Optimal. Leaf size=62 \[ \frac{c d-b e}{b^2 x}+\frac{c \log (x) (c d-b e)}{b^3}-\frac{c (c d-b e) \log (b+c x)}{b^3}-\frac{d}{2 b x^2} \]
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Rubi [A] time = 0.0483203, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{c d-b e}{b^2 x}+\frac{c \log (x) (c d-b e)}{b^3}-\frac{c (c d-b e) \log (b+c x)}{b^3}-\frac{d}{2 b x^2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{d+e x}{x^2 \left (b x+c x^2\right )} \, dx &=\int \left (\frac{d}{b x^3}+\frac{-c d+b e}{b^2 x^2}-\frac{c (-c d+b e)}{b^3 x}+\frac{c^2 (-c d+b e)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac{d}{2 b x^2}+\frac{c d-b e}{b^2 x}+\frac{c (c d-b e) \log (x)}{b^3}-\frac{c (c d-b e) \log (b+c x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0318751, size = 58, normalized size = 0.94 \[ \frac{-\frac{b (b d+2 b e x-2 c d x)}{x^2}+2 c \log (x) (c d-b e)+2 c (b e-c d) \log (b+c x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 75, normalized size = 1.2 \begin{align*} -{\frac{d}{2\,b{x}^{2}}}-{\frac{e}{bx}}+{\frac{cd}{{b}^{2}x}}-{\frac{c\ln \left ( x \right ) e}{{b}^{2}}}+{\frac{{c}^{2}\ln \left ( x \right ) d}{{b}^{3}}}+{\frac{c\ln \left ( cx+b \right ) e}{{b}^{2}}}-{\frac{{c}^{2}\ln \left ( cx+b \right ) d}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14581, size = 85, normalized size = 1.37 \begin{align*} -\frac{{\left (c^{2} d - b c e\right )} \log \left (c x + b\right )}{b^{3}} + \frac{{\left (c^{2} d - b c e\right )} \log \left (x\right )}{b^{3}} - \frac{b d - 2 \,{\left (c d - b e\right )} x}{2 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78322, size = 154, normalized size = 2.48 \begin{align*} -\frac{2 \,{\left (c^{2} d - b c e\right )} x^{2} \log \left (c x + b\right ) - 2 \,{\left (c^{2} d - b c e\right )} x^{2} \log \left (x\right ) + b^{2} d - 2 \,{\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.793768, size = 131, normalized size = 2.11 \begin{align*} - \frac{b d + x \left (2 b e - 2 c d\right )}{2 b^{2} x^{2}} - \frac{c \left (b e - c d\right ) \log{\left (x + \frac{b^{2} c e - b c^{2} d - b c \left (b e - c d\right )}{2 b c^{2} e - 2 c^{3} d} \right )}}{b^{3}} + \frac{c \left (b e - c d\right ) \log{\left (x + \frac{b^{2} c e - b c^{2} d + b c \left (b e - c d\right )}{2 b c^{2} e - 2 c^{3} d} \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24864, size = 105, normalized size = 1.69 \begin{align*} \frac{{\left (c^{2} d - b c e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{{\left (c^{3} d - b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} - \frac{b^{2} d - 2 \,{\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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