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

Optimal. Leaf size=52 \[ \frac{\left (\sqrt{a+x^2}+x\right )^{n+1}}{2 (n+1)}-\frac{a \left (\sqrt{a+x^2}+x\right )^{n-1}}{2 (1-n)} \]

[Out]

-(a*(x + Sqrt[a + x^2])^(-1 + n))/(2*(1 - n)) + (x + Sqrt[a + x^2])^(1 + n)/(2*(
1 + n))

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Rubi [A]  time = 0.0445983, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (\sqrt{a+x^2}+x\right )^{n+1}}{2 (n+1)}-\frac{a \left (\sqrt{a+x^2}+x\right )^{n-1}}{2 (1-n)} \]

Antiderivative was successfully verified.

[In]  Int[(x + Sqrt[a + x^2])^n,x]

[Out]

-(a*(x + Sqrt[a + x^2])^(-1 + n))/(2*(1 - n)) + (x + Sqrt[a + x^2])^(1 + n)/(2*(
1 + n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + sqrt(a + x**2))**n, x)

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Mathematica [A]  time = 0.0271864, size = 36, normalized size = 0.69 \[ \frac{\left (\sqrt{a+x^2}+x\right )^n \left (n \sqrt{a+x^2}-x\right )}{n^2-1} \]

Antiderivative was successfully verified.

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

[Out]

((x + Sqrt[a + x^2])^n*(-x + n*Sqrt[a + x^2]))/(-1 + n^2)

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Maple [B]  time = 0.012, size = 120, normalized size = 2.3 \[{\frac{n}{4\,\sqrt{\pi }}{a}^{{\frac{1}{2}}+{\frac{n}{2}}} \left ( 8\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n \left ( -2+2\,n \right ) } \left ({\frac{an}{{x}^{2}}}+n-1 \right ) \left ( \sqrt{{\frac{a}{{x}^{2}}}+1}+1 \right ) ^{-1+n}}+4\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n}\sqrt{{\frac{a}{{x}^{2}}}+1} \left ( \sqrt{{\frac{a}{{x}^{2}}}+1}+1 \right ) ^{-1+n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+(x^2+a)^(1/2))^n,x)

[Out]

1/4*a^(1/2+1/2*n)/Pi^(1/2)*n*(8*Pi^(1/2)/(1+n)/n*x^(1+n)*a^(-1/2-1/2*n)*(a/x^2*n
+n-1)/(-2+2*n)*((a/x^2+1)^(1/2)+1)^(-1+n)+4*Pi^(1/2)/(1+n)/n*x^(1+n)*a^(-1/2-1/2
*n)*(a/x^2+1)^(1/2)*((a/x^2+1)^(1/2)+1)^(-1+n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x + sqrt(x^2 + a))^n, x)

