Optimal. Leaf size=38 \[ -\frac{x^{-2 n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{b n (p+1)} \]
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Rubi [A] time = 0.0389941, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {2014} \[ -\frac{x^{-2 n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin{align*} \int x^{-1-n (1+2 p)} \left (b x^n+c x^{2 n}\right )^p \, dx &=-\frac{x^{-2 n (1+p)} \left (b x^n+c x^{2 n}\right )^{1+p}}{b n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0205104, size = 43, normalized size = 1.13 \[ -\frac{x^{-n (2 p+1)} \left (b+c x^n\right ) \left (x^n \left (b+c x^n\right )\right )^p}{b n (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-n \left ( 1+2\,p \right ) } \left ( b{x}^{n}+c{x}^{2\,n} \right ) ^{p}\, 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 (c x^{2 \, n} + b x^{n}\right )}^{p} x^{-n{\left (2 \, p + 1\right )} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56053, size = 124, normalized size = 3.26 \begin{align*} -\frac{{\left (c x x^{-2 \, n p - n - 1} x^{n} + b x x^{-2 \, n p - n - 1}\right )}{\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{b n p + b 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2 \, n} + b x^{n}\right )}^{p} x^{-n{\left (2 \, p + 1\right )} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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