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]

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

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [B]  time = 0.43, size = 42, normalized size = 1.91 \[ {\left (x + 1\right )} \sqrt {\frac {x}{x + 1}} - \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} - 1\right ) \]

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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giac [B]  time = 0.39, size = 35, normalized size = 1.59 \[ \frac {1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \mathrm {sgn}\left (x + 1\right ) + \sqrt {x^{2} + x} \mathrm {sgn}\left (x + 1\right ) \]

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)

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maple [B]  time = 0.01, size = 43, normalized size = 1.95 \[ -\frac {\sqrt {\frac {x}{x +1}}\, \left (x +1\right ) \left (\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )-2 \sqrt {x^{2}+x}\right )}{2 \sqrt {\left (x +1\right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.44, size = 51, normalized size = 2.32 \[ -\frac {\sqrt {\frac {x}{x + 1}}}{\frac {x}{x + 1} - 1} - \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} - 1\right ) \]

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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mupad [B]  time = 0.04, size = 35, normalized size = 1.59 \[ -\mathrm {atanh}\left (\sqrt {\frac {x}{x+1}}\right )-\frac {\sqrt {\frac {x}{x+1}}}{\frac {x}{x+1}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x/(x + 1))^(1/2),x)

[Out]

- atanh((x/(x + 1))^(1/2)) - (x/(x + 1))^(1/2)/(x/(x + 1) - 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {x}{x + 1}}\, dx \]

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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