Optimal. Leaf size=68 \[ \frac {(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac {(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac {A \log (x)}{b d} \]
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Rubi [A] time = 0.07, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac {(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac {A \log (x)}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )} \, dx &=\int \left (\frac {A}{b d x}-\frac {c (b B-A c)}{b (-c d+b e) (b+c x)}-\frac {e (B d-A e)}{d (c d-b e) (d+e x)}\right ) \, dx\\ &=\frac {A \log (x)}{b d}+\frac {(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac {(B d-A e) \log (d+e x)}{d (c d-b e)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.93 \begin {gather*} \frac {\log (b+c x) (A c d-b B d)+b (B d-A e) \log (d+e x)+A \log (x) (b e-c d)}{b d (b e-c d)} \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 {A+B x}{(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 = 1.54, size = 65, normalized size = 0.96 \begin {gather*} \frac {{\left (B b - A c\right )} d \log \left (c x + b\right ) - {\left (B b d - A b e\right )} \log \left (e x + d\right ) + {\left (A c d - A 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 = 0.16, size = 85, normalized size = 1.25 \begin {gather*} \frac {{\left (B b c - A c^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{2} d - b^{2} c e} - \frac {{\left (B d e - A e^{2}\right )} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2}} + \frac {A \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 = 0.06, size = 94, normalized size = 1.38 \begin {gather*} \frac {A c \ln \left (c x +b \right )}{\left (b e -c d \right ) b}-\frac {A e \ln \left (e x +d \right )}{\left (b e -c d \right ) d}-\frac {B \ln \left (c x +b \right )}{b e -c d}+\frac {B \ln \left (e x +d \right )}{b e -c d}+\frac {A \ln \relax (x )}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 68, normalized size = 1.00 \begin {gather*} \frac {{\left (B b - A c\right )} \log \left (c x + b\right )}{b c d - b^{2} e} - \frac {{\left (B d - A e\right )} \log \left (e x + d\right )}{c d^{2} - b d e} + \frac {A \log \relax (x)}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 67, normalized size = 0.99 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,e-B\,d\right )}{c\,d^2-b\,d\,e}+\frac {\ln \left (b+c\,x\right )\,\left (A\,c-B\,b\right )}{b^2\,e-b\,c\,d}+\frac {A\,\ln \relax (x)}{b\,d} \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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