Optimal. Leaf size=29 \[ \frac{\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
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Rubi [A] time = 0.0282607, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, 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 = 0.0354757, size = 28, normalized size = 0.97 \[ \frac{\left (x^n \left (b+c x^n\right )-a\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 45, normalized size = 1.6 \begin{align*} -{\frac{ \left ( -c \left ({x}^{n} \right ) ^{2}-b{x}^{n}+a \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 = 1.25317, size = 58, normalized size = 2. \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 = 1.05441, size = 82, normalized size = 2.83 \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 = 1.18868, size = 39, normalized size = 1.34 \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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