Optimal. Leaf size=65 \[ -\frac{c d-b e}{b^2 (b+c x)}-\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{d}{b^2 x} \]
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Rubi [A] time = 0.049319, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {631} \[ -\frac{c d-b e}{b^2 (b+c x)}-\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{d}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{d}{b^2 x^2}+\frac{-2 c d+b e}{b^3 x}-\frac{c (-c d+b e)}{b^2 (b+c x)^2}-\frac{c (-2 c d+b e)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac{d}{b^2 x}-\frac{c d-b e}{b^2 (b+c x)}-\frac{(2 c d-b e) \log (x)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.038353, size = 56, normalized size = 0.86 \[ \frac{\frac{b (b e-c d)}{b+c x}+\log (x) (b e-2 c d)+(2 c d-b e) \log (b+c x)-\frac{b d}{x}}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 78, normalized size = 1.2 \begin{align*} -{\frac{d}{{b}^{2}x}}+{\frac{e\ln \left ( x \right ) }{{b}^{2}}}-2\,{\frac{\ln \left ( x \right ) cd}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) e}{{b}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) cd}{{b}^{3}}}+{\frac{e}{b \left ( cx+b \right ) }}-{\frac{cd}{{b}^{2} \left ( cx+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09162, size = 93, normalized size = 1.43 \begin{align*} -\frac{b d +{\left (2 \, c d - b e\right )} x}{b^{2} c x^{2} + b^{3} x} + \frac{{\left (2 \, c d - b e\right )} \log \left (c x + b\right )}{b^{3}} - \frac{{\left (2 \, c d - b e\right )} \log \left (x\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94649, size = 227, normalized size = 3.49 \begin{align*} -\frac{b^{2} d +{\left (2 \, b c d - b^{2} e\right )} x -{\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (c x + b\right ) +{\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.893364, size = 128, normalized size = 1.97 \begin{align*} \frac{- b d + x \left (b e - 2 c d\right )}{b^{3} x + b^{2} c x^{2}} + \frac{\left (b e - 2 c d\right ) \log{\left (x + \frac{b^{2} e - 2 b c d - b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} - \frac{\left (b e - 2 c d\right ) \log{\left (x + \frac{b^{2} e - 2 b c d + b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12824, size = 104, normalized size = 1.6 \begin{align*} -\frac{{\left (2 \, c d - b e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c d x - b x e + b d}{{\left (c x^{2} + b x\right )} b^{2}} + \frac{{\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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