3.8.31 \(\int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx\) [731]

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

[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 = {52, 56, 221} \begin {gather*} \sqrt {x} \sqrt {x+1}-\sinh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/Sqrt[1 + x],x]

[Out]

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

Rule 52

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 56

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 221

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}-\text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(22)=44\).
time = 0.04, size = 48, normalized size = 2.18 \begin {gather*} \frac {\sqrt {\frac {x}{1+x}} \left (\sqrt {x} (1+x)-\sqrt {1+x} \tanh ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )\right )}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/Sqrt[1 + x],x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(16)=32\).
time = 0.38, size = 39, normalized size = 1.77

method	result	size
meijerg	\(\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {1+x}-\sqrt {\pi }\, \arcsinh \left (\sqrt {x}\right )}{\sqrt {\pi }}\)	\(27\)
default	\(\sqrt {x}\, \sqrt {1+x}-\frac {\sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{2 \sqrt {x}\, \sqrt {1+x}}\)	\(39\)
risch	\(\sqrt {x}\, \sqrt {1+x}-\frac {\sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{2 \sqrt {x}\, \sqrt {1+x}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (16) = 32\).
time = 0.28, size = 49, normalized size = 2.23 \begin {gather*} \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 {gather*}

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 = 0.34, size = 28, normalized size = 1.27 \begin {gather*} \sqrt {x + 1} \sqrt {x} + \frac {1}{2} \, \log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.83, size = 63, normalized size = 2.86 \begin {gather*} \begin {cases} \sqrt {x} \sqrt {x + 1} - \operatorname {acosh}{\left (\sqrt {x + 1} \right )} & \text {for}\: \left |{x + 1}\right | > 1 \\i \operatorname {asin}{\left (\sqrt {x + 1} \right )} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {- x}} + \frac {i \sqrt {x + 1}}{\sqrt {- x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.49, size = 22, normalized size = 1.00 \begin {gather*} \sqrt {x + 1} \sqrt {x} + \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \end {gather*}

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

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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