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 = 0.11, antiderivative size = 164, normalized size of antiderivative = 1.00, 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 270
Rule 2122
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*}
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Mathematica [A] time = 0.38, size = 138, normalized size = 0.84 \[ \frac {1}{32} \left (\sqrt {a+x^2}+x\right )^{n-5} \left (\frac {a^5}{n-5}+\frac {5 a^4 \left (\sqrt {a+x^2}+x\right )^2}{n-3}+\frac {10 a^3 \left (\sqrt {a+x^2}+x\right )^4}{n-1}+\frac {10 a^2 \left (\sqrt {a+x^2}+x\right )^6}{n+1}+\frac {\left (\sqrt {a+x^2}+x\right )^{10}}{n+5}+\frac {5 a \left (\sqrt {a+x^2}+x\right )^8}{n+3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 158, normalized size = 0.96 \[ -\frac {{\left (5 \, {\left (n^{4} - 10 \, n^{2} + 9\right )} x^{5} + 10 \, {\left (a n^{4} - 16 \, a n^{2} + 15 \, a\right )} x^{3} + 5 \, {\left (a^{2} n^{4} - 22 \, a^{2} n^{2} + 45 \, a^{2}\right )} x - {\left (a^{2} n^{5} - 30 \, a^{2} n^{3} + {\left (n^{5} - 10 \, n^{3} + 9 \, n\right )} x^{4} + 149 \, a^{2} n + 2 \, {\left (a n^{5} - 20 \, a n^{3} + 19 \, a n\right )} x^{2}\right )} \sqrt {x^{2} + a}\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{6} - 35 \, n^{4} + 259 \, n^{2} - 225} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{2} + a\right )}^{2} {\left (x + \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 216, normalized size = 1.32 \[ \frac {a 2^{n +1} x^{n +3} \hypergeom \left (\left [-\frac {n}{2}, -\frac {n}{2}+\frac {1}{2}, -\frac {n}{2}-\frac {3}{2}\right ], \left [-n +1, -\frac {n}{2}-\frac {1}{2}\right ], -\frac {a}{x^{2}}\right )}{n +3}+\frac {2^{n} x^{n +5} \hypergeom \left (\left [-\frac {n}{2}, -\frac {n}{2}-\frac {5}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \left [-n +1, -\frac {n}{2}-\frac {3}{2}\right ], -\frac {a}{x^{2}}\right )}{n +5}+\frac {\left (\frac {8 \sqrt {\pi }\, \left (n +\frac {a n}{x^{2}}-1\right ) a^{-\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (\sqrt {\frac {a}{x^{2}}+1}+1\right )^{n -1}}{\left (n +1\right ) \left (2 n -2\right ) n}+\frac {4 \sqrt {\pi }\, \sqrt {\frac {a}{x^{2}}+1}\, a^{-\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (\sqrt {\frac {a}{x^{2}}+1}+1\right )^{n -1}}{\left (n +1\right ) n}\right ) n \,a^{\frac {n}{2}+\frac {5}{2}}}{4 \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{2} + a\right )}^{2} {\left (x + \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (x^2+a\right )}^2\,{\left (x+\sqrt {x^2+a}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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