^2}+\frac{a \log (c+d x)}{(b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 61, normalized size = 1. \begin{align*}{\frac{a\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}}+{\frac{c}{d \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{a\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23646, size = 132, normalized size = 2.16 \begin{align*} -\frac{a \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{a \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{c}{b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31251, size = 224, normalized size = 3.67 \begin{align*} -\frac{b c^{2} - a c d +{\left (a d^{2} x + a c d\right )} \log \left (b x + a\right ) -{\left (a d^{2} x + a c d\right )} \log \left (d x + c\right )}{b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.17505, size = 238, normalized size = 3.9 \begin{align*} \frac{a \log{\left (x + \frac{- \frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d + \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{\left (a d - b c\right )^{2}} - \frac{a \log{\left (x + \frac{\frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d - \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{\left (a d - b c\right )^{2}} + \frac{c}{a c d^{2} - b c^{2} d + x \left (a d^{3} - b c d^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19232, size = 115, normalized size = 1.89 \begin{align*} -\frac{\frac{a d^{2} \log \left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac{c d}{{\left (b c d - a d^{2}\right )}{\left (d x + c\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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