Optimal. Leaf size=131 \[ -\frac{a^4 \left (\sqrt{a+x^2}+x\right )^{n-4}}{16 (4-n)}-\frac{a^3 \left (\sqrt{a+x^2}+x\right )^{n-2}}{4 (2-n)}+\frac{3 a^2 \left (\sqrt{a+x^2}+x\right )^n}{8 n}+\frac{a \left (\sqrt{a+x^2}+x\right )^{n+2}}{4 (n+2)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+4}}{16 (n+4)} \]
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Rubi [A] time = 0.094264, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2122, 270} \[ -\frac{a^4 \left (\sqrt{a+x^2}+x\right )^{n-4}}{16 (4-n)}-\frac{a^3 \left (\sqrt{a+x^2}+x\right )^{n-2}}{4 (2-n)}+\frac{3 a^2 \left (\sqrt{a+x^2}+x\right )^n}{8 n}+\frac{a \left (\sqrt{a+x^2}+x\right )^{n+2}}{4 (n+2)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+4}}{16 (n+4)} \]
Antiderivative was successfully verified.
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Rule 2122
Rule 270
Rubi steps
\begin{align*} \int \left (a+x^2\right )^{3/2} \left (x+\sqrt{a+x^2}\right )^n \, dx &=\frac{1}{16} \operatorname{Subst}\left (\int x^{-5+n} \left (a+x^2\right )^4 \, dx,x,x+\sqrt{a+x^2}\right )\\ &=\frac{1}{16} \operatorname{Subst}\left (\int \left (a^4 x^{-5+n}+4 a^3 x^{-3+n}+6 a^2 x^{-1+n}+4 a x^{1+n}+x^{3+n}\right ) \, dx,x,x+\sqrt{a+x^2}\right )\\ &=-\frac{a^4 \left (x+\sqrt{a+x^2}\right )^{-4+n}}{16 (4-n)}-\frac{a^3 \left (x+\sqrt{a+x^2}\right )^{-2+n}}{4 (2-n)}+\frac{3 a^2 \left (x+\sqrt{a+x^2}\right )^n}{8 n}+\frac{a \left (x+\sqrt{a+x^2}\right )^{2+n}}{4 (2+n)}+\frac{\left (x+\sqrt{a+x^2}\right )^{4+n}}{16 (4+n)}\\ \end{align*}
Mathematica [A] time = 0.222189, size = 111, normalized size = 0.85 \[ \frac{1}{16} \left (\sqrt{a+x^2}+x\right )^n \left (\frac{a^4}{(n-4) \left (\sqrt{a+x^2}+x\right )^4}+\frac{4 a^3}{(n-2) \left (\sqrt{a+x^2}+x\right )^2}+\frac{6 a^2}{n}+\frac{4 a \left (\sqrt{a+x^2}+x\right )^2}{n+2}+\frac{\left (\sqrt{a+x^2}+x\right )^4}{n+4}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{2}+a \right ) ^{{\frac{3}{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{\left (x^{2} + a\right )}^{\frac{3}{2}}{\left (x + \sqrt{x^{2} + a}\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3603, size = 244, normalized size = 1.86 \begin{align*} \frac{{\left (a^{2} n^{4} +{\left (n^{4} - 4 \, n^{2}\right )} x^{4} - 16 \, a^{2} n^{2} + 2 \,{\left (a n^{4} - 10 \, a n^{2}\right )} x^{2} + 24 \, a^{2} - 4 \,{\left ({\left (n^{3} - 4 \, n\right )} x^{3} +{\left (a n^{3} - 10 \, a n\right )} x\right )} \sqrt{x^{2} + a}\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{5} - 20 \, n^{3} + 64 \, n} \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 \left (a + x^{2}\right )^{\frac{3}{2}} \left (x + \sqrt{a + x^{2}}\right )^{n}\, 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{\left (x^{2} + a\right )}^{\frac{3}{2}}{\left (x + \sqrt{x^{2} + a}\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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