Optimal. Leaf size=50 \[ \frac {F^a (c+d x)^{m+1} \left (-\frac {b \log (F)}{c+d x}\right )^{m+1} \Gamma \left (-m-1,-\frac {b \log (F)}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ \frac {F^a (c+d x)^{m+1} \left (-\frac {b \log (F)}{c+d x}\right )^{m+1} \text {Gamma}\left (-m-1,-\frac {b \log (F)}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int F^{a+\frac {b}{c+d x}} (c+d x)^m \, dx &=\frac {F^a (c+d x)^{1+m} \Gamma \left (-1-m,-\frac {b \log (F)}{c+d x}\right ) \left (-\frac {b \log (F)}{c+d x}\right )^{1+m}}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 1.00 \[ \frac {F^a (c+d x)^{m+1} \left (-\frac {b \log (F)}{c+d x}\right )^{m+1} \Gamma \left (-m-1,-\frac {b \log (F)}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )}^{m} F^{\frac {a d x + a c + b}{d x + c}}, x\right ) \]
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 \[ \int {\left (d x + c\right )}^{m} F^{a + \frac {b}{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int F^{a +\frac {b}{d x +c}} \left (d x +c \right )^{m}\, dx \]
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 \[ \int {\left (d x + c\right )}^{m} F^{a + \frac {b}{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 73, normalized size = 1.46 \[ \frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \relax (F)}{2\,\left (c+d\,x\right )}}\,{\left (c+d\,x\right )}^{m+1}\,{\mathrm {M}}_{\frac {m}{2}+1,-\frac {m}{2}-\frac {1}{2}}\left (\frac {b\,\ln \relax (F)}{c+d\,x}\right )\,{\left (\frac {b\,\ln \relax (F)}{c+d\,x}\right )}^{m/2}}{d\,\left (m+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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