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

Optimal. Leaf size=131 \[ -\frac{a^4 \left (\sqrt{a+x^2}+x\right )^{n-4}}{16 (4-n)}-\frac{a^3 \left (\sqrt{a+x^2}+x\right )^{n-2}}{4 (2-n)}+\frac{3 a^2 \left (\sqrt{a+x^2}+x\right )^n}{8 n}+\frac{a \left (\sqrt{a+x^2}+x\right )^{n+2}}{4 (n+2)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+4}}{16 (n+4)} \]

[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 = 0.176048, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{a^4 \left (\sqrt{a+x^2}+x\right )^{n-4}}{16 (4-n)}-\frac{a^3 \left (\sqrt{a+x^2}+x\right )^{n-2}}{4 (2-n)}+\frac{3 a^2 \left (\sqrt{a+x^2}+x\right )^n}{8 n}+\frac{a \left (\sqrt{a+x^2}+x\right )^{n+2}}{4 (n+2)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+4}}{16 (n+4)} \]

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 = 4.06056, size = 355, normalized size = 2.71 \[ \frac{\sqrt{a+x^2} \left (\sqrt{a+x^2}+x\right )^n \left (\frac{a \sqrt{a+x^2} \left (a^2 \left (n^2-2\right )+a (n-2) x \left (2 (n+1) \sqrt{a+x^2}+(3 n+2) x\right )+2 (n-2) n x^3 \left (\sqrt{a+x^2}+x\right )\right )}{\left (n^2-4\right ) \left (x \left (\sqrt{a+x^2}+x\right )+a\right )^2}+\frac{\left (\sqrt{a+x^2}+x\right )^4 \left (2 a^4+a^3 (n-4) x \left ((n-4) x-2 \sqrt{a+x^2}\right )+a^2 (n-4) x^3 \left (4 (n-1) \sqrt{a+x^2}+(9 n-4) x\right )+8 (n-4) n x^7 \left (\sqrt{a+x^2}+x\right )+4 a (n-4) n x^5 \left (3 \sqrt{a+x^2}+4 x\right )\right )}{(n-4) (n+4) \left (a^4 \left (\sqrt{a+x^2}+8 x\right )+8 a^3 x^2 \left (4 \sqrt{a+x^2}+11 x\right )+16 a^2 x^4 \left (10 \sqrt{a+x^2}+17 x\right )+128 x^8 \left (\sqrt{a+x^2}+x\right )+64 a x^6 \left (4 \sqrt{a+x^2}+5 x\right )\right )}\right )}{n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + x^2)^(3/2)*(x + Sqrt[a + x^2])^n,x]

[Out]

(Sqrt[a + x^2]*(x + Sqrt[a + x^2])^n*(((x + Sqrt[a + x^2])^4*(2*a^4 + a^3*(-4 +
n)*x*((-4 + n)*x - 2*Sqrt[a + x^2]) + 8*(-4 + n)*n*x^7*(x + 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)*(128*x^8*(x + Sqrt[a + x^2]) + a^4*(8*x
+ Sqrt[a + x^2]) + 64*a*x^6*(5*x + 4*Sqrt[a + x^2]) + 8*a^3*x^2*(11*x + 4*Sqrt[
a + x^2]) + 16*a^2*x^4*(17*x + 10*Sqrt[a + x^2]))) + (a*Sqrt[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 = 0.024, 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 = 0.295498, size = 149, normalized size = 1.14 \[ \frac{{\left (a^{2} n^{4} +{\left (n^{4} - 4 \, n^{2}\right )} x^{4} - 16 \, a^{2} n^{2} + 2 \,{\left (a n^{4} - 10 \, a n^{2}\right )} x^{2} + 24 \, a^{2} - 4 \,{\left ({\left (n^{3} - 4 \, n\right )} x^{3} +{\left (a n^{3} - 10 \, a n\right )} x\right )} \sqrt{x^{2} + a}\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{5} - 20 \, n^{3} + 64 \, n} \]

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)