Optimal. Leaf size=54 \[ -\frac{F^a (c+d x)^3 \left (-b \log (F) (c+d x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b \log (F) (c+d x)^n\right )}{d n} \]
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Rubi [A] time = 0.0357216, antiderivative size = 54, 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} \[ -\frac{F^a (c+d x)^3 \left (-b \log (F) (c+d x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b \log (F) (c+d x)^n\right )}{d n} \]
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)^2 \, dx &=-\frac{F^a (c+d x)^3 \Gamma \left (\frac{3}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-3/n}}{d n}\\ \end{align*}
Mathematica [A] time = 0.0132768, size = 54, normalized size = 1. \[ -\frac{F^a (c+d x)^3 \left (-b \log (F) (c+d x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b \log (F) (c+d x)^n\right )}{d n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{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{\left (d x + c\right )}^{2} 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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} F^{{\left (d x + c\right )}^{n} b + a}, 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 )}^{2} 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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