3.2250 \(\int \frac {\sqrt {x}}{1+\sqrt {x}} \, dx\)

Optimal. Leaf size=19 \[ x-2 \sqrt {x}+2 \log \left (\sqrt {x}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ x-2 \sqrt {x}+2 \log \left (\sqrt {x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 + Sqrt[x]),x]

[Out]

-2*Sqrt[x] + x + 2*Log[1 + Sqrt[x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ x-2 \sqrt {x}+2 \log \left (\sqrt {x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(1 + Sqrt[x]),x]

[Out]

-2*Sqrt[x] + x + 2*Log[1 + Sqrt[x]]

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fricas [A]  time = 0.96, size = 15, normalized size = 0.79 \[ x - 2 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*sqrt(x) + 2*log(sqrt(x) + 1)

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giac [A]  time = 0.17, size = 15, normalized size = 0.79 \[ x - 2 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*sqrt(x) + 2*log(sqrt(x) + 1)

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maple [A]  time = 0.00, size = 16, normalized size = 0.84 \[ x +2 \ln \left (\sqrt {x}+1\right )-2 \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.87, size = 22, normalized size = 1.16 \[ {\left (\sqrt {x} + 1\right )}^{2} - 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 1\right ) - 4 \]

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)^2 - 4*sqrt(x) + 2*log(sqrt(x) + 1) - 4

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mupad [B]  time = 0.05, size = 15, normalized size = 0.79 \[ x+2\,\ln \left (\sqrt {x}+1\right )-2\,\sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(x^(1/2) + 1),x)

[Out]

x + 2*log(x^(1/2) + 1) - 2*x^(1/2)

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sympy [A]  time = 0.15, size = 17, normalized size = 0.89 \[ - 2 \sqrt {x} + x + 2 \log {\left (\sqrt {x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(x) + x + 2*log(sqrt(x) + 1)

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