Optimal. Leaf size=65 \[ \frac{1}{3} x^3 F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac{3}{n},-b \log (F);\frac{n+3}{n};-\frac{d x^n}{c}\right ) \]
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Rubi [A] time = 0.0492867, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2274, 12, 365, 364} \[ \frac{1}{3} x^3 F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac{3}{n},-b \log (F);\frac{n+3}{n};-\frac{d x^n}{c}\right ) \]
Antiderivative was successfully verified.
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Rule 2274
Rule 12
Rule 365
Rule 364
Rubi steps
\begin{align*} \int F^{a+b \log \left (c+d x^n\right )} x^2 \, dx &=\int F^a x^2 \left (c+d x^n\right )^{b \log (F)} \, dx\\ &=F^a \int x^2 \left (c+d x^n\right )^{b \log (F)} \, dx\\ &=\left (F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac{d x^n}{c}\right )^{-b \log (F)}\right ) \int x^2 \left (1+\frac{d x^n}{c}\right )^{b \log (F)} \, dx\\ &=\frac{1}{3} F^a x^3 \left (c+d x^n\right )^{b \log (F)} \left (1+\frac{d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (\frac{3}{n},-b \log (F);\frac{3+n}{n};-\frac{d x^n}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.140509, size = 85, normalized size = 1.31 \[ -\frac{x^3 \left (-\frac{d x^n}{c}\right )^{-3/n} \left (c+d x^n\right ) F^{a+b \log \left (c+d x^n\right )} \, _2F_1\left (\frac{n-3}{n},b \log (F)+1;b \log (F)+2;\frac{d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b\ln \left ( c+d{x}^{n} \right ) }{x}^{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 F^{b \log \left (d x^{n} + c\right ) + a} x^{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 (F^{b \log \left (d x^{n} + c\right ) + a} x^{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 F^{b \log \left (d x^{n} + c\right ) + a} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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