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 = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {2 \left (a^2+x^{n+1}\right )^{3/2}}{3 (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 18, normalized size = 0.82 \[ \frac {2 \, {\left (a^{2} + x^{n + 1}\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 18, normalized size = 0.82 \[ \frac {2 \, {\left (a^{2} + x^{n + 1}\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 19, normalized size = 0.86 \[ \frac {2 \left (a^{2}+x \,x^{n}\right )^{\frac {3}{2}}}{3 \left (n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 18, normalized size = 0.82 \[ \frac {2 \, {\left (a^{2} + x^{n + 1}\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 20, normalized size = 0.91 \[ \frac {2\,{\left (x^{n+1}+a^2\right )}^{3/2}}{3\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.39, size = 58, normalized size = 2.64 \[ \begin {cases} \frac {2 a^{2} \sqrt {a^{2} + x x^{n}}}{3 n + 3} + \frac {2 x x^{n} \sqrt {a^{2} + x x^{n}}}{3 n + 3} & \text {for}\: n \neq -1 \\\sqrt {a^{2} + 1} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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