Optimal. Leaf size=141 \[ \frac{a^4 \left (x-\sqrt{a+x^2}\right )^{n-4}}{16 (4-n)}+\frac{a^3 \left (x-\sqrt{a+x^2}\right )^{n-2}}{4 (2-n)}-\frac{3 a^2 \left (x-\sqrt{a+x^2}\right )^n}{8 n}-\frac{a \left (x-\sqrt{a+x^2}\right )^{n+2}}{4 (n+2)}-\frac{\left (x-\sqrt{a+x^2}\right )^{n+4}}{16 (n+4)} \]
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Rubi [A] time = 0.180091, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{a^4 \left (x-\sqrt{a+x^2}\right )^{n-4}}{16 (4-n)}+\frac{a^3 \left (x-\sqrt{a+x^2}\right )^{n-2}}{4 (2-n)}-\frac{3 a^2 \left (x-\sqrt{a+x^2}\right )^n}{8 n}-\frac{a \left (x-\sqrt{a+x^2}\right )^{n+2}}{4 (n+2)}-\frac{\left (x-\sqrt{a+x^2}\right )^{n+4}}{16 (n+4)} \]
Antiderivative was successfully verified.
[In] Int[(a + x^2)^(3/2)*(x - Sqrt[a + x^2])^n,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{\int ^{x - \sqrt{a + x^{2}}} \frac{x^{n} \left (a + x^{2}\right )^{4}}{x^{5}}\, dx}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+a)**(3/2)*(x-(x**2+a)**(1/2))**n,x)
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Mathematica [B] time = 3.19027, size = 366, normalized size = 2.6 \[ \frac{\left (x-\sqrt{a+x^2}\right )^n \left (\frac{a \left (a+x^2\right ) \left (-a^2 \left (n^2-2\right )+a (n-2) x \left (2 (n+1) \sqrt{a+x^2}-(3 n+2) x\right )+2 (n-2) n x^3 \left (\sqrt{a+x^2}-x\right )\right )}{\left (n^2-4\right ) \left (x \left (x-\sqrt{a+x^2}\right )+a\right )^2}+\frac{\sqrt{a+x^2} \left (x-\sqrt{a+x^2}\right )^4 \left (-2 a^4-a^3 (n-4) x \left (2 \sqrt{a+x^2}+(n-4) x\right )+a^2 (n-4) x^3 \left (4 (n-1) \sqrt{a+x^2}+(4-9 n) x\right )+8 (n-4) n x^7 \left (\sqrt{a+x^2}-x\right )+4 a (n-4) n x^5 \left (3 \sqrt{a+x^2}-4 x\right )\right )}{(n-4) (n+4) \left (a^4 \left (\sqrt{a+x^2}-8 x\right )+8 a^3 x^2 \left (4 \sqrt{a+x^2}-11 x\right )+16 a^2 x^4 \left (10 \sqrt{a+x^2}-17 x\right )+128 x^8 \left (\sqrt{a+x^2}-x\right )+64 a x^6 \left (4 \sqrt{a+x^2}-5 x\right )\right )}\right )}{n} \]
Antiderivative was successfully verified.
[In] Integrate[(a + x^2)^(3/2)*(x - Sqrt[a + x^2])^n,x]
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Maple [F] time = 0.04, size = 0, normalized size = 0. \[ \int \left ({x}^{2}+a \right ) ^{{\frac{3}{2}}} \left ( x-\sqrt{{x}^{2}+a} \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+a)^(3/2)*(x-(x^2+a)^(1/2))^n,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} + a\right )}^{\frac{3}{2}}{\left (x - \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + a)^(3/2)*(x - sqrt(x^2 + a))^n,x, algorithm="maxima")
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Fricas [A] time = 0.298217, size = 153, normalized size = 1.09 \[ -\frac{{\left (a^{2} n^{4} +{\left (n^{4} - 4 \, n^{2}\right )} x^{4} - 16 \, a^{2} n^{2} + 2 \,{\left (a n^{4} - 10 \, a n^{2}\right )} x^{2} + 24 \, a^{2} + 4 \,{\left ({\left (n^{3} - 4 \, n\right )} x^{3} +{\left (a n^{3} - 10 \, a n\right )} x\right )} \sqrt{x^{2} + a}\right )}{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{n^{5} - 20 \, n^{3} + 64 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + a)^(3/2)*(x - sqrt(x^2 + a))^n,x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+a)**(3/2)*(x-(x**2+a)**(1/2))**n,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} + a\right )}^{\frac{3}{2}}{\left (x - \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + a)^(3/2)*(x - sqrt(x^2 + a))^n,x, algorithm="giac")
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