Integrand size = 25, antiderivative size = 63 \[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx=-\frac {4 \left (x-\sqrt {a+x^2}\right )^{2+n} \operatorname {Hypergeometric2F1}\left (2,\frac {2+n}{2},\frac {4+n}{2},-\frac {\left (x-\sqrt {a+x^2}\right )^2}{a}\right )}{a^2 (2+n)} \]
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Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2147, 371} \[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx=-\frac {4 \left (x-\sqrt {a+x^2}\right )^{n+2} \operatorname {Hypergeometric2F1}\left (2,\frac {n+2}{2},\frac {n+4}{2},-\frac {\left (x-\sqrt {x^2+a}\right )^2}{a}\right )}{a^2 (n+2)} \]
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Rule 371
Rule 2147
Rubi steps \begin{align*} \text {integral}& = -\left (4 \text {Subst}\left (\int \frac {x^{1+n}}{\left (a+x^2\right )^2} \, dx,x,x-\sqrt {a+x^2}\right )\right ) \\ & = -\frac {4 \left (x-\sqrt {a+x^2}\right )^{2+n} \, _2F_1\left (2,\frac {2+n}{2};\frac {4+n}{2};-\frac {\left (x-\sqrt {a+x^2}\right )^2}{a}\right )}{a^2 (2+n)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx=-\frac {4 \left (x-\sqrt {a+x^2}\right )^{2+n} \operatorname {Hypergeometric2F1}\left (2,\frac {2+n}{2},1+\frac {2+n}{2},-\frac {\left (x-\sqrt {a+x^2}\right )^2}{a}\right )}{a^2 (2+n)} \]
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\[\int \frac {\left (x -\sqrt {x^{2}+a}\right )^{n}}{\left (x^{2}+a \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx=\int { \frac {{\left (x - \sqrt {x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx=\int \frac {\left (x - \sqrt {a + x^{2}}\right )^{n}}{\left (a + x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx=\int { \frac {{\left (x - \sqrt {x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx=\int { \frac {{\left (x - \sqrt {x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx=\int \frac {{\left (x-\sqrt {x^2+a}\right )}^n}{{\left (x^2+a\right )}^{3/2}} \,d x \]
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