3.7.8 \(\int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx\) [608]

Optimal. Leaf size=21 \[ 4 \sqrt {x}+x+4 \log \left (1-\sqrt {x}\right ) \]

[Out]

x+4*ln(1-x^(1/2))+4*x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {383, 78} \begin {gather*} x+4 \sqrt {x}+4 \log \left (1-\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])/(-1 + Sqrt[x]),x]

[Out]

4*Sqrt[x] + x + 4*Log[1 - Sqrt[x]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rubi steps

\begin {align*} \int \frac {1+\sqrt {x}}{-1+\sqrt {x}} \, dx &=2 \text {Subst}\left (\int \frac {x (1+x)}{-1+x} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (2+\frac {2}{-1+x}+x\right ) \, dx,x,\sqrt {x}\right )\\ &=4 \sqrt {x}+x+4 \log \left (1-\sqrt {x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 19, normalized size = 0.90 \begin {gather*} 4 \sqrt {x}+x+4 \log \left (-1+\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])/(-1 + Sqrt[x]),x]

[Out]

4*Sqrt[x] + x + 4*Log[-1 + Sqrt[x]]

________________________________________________________________________________________

Maple [A]
time = 0.33, size = 16, normalized size = 0.76

method	result	size
derivativedivides	\(x +4 \sqrt {x}+4 \ln \left (-1+\sqrt {x}\right )\)	\(16\)
default	\(x +4 \sqrt {x}+4 \ln \left (-1+\sqrt {x}\right )\)	\(16\)
trager	\(x -2+4 \sqrt {x}+2 \ln \left (-1-x +2 \sqrt {x}\right )\)	\(22\)
meijerg	\(2 \sqrt {x}+4 \ln \left (1-\sqrt {x}\right )+\frac {\sqrt {x}\, \left (3 \sqrt {x}+6\right )}{3}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+4*x^(1/2)+4*ln(-1+x^(1/2))

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 15, normalized size = 0.71 \begin {gather*} x + 4 \, \sqrt {x} + 4 \, \log \left (\sqrt {x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 4*sqrt(x) + 4*log(sqrt(x) - 1)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 15, normalized size = 0.71 \begin {gather*} x + 4 \, \sqrt {x} + 4 \, \log \left (\sqrt {x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 4*sqrt(x) + 4*log(sqrt(x) - 1)

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 17, normalized size = 0.81 \begin {gather*} 4 \sqrt {x} + x + 4 \log {\left (\sqrt {x} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(x) + x + 4*log(sqrt(x) - 1)

________________________________________________________________________________________

Giac [A]
time = 3.35, size = 16, normalized size = 0.76 \begin {gather*} x + 4 \, \sqrt {x} + 4 \, \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 4*sqrt(x) + 4*log(abs(sqrt(x) - 1))

________________________________________________________________________________________

Mupad [B]
time = 3.01, size = 15, normalized size = 0.71 \begin {gather*} x+4\,\ln \left (\sqrt {x}-1\right )+4\,\sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2) + 1)/(x^(1/2) - 1),x)

[Out]

x + 4*log(x^(1/2) - 1) + 4*x^(1/2)

________________________________________________________________________________________