Optimal. Leaf size=61 \[ \frac{F^a (c+d x)^{m+1} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{1}{2} (-m-1),-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]
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Rubi [A] time = 0.0536999, antiderivative size = 61, normalized size of antiderivative = 1., 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} \[ \frac{F^a (c+d x)^{m+1} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{1}{2} (-m-1),-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int F^{a+\frac{b}{(c+d x)^2}} (c+d x)^m \, dx &=\frac{F^a (c+d x)^{1+m} \Gamma \left (\frac{1}{2} (-1-m),-\frac{b \log (F)}{(c+d x)^2}\right ) \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{\frac{1+m}{2}}}{2 d}\\ \end{align*}
Mathematica [A] time = 0.034179, size = 61, normalized size = 1. \[ \frac{F^a (c+d x)^{m+1} \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{1}{2} (-m-1),-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{2}}}} \left ( dx+c \right ) ^{m}\, dx \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 )}^{m} F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x + c\right )}^{m} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}, x\right ) \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 )}^{m} F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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