Optimal. Leaf size=75 \[ -\frac {a^2 \left (x+\sqrt {a+x^2}\right )^{-2+n}}{4 (2-n)}+\frac {a \left (x+\sqrt {a+x^2}\right )^n}{2 n}+\frac {\left (x+\sqrt {a+x^2}\right )^{2+n}}{4 (2+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2147, 276}
\begin {gather*} -\frac {a^2 \left (\sqrt {a+x^2}+x\right )^{n-2}}{4 (2-n)}+\frac {a \left (\sqrt {a+x^2}+x\right )^n}{2 n}+\frac {\left (\sqrt {a+x^2}+x\right )^{n+2}}{4 (n+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2147
Rubi steps
\begin {align*} \int \sqrt {a+x^2} \left (x+\sqrt {a+x^2}\right )^n \, dx &=\frac {1}{4} \text {Subst}\left (\int x^{-3+n} \left (a+x^2\right )^2 \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (a^2 x^{-3+n}+2 a x^{-1+n}+x^{1+n}\right ) \, dx,x,x+\sqrt {a+x^2}\right )\\ &=-\frac {a^2 \left (x+\sqrt {a+x^2}\right )^{-2+n}}{4 (2-n)}+\frac {a \left (x+\sqrt {a+x^2}\right )^n}{2 n}+\frac {\left (x+\sqrt {a+x^2}\right )^{2+n}}{4 (2+n)}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 65, normalized size = 0.87 \begin {gather*} \frac {1}{4} \left (x+\sqrt {a+x^2}\right )^n \left (\frac {2 a}{n}+\frac {a^2}{(-2+n) \left (x+\sqrt {a+x^2}\right )^2}+\frac {\left (x+\sqrt {a+x^2}\right )^2}{2+n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \sqrt {x^{2}+a}\, \left (x +\sqrt {x^{2}+a}\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 48, normalized size = 0.64 \begin {gather*} \frac {{\left (n^{2} x^{2} + a n^{2} - 2 \, \sqrt {x^{2} + a} n x - 2 \, a\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{3} - 4 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + x^{2}} \left (x + \sqrt {a + x^{2}}\right )^{n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {x^2+a}\,{\left (x+\sqrt {x^2+a}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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