Optimal. Leaf size=88 \[ -\frac {\log (x) (3 c d-b e)}{b^4}+\frac {(3 c d-b e) \log (b+c x)}{b^4}-\frac {2 c d-b e}{b^3 (b+c x)}-\frac {d}{b^3 x}-\frac {c d-b e}{2 b^2 (b+c x)^2} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {765} \[ -\frac {2 c d-b e}{b^3 (b+c x)}-\frac {c d-b e}{2 b^2 (b+c x)^2}-\frac {\log (x) (3 c d-b e)}{b^4}+\frac {(3 c d-b e) \log (b+c x)}{b^4}-\frac {d}{b^3 x} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x (d+e x)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {d}{b^3 x^2}+\frac {-3 c d+b e}{b^4 x}-\frac {c (-c d+b e)}{b^2 (b+c x)^3}-\frac {c (-2 c d+b e)}{b^3 (b+c x)^2}-\frac {c (-3 c d+b e)}{b^4 (b+c x)}\right ) \, dx\\ &=-\frac {d}{b^3 x}-\frac {c d-b e}{2 b^2 (b+c x)^2}-\frac {2 c d-b e}{b^3 (b+c x)}-\frac {(3 c d-b e) \log (x)}{b^4}+\frac {(3 c d-b e) \log (b+c x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 81, normalized size = 0.92 \[ \frac {\frac {b^2 (b e-c d)}{(b+c x)^2}+\frac {2 b (b e-2 c d)}{b+c x}+2 \log (x) (b e-3 c d)+2 (3 c d-b e) \log (b+c x)-\frac {2 b d}{x}}{2 b^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 195, normalized size = 2.22 \[ -\frac {2 \, b^{3} d + 2 \, {\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + 3 \, {\left (3 \, b^{2} c d - b^{3} e\right )} x - 2 \, {\left ({\left (3 \, c^{3} d - b c^{2} e\right )} x^{3} + 2 \, {\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + {\left (3 \, b^{2} c d - b^{3} e\right )} x\right )} \log \left (c x + b\right ) + 2 \, {\left ({\left (3 \, c^{3} d - b c^{2} e\right )} x^{3} + 2 \, {\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + {\left (3 \, b^{2} c d - b^{3} e\right )} x\right )} \log \relax (x)}{2 \, {\left (b^{4} c^{2} x^{3} + 2 \, b^{5} c x^{2} + b^{6} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 107, normalized size = 1.22 \[ -\frac {{\left (3 \, c d - b e\right )} \log \left ({\left | x \right |}\right )}{b^{4}} + \frac {{\left (3 \, c^{2} d - b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac {2 \, b^{3} d + 2 \, {\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + 3 \, {\left (3 \, b^{2} c d - b^{3} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 105, normalized size = 1.19 \[ \frac {e}{2 \left (c x +b \right )^{2} b}-\frac {c d}{2 \left (c x +b \right )^{2} b^{2}}+\frac {e}{\left (c x +b \right ) b^{2}}-\frac {2 c d}{\left (c x +b \right ) b^{3}}+\frac {e \ln \relax (x )}{b^{3}}-\frac {e \ln \left (c x +b \right )}{b^{3}}-\frac {3 c d \ln \relax (x )}{b^{4}}+\frac {3 c d \ln \left (c x +b \right )}{b^{4}}-\frac {d}{b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 104, normalized size = 1.18 \[ -\frac {2 \, b^{2} d + 2 \, {\left (3 \, c^{2} d - b c e\right )} x^{2} + 3 \, {\left (3 \, b c d - b^{2} e\right )} x}{2 \, {\left (b^{3} c^{2} x^{3} + 2 \, b^{4} c x^{2} + b^{5} x\right )}} + \frac {{\left (3 \, c d - b e\right )} \log \left (c x + b\right )}{b^{4}} - \frac {{\left (3 \, c d - b e\right )} \log \relax (x)}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 84, normalized size = 0.95 \[ \frac {\frac {3\,x\,\left (b\,e-3\,c\,d\right )}{2\,b^2}-\frac {d}{b}+\frac {c\,x^2\,\left (b\,e-3\,c\,d\right )}{b^3}}{b^2\,x+2\,b\,c\,x^2+c^2\,x^3}-\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )\,\left (b\,e-3\,c\,d\right )}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.64, size = 168, normalized size = 1.91 \[ \frac {- 2 b^{2} d + x^{2} \left (2 b c e - 6 c^{2} d\right ) + x \left (3 b^{2} e - 9 b c d\right )}{2 b^{5} x + 4 b^{4} c x^{2} + 2 b^{3} c^{2} x^{3}} + \frac {\left (b e - 3 c d\right ) \log {\left (x + \frac {b^{2} e - 3 b c d - b \left (b e - 3 c d\right )}{2 b c e - 6 c^{2} d} \right )}}{b^{4}} - \frac {\left (b e - 3 c d\right ) \log {\left (x + \frac {b^{2} e - 3 b c d + b \left (b e - 3 c d\right )}{2 b c e - 6 c^{2} d} \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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