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]

-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 = 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*}

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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]/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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fricas [A]  time = 0.44, size = 28, normalized size = 1.27 \[ \sqrt {x + 1} \sqrt {x} + \frac {1}{2} \, \log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) \]

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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giac [A]  time = 0.39, size = 22, normalized size = 1.00 \[ \sqrt {x + 1} \sqrt {x} + \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \]

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(sqrt(x + 1) - sqrt(x))

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maple [B]  time = 0.00, size = 39, normalized size = 1.77 \[ \sqrt {x +1}\, \sqrt {x}-\frac {\sqrt {\left (x +1\right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{2 \sqrt {x +1}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.87, size = 49, normalized size = 2.23 \[ \frac {\sqrt {x + 1}}{\sqrt {x} {\left (\frac {x + 1}{x} - 1\right )}} - \frac {1}{2} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} + 1\right ) + \frac {1}{2} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} - 1\right ) \]

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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mupad [B]  time = 3.72, size = 26, normalized size = 1.18 \[ \sqrt {x}\,\sqrt {x+1}-2\,\mathrm {atanh}\left (\frac {\sqrt {x}}{\sqrt {x+1}-1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(x + 1)^(1/2),x)

[Out]

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

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sympy [A]  time = 1.48, size = 60, normalized size = 2.73 \[ \begin {cases} - \operatorname {acosh}{\left (\sqrt {x + 1} \right )} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x}} - \frac {\sqrt {x + 1}}{\sqrt {x}} & \text {for}\: \left |{x + 1}\right | > 1 \\i \sqrt {- x} \sqrt {x + 1} + i \operatorname {asin}{\left (\sqrt {x + 1} \right )} & \text {otherwise} \end {cases} \]

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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