Optimal. Leaf size=20 \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]
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Rubi [A] time = 0.0047202, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int x^n \sqrt{1+x^{1+n}} \, dx &=\frac{2 \left (1+x^{1+n}\right )^{3/2}}{3 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0077528, size = 20, normalized size = 1. \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 17, normalized size = 0.9 \begin{align*}{\frac{2}{3+3\,n} \left ( 1+x{x}^{n} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07326, size = 22, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30274, size = 47, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.00389, size = 48, normalized size = 2.4 \begin{align*} \begin{cases} \frac{2 x x^{n} \sqrt{x x^{n} + 1}}{3 n + 3} + \frac{2 \sqrt{x x^{n} + 1}}{3 n + 3} & \text{for}\: n \neq -1 \\\sqrt{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10981, size = 22, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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