Optimal. Leaf size=25 \[ \frac {(c d-b e) \log (b+c x)}{c^2}+\frac {e x}{c} \]
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Rubi [A] time = 0.02, antiderivative size = 25, 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 {(c d-b e) \log (b+c x)}{c^2}+\frac {e x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x (d+e x)}{b x+c x^2} \, dx &=\int \left (\frac {e}{c}+\frac {c d-b e}{c (b+c x)}\right ) \, dx\\ &=\frac {e x}{c}+\frac {(c d-b e) \log (b+c x)}{c^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {(c d-b e) \log (b+c x)}{c^2}+\frac {e x}{c} \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)}{b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 24, normalized size = 0.96 \begin {gather*} \frac {c e x + {\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 28, normalized size = 1.12 \begin {gather*} \frac {x e}{c} + \frac {{\left (c d - b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.28 \begin {gather*} -\frac {b e \ln \left (c x +b \right )}{c^{2}}+\frac {d \ln \left (c x +b \right )}{c}+\frac {e x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 25, normalized size = 1.00 \begin {gather*} \frac {e x}{c} + \frac {{\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 26, normalized size = 1.04 \begin {gather*} \frac {e\,x}{c}-\frac {\ln \left (b+c\,x\right )\,\left (b\,e-c\,d\right )}{c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 20, normalized size = 0.80 \begin {gather*} \frac {e x}{c} - \frac {\left (b e - c d\right ) \log {\left (b + c x \right )}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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