Optimal. Leaf size=30 \[ \frac {d \log (x)}{b}-\frac {(c d-b e) \log (b+c x)}{b c} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {631} \begin {gather*} \frac {d \log (x)}{b}-\frac {(c d-b e) \log (b+c x)}{b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int \frac {d+e x}{b x+c x^2} \, dx &=\int \left (\frac {d}{b x}+\frac {-c d+b e}{b (b+c x)}\right ) \, dx\\ &=\frac {d \log (x)}{b}-\frac {(c d-b e) \log (b+c x)}{b c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.97 \begin {gather*} \frac {(b e-c d) \log (b+c x)}{b c}+\frac {d \log (x)}{b} \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 {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 = 0.41, size = 29, normalized size = 0.97 \begin {gather*} \frac {c d \log \relax (x) - {\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 33, normalized size = 1.10 \begin {gather*} \frac {d \log \left ({\left | x \right |}\right )}{b} - \frac {{\left (c d - b e\right )} \log \left ({\left | c x + b \right |}\right )}{b c} \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 = 1.07 \begin {gather*} \frac {d \ln \relax (x )}{b}-\frac {d \ln \left (c x +b \right )}{b}+\frac {e \ln \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 = 1.33, size = 30, normalized size = 1.00 \begin {gather*} \frac {d \log \relax (x)}{b} - \frac {{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 28, normalized size = 0.93 \begin {gather*} \frac {d\,\ln \relax (x)}{b}-\ln \left (b+c\,x\right )\,\left (\frac {d}{b}-\frac {e}{c}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 41, normalized size = 1.37 \begin {gather*} \frac {d \log {\relax (x )}}{b} + \frac {\left (b e - c d\right ) \log {\left (x + \frac {- b d + \frac {b \left (b e - c d\right )}{c}}{b e - 2 c d} \right )}}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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