### 3.219 $$\int \frac{\sqrt{4+x}}{x} \, dx$$

Optimal. Leaf size=24 $2 \sqrt{x+4}-4 \tanh ^{-1}\left (\frac{\sqrt{x+4}}{2}\right )$

[Out]

2*Sqrt[4 + x] - 4*ArcTanh[Sqrt[4 + x]/2]

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Rubi [A]  time = 0.0054734, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.273, Rules used = {50, 63, 207} $2 \sqrt{x+4}-4 \tanh ^{-1}\left (\frac{\sqrt{x+4}}{2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[4 + x]/x,x]

[Out]

2*Sqrt[4 + x] - 4*ArcTanh[Sqrt[4 + x]/2]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{4+x}}{x} \, dx &=2 \sqrt{4+x}+4 \int \frac{1}{x \sqrt{4+x}} \, dx\\ &=2 \sqrt{4+x}+8 \operatorname{Subst}\left (\int \frac{1}{-4+x^2} \, dx,x,\sqrt{4+x}\right )\\ &=2 \sqrt{4+x}-4 \tanh ^{-1}\left (\frac{\sqrt{4+x}}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0045924, size = 24, normalized size = 1. $2 \sqrt{x+4}-4 \tanh ^{-1}\left (\frac{\sqrt{x+4}}{2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[4 + x]/x,x]

[Out]

2*Sqrt[4 + x] - 4*ArcTanh[Sqrt[4 + x]/2]

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Maple [A]  time = 0.007, size = 29, normalized size = 1.2 \begin{align*} 2\,\sqrt{4+x}-2\,\ln \left ( \sqrt{4+x}+2 \right ) +2\,\ln \left ( \sqrt{4+x}-2 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((4+x)^(1/2)/x,x)

[Out]

2*(4+x)^(1/2)-2*ln((4+x)^(1/2)+2)+2*ln((4+x)^(1/2)-2)

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Maxima [A]  time = 0.940056, size = 38, normalized size = 1.58 \begin{align*} 2 \, \sqrt{x + 4} - 2 \, \log \left (\sqrt{x + 4} + 2\right ) + 2 \, \log \left (\sqrt{x + 4} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+x)^(1/2)/x,x, algorithm="maxima")

[Out]

2*sqrt(x + 4) - 2*log(sqrt(x + 4) + 2) + 2*log(sqrt(x + 4) - 2)

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Fricas [A]  time = 1.79127, size = 88, normalized size = 3.67 \begin{align*} 2 \, \sqrt{x + 4} - 2 \, \log \left (\sqrt{x + 4} + 2\right ) + 2 \, \log \left (\sqrt{x + 4} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+x)^(1/2)/x,x, algorithm="fricas")

[Out]

2*sqrt(x + 4) - 2*log(sqrt(x + 4) + 2) + 2*log(sqrt(x + 4) - 2)

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Sympy [A]  time = 0.716193, size = 44, normalized size = 1.83 \begin{align*} \begin{cases} 2 \sqrt{x + 4} - 4 \operatorname{acoth}{\left (\frac{\sqrt{x + 4}}{2} \right )} & \text{for}\: \frac{\left |{x + 4}\right |}{4} > 1 \\2 \sqrt{x + 4} - 4 \operatorname{atanh}{\left (\frac{\sqrt{x + 4}}{2} \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+x)**(1/2)/x,x)

[Out]

Piecewise((2*sqrt(x + 4) - 4*acoth(sqrt(x + 4)/2), Abs(x + 4)/4 > 1), (2*sqrt(x + 4) - 4*atanh(sqrt(x + 4)/2),
True))

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Giac [A]  time = 1.05978, size = 39, normalized size = 1.62 \begin{align*} 2 \, \sqrt{x + 4} - 2 \, \log \left (\sqrt{x + 4} + 2\right ) + 2 \, \log \left ({\left | \sqrt{x + 4} - 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+x)^(1/2)/x,x, algorithm="giac")

[Out]

2*sqrt(x + 4) - 2*log(sqrt(x + 4) + 2) + 2*log(abs(sqrt(x + 4) - 2))