Optimal. Leaf size=42 \[ -\frac {d \log (b+c x)}{b^2}+\frac {d \log (x)}{b^2}+\frac {c d-b e}{b c (b+c x)} \]
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Rubi [A] time = 0.03, antiderivative size = 42, 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} \begin {gather*} -\frac {d \log (b+c x)}{b^2}+\frac {d \log (x)}{b^2}+\frac {c d-b e}{b c (b+c x)} \end {gather*}
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 )^2} \, dx &=\int \left (\frac {d}{b^2 x}+\frac {-c d+b e}{b (b+c x)^2}-\frac {c d}{b^2 (b+c x)}\right ) \, dx\\ &=\frac {c d-b e}{b c (b+c x)}+\frac {d \log (x)}{b^2}-\frac {d \log (b+c x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.90 \begin {gather*} \frac {\frac {b (c d-b e)}{c (b+c x)}-d \log (b+c x)+d \log (x)}{b^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 {x (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 [A] time = 0.39, size = 61, normalized size = 1.45 \begin {gather*} \frac {b c d - b^{2} e - {\left (c^{2} d x + b c d\right )} \log \left (c x + b\right ) + {\left (c^{2} d x + b c d\right )} \log \relax (x)}{b^{2} c^{2} x + b^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 1.14 \begin {gather*} -\frac {d \log \left ({\left | c x + b \right |}\right )}{b^{2}} + \frac {d \log \left ({\left | x \right |}\right )}{b^{2}} + \frac {b c d - b^{2} e}{{\left (c x + b\right )} b^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 46, normalized size = 1.10 \begin {gather*} \frac {d}{\left (c x +b \right ) b}+\frac {d \ln \relax (x )}{b^{2}}-\frac {d \ln \left (c x +b \right )}{b^{2}}-\frac {e}{\left (c x +b \right ) c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 43, normalized size = 1.02 \begin {gather*} \frac {c d - b e}{b c^{2} x + b^{2} c} - \frac {d \log \left (c x + b\right )}{b^{2}} + \frac {d \log \relax (x)}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 40, normalized size = 0.95 \begin {gather*} -\frac {2\,d\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b^2}-\frac {b\,e-c\,d}{b\,c\,\left (b+c\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 32, normalized size = 0.76 \begin {gather*} \frac {- b e + c d}{b^{2} c + b c^{2} x} + \frac {d \left (\log {\relax (x )} - \log {\left (\frac {b}{c} + x \right )}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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