\(\int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx\) [235]

Optimal result

Integrand size = 21, antiderivative size = 201 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=6 \sqrt [6]{x}+x-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}+4 \sqrt [6]{x}}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}+4 \sqrt [6]{x}\right )\right )+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right ) \]

[Out]

6*x^(1/6)+x+6/5*ln(1-x^(1/6))-3/10*ln(2+x^(1/6)+2*x^(1/3)-x^(1/6)*5^(1/2))*(-5^(1/2)+1)-3/10*ln(2+x^(1/6)+2*x^
(1/3)+x^(1/6)*5^(1/2))*(5^(1/2)+1)-3/5*arctan(1/20*(1+4*x^(1/6)+5^(1/2))*(50+10*5^(1/2))^(1/2))*(10-2*5^(1/2))
^(1/2)-3/5*arctan((1+4*x^(1/6)-5^(1/2))/(10+2*5^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1598, 348, 308, 208, 648, 632, 210, 642, 31} \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 \sqrt [6]{x}-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (4 \sqrt [6]{x}+\sqrt {5}+1\right )\right )+x+6 \sqrt [6]{x}+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt {5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt {5} \sqrt [6]{x}+\sqrt [6]{x}+2\right ) \]

[In]

Int[Sqrt[x]/(-x^(-1/3) + Sqrt[x]),x]

[Out]

6*x^(1/6) + x - (3*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6))/Sqrt[2*(5 + Sqrt[5])]])/5 - (3*Sqrt[
2*(5 - Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*x^(1/6)))/2])/5 + (6*Log[1 - x^(1/6)])/5 - (3
*(1 - Sqrt[5])*Log[2 + x^(1/6) - Sqrt[5]*x^(1/6) + 2*x^(1/3)])/10 - (3*(1 + Sqrt[5])*Log[2 + x^(1/6) + Sqrt[5]
*x^(1/6) + 2*x^(1/3)])/10

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 208

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n
]], k, u}, Simp[u = Int[(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; (
r/(a*n))*Int[1/(r - s*x), x] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGt
Q[(n - 3)/2, 0] && NegQ[a/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 348

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{5/6}}{-1+x^{5/6}} \, dx \\ & = 6 \text {Subst}\left (\int \frac {x^{10}}{-1+x^5} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \text {Subst}\left (\int \left (1+x^5+\frac {1}{-1+x^5}\right ) \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x+6 \text {Subst}\left (\int \frac {1}{-1+x^5} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x-\frac {6}{5} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [6]{x}\right )-\frac {12}{5} \text {Subst}\left (\int \frac {1+\frac {1}{4} \left (1-\sqrt {5}\right ) x}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {12}{5} \text {Subst}\left (\int \frac {1+\frac {1}{4} \left (1+\sqrt {5}\right ) x}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (1-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (1-\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (1+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (1+\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )+\frac {1}{5} \left (3 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5+\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (1+\sqrt {5}\right )+2 \sqrt [6]{x}\right )+\frac {1}{5} \left (3 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5-\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (1-\sqrt {5}\right )+2 \sqrt [6]{x}\right ) \\ & = 6 \sqrt [6]{x}+x-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}+4 \sqrt [6]{x}}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}+4 \sqrt [6]{x}\right )\right )+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=6 \sqrt [6]{x}+x+\frac {6}{5} \log \left (-1+\sqrt [6]{x}\right )-\frac {6}{5} \text {RootSum}\left [1+\text {$\#$1}+\text {$\#$1}^2+\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {4 \log \left (\sqrt [6]{x}-\text {$\#$1}\right )+3 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{1+2 \text {$\#$1}+3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[In]

Integrate[Sqrt[x]/(-x^(-1/3) + Sqrt[x]),x]

[Out]

6*x^(1/6) + x + (6*Log[-1 + x^(1/6)])/5 - (6*RootSum[1 + #1 + #1^2 + #1^3 + #1^4 & , (4*Log[x^(1/6) - #1] + 3*
Log[x^(1/6) - #1]*#1 + 2*Log[x^(1/6) - #1]*#1^2 + Log[x^(1/6) - #1]*#1^3)/(1 + 2*#1 + 3*#1^2 + 4*#1^3) & ])/5

Maple [A] (warning: unable to verify)

Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.66

method	result	size
meijerg	\(\frac {6 \left (-1\right )^{\frac {4}{5}} \left (-\frac {5 x^{\frac {1}{6}} \left (-1\right )^{\frac {1}{5}} \left (11 x^{\frac {5}{6}}+66\right )}{66}-\left (-1\right )^{\frac {1}{5}} \left (\ln \left (1-x^{\frac {1}{6}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}}{1-\cos \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}}{1+\cos \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}}\right )\right )\right )}{5}\)	\(132\)
derivativedivides	\(x +6 x^{\frac {1}{6}}-\frac {3 \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {5}\right ) \left (-\sqrt {5}+1\right )}{10}-\frac {12 \left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {3 \left (-\sqrt {5}-1\right ) \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {5}\right )}{10}+\frac {12 \left (-4-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}\)	\(166\)
default	\(x +6 x^{\frac {1}{6}}-\frac {3 \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {5}\right ) \left (-\sqrt {5}+1\right )}{10}-\frac {12 \left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {3 \left (-\sqrt {5}-1\right ) \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {5}\right )}{10}+\frac {12 \left (-4-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}\)	\(166\)

