Optimal. Leaf size=52 \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]
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Rubi [A] time = 0.0345223, antiderivative size = 52, 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} \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int \frac{F^{a+b (c+d x)^n}}{(c+d x)^2} \, dx &=-\frac{F^a \Gamma \left (-\frac{1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{\frac{1}{n}}}{d n (c+d x)}\\ \end{align*}
Mathematica [A] time = 0.0120298, size = 52, normalized size = 1. \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{n}}}{ \left ( dx+c \right ) ^{2}}}\, 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 \frac{F^{{\left (d x + c\right )}^{n} b + a}}{{\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 (\frac{F^{{\left (d x + c\right )}^{n} b + a}}{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 \frac{F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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