Optimal. Leaf size=110 \[ \frac {c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}-\frac {\log (x) (b e+2 c d)}{b^3 d^2}-\frac {c^2}{b^2 (b+c x) (c d-b e)}-\frac {1}{b^2 d x}+\frac {e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
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Rubi [A] time = 0.12, antiderivative size = 110, 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 {c^2}{b^2 (b+c x) (c d-b e)}+\frac {c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}-\frac {\log (x) (b e+2 c d)}{b^3 d^2}-\frac {1}{b^2 d x}+\frac {e^3 \log (d+e x)}{d^2 (c d-b e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {1}{b^2 d x^2}+\frac {-2 c d-b e}{b^3 d^2 x}-\frac {c^3}{b^2 (-c d+b e) (b+c x)^2}-\frac {c^3 (-2 c d+3 b e)}{b^3 (-c d+b e)^2 (b+c x)}+\frac {e^4}{d^2 (c d-b e)^2 (d+e x)}\right ) \, dx\\ &=-\frac {1}{b^2 d x}-\frac {c^2}{b^2 (c d-b e) (b+c x)}-\frac {(2 c d+b e) \log (x)}{b^3 d^2}+\frac {c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}+\frac {e^3 \log (d+e x)}{d^2 (c d-b e)^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 111, normalized size = 1.01 \begin {gather*} \frac {\left (2 c^3 d-3 b c^2 e\right ) \log (b+c x)}{b^3 (b e-c d)^2}+\frac {\log (x) (-b e-2 c d)}{b^3 d^2}+\frac {c^2}{b^2 (b+c x) (b e-c d)}-\frac {1}{b^2 d x}+\frac {e^3 \log (d+e x)}{d^2 (c d-b e)^2} \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) \left (b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 4.28, size = 288, normalized size = 2.62 \begin {gather*} -\frac {b^{2} c^{2} d^{3} - 2 \, b^{3} c d^{2} e + b^{4} d e^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{3} c d e^{2}\right )} x - {\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e\right )} x^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e\right )} x\right )} \log \left (c x + b\right ) - {\left (b^{3} c e^{3} x^{2} + b^{4} e^{3} x\right )} \log \left (e x + d\right ) + {\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{3} c e^{3}\right )} x^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{4} e^{3}\right )} x\right )} \log \relax (x)}{{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e + b^{5} c d^{2} e^{2}\right )} x^{2} + {\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 202, normalized size = 1.84 \begin {gather*} \frac {{\left (2 \, c^{4} d - 3 \, b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3} d^{2} - 2 \, b^{4} c^{2} d e + b^{5} c e^{2}} + \frac {e^{4} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}} - \frac {{\left (2 \, c d + b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{2}} - \frac {b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2}\right )} x}{{\left (c d - b e\right )}^{2} {\left (c x + b\right )} b^{2} d^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 132, normalized size = 1.20 \begin {gather*} -\frac {3 c^{2} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{2} b^{2}}+\frac {2 c^{3} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{2} b^{3}}+\frac {e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{2} d^{2}}+\frac {c^{2}}{\left (b e -c d \right ) \left (c x +b \right ) b^{2}}-\frac {e \ln \relax (x )}{b^{2} d^{2}}-\frac {2 c \ln \relax (x )}{b^{3} d}-\frac {1}{b^{2} d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 177, normalized size = 1.61 \begin {gather*} \frac {e^{3} \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}} + \frac {{\left (2 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}} - \frac {b c d - b^{2} e + {\left (2 \, c^{2} d - b c e\right )} x}{{\left (b^{2} c^{2} d^{2} - b^{3} c d e\right )} x^{2} + {\left (b^{3} c d^{2} - b^{4} d e\right )} x} - \frac {{\left (2 \, c d + b e\right )} \log \relax (x)}{b^{3} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 143, normalized size = 1.30 \begin {gather*} \frac {\ln \left (b+c\,x\right )\,\left (2\,c^3\,d-3\,b\,c^2\,e\right )}{b^5\,e^2-2\,b^4\,c\,d\,e+b^3\,c^2\,d^2}-\frac {\frac {1}{b\,d}-\frac {x\,\left (2\,c^2\,d-b\,c\,e\right )}{b^2\,d\,\left (b\,e-c\,d\right )}}{c\,x^2+b\,x}+\frac {e^3\,\ln \left (d+e\,x\right )}{d^2\,{\left (b\,e-c\,d\right )}^2}-\frac {\ln \relax (x)\,\left (b\,e+2\,c\,d\right )}{b^3\,d^2} \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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