Optimal. Leaf size=22 \[ \frac{2 \left (a^2+x^{n+1}\right )^{3/2}}{3 (n+1)} \]
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Rubi [A] time = 0.0049375, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \frac{2 \left (a^2+x^{n+1}\right )^{3/2}}{3 (n+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int x^n \sqrt{a^2+x^{1+n}} \, dx &=\frac{2 \left (a^2+x^{1+n}\right )^{3/2}}{3 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.00849, size = 22, normalized size = 1. \[ \frac{2 \left (a^2+x^{n+1}\right )^{3/2}}{3 (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 19, normalized size = 0.9 \begin{align*}{\frac{2}{3+3\,n} \left ({a}^{2}+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 = 0.980573, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (a^{2} + x^{n + 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 = 1.38658, size = 50, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (a^{2} + x^{n + 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 = 5.14821, size = 58, normalized size = 2.64 \begin{align*} \begin{cases} \frac{2 a^{2} \sqrt{a^{2} + x x^{n}}}{3 n + 3} + \frac{2 x x^{n} \sqrt{a^{2} + x x^{n}}}{3 n + 3} & \text{for}\: n \neq -1 \\\sqrt{a^{2} + 1} \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 = 1.16228, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (a^{2} + x^{n + 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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