Optimal. Leaf size=31 \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]
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Rubi [A] time = 0.0208401, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{(c+d x)^2} \, dx &=\int \left (\frac{-b c+a d}{d (c+d x)^2}+\frac{b}{d (c+d x)}\right ) \, dx\\ &=\frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0101126, size = 31, normalized size = 1. \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 39, normalized size = 1.3 \begin{align*} -{\frac{a}{d \left ( dx+c \right ) }}+{\frac{bc}{{d}^{2} \left ( dx+c \right ) }}+{\frac{b\ln \left ( dx+c \right ) }{{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943292, size = 46, normalized size = 1.48 \begin{align*} \frac{b c - a d}{d^{3} x + c d^{2}} + \frac{b \log \left (d x + c\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62261, size = 78, normalized size = 2.52 \begin{align*} \frac{b c - a d +{\left (b d x + b c\right )} \log \left (d x + c\right )}{d^{3} x + c d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.336933, size = 27, normalized size = 0.87 \begin{align*} \frac{b \log{\left (c + d x \right )}}{d^{2}} - \frac{a d - b c}{c d^{2} + d^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0497, size = 77, normalized size = 2.48 \begin{align*} -\frac{b{\left (\frac{\log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d} - \frac{c}{{\left (d x + c\right )} d}\right )}}{d} - \frac{a}{{\left (d x + c\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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