Optimal. Leaf size=187 \[ -\frac{a^6 \left (\sqrt{a+x^2}+x\right )^{n-6}}{64 (6-n)}-\frac{3 a^5 \left (\sqrt{a+x^2}+x\right )^{n-4}}{32 (4-n)}-\frac{15 a^4 \left (\sqrt{a+x^2}+x\right )^{n-2}}{64 (2-n)}+\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^n}{16 n}+\frac{15 a^2 \left (\sqrt{a+x^2}+x\right )^{n+2}}{64 (n+2)}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^{n+4}}{32 (n+4)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+6}}{64 (n+6)} \]
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Rubi [A] time = 0.117178, antiderivative size = 187, 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} \[ -\frac{a^6 \left (\sqrt{a+x^2}+x\right )^{n-6}}{64 (6-n)}-\frac{3 a^5 \left (\sqrt{a+x^2}+x\right )^{n-4}}{32 (4-n)}-\frac{15 a^4 \left (\sqrt{a+x^2}+x\right )^{n-2}}{64 (2-n)}+\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^n}{16 n}+\frac{15 a^2 \left (\sqrt{a+x^2}+x\right )^{n+2}}{64 (n+2)}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^{n+4}}{32 (n+4)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+6}}{64 (n+6)} \]
Antiderivative was successfully verified.
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Rule 2122
Rule 270
Rubi steps
\begin{align*} \int \left (a+x^2\right )^{5/2} \left (x+\sqrt{a+x^2}\right )^n \, dx &=\frac{1}{64} \operatorname{Subst}\left (\int x^{-7+n} \left (a+x^2\right )^6 \, dx,x,x+\sqrt{a+x^2}\right )\\ &=\frac{1}{64} \operatorname{Subst}\left (\int \left (a^6 x^{-7+n}+6 a^5 x^{-5+n}+15 a^4 x^{-3+n}+20 a^3 x^{-1+n}+15 a^2 x^{1+n}+6 a x^{3+n}+x^{5+n}\right ) \, dx,x,x+\sqrt{a+x^2}\right )\\ &=-\frac{a^6 \left (x+\sqrt{a+x^2}\right )^{-6+n}}{64 (6-n)}-\frac{3 a^5 \left (x+\sqrt{a+x^2}\right )^{-4+n}}{32 (4-n)}-\frac{15 a^4 \left (x+\sqrt{a+x^2}\right )^{-2+n}}{64 (2-n)}+\frac{5 a^3 \left (x+\sqrt{a+x^2}\right )^n}{16 n}+\frac{15 a^2 \left (x+\sqrt{a+x^2}\right )^{2+n}}{64 (2+n)}+\frac{3 a \left (x+\sqrt{a+x^2}\right )^{4+n}}{32 (4+n)}+\frac{\left (x+\sqrt{a+x^2}\right )^{6+n}}{64 (6+n)}\\ \end{align*}
Mathematica [A] time = 0.328109, size = 157, normalized size = 0.84 \[ \frac{1}{64} \left (\sqrt{a+x^2}+x\right )^n \left (\frac{a^6}{(n-6) \left (\sqrt{a+x^2}+x\right )^6}+\frac{6 a^5}{(n-4) \left (\sqrt{a+x^2}+x\right )^4}+\frac{15 a^4}{(n-2) \left (\sqrt{a+x^2}+x\right )^2}+\frac{15 a^2 \left (\sqrt{a+x^2}+x\right )^2}{n+2}+\frac{20 a^3}{n}+\frac{6 a \left (\sqrt{a+x^2}+x\right )^4}{n+4}+\frac{\left (\sqrt{a+x^2}+x\right )^6}{n+6}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.015, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{2}+a \right ) ^{{\frac{5}{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{5}{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 = 1.34751, size = 459, normalized size = 2.45 \begin{align*} \frac{{\left (a^{3} n^{6} - 50 \, a^{3} n^{4} +{\left (n^{6} - 20 \, n^{4} + 64 \, n^{2}\right )} x^{6} + 544 \, a^{3} n^{2} + 3 \,{\left (a n^{6} - 30 \, a n^{4} + 104 \, a n^{2}\right )} x^{4} - 720 \, a^{3} + 3 \,{\left (a^{2} n^{6} - 40 \, a^{2} n^{4} + 264 \, a^{2} n^{2}\right )} x^{2} - 6 \,{\left ({\left (n^{5} - 20 \, n^{3} + 64 \, n\right )} x^{5} + 2 \,{\left (a n^{5} - 30 \, a n^{3} + 104 \, a n\right )} x^{3} +{\left (a^{2} n^{5} - 40 \, a^{2} n^{3} + 264 \, a^{2} n\right )} x\right )} \sqrt{x^{2} + a}\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{7} - 56 \, n^{5} + 784 \, n^{3} - 2304 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{5}{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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