[In]

int(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

6/5*(-1)^(4/5)*(-5/66*x^(1/6)*(-1)^(1/5)*(11*x^(5/6)+66)-(-1)^(1/5)*(ln(1-x^(1/6))+cos(2/5*Pi)*ln(1-2*cos(2/5*
Pi)*x^(1/6)+x^(1/3))-2*sin(2/5*Pi)*arctan(sin(2/5*Pi)*x^(1/6)/(1-cos(2/5*Pi)*x^(1/6)))-cos(1/5*Pi)*ln(1+2*cos(
1/5*Pi)*x^(1/6)+x^(1/3))-2*sin(1/5*Pi)*arctan(sin(1/5*Pi)*x^(1/6)/(1+cos(1/5*Pi)*x^(1/6)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (134) = 268\).

Time = 0.91 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3}{10} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )} \log \left (\frac {3}{2} \, \sqrt {2} \sqrt {\sqrt {5} - 5} + \frac {3}{2} \, \sqrt {5} + 6 \, x^{\frac {1}{6}} + \frac {3}{2}\right ) + \frac {3}{10} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} \log \left (-\frac {3}{2} \, \sqrt {2} \sqrt {\sqrt {5} - 5} + \frac {3}{2} \, \sqrt {5} + 6 \, x^{\frac {1}{6}} + \frac {3}{2}\right ) + \frac {1}{10} \, {\left (3 \, \sqrt {5} - \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} - 3\right )} \log \left (-3 \, \sqrt {5} + \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} + 12 \, x^{\frac {1}{6}} + 3\right ) + \frac {1}{10} \, {\left (3 \, \sqrt {5} + \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} - 3\right )} \log \left (-3 \, \sqrt {5} - \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} + 12 \, x^{\frac {1}{6}} + 3\right ) + x + 6 \, x^{\frac {1}{6}} + \frac {6}{5} \, \log \left (x^{\frac {1}{6}} - 1\right ) \]

[In]

integrate(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x, algorithm="fricas")

[Out]

-3/10*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*log(3/2*sqrt(2)*sqrt(sqrt(5) - 5) + 3/2*sqrt(5) + 6*x^(1/6) +
3/2) + 3/10*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)*log(-3/2*sqrt(2)*sqrt(sqrt(5) - 5) + 3/2*sqrt(5) + 6*x^(
1/6) + 3/2) + 1/10*(3*sqrt(5) - sqrt(-27/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(sqr
t(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5)
- 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) - 3)*log(-3*sqrt(5) + sqrt(-27/4*(sqrt(2)*sqrt(sqrt(
5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5)
- 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) + 12
*x^(1/6) + 3) + 1/10*(3*sqrt(5) + sqrt(-27/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(s
qrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(
5) - 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) - 3)*log(-3*sqrt(5) - sqrt(-27/4*(sqrt(2)*sqrt(sqr
t(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5
) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) +
12*x^(1/6) + 3) + x + 6*x^(1/6) + 6/5*log(x^(1/6) - 1)

Sympy [F]

\[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=\int \frac {x^{\frac {5}{6}}}{\left (\sqrt [6]{x} - 1\right ) \left (\sqrt [6]{x} + x^{\frac {2}{3}} + \sqrt [3]{x} + \sqrt {x} + 1\right )}\, dx \]

[In]

integrate(x**(1/2)/(-1/x**(1/3)+x**(1/2)),x)

[Out]

Integral(x**(5/6)/((x**(1/6) - 1)*(x**(1/6) + x**(2/3) + x**(1/3) + sqrt(x) + 1)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (134) = 268\).

Time = 0.32 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3 \, \sqrt {5} \left (-1\right )^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {2 \, \sqrt {5} - 10}} - \frac {3 \, \sqrt {5} \left (-1\right )^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} - 10}} - \frac {6}{5} \, \left (-1\right )^{\frac {1}{5}} \log \left (\left (-1\right )^{\frac {1}{5}} + x^{\frac {1}{6}}\right ) + x - \frac {3 \, {\left (\sqrt {5} + 3\right )} \log \left (-x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {4}{5}} + \left (-1\right )^{\frac {4}{5}}\right )}} - \frac {3 \, {\left (\sqrt {5} - 3\right )} \log \left (x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {4}{5}} - \left (-1\right )^{\frac {4}{5}}\right )}} + 6 \, x^{\frac {1}{6}} \]

