Optimal. Leaf size=164 \[ -\frac{a^5 \left (\sqrt{a+x^2}+x\right )^{n-5}}{32 (5-n)}-\frac{5 a^4 \left (\sqrt{a+x^2}+x\right )^{n-3}}{32 (3-n)}-\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^{n-1}}{16 (1-n)}+\frac{5 a^2 \left (\sqrt{a+x^2}+x\right )^{n+1}}{16 (n+1)}+\frac{5 a \left (\sqrt{a+x^2}+x\right )^{n+3}}{32 (n+3)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+5}}{32 (n+5)} \]
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Rubi [A] time = 0.110497, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2122, 270} \[ -\frac{a^5 \left (\sqrt{a+x^2}+x\right )^{n-5}}{32 (5-n)}-\frac{5 a^4 \left (\sqrt{a+x^2}+x\right )^{n-3}}{32 (3-n)}-\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^{n-1}}{16 (1-n)}+\frac{5 a^2 \left (\sqrt{a+x^2}+x\right )^{n+1}}{16 (n+1)}+\frac{5 a \left (\sqrt{a+x^2}+x\right )^{n+3}}{32 (n+3)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+5}}{32 (n+5)} \]
Antiderivative was successfully verified.
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Rule 2122
Rule 270
Rubi steps
\begin{align*} \int \left (a+x^2\right )^2 \left (x+\sqrt{a+x^2}\right )^n \, dx &=\frac{1}{32} \operatorname{Subst}\left (\int x^{-6+n} \left (a+x^2\right )^5 \, dx,x,x+\sqrt{a+x^2}\right )\\ &=\frac{1}{32} \operatorname{Subst}\left (\int \left (a^5 x^{-6+n}+5 a^4 x^{-4+n}+10 a^3 x^{-2+n}+10 a^2 x^n+5 a x^{2+n}+x^{4+n}\right ) \, dx,x,x+\sqrt{a+x^2}\right )\\ &=-\frac{a^5 \left (x+\sqrt{a+x^2}\right )^{-5+n}}{32 (5-n)}-\frac{5 a^4 \left (x+\sqrt{a+x^2}\right )^{-3+n}}{32 (3-n)}-\frac{5 a^3 \left (x+\sqrt{a+x^2}\right )^{-1+n}}{16 (1-n)}+\frac{5 a^2 \left (x+\sqrt{a+x^2}\right )^{1+n}}{16 (1+n)}+\frac{5 a \left (x+\sqrt{a+x^2}\right )^{3+n}}{32 (3+n)}+\frac{\left (x+\sqrt{a+x^2}\right )^{5+n}}{32 (5+n)}\\ \end{align*}
Mathematica [A] time = 0.342343, size = 138, normalized size = 0.84 \[ \frac{1}{32} \left (\sqrt{a+x^2}+x\right )^{n-5} \left (\frac{10 a^2 \left (\sqrt{a+x^2}+x\right )^6}{n+1}+\frac{10 a^3 \left (\sqrt{a+x^2}+x\right )^4}{n-1}+\frac{5 a^4 \left (\sqrt{a+x^2}+x\right )^2}{n-3}+\frac{a^5}{n-5}+\frac{\left (\sqrt{a+x^2}+x\right )^{10}}{n+5}+\frac{5 a \left (\sqrt{a+x^2}+x\right )^8}{n+3}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.044, size = 216, normalized size = 1.3 \begin{align*}{\frac{{2}^{n}{x}^{5+n}}{5+n}{\mbox{$_3$F$_2$}(-{\frac{n}{2}},{\frac{1}{2}}-{\frac{n}{2}},-{\frac{5}{2}}-{\frac{n}{2}};\,1-n,-{\frac{3}{2}}-{\frac{n}{2}};\,-{\frac{a}{{x}^{2}}})}}+{\frac{{2}^{1+n}a{x}^{3+n}}{3+n}{\mbox{$_3$F$_2$}(-{\frac{n}{2}},{\frac{1}{2}}-{\frac{n}{2}},-{\frac{3}{2}}-{\frac{n}{2}};\,1-n,-{\frac{1}{2}}-{\frac{n}{2}};\,-{\frac{a}{{x}^{2}}})}}+{\frac{n}{4\,\sqrt{\pi }}{a}^{{\frac{5}{2}}+{\frac{n}{2}}} \left ( 8\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n \left ( -2+2\,n \right ) } \left ({\frac{an}{{x}^{2}}}+n-1 \right ) \left ( \sqrt{1+{\frac{a}{{x}^{2}}}}+1 \right ) ^{-1+n}}+4\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n}\sqrt{1+{\frac{a}{{x}^{2}}}} \left ( \sqrt{1+{\frac{a}{{x}^{2}}}}+1 \right ) ^{-1+n}} \right ) } \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 )}^{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 = 1.3776, size = 363, normalized size = 2.21 \begin{align*} -\frac{{\left (5 \,{\left (n^{4} - 10 \, n^{2} + 9\right )} x^{5} + 10 \,{\left (a n^{4} - 16 \, a n^{2} + 15 \, a\right )} x^{3} + 5 \,{\left (a^{2} n^{4} - 22 \, a^{2} n^{2} + 45 \, a^{2}\right )} x -{\left (a^{2} n^{5} - 30 \, a^{2} n^{3} +{\left (n^{5} - 10 \, n^{3} + 9 \, n\right )} x^{4} + 149 \, a^{2} n + 2 \,{\left (a n^{5} - 20 \, a n^{3} + 19 \, a n\right )} x^{2}\right )} \sqrt{x^{2} + a}\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{6} - 35 \, n^{4} + 259 \, n^{2} - 225} \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 )}^{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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