Optimal. Leaf size=59 \[ \frac{8 \left (\sqrt{a+x^2}+x\right )^{n+3} \, _2F_1\left (3,\frac{n+3}{2};\frac{n+5}{2};-\frac{\left (x+\sqrt{x^2+a}\right )^2}{a}\right )}{a^3 (n+3)} \]
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Rubi [A] time = 0.0661979, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2122, 364} \[ \frac{8 \left (\sqrt{a+x^2}+x\right )^{n+3} \, _2F_1\left (3,\frac{n+3}{2};\frac{n+5}{2};-\frac{\left (x+\sqrt{x^2+a}\right )^2}{a}\right )}{a^3 (n+3)} \]
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 )^2} \, dx &=8 \operatorname{Subst}\left (\int \frac{x^{2+n}}{\left (a+x^2\right )^3} \, dx,x,x+\sqrt{a+x^2}\right )\\ &=\frac{8 \left (x+\sqrt{a+x^2}\right )^{3+n} \, _2F_1\left (3,\frac{3+n}{2};\frac{5+n}{2};-\frac{\left (x+\sqrt{a+x^2}\right )^2}{a}\right )}{a^3 (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0302536, size = 61, normalized size = 1.03 \[ \frac{8 \left (\sqrt{a+x^2}+x\right )^{n+3} \, _2F_1\left (3,\frac{n+3}{2};\frac{n+3}{2}+1;-\frac{\left (x+\sqrt{x^2+a}\right )^2}{a}\right )}{a^3 (n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({x}^{2}+a \right ) ^{2}} \left ( x+\sqrt{{x}^{2}+a} \right ) ^{n}}\, 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 )}^{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{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{x^{4} + 2 \, a x^{2} + a^{2}}, 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 )^{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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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