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

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

[Out]

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

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Rubi [A]  time = 0.0466202, 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 \[ \frac{\left (\sqrt{a+x^2}+x\right )^{b+1}}{2 (b+1)}-\frac{a \left (\sqrt{a+x^2}+x\right )^{b-1}}{2 (1-b)} \]

Antiderivative was successfully verified.

[In]  Int[(x + Sqrt[a + x^2])^b,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + sqrt(a + x**2))**b, x)

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Mathematica [A]  time = 0.040084, size = 36, normalized size = 0.69 \[ \frac{\left (\sqrt{a+x^2}+x\right )^b \left (b \sqrt{a+x^2}-x\right )}{b^2-1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.045, size = 120, normalized size = 2.3 \[{\frac{b}{4\,\sqrt{\pi }}{a}^{{\frac{b}{2}}+{\frac{1}{2}}} \left ( 8\,{\frac{\sqrt{\pi }{x}^{1+b}{a}^{-b/2-1/2}}{ \left ( 1+b \right ) b \left ( -2+2\,b \right ) } \left ({\frac{ab}{{x}^{2}}}+b-1 \right ) \left ( \sqrt{{\frac{a}{{x}^{2}}}+1}+1 \right ) ^{-1+b}}+4\,{\frac{\sqrt{\pi }{x}^{1+b}{a}^{-b/2-1/2}}{ \left ( 1+b \right ) b}\sqrt{{\frac{a}{{x}^{2}}}+1} \left ( \sqrt{{\frac{a}{{x}^{2}}}+1}+1 \right ) ^{-1+b}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+(x^2+a)^(1/2))^b,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x + sqrt(x^2 + a))^b, x)

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Fricas [A]  time = 0.229006, size = 43, normalized size = 0.83 \[ \frac{{\left (\sqrt{x^{2} + a} b - x\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{b}}{b^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.97977, 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))**b,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x + sqrt(x^2 + a))^b, x)