Optimal. Leaf size=31 \[ \frac {F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{e} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2209}
\begin {gather*} \frac {F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rubi steps
\begin {align*} \int \frac {F^{c (a+b x)}}{d+e x} \, dx &=\frac {F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{e}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 31, normalized size = 1.00 \begin {gather*} \frac {F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 56, normalized size = 1.81
method | result | size |
risch | \(-\frac {F^{\frac {c \left (a e -b d \right )}{e}} \expIntegral \left (1, -b c x \ln \left (F \right )-c a \ln \left (F \right )-\frac {-\ln \left (F \right ) a c e +\ln \left (F \right ) b c d}{e}\right )}{e}\) | \(56\) |
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 = 0.37, size = 38, normalized size = 1.23 \begin {gather*} \frac {{\rm Ei}\left ({\left (b c x e + b c d\right )} e^{\left (-1\right )} \log \left (F\right )\right ) e^{\left (-1\right )}}{F^{{\left (b c d - a c e\right )} e^{\left (-1\right )}}} \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*} \int \frac {F^{c \left (a + b x\right )}}{d + e x}\, dx \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 = -0.03 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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