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

Optimal. Leaf size=19 \[ 2 \sqrt {\sqrt {a^2+x^2}+x} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2122, 30} \[ 2 \sqrt {\sqrt {a^2+x^2}+x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a^2 + x^2],x]

[Out]

2*Sqrt[x + Sqrt[a^2 + x^2]]

Rule 30

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

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ 2 \sqrt {\sqrt {a^2+x^2}+x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a^2 + x^2],x]

[Out]

2*Sqrt[x + Sqrt[a^2 + x^2]]

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fricas [A]  time = 1.02, size = 15, normalized size = 0.79 \[ 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + sqrt(a^2 + x^2))

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giac [A]  time = 0.79, size = 15, normalized size = 0.79 \[ 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + sqrt(a^2 + x^2))

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x +\sqrt {a^{2}+x^{2}}}}{\sqrt {a^{2}+x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{\sqrt {a^{2} + x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.42, size = 15, normalized size = 0.79 \[ 2\,\sqrt {x+\sqrt {a^2+x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x + (a^2 + x^2)^(1/2))^(1/2)

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sympy [A]  time = 0.23, size = 15, normalized size = 0.79 \[ 2 \sqrt {x + \sqrt {a^{2} + x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + sqrt(a**2 + x**2))

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