Optimal. Leaf size=31 \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^n\right )}{d n} \]
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Rubi [A] time = 0.0392378, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2218} \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^n\right )}{d n} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1-5 n} \, dx &=\frac{b^5 F^a \Gamma \left (-5,-b (c+d x)^n \log (F)\right ) \log ^5(F)}{d n}\\ \end{align*}
Mathematica [A] time = 0.0072041, size = 31, normalized size = 1. \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^n\right )}{d n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.095, size = 213, normalized size = 6.9 \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{5\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{5}}}-{\frac{b\ln \left ( F \right ){F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{20\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{4}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{60\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{3}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{120\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{2}}}-{\frac{{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{120\,dn \left ( dx+c \right ) ^{n}}}-{\frac{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}{F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) \right ) }{120\,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 )}^{-5 \, 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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 )}^{-5 \, 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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