Optimal. Leaf size=59 \[ \frac {4 \left (\sqrt {a+x^2}+x\right )^{n+2} \, _2F_1\left (2,\frac {n+2}{2};\frac {n+4}{2};-\frac {\left (x+\sqrt {x^2+a}\right )^2}{a}\right )}{a^2 (n+2)} \]
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Rubi [A] time = 0.07, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2122, 364} \[ \frac {4 \left (\sqrt {a+x^2}+x\right )^{n+2} \, _2F_1\left (2,\frac {n+2}{2};\frac {n+4}{2};-\frac {\left (x+\sqrt {x^2+a}\right )^2}{a}\right )}{a^2 (n+2)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2122
Rubi steps
\begin {align*} \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx &=4 \operatorname {Subst}\left (\int \frac {x^{1+n}}{\left (a+x^2\right )^2} \, dx,x,x+\sqrt {a+x^2}\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*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 1.03 \[ \frac {4 \left (\sqrt {a+x^2}+x\right )^{n+2} \, _2F_1\left (2,\frac {n+2}{2};\frac {n+2}{2}+1;-\frac {\left (x+\sqrt {x^2+a}\right )^2}{a}\right )}{a^2 (n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} + a} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{x^{4} + 2 \, a x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (x + \sqrt {x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (x +\sqrt {x^{2}+a}\right )^{n}}{\left (x^{2}+a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (x + \sqrt {x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (x+\sqrt {x^2+a}\right )}^n}{{\left (x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (x + \sqrt {a + x^{2}}\right )^{n}}{\left (a + x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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