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

Optimal. Leaf size=75 \[ -\frac {a^2 \left (\sqrt {a+x^2}+x\right )^{n-2}}{4 (2-n)}+\frac {a \left (\sqrt {a+x^2}+x\right )^n}{2 n}+\frac {\left (\sqrt {a+x^2}+x\right )^{n+2}}{4 (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2122, 270} \[ -\frac {a^2 \left (\sqrt {a+x^2}+x\right )^{n-2}}{4 (2-n)}+\frac {a \left (\sqrt {a+x^2}+x\right )^n}{2 n}+\frac {\left (\sqrt {a+x^2}+x\right )^{n+2}}{4 (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*(x + Sqrt[a + x^2])^(-2 + n))/(4*(2 - n)) + (a*(x + Sqrt[a + x^2])^n)/(2*n) + (x + Sqrt[a + x^2])^(2 + n
)/(4*(2 + n))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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])

Rubi steps

\begin {align*} \int \sqrt {a+x^2} \left (x+\sqrt {a+x^2}\right )^n \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x^{-3+n} \left (a+x^2\right )^2 \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (a^2 x^{-3+n}+2 a x^{-1+n}+x^{1+n}\right ) \, dx,x,x+\sqrt {a+x^2}\right )\\ &=-\frac {a^2 \left (x+\sqrt {a+x^2}\right )^{-2+n}}{4 (2-n)}+\frac {a \left (x+\sqrt {a+x^2}\right )^n}{2 n}+\frac {\left (x+\sqrt {a+x^2}\right )^{2+n}}{4 (2+n)}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 65, normalized size = 0.87 \[ \frac {1}{4} \left (\sqrt {a+x^2}+x\right )^n \left (\frac {a^2}{(n-2) \left (\sqrt {a+x^2}+x\right )^2}+\frac {\left (\sqrt {a+x^2}+x\right )^2}{n+2}+\frac {2 a}{n}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((x + Sqrt[a + x^2])^n*((2*a)/n + a^2/((-2 + n)*(x + Sqrt[a + x^2])^2) + (x + Sqrt[a + x^2])^2/(2 + n)))/4

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fricas [A]  time = 0.46, size = 48, normalized size = 0.64 \[ \frac {{\left (n^{2} x^{2} + a n^{2} - 2 \, \sqrt {x^{2} + a} n x - 2 \, a\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{3} - 4 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(n^2*x^2 + a*n^2 - 2*sqrt(x^2 + a)*n*x - 2*a)*(x + sqrt(x^2 + a))^n/(n^3 - 4*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \sqrt {x^{2}+a}\, \left (x +\sqrt {x^{2}+a}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {x^2+a}\,{\left (x+\sqrt {x^2+a}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + x**2)*(x + sqrt(a + x**2))**n, x)

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