Optimal. Leaf size=141 \[ \frac {a^4 \left (x-\sqrt {a+x^2}\right )^{n-4}}{16 (4-n)}+\frac {a^3 \left (x-\sqrt {a+x^2}\right )^{n-2}}{4 (2-n)}-\frac {3 a^2 \left (x-\sqrt {a+x^2}\right )^n}{8 n}-\frac {a \left (x-\sqrt {a+x^2}\right )^{n+2}}{4 (n+2)}-\frac {\left (x-\sqrt {a+x^2}\right )^{n+4}}{16 (n+4)} \]
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Rubi [A] time = 0.09, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2122, 270} \[ \frac {a^4 \left (x-\sqrt {a+x^2}\right )^{n-4}}{16 (4-n)}+\frac {a^3 \left (x-\sqrt {a+x^2}\right )^{n-2}}{4 (2-n)}-\frac {3 a^2 \left (x-\sqrt {a+x^2}\right )^n}{8 n}-\frac {a \left (x-\sqrt {a+x^2}\right )^{n+2}}{4 (n+2)}-\frac {\left (x-\sqrt {a+x^2}\right )^{n+4}}{16 (n+4)} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2122
Rubi steps
\begin {align*} \int \left (a+x^2\right )^{3/2} \left (x-\sqrt {a+x^2}\right )^n \, dx &=-\left (\frac {1}{16} \operatorname {Subst}\left (\int x^{-5+n} \left (a+x^2\right )^4 \, dx,x,x-\sqrt {a+x^2}\right )\right )\\ &=-\left (\frac {1}{16} \operatorname {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 )\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 = 0.25, size = 123, normalized size = 0.87 \[ \frac {1}{16} \left (x-\sqrt {a+x^2}\right )^n \left (-\frac {a^4}{(n-4) \left (x-\sqrt {a+x^2}\right )^4}-\frac {4 a^3}{(n-2) \left (x-\sqrt {a+x^2}\right )^2}-\frac {6 a^2}{n}-\frac {4 a \left (x-\sqrt {a+x^2}\right )^2}{n+2}-\frac {\left (x-\sqrt {a+x^2}\right )^4}{n+4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 113, normalized size = 0.80 \[ -\frac {{\left (a^{2} n^{4} + {\left (n^{4} - 4 \, n^{2}\right )} x^{4} - 16 \, a^{2} n^{2} + 2 \, {\left (a n^{4} - 10 \, a n^{2}\right )} x^{2} + 24 \, a^{2} + 4 \, {\left ({\left (n^{3} - 4 \, n\right )} x^{3} + {\left (a n^{3} - 10 \, a n\right )} x\right )} \sqrt {x^{2} + a}\right )} {\left (x - \sqrt {x^{2} + a}\right )}^{n}}{n^{5} - 20 \, n^{3} + 64 \, n} \]
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 \[ \int {\left (x^{2} + a\right )}^{\frac {3}{2}} {\left (x - \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, 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 \[ \int {\left (x^{2} + a\right )}^{\frac {3}{2}} {\left (x - \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]
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 \[ \int {\left (x-\sqrt {x^2+a}\right )}^n\,{\left (x^2+a\right )}^{3/2} \,d x \]
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 \[ \int \left (a + x^{2}\right )^{\frac {3}{2}} \left (x - \sqrt {a + x^{2}}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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