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Fricas [A]  time = 0.324031, size = 43, normalized size = 0.83 \[ \frac{{\left (\sqrt{x^{2} + a} n - x\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.1818, size = 2147, normalized size = 41.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a**(9/2)*a**(n/2)*n**2*x*sqrt(a/x**2 + 1)*sinh(n*asinh(x/sqrt(a)))*g
amma(-n/2)/(2*a**(9/2)*n**2*gamma(-n/2 + 1) - 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**
(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2)*x**2*gamma(-n/2 + 1)) + a**(9/2)*a*
*(n/2)*n*x*cosh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*gamma(-n/2 + 1)
- 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2
)*x**2*gamma(-n/2 + 1)) - a**(7/2)*a**(n/2)*n**2*x**3*sqrt(a/x**2 + 1)*sinh(n*as
inh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*gamma(-n/2 + 1) - 2*a**(9/2)*gamma(
-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2)*x**2*gamma(-n/2 +
1)) + a**(7/2)*a**(n/2)*n*x**3*cosh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*
n**2*gamma(-n/2 + 1) - 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-
n/2 + 1) - 2*a**(7/2)*x**2*gamma(-n/2 + 1)) + 2*a**5*a**(n/2)*n*cosh(n*asinh(x/s
qrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*gamma(-n/2 + 1) - 2
*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2)*x*
*2*gamma(-n/2 + 1)) - 2*a**5*a**(n/2)*n*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*gamma(-
n/2 + 1) - 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2
*a**(7/2)*x**2*gamma(-n/2 + 1)) - 2*a**4*a**(n/2)*n*x**2*sqrt(a/x**2 + 1)*sinh(n
*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*gamma(-n/
2 + 1) - 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a
**(7/2)*x**2*gamma(-n/2 + 1)) + 4*a**4*a**(n/2)*n*x**2*cosh(n*asinh(x/sqrt(a)) +
asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*gamma(-n/2 + 1) - 2*a**(9/2)
*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2)*x**2*gamma(
-n/2 + 1)) - 2*a**4*a**(n/2)*n*x**2*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*gamma(-n/2
+ 1) - 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**
(7/2)*x**2*gamma(-n/2 + 1)) - 2*a**4*a**(n/2)*x**2*sqrt(a/x**2 + 1)*sinh(n*asinh
(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*gamma(-n/2 + 1)
- 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2
)*x**2*gamma(-n/2 + 1)) + 2*a**4*a**(n/2)*x**2*cosh(n*asinh(x/sqrt(a)) + asinh(x
/sqrt(a)))*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*gamma(-n/2 + 1) - 2*a**(9/2)*gamma(-
n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2)*x**2*gamma(-n/2 + 1
)) - 2*a**3*a**(n/2)*n*x**4*sqrt(a/x**2 + 1)*sinh(n*asinh(x/sqrt(a)) + asinh(x/s
qrt(a)))*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*gamma(-n/2 + 1) - 2*a**(9/2)*gamma(-n/
2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2)*x**2*gamma(-n/2 + 1))
+ 2*a**3*a**(n/2)*n*x**4*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2
+ 1)/(2*a**(9/2)*n**2*gamma(-n/2 + 1) - 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)
*n**2*x**2*gamma(-n/2 + 1) - 2*a**(7/2)*x**2*gamma(-n/2 + 1)) - 2*a**3*a**(n/2)*
x**4*sqrt(a/x**2 + 1)*sinh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1
)/(2*a**(9/2)*n**2*gamma(-n/2 + 1) - 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**
2*x**2*gamma(-n/2 + 1) - 2*a**(7/2)*x**2*gamma(-n/2 + 1)) + 2*a**3*a**(n/2)*x**4
*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(2*a**(9/2)*n**2*ga
mma(-n/2 + 1) - 2*a**(9/2)*gamma(-n/2 + 1) + 2*a**(7/2)*n**2*x**2*gamma(-n/2 + 1
) - 2*a**(7/2)*x**2*gamma(-n/2 + 1)), Abs(x**2/a) > 1), (-2*a**(5/2)*a**(n/2)*n*
x*sqrt(1 + x**2/a)*sinh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(
2*a**(5/2)*n**2*gamma(-n/2 + 1) - 2*a**(5/2)*gamma(-n/2 + 1)) + a**(5/2)*a**(n/2
)*n*x*cosh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(5/2)*n**2*gamma(-n/2 + 1) - 2*
a**(5/2)*gamma(-n/2 + 1)) - 2*a**(5/2)*a**(n/2)*x*sqrt(1 + x**2/a)*sinh(n*asinh(
x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(2*a**(5/2)*n**2*gamma(-n/2 + 1)
- 2*a**(5/2)*gamma(-n/2 + 1)) - a**3*a**(n/2)*n**2*sqrt(1 + x**2/a)*sinh(n*asinh
(x/sqrt(a)))*gamma(-n/2)/(2*a**(5/2)*n**2*gamma(-n/2 + 1) - 2*a**(5/2)*gamma(-n/
2 + 1)) + 2*a**3*a**(n/2)*n*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n
/2 + 1)/(2*a**(5/2)*n**2*gamma(-n/2 + 1) - 2*a**(5/2)*gamma(-n/2 + 1)) + 2*a**2*
a**(n/2)*n*x**2*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(2*a
**(5/2)*n**2*gamma(-n/2 + 1) - 2*a**(5/2)*gamma(-n/2 + 1)) + 2*a**2*a**(n/2)*x**
2*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(-n/2 + 1)/(2*a**(5/2)*n**2*g
amma(-n/2 + 1) - 2*a**(5/2)*gamma(-n/2 + 1)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x + sqrt(x^2 + a))^n, x)