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*Sqrt[x + Sqrt[a^2 + x^2]]

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Rubi [A]  time = 0.0651093, antiderivative size = 19, normalized size of antiderivative = 1., 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 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{\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*}

Mathematica [A]  time = 0.0100317, size = 19, normalized size = 1. \[ 2 \sqrt{\sqrt{a^2+x^2}+x} \]

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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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{x+\sqrt{{a}^{2}+{x}^{2}}}{\frac{1}{\sqrt{{a}^{2}+{x}^{2}}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a^{2} + x^{2}}}\,{d x} \end{align*}

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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Fricas [A]  time = 2.02462, size = 39, normalized size = 2.05 \begin{align*} 2 \, \sqrt{x + \sqrt{a^{2} + x^{2}}} \end{align*}

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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Sympy [A]  time = 0.206048, size = 15, normalized size = 0.79 \begin{align*} 2 \sqrt{x + \sqrt{a^{2} + x^{2}}} \end{align*}

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

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]

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