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]

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

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Rubi [A]  time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 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]]

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.46, size = 32, normalized size = 0.62 \[ \frac {{\left (\sqrt {x^{2} + a} n - x\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{2} - 1} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x + \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]

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)

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maple [B]  time = 0.02, size = 120, normalized size = 2.31 \[ \frac {\left (\frac {8 \sqrt {\pi }\, \left (n +\frac {a n}{x^{2}}-1\right ) a^{-\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (\sqrt {\frac {a}{x^{2}}+1}+1\right )^{n -1}}{\left (n +1\right ) \left (2 n -2\right ) n}+\frac {4 \sqrt {\pi }\, \sqrt {\frac {a}{x^{2}}+1}\, a^{-\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (\sqrt {\frac {a}{x^{2}}+1}+1\right )^{n -1}}{\left (n +1\right ) n}\right ) n \,a^{\frac {n}{2}+\frac {1}{2}}}{4 \sqrt {\pi }} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x + \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (x+\sqrt {x^2+a}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + (a + x^2)^(1/2))^n,x)

[Out]

int((x + (a + x^2)^(1/2))^n, x)

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sympy [B]  time = 2.75, 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))**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/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*as
inh(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)) + as
inh(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)*g
amma(1 - n/2)), True))

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