Optimal. Leaf size=61 \[ \frac{x^{\frac{(1-n) (p+1)}{p}} \left (a x^{-\frac{1-n}{p}}+b x^{n-\frac{1-n}{p}}\right )^{p+1}}{b n (p+1)} \]
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Rubi [A] time = 0.0248845, antiderivative size = 61, 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} \[ \frac{x^{\frac{(1-n) (p+1)}{p}} \left (a x^{-\frac{1-n}{p}}+b x^{n-\frac{1-n}{p}}\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2000
Rubi steps
\begin{align*} \int \left (x^{\frac{-1+n}{p}} \left (a+b x^n\right )\right )^p \, dx &=\int \left (b x^{n+\frac{-1+n}{p}}+a x^{\frac{-1+n}{p}}\right )^p \, dx\\ &=\frac{x^{\frac{(1-n) (1+p)}{p}} \left (b x^{n-\frac{1-n}{p}}+a x^{-\frac{1-n}{p}}\right )^{1+p}}{b n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0250275, size = 45, normalized size = 0.74 \[ \frac{x^{1-n} \left (a+b x^n\right ) \left (x^{\frac{n-1}{p}} \left (a+b x^n\right )\right )^p}{b n (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.395, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{{\frac{-1+n}{p}}} \left ( a+b{x}^{n} \right ) \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 ({\left (b x^{n} + a\right )} x^{\frac{n - 1}{p}}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.907711, size = 107, normalized size = 1.75 \begin{align*} \frac{{\left (b x x^{n} + a x\right )}{\left (b x^{n} x^{\frac{n - 1}{p}} + a x^{\frac{n - 1}{p}}\right )}^{p}}{{\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^{\frac{n - 1}{p}}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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