Optimal. Leaf size=59 \[ \frac{4 \left (\sqrt{a+x^2}+x\right )^{n+2} \, _2F_1\left (2,\frac{n+2}{2};\frac{n+4}{2};-\frac{\left (x+\sqrt{x^2+a}\right )^2}{a}\right )}{a^2 (n+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0717657, antiderivative size = 59, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 2122
Rule 364
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*}
Mathematica [A] time = 0.0245902, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{ \left ( x+\sqrt{{x}^{2}+a} \right ) ^{n} \left ({x}^{2}+a \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{4} + 2 \, a x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]