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

Optimal. Leaf size=187 \[ -\frac{a^6 \left (\sqrt{a+x^2}+x\right )^{n-6}}{64 (6-n)}-\frac{3 a^5 \left (\sqrt{a+x^2}+x\right )^{n-4}}{32 (4-n)}-\frac{15 a^4 \left (\sqrt{a+x^2}+x\right )^{n-2}}{64 (2-n)}+\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^n}{16 n}+\frac{15 a^2 \left (\sqrt{a+x^2}+x\right )^{n+2}}{64 (n+2)}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^{n+4}}{32 (n+4)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+6}}{64 (n+6)} \]

[Out]

-(a^6*(x + Sqrt[a + x^2])^(-6 + n))/(64*(6 - n)) - (3*a^5*(x + Sqrt[a + x^2])^(-
4 + n))/(32*(4 - n)) - (15*a^4*(x + Sqrt[a + x^2])^(-2 + n))/(64*(2 - n)) + (5*a
^3*(x + Sqrt[a + x^2])^n)/(16*n) + (15*a^2*(x + Sqrt[a + x^2])^(2 + n))/(64*(2 +
n)) + (3*a*(x + Sqrt[a + x^2])^(4 + n))/(32*(4 + n)) + (x + Sqrt[a + x^2])^(6 +
n)/(64*(6 + n))

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Rubi [A]  time = 0.223775, antiderivative size = 187, 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 \[ -\frac{a^6 \left (\sqrt{a+x^2}+x\right )^{n-6}}{64 (6-n)}-\frac{3 a^5 \left (\sqrt{a+x^2}+x\right )^{n-4}}{32 (4-n)}-\frac{15 a^4 \left (\sqrt{a+x^2}+x\right )^{n-2}}{64 (2-n)}+\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^n}{16 n}+\frac{15 a^2 \left (\sqrt{a+x^2}+x\right )^{n+2}}{64 (n+2)}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^{n+4}}{32 (n+4)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+6}}{64 (n+6)} \]

Antiderivative was successfully verified.

[In]  Int[(a + x^2)^(5/2)*(x + Sqrt[a + x^2])^n,x]

[Out]

-(a^6*(x + Sqrt[a + x^2])^(-6 + n))/(64*(6 - n)) - (3*a^5*(x + Sqrt[a + x^2])^(-
4 + n))/(32*(4 - n)) - (15*a^4*(x + Sqrt[a + x^2])^(-2 + n))/(64*(2 - n)) + (5*a
^3*(x + Sqrt[a + x^2])^n)/(16*n) + (15*a^2*(x + Sqrt[a + x^2])^(2 + n))/(64*(2 +
n)) + (3*a*(x + Sqrt[a + x^2])^(4 + n))/(32*(4 + n)) + (x + Sqrt[a + x^2])^(6 +
n)/(64*(6 + n))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x + \sqrt{a + x^{2}}} \frac{x^{n} \left (a + x^{2}\right )^{6}}{x^{7}}\, dx}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+a)**(5/2)*(x+(x**2+a)**(1/2))**n,x)

[Out]

Integral(x**n*(a + x**2)**6/x**7, (x, x + sqrt(a + x**2)))/64

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Mathematica [B]  time = 15.851, size = 659, normalized size = 3.52 \[ \frac{a^2 \left (a+x^2\right ) \left (a^2 \left (n^2-2\right )+a (n-2) x \left (2 (n+1) \sqrt{a+x^2}+(3 n+2) x\right )+2 (n-2) n x^3 \left (\sqrt{a+x^2}+x\right )\right ) \left (\sqrt{a+x^2}+x\right )^n}{n \left (n^2-4\right ) \left (x \left (\sqrt{a+x^2}+x\right )+a\right )^2}+\frac{2 a \sqrt{a+x^2} \left (2 a^4+a^3 (n-4) x \left ((n-4) x-2 \sqrt{a+x^2}\right )+a^2 (n-4) x^3 \left (4 (n-1) \sqrt{a+x^2}+(9 n-4) x\right )+8 (n-4) n x^7 \left (\sqrt{a+x^2}+x\right )+4 a (n-4) n x^5 \left (3 \sqrt{a+x^2}+4 x\right )\right ) \left (\sqrt{a+x^2}+x\right )^{n+4}}{(n-4) n (n+4) \left (a^4 \left (\sqrt{a+x^2}+8 x\right )+8 a^3 x^2 \left (4 \sqrt{a+x^2}+11 x\right )+16 a^2 x^4 \left (10 \sqrt{a+x^2}+17 x\right )+128 x^8 \left (\sqrt{a+x^2}+x\right )+64 a x^6 \left (4 \sqrt{a+x^2}+5 x\right )\right )}+\frac{\left (x \left (\sqrt{a+x^2}+x\right )+a\right ) \left (\frac{a^6}{(n-6) \left (\sqrt{a+x^2}+x\right )^6}-\frac{2 a^5}{(n-4) \left (\sqrt{a+x^2}+x\right )^4}-\frac{a^4}{(n-2) \left (\sqrt{a+x^2}+x\right )^2}+\frac{4 a^3}{n}-\frac{a^2 \left (\sqrt{a+x^2}+x\right )^2}{n+2}-\frac{2 a \left (\sqrt{a+x^2}+x\right )^4}{n+4}+\frac{\left (\sqrt{a+x^2}+x\right )^6}{n+6}\right ) \left (\sqrt{a+x^2}+x\right )^{n+9}}{64 \left (a^5 \left (\sqrt{a+x^2}+10 x\right )+10 a^4 x^2 \left (5 \sqrt{a+x^2}+17 x\right )+16 a^3 x^4 \left (25 \sqrt{a+x^2}+52 x\right )+32 a^2 x^6 \left (35 \sqrt{a+x^2}+53 x\right )+512 x^{10} \left (\sqrt{a+x^2}+x\right )+256 a x^8 \left (5 \sqrt{a+x^2}+6 x\right )\right )} \]

