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

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

[Out]

(x + Sqrt[a + x^2])^n/n

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Rubi [A]  time = 0.0895184, 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 \[ \frac{\left (\sqrt{a+x^2}+x\right )^n}{n} \]

Antiderivative was successfully verified.

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

[Out]

(x + Sqrt[a + x^2])^n/n

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Rubi in Sympy [A]  time = 9.21112, size = 12, normalized size = 0.71 \[ \frac{\left (x + \sqrt{a + x^{2}}\right )^{n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + sqrt(a + x**2))**n/n

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Mathematica [A]  time = 0.0291316, size = 17, normalized size = 1. \[ \frac{\left (\sqrt{a+x^2}+x\right )^n}{n} \]

Antiderivative was successfully verified.

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

[Out]

(x + Sqrt[a + x^2])^n/n

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{1 \left ( x+\sqrt{{x}^{2}+a} \right ) ^{n}{\frac{1}{\sqrt{{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.727793, size = 20, normalized size = 1.18 \[ \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + sqrt(x^2 + a))^n/n

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Fricas [A]  time = 0.293211, size = 20, normalized size = 1.18 \[ \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + sqrt(x^2 + a))^n/n

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Sympy [A]  time = 14.3653, size = 311, normalized size = 18.29 \[ \begin{cases} - \frac{\sqrt{a} a^{\frac{n}{2}} \sinh{\left (- n \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{n x \sqrt{\frac{a}{x^{2}} + 1}} - \frac{2 a^{\frac{n}{2}} \cosh{\left (n \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )} \Gamma \left (- \frac{n}{2} + 1\right )}{n^{2} \Gamma \left (- \frac{n}{2}\right )} + \frac{a^{\frac{n}{2}} x \cosh{\left (- n \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} n} - \frac{a^{\frac{n}{2}} x \sinh{\left (- n \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} n \sqrt{\frac{a}{x^{2}} + 1}} & \text{for}\: \left |{\frac{x^{2}}{a}}\right | > 1 \\- \frac{a^{\frac{n}{2}} \sinh{\left (- n \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{n \sqrt{1 + \frac{x^{2}}{a}}} - \frac{2 a^{\frac{n}{2}} \cosh{\left (n \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )} \Gamma \left (- \frac{n}{2} + 1\right )}{n^{2} \Gamma \left (- \frac{n}{2}\right )} - \frac{a^{\frac{n}{2}} x^{2} \sinh{\left (- n \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{a n \sqrt{1 + \frac{x^{2}}{a}}} + \frac{a^{\frac{n}{2}} x \cosh{\left (- n \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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