Optimal. Leaf size=63 \[ -\frac{16 \left (x-\sqrt{a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac{n+4}{2};\frac{n+6}{2};-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]
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Rubi [A] time = 0.07218, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2122, 364} \[ -\frac{16 \left (x-\sqrt{a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac{n+4}{2};\frac{n+6}{2};-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]
Antiderivative was successfully verified.
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Rule 2122
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (x-\sqrt{a+x^2}\right )^n}{\left (a+x^2\right )^{5/2}} \, dx &=-\left (16 \operatorname{Subst}\left (\int \frac{x^{3+n}}{\left (a+x^2\right )^4} \, dx,x,x-\sqrt{a+x^2}\right )\right )\\ &=-\frac{16 \left (x-\sqrt{a+x^2}\right )^{4+n} \, _2F_1\left (4,\frac{4+n}{2};\frac{6+n}{2};-\frac{\left (x-\sqrt{a+x^2}\right )^2}{a}\right )}{a^4 (4+n)}\\ \end{align*}
Mathematica [A] time = 0.0303316, size = 65, normalized size = 1.03 \[ -\frac{16 \left (x-\sqrt{a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac{n+4}{2};\frac{n+4}{2}+1;-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{ \left ( x-\sqrt{{x}^{2}+a} \right ) ^{n} \left ({x}^{2}+a \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{2} + a}{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{x^{6} + 3 \, a x^{4} + 3 \, a^{2} x^{2} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x - \sqrt{a + x^{2}}\right )^{n}}{\left (a + x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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