Antiderivative was successfully verified.

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

[Out]

((x + Sqrt[a + x^2])^(9 + n)*(a + x*(x + Sqrt[a + x^2]))*((4*a^3)/n + a^6/((-6 +
n)*(x + Sqrt[a + x^2])^6) - (2*a^5)/((-4 + n)*(x + Sqrt[a + x^2])^4) - a^4/((-2
+ n)*(x + Sqrt[a + x^2])^2) - (a^2*(x + Sqrt[a + x^2])^2)/(2 + n) - (2*a*(x + S
qrt[a + x^2])^4)/(4 + n) + (x + Sqrt[a + x^2])^6/(6 + n)))/(64*(512*x^10*(x + Sq
rt[a + x^2]) + a^5*(10*x + Sqrt[a + x^2]) + 256*a*x^8*(6*x + 5*Sqrt[a + x^2]) +
10*a^4*x^2*(17*x + 5*Sqrt[a + x^2]) + 16*a^3*x^4*(52*x + 25*Sqrt[a + x^2]) + 32*
a^2*x^6*(53*x + 35*Sqrt[a + x^2]))) + (2*a*Sqrt[a + x^2]*(x + Sqrt[a + x^2])^(4
+ n)*(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)*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^2*(a + x^2)*(x + Sqrt[a + x^2])^n*(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])))/(n*(-4 + n^2
)*(a + x*(x + Sqrt[a + x^2]))^2)

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \[ \int \left ({x}^{2}+a \right ) ^{{\frac{5}{2}}} \left ( x+\sqrt{{x}^{2}+a} \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+a)^(5/2)*(x+(x^2+a)^(1/2))^n,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} + a\right )}^{\frac{5}{2}}{\left (x + \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + a)^(5/2)*(x + sqrt(x^2 + a))^n,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.298639, size = 271, normalized size = 1.45 \[ \frac{{\left (a^{3} n^{6} - 50 \, a^{3} n^{4} +{\left (n^{6} - 20 \, n^{4} + 64 \, n^{2}\right )} x^{6} + 544 \, a^{3} n^{2} + 3 \,{\left (a n^{6} - 30 \, a n^{4} + 104 \, a n^{2}\right )} x^{4} - 720 \, a^{3} + 3 \,{\left (a^{2} n^{6} - 40 \, a^{2} n^{4} + 264 \, a^{2} n^{2}\right )} x^{2} - 6 \,{\left ({\left (n^{5} - 20 \, n^{3} + 64 \, n\right )} x^{5} + 2 \,{\left (a n^{5} - 30 \, a n^{3} + 104 \, a n\right )} x^{3} +{\left (a^{2} n^{5} - 40 \, a^{2} n^{3} + 264 \, a^{2} n\right )} x\right )} \sqrt{x^{2} + a}\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{7} - 56 \, n^{5} + 784 \, n^{3} - 2304 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + a)^(5/2)*(x + sqrt(x^2 + a))^n,x, algorithm="fricas")

[Out]

(a^3*n^6 - 50*a^3*n^4 + (n^6 - 20*n^4 + 64*n^2)*x^6 + 544*a^3*n^2 + 3*(a*n^6 - 3
0*a*n^4 + 104*a*n^2)*x^4 - 720*a^3 + 3*(a^2*n^6 - 40*a^2*n^4 + 264*a^2*n^2)*x^2
- 6*((n^5 - 20*n^3 + 64*n)*x^5 + 2*(a*n^5 - 30*a*n^3 + 104*a*n)*x^3 + (a^2*n^5 -
40*a^2*n^3 + 264*a^2*n)*x)*sqrt(x^2 + a))*(x + sqrt(x^2 + a))^n/(n^7 - 56*n^5 +
784*n^3 - 2304*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)**(5/2)*(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. \[ \int{\left (x^{2} + a\right )}^{\frac{5}{2}}{\left (x + \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + a)^(5/2)*(x + sqrt(x^2 + a))^n,x, algorithm="giac")

[Out]

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