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

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 = 0.0033112, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {50, 54, 215} \[ \sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/Sqrt[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]

Rubi steps

\begin{align*} \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 = 0.0161803, 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]/Sqrt[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 = 0.003, size = 39, normalized size = 1.8 \begin{align*} \sqrt{x}\sqrt{1+x}-{\frac{1}{2}\sqrt{x \left ( 1+x \right ) }\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.983564, size = 66, normalized size = 3. \begin{align*} \frac{\sqrt{x + 1}}{\sqrt{x}{\left (\frac{x + 1}{x} - 1\right )}} - \frac{1}{2} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} + 1\right ) + \frac{1}{2} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78441, size = 86, normalized size = 3.91 \begin{align*} \sqrt{x + 1} \sqrt{x} + \frac{1}{2} \, \log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.54248, size = 60, normalized size = 2.73 \begin{align*} \begin{cases} - \operatorname{acosh}{\left (\sqrt{x + 1} \right )} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x}} - \frac{\sqrt{x + 1}}{\sqrt{x}} & \text{for}\: \left |{x + 1}\right | > 1 \\i \sqrt{- x} \sqrt{x + 1} + i \operatorname{asin}{\left (\sqrt{x + 1} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-acosh(sqrt(x + 1)) + (x + 1)**(3/2)/sqrt(x) - sqrt(x + 1)/sqrt(x), Abs(x + 1) > 1), (I*sqrt(-x)*sq
rt(x + 1) + I*asin(sqrt(x + 1)), True))

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Giac [A]  time = 1.2386, size = 31, normalized size = 1.41 \begin{align*} \sqrt{x + 1} \sqrt{x} + \log \left ({\left | -\sqrt{x + 1} + \sqrt{x} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x + 1)*sqrt(x) + log(abs(-sqrt(x + 1) + sqrt(x)))