Optimal. Leaf size=57 \[ -\frac{1}{(a+b x) (b c-a d)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2} \]
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Rubi [A] time = 0.0301914, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {7, 44} \[ -\frac{1}{(a+b x) (b c-a d)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 7
Rule 44
Rubi steps
\begin{align*} \int \frac{(c+d x)^{1+2 n-2 (1+n)}}{(a+b x)^2} \, dx &=\int \frac{1}{(a+b x)^2 (c+d x)} \, dx\\ &=\int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx\\ &=-\frac{1}{(b c-a d) (a+b x)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.0256779, size = 53, normalized size = 0.93 \[ \frac{d (a+b x) \log (c+d x)-d (a+b x) \log (a+b x)+a d-b c}{(a+b x) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 57, normalized size = 1. \begin{align*}{\frac{d\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}}+{\frac{1}{ \left ( ad-bc \right ) \left ( bx+a \right ) }}-{\frac{d\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 = 0.990379, size = 124, normalized size = 2.18 \begin{align*} -\frac{d \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{d \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{1}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28794, size = 200, normalized size = 3.51 \begin{align*} -\frac{b c - a d +{\left (b d x + a d\right )} \log \left (b x + a\right ) -{\left (b d x + a d\right )} \log \left (d x + c\right )}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.814668, size = 233, normalized size = 4.09 \begin{align*} \frac{d \log{\left (x + \frac{- \frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{\left (a d - b c\right )^{2}} - \frac{d \log{\left (x + \frac{\frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{\left (a d - b c\right )^{2}} + \frac{1}{a^{2} d - a b c + x \left (a b d - b^{2} c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06661, size = 105, normalized size = 1.84 \begin{align*} \frac{b d \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac{b}{{\left (b^{2} c - a b d\right )}{\left (b x + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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