Optimal. Leaf size=71 \[ \frac {e^{-\frac {a}{b}} i \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^2}+\frac {(f h-e i) \log (a+b \log (c (e+f x)))}{b d f^2} \]
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Rubi [A]
time = 0.15, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2458, 12, 2395,
2336, 2209, 2339, 29} \begin {gather*} \frac {i e^{-\frac {a}{b}} \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^2}+\frac {(f h-e i) \log (a+b \log (c (e+f x)))}{b d f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2209
Rule 2336
Rule 2339
Rule 2395
Rule 2458
Rubi steps
\begin {align*} \int \frac {h+194 x}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac {\text {Subst}\left (\int \frac {\frac {-194 e+f h}{f}+\frac {194 x}{f}}{d x (a+b \log (c x))} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {\frac {-194 e+f h}{f}+\frac {194 x}{f}}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\text {Subst}\left (\int \left (\frac {194}{f (a+b \log (c x))}+\frac {-194 e+f h}{f x (a+b \log (c x))}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac {194 \text {Subst}\left (\int \frac {1}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^2}-\frac {(194 e-f h) \text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac {194 \text {Subst}\left (\int \frac {e^x}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c d f^2}-\frac {(194 e-f h) \text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d f^2}\\ &=\frac {194 e^{-\frac {a}{b}} \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^2}-\frac {(194 e-f h) \log (a+b \log (c (e+f x)))}{b d f^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 76, normalized size = 1.07 \begin {gather*} \frac {e^{-\frac {a}{b}} i \text {Ei}\left (\frac {a}{b}+\log (c (e+f x))\right )-c e i \log (a+b \log (c (e+f x)))+c f h \log (f (a+b \log (c (e+f x))))}{b c d f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.04, size = 88, normalized size = 1.24
method | result | size |
derivativedivides | \(-\frac {\frac {i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {h c f \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c e i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c \,f^{2} d}\) | \(88\) |
default | \(-\frac {\frac {i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {h c f \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c e i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c \,f^{2} d}\) | \(88\) |
risch | \(-\frac {i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{2} c b}+\frac {h \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d f b}-\frac {e i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{2} b}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 76, normalized size = 1.07 \begin {gather*} \frac {{\left ({\left (c f h - i \, c e\right )} e^{\frac {a}{b}} \log \left (\frac {b \log \left (c f x + c e\right ) + a}{b}\right ) + i \, \operatorname {log\_integral}\left ({\left (c f x + c e\right )} e^{\frac {a}{b}}\right )\right )} e^{\left (-\frac {a}{b}\right )}}{b c d f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {h}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {i x}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {h+i\,x}{\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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