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

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 = 0.0211651, 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, Rules used = {2117, 14} \[ \frac{\left (\sqrt{a+x^2}+x\right )^{n+1}}{2 (n+1)}-\frac{a \left (\sqrt{a+x^2}+x\right )^{n-1}}{2 (1-n)} \]

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))

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0346207, size = 43, normalized size = 0.83 \[ \frac{\left (\sqrt{a+x^2}+x\right )^{n-1} \left ((n-1) x \left (\sqrt{a+x^2}+x\right )+a n\right )}{n^2-1} \]

Antiderivative was successfully verified.

[In]

Integrate[(x + Sqrt[a + x^2])^n,x]

[Out]

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

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Maple [B]  time = 0.007, size = 120, normalized size = 2.3 \begin{align*}{\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{1+{\frac{a}{{x}^{2}}}}+1 \right ) ^{-1+n}}+4\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n}\sqrt{1+{\frac{a}{{x}^{2}}}} \left ( \sqrt{1+{\frac{a}{{x}^{2}}}}+1 \right ) ^{-1+n}} \right ) } \end{align*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x^2+a)^(1/2))^n,x, algorithm="maxima")

[Out]

integrate((x + sqrt(x^2 + a))^n, x)

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Fricas [A]  time = 1.38304, size = 74, normalized size = 1.42 \begin{align*} \frac{{\left (\sqrt{x^{2} + a} n - x\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x^2+a)^(1/2))^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 [B]  time = 3.51579, size = 2149, normalized size = 41.33 \begin{align*} \text{result too large to display} \end{align*}

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)))*gamma(-n/2)/(2*a**(9/2)*n**2*ga
mma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2
)) + a**(9/2)*a**(n/2)*n*x*cosh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*g
amma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - a**(7/2)*a**(n/2)*n**2
*x**3*sqrt(a/x**2 + 1)*sinh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma
(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + a**(7/2)*a**(n/2)*n*x**3*c
osh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n
**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + 2*a**5*a**(n/2)*n*cosh(n*asinh(x/sqrt(a)) + asinh(
x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*
gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*a**5*a**(n/2)*n*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 -
n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 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(1 - n/2)/(2*a**(9/2)*n
**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1
- n/2)) + 4*a**4*a**(n/2)*n*x**2*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*
gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n
/2)) - 2*a**4*a**(n/2)*n*x**2*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a
**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*a**4*a**(n/2)*x**2*sqrt(a/x**2 + 1)*sin
h(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 -
n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + 2*a**4*a**(n/2)*x**2*cosh(n*as
inh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2)
+ 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*a**3*a**(n/2)*n*x**4*sqrt(a/x**2 +
1)*sinh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*ga
mma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + 2*a**3*a**(n/2)*n*x**4*
cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(
1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*a**3*a**(n/2)*x**4*sqrt(a
/x**2 + 1)*sinh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(
9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + 2*a**3*a**(n/2)*
x**4*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*g
amma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)), Abs(x**2)/Abs(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(1 - n/2)/(2*a**(5
/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2)) + a**(5/2)*a**(n/2)*n*x*cosh(n*asinh(x/sqrt(a)))*gamma(-n
/2)/(2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2)) - 2*a**(5/2)*a**(n/2)*x*sqrt(1 + x**2/a)*sinh
(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 -
n/2)) - 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(1 - n/
2) - 2*a**(5/2)*gamma(1 - n/2)) + 2*a**3*a**(n/2)*n*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)
/(2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2)) + 2*a**2*a**(n/2)*n*x**2*cosh(n*asinh(x/sqrt(a))
+ asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2)) + 2*a**2*a**(
n/2)*x**2*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5
/2)*gamma(1 - n/2)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x^2+a)^(1/2))^n,x, algorithm="giac")

[Out]

integrate((x + sqrt(x^2 + a))^n, x)