Optimal. Leaf size=27 \[ \frac{x^{n (p+1)} \left (b+c x^n\right )^{p+1}}{n (p+1)} \]
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Rubi [A] time = 0.0124291, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {449} \[ \frac{x^{n (p+1)} \left (b+c x^n\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin{align*} \int x^{-1+n (1+p)} \left (b+c x^n\right )^p \left (b+2 c x^n\right ) \, dx &=\frac{x^{n (1+p)} \left (b+c x^n\right )^{1+p}}{n (1+p)}\\ \end{align*}
Mathematica [C] time = 0.12544, size = 101, normalized size = 3.74 \[ \frac{\left (b+c x^n\right )^p \left (\frac{c x^n}{b}+1\right )^{-p} \left (b (p+2) x^{n (p+1)} \, _2F_1\left (-p,p+1;p+2;-\frac{c x^n}{b}\right )+2 c (p+1) x^{n (p+2)} \, _2F_1\left (-p,p+2;p+3;-\frac{c x^n}{b}\right )\right )}{n (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1+n \left ( 1+p \right ) } \left ( b+c{x}^{n} \right ) ^{p} \left ( b+2\,c{x}^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16396, size = 53, normalized size = 1.96 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (n p \log \left (x\right ) + p \log \left (c x^{n} + b\right )\right )}}{n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07237, size = 77, normalized size = 2.85 \begin{align*} \frac{{\left (c x x^{n} + b x\right )}{\left (c x^{n} + b\right )}^{p} x^{n p + n - 1}}{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 [B] time = 1.09821, size = 89, normalized size = 3.3 \begin{align*} \frac{{\left (c x^{n} + b\right )}^{p} c x x^{n} e^{\left (n p \log \left (x\right ) + n \log \left (x\right ) - \log \left (x\right )\right )} +{\left (c x^{n} + b\right )}^{p} b x e^{\left (n p \log \left (x\right ) + n \log \left (x\right ) - \log \left (x\right )\right )}}{n p + n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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