Optimal. Leaf size=134 \[ \frac {e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}-\frac {c^3 \log (b+c x)}{b (c d-b e)^3}-\frac {e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}-\frac {e}{2 d (d+e x)^2 (c d-b e)}+\frac {\log (x)}{b d^3} \]
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Rubi [A] time = 0.12, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \begin {gather*} \frac {e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}-\frac {c^3 \log (b+c x)}{b (c d-b e)^3}-\frac {e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}-\frac {e}{2 d (d+e x)^2 (c d-b e)}+\frac {\log (x)}{b d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx &=\int \left (\frac {1}{b d^3 x}+\frac {c^4}{b (-c d+b e)^3 (b+c x)}+\frac {e^2}{d (c d-b e) (d+e x)^3}+\frac {e^2 (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)^2}+\frac {e^2 \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {e}{2 d (c d-b e) (d+e x)^2}-\frac {e (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {\log (x)}{b d^3}-\frac {c^3 \log (b+c x)}{b (c d-b e)^3}+\frac {e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 116, normalized size = 0.87 \begin {gather*} \frac {\frac {e \left (\frac {d (c d-b e) (c d (5 d+4 e x)-b e (3 d+2 e x))}{(d+e x)^2}-2 \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)\right )}{d^3}+\frac {2 c^3 \log (b+c x)}{b}}{2 (b e-c d)^3}+\frac {\log (x)}{b d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d+e x)^3 \left (b x+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 7.41, size = 506, normalized size = 3.78 \begin {gather*} -\frac {5 \, b c^{2} d^{4} e - 8 \, b^{2} c d^{3} e^{2} + 3 \, b^{3} d^{2} e^{3} + 2 \, {\left (2 \, b c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{2} e^{3} + b^{3} d e^{4}\right )} x + 2 \, {\left (c^{3} d^{3} e^{2} x^{2} + 2 \, c^{3} d^{4} e x + c^{3} d^{5}\right )} \log \left (c x + b\right ) - 2 \, {\left (3 \, b c^{2} d^{4} e - 3 \, b^{2} c d^{3} e^{2} + b^{3} d^{2} e^{3} + {\left (3 \, b c^{2} d^{2} e^{3} - 3 \, b^{2} c d e^{4} + b^{3} e^{5}\right )} x^{2} + 2 \, {\left (3 \, b c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{2} e^{3} + b^{3} d e^{4}\right )} x\right )} \log \left (e x + d\right ) - 2 \, {\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3} + {\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \log \relax (x)}{2 \, {\left (b c^{3} d^{8} - 3 \, b^{2} c^{2} d^{7} e + 3 \, b^{3} c d^{6} e^{2} - b^{4} d^{5} e^{3} + {\left (b c^{3} d^{6} e^{2} - 3 \, b^{2} c^{2} d^{5} e^{3} + 3 \, b^{3} c d^{4} e^{4} - b^{4} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (b c^{3} d^{7} e - 3 \, b^{2} c^{2} d^{6} e^{2} + 3 \, b^{3} c d^{5} e^{3} - b^{4} d^{4} e^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 227, normalized size = 1.69 \begin {gather*} -\frac {c^{4} \log \left ({\left | c x + b \right |}\right )}{b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} - b^{4} c e^{3}} + \frac {{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac {\log \left ({\left | x \right |}\right )}{b d^{3}} - \frac {5 \, c^{2} d^{4} e - 8 \, b c d^{3} e^{2} + 3 \, b^{2} d^{2} e^{3} + 2 \, {\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (x e + d\right )}^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 184, normalized size = 1.37 \begin {gather*} -\frac {b^{2} e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}+\frac {3 b c \,e^{2} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{2}}+\frac {c^{3} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b}-\frac {3 c^{2} e \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d}+\frac {b \,e^{2}}{\left (b e -c d \right )^{2} \left (e x +d \right ) d^{2}}-\frac {2 c e}{\left (b e -c d \right )^{2} \left (e x +d \right ) d}+\frac {e}{2 \left (b e -c d \right ) \left (e x +d \right )^{2} d}+\frac {\ln \relax (x )}{b \,d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.52, size = 266, normalized size = 1.99 \begin {gather*} -\frac {c^{3} \log \left (c x + b\right )}{b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}} + \frac {{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {5 \, c d^{2} e - 3 \, b d e^{2} + 2 \, {\left (2 \, c d e^{2} - b e^{3}\right )} x}{2 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}} + \frac {\log \relax (x)}{b d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 235, normalized size = 1.75 \begin {gather*} \frac {\frac {3\,b\,e^2-5\,c\,d\,e}{2\,d\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {e^2\,x\,\left (b\,e-2\,c\,d\right )}{d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{d^2+2\,d\,e\,x+e^2\,x^2}+\frac {c^3\,\ln \left (b+c\,x\right )}{b^4\,e^3-3\,b^3\,c\,d\,e^2+3\,b^2\,c^2\,d^2\,e-b\,c^3\,d^3}+\frac {\ln \left (d+e\,x\right )\,\left (b^2\,e^3-3\,b\,c\,d\,e^2+3\,c^2\,d^2\,e\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}+\frac {\ln \relax (x)}{b\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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