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 = 0.12, antiderivative size = 187, normalized size of antiderivative = 1.00, 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 270
Rule 2122
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*}
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Mathematica [A] time = 0.38, size = 157, normalized size = 0.84 \[ \frac {1}{64} \left (\sqrt {a+x^2}+x\right )^n \left (\frac {a^6}{(n-6) \left (\sqrt {a+x^2}+x\right )^6}+\frac {6 a^5}{(n-4) \left (\sqrt {a+x^2}+x\right )^4}+\frac {15 a^4}{(n-2) \left (\sqrt {a+x^2}+x\right )^2}+\frac {20 a^3}{n}+\frac {15 a^2 \left (\sqrt {a+x^2}+x\right )^2}{n+2}+\frac {6 a \left (\sqrt {a+x^2}+x\right )^4}{n+4}+\frac {\left (\sqrt {a+x^2}+x\right )^6}{n+6}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 201, normalized size = 1.07 \[ \frac {{\left (a^{3} n^{6} - 50 \, a^{3} n^{4} + {\left (n^{6} - 20 \, n^{4} + 64 \, n^{2}\right )} x^{6} + 544 \, a^{3} n^{2} + 3 \, {\left (a n^{6} - 30 \, a n^{4} + 104 \, a n^{2}\right )} x^{4} - 720 \, a^{3} + 3 \, {\left (a^{2} n^{6} - 40 \, a^{2} n^{4} + 264 \, a^{2} n^{2}\right )} x^{2} - 6 \, {\left ({\left (n^{5} - 20 \, n^{3} + 64 \, n\right )} x^{5} + 2 \, {\left (a n^{5} - 30 \, a n^{3} + 104 \, a n\right )} x^{3} + {\left (a^{2} n^{5} - 40 \, a^{2} n^{3} + 264 \, a^{2} n\right )} x\right )} \sqrt {x^{2} + a}\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{7} - 56 \, n^{5} + 784 \, n^{3} - 2304 \, n} \]
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 \[ \int {\left (x^{2} + a\right )}^{\frac {5}{2}} {\left (x + \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (x^{2}+a \right )^{\frac {5}{2}} \left (x +\sqrt {x^{2}+a}\right )^{n}\, dx \]
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 \[ \int {\left (x^{2} + a\right )}^{\frac {5}{2}} {\left (x + \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]
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 \[ \int {\left (x^2+a\right )}^{5/2}\,{\left (x+\sqrt {x^2+a}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + x^{2}\right )^{\frac {5}{2}} \left (x + \sqrt {a + x^{2}}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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