3.345 \(\int \frac{\left (x-\sqrt{a+x^2}\right )^n}{\left (a+x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ -\frac{16 \left (x-\sqrt{a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac{n+4}{2};\frac{n+6}{2};-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]

[Out]

(-16*(x - Sqrt[a + x^2])^(4 + n)*Hypergeometric2F1[4, (4 + n)/2, (6 + n)/2, -((x
- Sqrt[a + x^2])^2/a)])/(a^4*(4 + n))

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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]

(-16*(x - Sqrt[a + x^2])^(4 + n)*Hypergeometric2F1[4, (4 + n)/2, (6 + n)/2, -((x
- Sqrt[a + x^2])^2/a)])/(a^4*(4 + n))

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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]

-16*Integral(x**3*x**n/(a + x**2)**4, (x, x - sqrt(a + x**2)))

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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]

Integrate[(x - Sqrt[a + x^2])^n/(a + x^2)^(5/2), x]

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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]

int((x-(x^2+a)^(1/2))^n/(x^2+a)^(5/2),x)

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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]

integrate((x - sqrt(x^2 + a))^n/(x^2 + a)^(5/2), x)

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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]

integral((x - sqrt(x^2 + a))^n/((x^4 + 2*a*x^2 + a^2)*sqrt(x^2 + a)), x)

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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]

Timed 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]

integrate((x - sqrt(x^2 + a))^n/(x^2 + a)^(5/2), x)