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

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 = 0.0536436, 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, Rules used = {2122, 30} \[ \frac{\left (\sqrt{a+x^2}+x\right )^n}{n} \]

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

Rule 2122

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1*(i/c)^m)/(2^(2*m + 1)*e*f^(2*m)), Subst[Int[(x^n*(d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1))/(-d + x)^(2*(m +
1)), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && E
qQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (x+\sqrt{a+x^2}\right )^n}{\sqrt{a+x^2}} \, dx &=\operatorname{Subst}\left (\int x^{-1+n} \, dx,x,x+\sqrt{a+x^2}\right )\\ &=\frac{\left (x+\sqrt{a+x^2}\right )^n}{n}\\ \end{align*}

Mathematica [A]  time = 0.0061938, 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 = 0.018, size = 0, normalized size = 0. \begin{align*} \int{ \left ( x+\sqrt{{x}^{2}+a} \right ) ^{n}{\frac{1}{\sqrt{{x}^{2}+a}}}}\, dx \end{align*}

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

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45115, size = 34, normalized size = 2. \begin{align*} \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n} \end{align*}

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, algorithm="fricas")

[Out]

(x + sqrt(x^2 + a))^n/n

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Sympy [B]  time = 3.74217, size = 313, normalized size = 18.41 \begin{align*} \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 (1 - \frac{n}{2}\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}\: \frac{\left |{x^{2}}\right |}{\left |{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 (1 - \frac{n}{2}\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} \end{align*}

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*sqrt(a/x**2 + 1)) - 2*a**(n/2)*
cosh(n*asinh(x/sqrt(a)))*gamma(1 - n/2)/(n**2*gamma(-n/2)) + a**(n/2)*x*cosh(-n*asinh(x/sqrt(a)) + asinh(x/sqr
t(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)/Abs(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(1 - n/2)/(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)))/(sqrt(a)*n), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{\sqrt{x^{2} + a}}\,{d x} \end{align*}

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, algorithm="giac")

[Out]

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