Optimal. Leaf size=37 \[ \frac{x^{-n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{c n (p+1)} \]
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Rubi [A] time = 0.0395401, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {2014} \[ \frac{x^{-n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{c n (p+1)} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin{align*} \int x^{-1-n (-1+p)} \left (b x^n+c x^{2 n}\right )^p \, dx &=\frac{x^{-n (1+p)} \left (b x^n+c x^{2 n}\right )^{1+p}}{c n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0170824, size = 38, normalized size = 1.03 \[ \frac{x^{-n p} \left (b+c x^n\right ) \left (x^n \left (b+c x^n\right )\right )^p}{c n (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.097, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-n \left ( -1+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 [A] time = 1.08022, size = 58, normalized size = 1.57 \begin{align*} \frac{{\left (c x^{n} + b\right )} e^{\left (-n p \log \left (x\right ) + p \log \left (c x^{n} + b\right ) + p \log \left (x^{n}\right )\right )}}{c n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58892, size = 126, normalized size = 3.41 \begin{align*} \frac{{\left (c x x^{-n p + n - 1} x^{n} + b x x^{-n p + n - 1}\right )}{\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{{\left (c n p + c n\right )} x^{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 (p - 1\right )} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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