3.341 \(\int \left (a+x^2\right )^{3/2} \left (x-\sqrt{a+x^2}\right )^n \, dx\)

Optimal. Leaf size=141 \[ \frac{a^4 \left (x-\sqrt{a+x^2}\right )^{n-4}}{16 (4-n)}+\frac{a^3 \left (x-\sqrt{a+x^2}\right )^{n-2}}{4 (2-n)}-\frac{3 a^2 \left (x-\sqrt{a+x^2}\right )^n}{8 n}-\frac{a \left (x-\sqrt{a+x^2}\right )^{n+2}}{4 (n+2)}-\frac{\left (x-\sqrt{a+x^2}\right )^{n+4}}{16 (n+4)} \]

[Out]

(a^4*(x - Sqrt[a + x^2])^(-4 + n))/(16*(4 - n)) + (a^3*(x - Sqrt[a + x^2])^(-2 +
n))/(4*(2 - n)) - (3*a^2*(x - Sqrt[a + x^2])^n)/(8*n) - (a*(x - Sqrt[a + x^2])^
(2 + n))/(4*(2 + n)) - (x - Sqrt[a + x^2])^(4 + n)/(16*(4 + n))

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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]

[Out]

(a^4*(x - Sqrt[a + x^2])^(-4 + n))/(16*(4 - n)) + (a^3*(x - Sqrt[a + x^2])^(-2 +
n))/(4*(2 - n)) - (3*a^2*(x - Sqrt[a + x^2])^n)/(8*n) - (a*(x - Sqrt[a + x^2])^
(2 + n))/(4*(2 + n)) - (x - Sqrt[a + x^2])^(4 + n)/(16*(4 + n))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\int ^{x - \sqrt{a + x^{2}}} \frac{x^{n} \left (a + x^{2}\right )^{4}}{x^{5}}\, dx}{16} \]

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)

[Out]

-Integral(x**n*(a + x**2)**4/x**5, (x, x - sqrt(a + x**2)))/16

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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]

[Out]

((x - Sqrt[a + x^2])^n*((Sqrt[a + x^2]*(x - Sqrt[a + x^2])^4*(-2*a^4 + 8*(-4 + n
)*n*x^7*(-x + Sqrt[a + x^2]) - a^3*(-4 + n)*x*((-4 + n)*x + 2*Sqrt[a + x^2]) + 4
*a*(-4 + n)*n*x^5*(-4*x + 3*Sqrt[a + x^2]) + a^2*(-4 + n)*x^3*((4 - 9*n)*x + 4*(
-1 + n)*Sqrt[a + x^2])))/((-4 + n)*(4 + n)*(a^4*(-8*x + Sqrt[a + x^2]) + 128*x^8
*(-x + Sqrt[a + x^2]) + 8*a^3*x^2*(-11*x + 4*Sqrt[a + x^2]) + 64*a*x^6*(-5*x + 4
*Sqrt[a + x^2]) + 16*a^2*x^4*(-17*x + 10*Sqrt[a + x^2]))) + (a*(a + x^2)*(-(a^2*
(-2 + n^2)) + 2*(-2 + n)*n*x^3*(-x + Sqrt[a + x^2]) + a*(-2 + n)*x*(-((2 + 3*n)*
x) + 2*(1 + n)*Sqrt[a + x^2])))/((-4 + n^2)*(a + x*(x - Sqrt[a + x^2]))^2)))/n

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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)

[Out]

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")

[Out]

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

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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")

[Out]

-(a^2*n^4 + (n^4 - 4*n^2)*x^4 - 16*a^2*n^2 + 2*(a*n^4 - 10*a*n^2)*x^2 + 24*a^2 +
4*((n^3 - 4*n)*x^3 + (a*n^3 - 10*a*n)*x)*sqrt(x^2 + a))*(x - sqrt(x^2 + a))^n/(
n^5 - 20*n^3 + 64*n)

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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)

[Out]

Timed out

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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")

[Out]

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