Integrand size = 21, antiderivative size = 164 \[ \int \left (a+x^2\right )^2 \left (x+\sqrt {a+x^2}\right )^n \, dx=-\frac {a^5 \left (x+\sqrt {a+x^2}\right )^{-5+n}}{32 (5-n)}-\frac {5 a^4 \left (x+\sqrt {a+x^2}\right )^{-3+n}}{32 (3-n)}-\frac {5 a^3 \left (x+\sqrt {a+x^2}\right )^{-1+n}}{16 (1-n)}+\frac {5 a^2 \left (x+\sqrt {a+x^2}\right )^{1+n}}{16 (1+n)}+\frac {5 a \left (x+\sqrt {a+x^2}\right )^{3+n}}{32 (3+n)}+\frac {\left (x+\sqrt {a+x^2}\right )^{5+n}}{32 (5+n)} \]
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Time = 0.08 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2147, 276} \[ \int \left (a+x^2\right )^2 \left (x+\sqrt {a+x^2}\right )^n \, dx=-\frac {a^5 \left (\sqrt {a+x^2}+x\right )^{n-5}}{32 (5-n)}-\frac {5 a^4 \left (\sqrt {a+x^2}+x\right )^{n-3}}{32 (3-n)}-\frac {5 a^3 \left (\sqrt {a+x^2}+x\right )^{n-1}}{16 (1-n)}+\frac {5 a^2 \left (\sqrt {a+x^2}+x\right )^{n+1}}{16 (n+1)}+\frac {5 a \left (\sqrt {a+x^2}+x\right )^{n+3}}{32 (n+3)}+\frac {\left (\sqrt {a+x^2}+x\right )^{n+5}}{32 (n+5)} \]
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Rule 276
Rule 2147
Rubi steps \begin{align*} \text {integral}& = \frac {1}{32} \text {Subst}\left (\int x^{-6+n} \left (a+x^2\right )^5 \, dx,x,x+\sqrt {a+x^2}\right ) \\ & = \frac {1}{32} \text {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*}
Time = 0.40 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.84 \[ \int \left (a+x^2\right )^2 \left (x+\sqrt {a+x^2}\right )^n \, dx=\frac {1}{32} \left (x+\sqrt {a+x^2}\right )^{-5+n} \left (\frac {a^5}{-5+n}+\frac {5 a^4 \left (x+\sqrt {a+x^2}\right )^2}{-3+n}+\frac {10 a^3 \left (x+\sqrt {a+x^2}\right )^4}{-1+n}+\frac {10 a^2 \left (x+\sqrt {a+x^2}\right )^6}{1+n}+\frac {5 a \left (x+\sqrt {a+x^2}\right )^8}{3+n}+\frac {\left (x+\sqrt {a+x^2}\right )^{10}}{5+n}\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.96 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.32
method | result | size |
meijerg | \(\frac {2^{n} x^{5+n} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (-\frac {n}{2},-\frac {5}{2}-\frac {n}{2},\frac {1}{2}-\frac {n}{2};1-n ,-\frac {3}{2}-\frac {n}{2};-\frac {a}{x^{2}}\right )}{5+n}+\frac {2^{1+n} a \,x^{3+n} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (-\frac {n}{2},-\frac {3}{2}-\frac {n}{2},\frac {1}{2}-\frac {n}{2};1-n ,-\frac {1}{2}-\frac {n}{2};-\frac {a}{x^{2}}\right )}{3+n}+\frac {a^{\frac {5}{2}+\frac {n}{2}} n \left (\frac {8 \sqrt {\pi }\, x^{1+n} a^{-\frac {1}{2}-\frac {n}{2}} \left (\frac {a n}{x^{2}}+n -1\right ) {\left (\sqrt {1+\frac {a}{x^{2}}}+1\right )}^{-1+n}}{\left (1+n \right ) n \left (-2+2 n \right )}+\frac {4 \sqrt {\pi }\, x^{1+n} a^{-\frac {1}{2}-\frac {n}{2}} \sqrt {1+\frac {a}{x^{2}}}\, {\left (\sqrt {1+\frac {a}{x^{2}}}+1\right )}^{-1+n}}{\left (1+n \right ) n}\right )}{4 \sqrt {\pi }}\) | \(216\) |
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none
Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.96 \[ \int \left (a+x^2\right )^2 \left (x+\sqrt {a+x^2}\right )^n \, dx=-\frac {{\left (5 \, {\left (n^{4} - 10 \, n^{2} + 9\right )} x^{5} + 10 \, {\left (a n^{4} - 16 \, a n^{2} + 15 \, a\right )} x^{3} + 5 \, {\left (a^{2} n^{4} - 22 \, a^{2} n^{2} + 45 \, a^{2}\right )} x - {\left (a^{2} n^{5} - 30 \, a^{2} n^{3} + {\left (n^{5} - 10 \, n^{3} + 9 \, n\right )} x^{4} + 149 \, a^{2} n + 2 \, {\left (a n^{5} - 20 \, a n^{3} + 19 \, a n\right )} x^{2}\right )} \sqrt {x^{2} + a}\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{6} - 35 \, n^{4} + 259 \, n^{2} - 225} \]
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Timed out. \[ \int \left (a+x^2\right )^2 \left (x+\sqrt {a+x^2}\right )^n \, dx=\text {Timed out} \]
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\[ \int \left (a+x^2\right )^2 \left (x+\sqrt {a+x^2}\right )^n \, dx=\int { {\left (x^{2} + a\right )}^{2} {\left (x + \sqrt {x^{2} + a}\right )}^{n} \,d x } \]
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\[ \int \left (a+x^2\right )^2 \left (x+\sqrt {a+x^2}\right )^n \, dx=\int { {\left (x^{2} + a\right )}^{2} {\left (x + \sqrt {x^{2} + a}\right )}^{n} \,d x } \]
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Timed out. \[ \int \left (a+x^2\right )^2 \left (x+\sqrt {a+x^2}\right )^n \, dx=\int {\left (x^2+a\right )}^2\,{\left (x+\sqrt {x^2+a}\right )}^n \,d x \]
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