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

Optimal. Leaf size=17 \[ \frac{\left (\sqrt{a+x^2}+x\right )^b}{b} \]

[Out]

(x + Sqrt[a + x^2])^b/b

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Rubi [A]  time = 0.0872216, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\left (\sqrt{a+x^2}+x\right )^b}{b} \]

Antiderivative was successfully verified.

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

[Out]

(x + Sqrt[a + x^2])^b/b

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Rubi in Sympy [A]  time = 4.05421, size = 12, normalized size = 0.71 \[ \frac{\left (x + \sqrt{a + x^{2}}\right )^{b}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + sqrt(a + x**2))**b/b

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Mathematica [A]  time = 0.027324, size = 17, normalized size = 1. \[ \frac{\left (\sqrt{a+x^2}+x\right )^b}{b} \]

Antiderivative was successfully verified.

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

[Out]

(x + Sqrt[a + x^2])^b/b

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{1 \left ( x+\sqrt{{x}^{2}+a} \right ) ^{b}{\frac{1}{\sqrt{{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+(x^2+a)^(1/2))^b/(x^2+a)^(1/2),x)

[Out]

int((x+(x^2+a)^(1/2))^b/(x^2+a)^(1/2),x)

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Maxima [A]  time = 1.39616, size = 20, normalized size = 1.18 \[ \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{b}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + sqrt(x^2 + a))^b/b

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Fricas [A]  time = 0.230442, size = 20, normalized size = 1.18 \[ \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{b}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + sqrt(x^2 + a))^b/b

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Sympy [A]  time = 5.35941, size = 311, normalized size = 18.29 \[ \begin{cases} - \frac{\sqrt{a} a^{\frac{b}{2}} \sinh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{b x \sqrt{\frac{a}{x^{2}} + 1}} - \frac{2 a^{\frac{b}{2}} \cosh{\left (b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )} \Gamma \left (- \frac{b}{2} + 1\right )}{b^{2} \Gamma \left (- \frac{b}{2}\right )} + \frac{a^{\frac{b}{2}} x \cosh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} b} - \frac{a^{\frac{b}{2}} x \sinh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} b \sqrt{\frac{a}{x^{2}} + 1}} & \text{for}\: \left |{\frac{x^{2}}{a}}\right | > 1 \\- \frac{a^{\frac{b}{2}} \sinh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{b \sqrt{1 + \frac{x^{2}}{a}}} - \frac{2 a^{\frac{b}{2}} \cosh{\left (b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )} \Gamma \left (- \frac{b}{2} + 1\right )}{b^{2} \Gamma \left (- \frac{b}{2}\right )} - \frac{a^{\frac{b}{2}} x^{2} \sinh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{a b \sqrt{1 + \frac{x^{2}}{a}}} + \frac{a^{\frac{b}{2}} x \cosh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-sqrt(a)*a**(b/2)*sinh(-b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))/(b*x*s
qrt(a/x**2 + 1)) - 2*a**(b/2)*cosh(b*asinh(x/sqrt(a)))*gamma(-b/2 + 1)/(b**2*gam
ma(-b/2)) + a**(b/2)*x*cosh(-b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))/(sqrt(a)*b)
- a**(b/2)*x*sinh(-b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))/(sqrt(a)*b*sqrt(a/x**2
+ 1)), Abs(x**2/a) > 1), (-a**(b/2)*sinh(-b*asinh(x/sqrt(a)) + asinh(x/sqrt(a))
)/(b*sqrt(1 + x**2/a)) - 2*a**(b/2)*cosh(b*asinh(x/sqrt(a)))*gamma(-b/2 + 1)/(b*
*2*gamma(-b/2)) - a**(b/2)*x**2*sinh(-b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))/(a*
b*sqrt(1 + x**2/a)) + a**(b/2)*x*cosh(-b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))/(s
qrt(a)*b), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x + sqrt(x^2 + a))^b/sqrt(x^2 + a), x)