3.1.25 \(\int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx\) [25]

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

[Out]

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

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Rubi [A]
time = 0.04, 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 = {2147, 30} \begin {gather*} 2 \sqrt {\sqrt {a^2+x^2}+x} \end {gather*}

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 2147

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^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 &=\text {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 = 0.03, size = 19, normalized size = 1.00 \begin {gather*} 2 \sqrt {x+\sqrt {a^2+x^2}} \end {gather*}

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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Mathics [A]
time = 1.76, size = 15, normalized size = 0.79 \begin {gather*} 2 \sqrt {x+\sqrt {a^2+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('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 = 0.02, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.31, size = 15, normalized size = 0.79 \begin {gather*} 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}} \end {gather*}

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 = 0.10, size = 15, normalized size = 0.79 \begin {gather*} 2 \sqrt {x + \sqrt {a^{2} + x^{2}}} \end {gather*}

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 [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} 2 \sqrt {x+\sqrt {a^{2}+x^{2}}} \end {gather*}

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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Mupad [B]
time = 0.42, size = 15, normalized size = 0.79 \begin {gather*} 2\,\sqrt {x+\sqrt {a^2+x^2}} \end {gather*}

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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