Optimal. Leaf size=61 \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b \log (F) (c+d x)^2\right )}{2 d} \]
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Rubi [A] time = 0.0650038, 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} \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b \log (F) (c+d x)^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int F^{a+b (c+d x)^2} (c+d x)^m \, dx &=-\frac{F^a (c+d x)^{1+m} \Gamma \left (\frac{1+m}{2},-b (c+d x)^2 \log (F)\right ) \left (-b (c+d x)^2 \log (F)\right )^{\frac{1}{2} (-1-m)}}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0357247, size = 61, normalized size = 1. \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^2\right )^{\frac{1}{2} (-m-1)} \text{Gamma}\left (\frac{m+1}{2},-b \log (F) (c+d x)^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+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^{{\left (d x + c\right )}^{2} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58731, size = 163, normalized size = 2.67 \begin{align*} \frac{e^{\left (-\frac{1}{2} \,{\left (m - 1\right )} \log \left (-b \log \left (F\right )\right ) + a \log \left (F\right )\right )} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, -{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right )}{2 \, b d \log \left (F\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^{{\left (d x + c\right )}^{2} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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