3.978 \(\int \sqrt{\frac{x}{1+x}} \, dx\)

Optimal. Leaf size=22 \[ \sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

Sqrt[x]*Sqrt[1 + x] - ArcSinh[Sqrt[x]]

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Rubi [A]  time = 0.004786, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1958, 50, 54, 215} \[ \sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x/(1 + x)],x]

[Out]

Sqrt[x]*Sqrt[1 + x] - ArcSinh[Sqrt[x]]

Rule 1958

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[(u*(e*(a + b*x
^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]

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 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.015082, size = 42, normalized size = 1.91 \[ \frac{\sqrt{\frac{x}{x+1}} \left (\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x/(1 + x)],x]

[Out]

(Sqrt[x/(1 + x)]*(Sqrt[x]*(1 + x) - Sqrt[1 + x]*ArcSinh[Sqrt[x]]))/Sqrt[x]

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Maple [B]  time = 0., size = 45, normalized size = 2.1 \begin{align*}{\frac{1+x}{2}\sqrt{{\frac{x}{1+x}}} \left ( 2\,\sqrt{{x}^{2}+x}-\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.11973, size = 69, normalized size = 3.14 \begin{align*} -\frac{\sqrt{\frac{x}{x + 1}}}{\frac{x}{x + 1} - 1} - \frac{1}{2} \, \log \left (\sqrt{\frac{x}{x + 1}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x}{x + 1}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x/(x + 1))/(x/(x + 1) - 1) - 1/2*log(sqrt(x/(x + 1)) + 1) + 1/2*log(sqrt(x/(x + 1)) - 1)

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Fricas [B]  time = 1.43171, size = 117, normalized size = 5.32 \begin{align*}{\left (x + 1\right )} \sqrt{\frac{x}{x + 1}} - \frac{1}{2} \, \log \left (\sqrt{\frac{x}{x + 1}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x}{x + 1}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*sqrt(x/(x + 1)) - 1/2*log(sqrt(x/(x + 1)) + 1) + 1/2*log(sqrt(x/(x + 1)) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x/(x + 1)), x)

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Giac [B]  time = 1.25616, size = 47, normalized size = 2.14 \begin{align*} \frac{1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ) \mathrm{sgn}\left (x + 1\right ) + \sqrt{x^{2} + x} \mathrm{sgn}\left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(-2*x + 2*sqrt(x^2 + x) - 1))*sgn(x + 1) + sqrt(x^2 + x)*sgn(x + 1)