Optimal. Leaf size=53 \[ -\frac {c \log (b+c x)}{b (c d-b e)}+\frac {e \log (d+e x)}{d (c d-b e)}+\frac {\log (x)}{b d} \]
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Rubi [A] time = 0.05, antiderivative size = 53, 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 \log (b+c x)}{b (c d-b e)}+\frac {e \log (d+e x)}{d (c d-b e)}+\frac {\log (x)}{b d} \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 )} \, dx &=\int \left (\frac {1}{b d x}+\frac {c^2}{b (-c d+b e) (b+c x)}+\frac {e^2}{d (c d-b e) (d+e x)}\right ) \, dx\\ &=\frac {\log (x)}{b d}-\frac {c \log (b+c x)}{b (c d-b e)}+\frac {e \log (d+e x)}{d (c d-b e)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.91 \begin {gather*} \frac {-c d \log (b+c x)+b e \log (d+e x)-b e \log (x)+c d \log (x)}{b c d^2-b^2 d e} \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 )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.52, size = 50, normalized size = 0.94 \begin {gather*} -\frac {c d \log \left (c x + b\right ) - b e \log \left (e x + d\right ) - {\left (c d - b e\right )} \log \relax (x)}{b c d^{2} - b^{2} d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 67, normalized size = 1.26 \begin {gather*} -\frac {c^{2} \log \left ({\left | c x + b \right |}\right )}{b c^{2} d - b^{2} c e} + \frac {e^{2} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2}} + \frac {\log \left ({\left | x \right |}\right )}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 1.02 \begin {gather*} \frac {c \ln \left (c x +b \right )}{\left (b e -c d \right ) b}-\frac {e \ln \left (e x +d \right )}{\left (b e -c d \right ) d}+\frac {\ln \relax (x )}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 53, normalized size = 1.00 \begin {gather*} -\frac {c \log \left (c x + b\right )}{b c d - b^{2} e} + \frac {e \log \left (e x + d\right )}{c d^{2} - b d e} + \frac {\log \relax (x)}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 93, normalized size = 1.75 \begin {gather*} \frac {e\,\ln \left (\frac {{\left (d+e\,x\right )}^2}{x\,\left (b+c\,x\right )}\right )}{2\,c\,d^2-2\,b\,d\,e}-\frac {\ln \left (\frac {b-\sqrt {b^2}+2\,c\,x}{b+\sqrt {b^2}+2\,c\,x}\right )\,\left (b\,e-2\,c\,d\right )}{\left (2\,c\,d^2-2\,b\,d\,e\right )\,\sqrt {b^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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