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 = 0.0254543, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 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 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx &=\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 = 0.0101975, 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 = 0.044, size = 39, normalized size = 1.3 \begin{align*}{\frac{b\ln \left ( dx+c \right ) }{{d}^{2}}}-{\frac{a}{d \left ( dx+c \right ) }}+{\frac{bc}{{d}^{2} \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04856, 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 = 1.58518, 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 = 0.44771, 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 = 1.19807, size = 43, normalized size = 1.39 \begin{align*} \frac{b \log \left ({\left | d x + c \right |}\right )}{d^{2}} + \frac{b c - a d}{{\left (d x + c\right )} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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