Optimal. Leaf size=63 \[ -\frac{16 \left (x-\sqrt{a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac{n+4}{2};\frac{n+6}{2};-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]
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Rubi [A] time = 0.12499, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{16 \left (x-\sqrt{a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac{n+4}{2};\frac{n+6}{2};-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]
Antiderivative was successfully verified.
[In] Int[(x - Sqrt[a + x^2])^n/(a + x^2)^(5/2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - 16 \int ^{x - \sqrt{a + x^{2}}} \frac{x^{3} x^{n}}{\left (a + x^{2}\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x-(x**2+a)**(1/2))**n/(x**2+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0502313, size = 0, normalized size = 0. \[ \int \frac{\left (x-\sqrt{a+x^2}\right )^n}{\left (a+x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(x - Sqrt[a + x^2])^n/(a + x^2)^(5/2),x]
[Out]
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Maple [F] time = 0.039, size = 0, normalized size = 0. \[ \int{1 \left ( x-\sqrt{{x}^{2}+a} \right ) ^{n} \left ({x}^{2}+a \right ) ^{-{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x-(x^2+a)^(1/2))^n/(x^2+a)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^n/(x^2 + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{{\left (x^{4} + 2 \, a x^{2} + a^{2}\right )} \sqrt{x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^n/(x^2 + a)^(5/2),x, algorithm="fricas")
[Out]
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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-(x**2+a)**(1/2))**n/(x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^n/(x^2 + a)^(5/2),x, algorithm="giac")
[Out]