3.342 \(\int \sqrt{a+x^2} \left (x-\sqrt{a+x^2}\right )^n \, dx\)

Optimal. Leaf size=81 \[ \frac{a^2 \left (x-\sqrt{a+x^2}\right )^{n-2}}{4 (2-n)}-\frac{a \left (x-\sqrt{a+x^2}\right )^n}{2 n}-\frac{\left (x-\sqrt{a+x^2}\right )^{n+2}}{4 (n+2)} \]

[Out]

(a^2*(x - Sqrt[a + x^2])^(-2 + n))/(4*(2 - n)) - (a*(x - Sqrt[a + x^2])^n)/(2*n)
- (x - Sqrt[a + x^2])^(2 + n)/(4*(2 + n))

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Rubi [A]  time = 0.139141, antiderivative size = 81, 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 \[ \frac{a^2 \left (x-\sqrt{a+x^2}\right )^{n-2}}{4 (2-n)}-\frac{a \left (x-\sqrt{a+x^2}\right )^n}{2 n}-\frac{\left (x-\sqrt{a+x^2}\right )^{n+2}}{4 (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + x^2]*(x - Sqrt[a + x^2])^n,x]

[Out]

(a^2*(x - Sqrt[a + x^2])^(-2 + n))/(4*(2 - n)) - (a*(x - Sqrt[a + x^2])^n)/(2*n)
- (x - Sqrt[a + x^2])^(2 + n)/(4*(2 + n))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\int ^{x - \sqrt{a + x^{2}}} \frac{x^{n} \left (a + x^{2}\right )^{2}}{x^{3}}\, dx}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+a)**(1/2)*(x-(x**2+a)**(1/2))**n,x)

[Out]

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

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Mathematica [A]  time = 0.849468, size = 112, normalized size = 1.38 \[ \frac{\left (a+x^2\right ) \left (x-\sqrt{a+x^2}\right )^n \left (-a^2 \left (n^2-2\right )+a (n-2) x \left (2 (n+1) \sqrt{a+x^2}-(3 n+2) x\right )+2 (n-2) n x^3 \left (\sqrt{a+x^2}-x\right )\right )}{n \left (n^2-4\right ) \left (x \left (x-\sqrt{a+x^2}\right )+a\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + x^2]*(x - Sqrt[a + x^2])^n,x]

[Out]

((a + x^2)*(x - Sqrt[a + x^2])^n*(-(a^2*(-2 + n^2)) + 2*(-2 + n)*n*x^3*(-x + Sqr
t[a + x^2]) + a*(-2 + n)*x*(-((2 + 3*n)*x) + 2*(1 + n)*Sqrt[a + x^2])))/(n*(-4 +
n^2)*(a + x*(x - Sqrt[a + x^2]))^2)

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int \sqrt{{x}^{2}+a} \left ( x-\sqrt{{x}^{2}+a} \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+a)^(1/2)*(x-(x^2+a)^(1/2))^n,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{2} + a}{\left (x - \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + a)*(x - sqrt(x^2 + a))^n,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.323089, size = 69, normalized size = 0.85 \[ -\frac{{\left (n^{2} x^{2} + a n^{2} + 2 \, \sqrt{x^{2} + a} n x - 2 \, a\right )}{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{n^{3} - 4 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + a)*(x - sqrt(x^2 + a))^n,x, algorithm="fricas")

[Out]

-(n^2*x^2 + a*n^2 + 2*sqrt(x^2 + a)*n*x - 2*a)*(x - sqrt(x^2 + a))^n/(n^3 - 4*n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + x^{2}} \left (x - \sqrt{a + x^{2}}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+a)**(1/2)*(x-(x**2+a)**(1/2))**n,x)

[Out]

Integral(sqrt(a + x**2)*(x - sqrt(a + x**2))**n, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{2} + a}{\left (x - \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + a)*(x - sqrt(x^2 + a))^n,x, algorithm="giac")

[Out]

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