Optimal. Leaf size=56 \[ \frac{b F^a \log (F) \text{Ei}\left (b (c+d x)^n \log (F)\right )}{d n}-\frac{(c+d x)^{-n} F^{a+b (c+d x)^n}}{d n} \]
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Rubi [A] time = 0.0725984, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2215, 2210} \[ \frac{b F^a \log (F) \text{Ei}\left (b (c+d x)^n \log (F)\right )}{d n}-\frac{(c+d x)^{-n} F^{a+b (c+d x)^n}}{d n} \]
Antiderivative was successfully verified.
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Rule 2215
Rule 2210
Rubi steps
\begin{align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx &=-\frac{F^{a+b (c+d x)^n} (c+d x)^{-n}}{d n}+(b \log (F)) \int \frac{F^{a+b (c+d x)^n}}{c+d x} \, dx\\ &=-\frac{F^{a+b (c+d x)^n} (c+d x)^{-n}}{d n}+\frac{b F^a \text{Ei}\left (b (c+d x)^n \log (F)\right ) \log (F)}{d n}\\ \end{align*}
Mathematica [A] time = 0.0063399, size = 27, normalized size = 0.48 \[ \frac{b F^a \log (F) \text{Gamma}\left (-1,-b \log (F) (c+d x)^n\right )}{d n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.12, size = 61, normalized size = 1.1 \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{dn \left ( dx+c \right ) ^{n}}}-{\frac{b\ln \left ( F \right ){F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) \right ) }{dn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{-n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50631, size = 147, normalized size = 2.62 \begin{align*} \frac{{\left (d x + c\right )}^{n} F^{a} b{\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right ) - e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{{\left (d x + c\right )}^{n} d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{-n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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