Optimal. Leaf size=131 \[ -\frac {a^4 \left (x+\sqrt {a+x^2}\right )^{-4+n}}{16 (4-n)}-\frac {a^3 \left (x+\sqrt {a+x^2}\right )^{-2+n}}{4 (2-n)}+\frac {3 a^2 \left (x+\sqrt {a+x^2}\right )^n}{8 n}+\frac {a \left (x+\sqrt {a+x^2}\right )^{2+n}}{4 (2+n)}+\frac {\left (x+\sqrt {a+x^2}\right )^{4+n}}{16 (4+n)} \]
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Rubi [A]
time = 0.07, antiderivative size = 131, 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^4 \left (\sqrt {a+x^2}+x\right )^{n-4}}{16 (4-n)}-\frac {a^3 \left (\sqrt {a+x^2}+x\right )^{n-2}}{4 (2-n)}+\frac {3 a^2 \left (\sqrt {a+x^2}+x\right )^n}{8 n}+\frac {a \left (\sqrt {a+x^2}+x\right )^{n+2}}{4 (n+2)}+\frac {\left (\sqrt {a+x^2}+x\right )^{n+4}}{16 (n+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2147
Rubi steps
\begin {align*} \int \left (a+x^2\right )^{3/2} \left (x+\sqrt {a+x^2}\right )^n \, dx &=\frac {1}{16} \text {Subst}\left (\int x^{-5+n} \left (a+x^2\right )^4 \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {1}{16} \text {Subst}\left (\int \left (a^4 x^{-5+n}+4 a^3 x^{-3+n}+6 a^2 x^{-1+n}+4 a x^{1+n}+x^{3+n}\right ) \, dx,x,x+\sqrt {a+x^2}\right )\\ &=-\frac {a^4 \left (x+\sqrt {a+x^2}\right )^{-4+n}}{16 (4-n)}-\frac {a^3 \left (x+\sqrt {a+x^2}\right )^{-2+n}}{4 (2-n)}+\frac {3 a^2 \left (x+\sqrt {a+x^2}\right )^n}{8 n}+\frac {a \left (x+\sqrt {a+x^2}\right )^{2+n}}{4 (2+n)}+\frac {\left (x+\sqrt {a+x^2}\right )^{4+n}}{16 (4+n)}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 111, normalized size = 0.85 \begin {gather*} \frac {1}{16} \left (x+\sqrt {a+x^2}\right )^n \left (\frac {6 a^2}{n}+\frac {a^4}{(-4+n) \left (x+\sqrt {a+x^2}\right )^4}+\frac {4 a^3}{(-2+n) \left (x+\sqrt {a+x^2}\right )^2}+\frac {4 a \left (x+\sqrt {a+x^2}\right )^2}{2+n}+\frac {\left (x+\sqrt {a+x^2}\right )^4}{4+n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (x^{2}+a \right )^{\frac {3}{2}} \left (x +\sqrt {x^{2}+a}\right )^{n}\, dx\]
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 = 110, normalized size = 0.84 \begin {gather*} \frac {{\left (a^{2} n^{4} + {\left (n^{4} - 4 \, n^{2}\right )} x^{4} - 16 \, a^{2} n^{2} + 2 \, {\left (a n^{4} - 10 \, a n^{2}\right )} x^{2} + 24 \, a^{2} - 4 \, {\left ({\left (n^{3} - 4 \, n\right )} x^{3} + {\left (a n^{3} - 10 \, a n\right )} x\right )} \sqrt {x^{2} + a}\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{5} - 20 \, n^{3} + 64 \, 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 \left (a + x^{2}\right )^{\frac {3}{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 {\left (x^2+a\right )}^{3/2}\,{\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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