Optimal. Leaf size=57 \[ \frac{p x^{(1-n) (p+1)} \left (a x^{-(1-n) p}+b x^{n-(1-n) p}\right )^{\frac{1}{p}+1}}{b n (p+1)} \]
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Rubi [A] time = 0.0277425, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1979, 2000} \[ \frac{p x^{(1-n) (p+1)} \left (a x^{-(1-n) p}+b x^{n-(1-n) p}\right )^{\frac{1}{p}+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2000
Rubi steps
\begin{align*} \int \left (x^{(-1+n) p} \left (a+b x^n\right )\right )^{\frac{1}{p}} \, dx &=\int \left (a x^{(-1+n) p}+b x^{n+(-1+n) p}\right )^{\frac{1}{p}} \, dx\\ &=\frac{p x^{(1-n) (1+p)} \left (a x^{-(1-n) p}+b x^{n-(1-n) p}\right )^{1+\frac{1}{p}}}{b n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.021195, size = 47, normalized size = 0.82 \[ \frac{x^{1-n} \left (a+b x^n\right ) \left (x^{(n-1) p} \left (a+b x^n\right )\right )^{\frac{1}{p}}}{b n \left (\frac{1}{p}+1\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.435, size = 0, normalized size = 0. \begin{align*} \int \sqrt [p]{{x}^{ \left ( -1+n \right ) p} \left ( a+b{x}^{n} \right ) }\, 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 ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac{1}{p}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.918842, size = 101, normalized size = 1.77 \begin{align*} \frac{{\left (b p x x^{n} + a p x\right )} \left ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac{1}{p}\right )}}{{\left (b n p + b 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 ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac{1}{p}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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