Optimal. Leaf size=27 \[ \frac{\left (a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
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Rubi [A] time = 0.0284039, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1468, 629} \[ \frac{\left (a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
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Rule 1468
Rule 629
Rubi steps
\begin{align*} \int x^{-1+n} \left (b+2 c x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int (b+2 c x) \left (a+b x+c x^2\right )^p \, dx,x,x^n\right )}{n}\\ &=\frac{\left (a+b x^n+c x^{2 n}\right )^{1+p}}{n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0338035, size = 26, normalized size = 0.96 \[ \frac{\left (a+x^n \left (b+c x^n\right )\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 40, normalized size = 1.5 \begin{align*}{\frac{ \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{p}}{n \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28956, size = 53, normalized size = 1.96 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n} + a\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18037, size = 82, normalized size = 3.04 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n} + a\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{n p + n} \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 [A] time = 1.15221, size = 36, normalized size = 1.33 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p + 1}}{n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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