[In]

integrate(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x, algorithm="maxima")

[Out]

-3/5*sqrt(5)*(-1)^(1/5)*(sqrt(5) - 1)*log((sqrt(5)*(-1)^(1/5) + (-1)^(1/5)*sqrt(2*sqrt(5) - 10) + (-1)^(1/5) -
4*x^(1/6))/(sqrt(5)*(-1)^(1/5) - (-1)^(1/5)*sqrt(2*sqrt(5) - 10) + (-1)^(1/5) - 4*x^(1/6)))/sqrt(2*sqrt(5) -
10) - 3/5*sqrt(5)*(-1)^(1/5)*(sqrt(5) + 1)*log((sqrt(5)*(-1)^(1/5) - (-1)^(1/5)*sqrt(-2*sqrt(5) - 10) - (-1)^(
1/5) + 4*x^(1/6))/(sqrt(5)*(-1)^(1/5) + (-1)^(1/5)*sqrt(-2*sqrt(5) - 10) - (-1)^(1/5) + 4*x^(1/6)))/sqrt(-2*sq
rt(5) - 10) - 6/5*(-1)^(1/5)*log((-1)^(1/5) + x^(1/6)) + x - 3/5*(sqrt(5) + 3)*log(-x^(1/6)*(sqrt(5)*(-1)^(1/5
) + (-1)^(1/5)) + 2*(-1)^(2/5) + 2*x^(1/3))/(sqrt(5)*(-1)^(4/5) + (-1)^(4/5)) - 3/5*(sqrt(5) - 3)*log(x^(1/6)*
(sqrt(5)*(-1)^(1/5) - (-1)^(1/5)) + 2*(-1)^(2/5) + 2*x^(1/3))/(sqrt(5)*(-1)^(4/5) - (-1)^(4/5)) + 6*x^(1/6)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3}{5} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {\sqrt {5} - 4 \, x^{\frac {1}{6}} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) - \frac {3}{5} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {\sqrt {5} + 4 \, x^{\frac {1}{6}} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {3}{10} \, \sqrt {5} \log \left (\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} + 1\right )} + x^{\frac {1}{3}} + 1\right ) + \frac {3}{10} \, \sqrt {5} \log \left (-\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} - 1\right )} + x^{\frac {1}{3}} + 1\right ) + x + 6 \, x^{\frac {1}{6}} - \frac {3}{10} \, \log \left (x^{\frac {2}{3}} + \sqrt {x} + x^{\frac {1}{3}} + x^{\frac {1}{6}} + 1\right ) + \frac {6}{5} \, \log \left ({\left | x^{\frac {1}{6}} - 1 \right |}\right ) \]

[In]

integrate(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x, algorithm="giac")

[Out]

-3/5*sqrt(2*sqrt(5) + 10)*arctan(-(sqrt(5) - 4*x^(1/6) - 1)/sqrt(2*sqrt(5) + 10)) - 3/5*sqrt(-2*sqrt(5) + 10)*
arctan((sqrt(5) + 4*x^(1/6) + 1)/sqrt(-2*sqrt(5) + 10)) - 3/10*sqrt(5)*log(1/2*x^(1/6)*(sqrt(5) + 1) + x^(1/3)
+ 1) + 3/10*sqrt(5)*log(-1/2*x^(1/6)*(sqrt(5) - 1) + x^(1/3) + 1) + x + 6*x^(1/6) - 3/10*log(x^(2/3) + sqrt(x
) + x^(1/3) + x^(1/6) + 1) + 6/5*log(abs(x^(1/6) - 1))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=x+\frac {6\,\ln \left (1296\,x^{1/6}-1296\right )}{5}-\ln \left (270\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}-270\,\sqrt {5}+1080\,x^{1/6}+270\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}-\frac {3\,\sqrt {5}}{10}+\frac {3}{10}\right )+\ln \left (270\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}+270\,\sqrt {5}-1080\,x^{1/6}-270\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}+\frac {3\,\sqrt {5}}{10}-\frac {3}{10}\right )+6\,x^{1/6}-\ln \left (270\,\sqrt {5}+1080\,x^{1/6}-270\,\sqrt {2}\,\sqrt {\sqrt {5}-5}+270\right )\,\left (\frac {3\,\sqrt {5}}{10}-\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )-\ln \left (270\,\sqrt {5}+1080\,x^{1/6}+270\,\sqrt {2}\,\sqrt {\sqrt {5}-5}+270\right )\,\left (\frac {3\,\sqrt {5}}{10}+\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right ) \]

[In]

int(x^(1/2)/(x^(1/2) - 1/x^(1/3)),x)

[Out]

x + (6*log(1296*x^(1/6) - 1296))/5 - log(270*2^(1/2)*(- 5^(1/2) - 5)^(1/2) - 270*5^(1/2) + 1080*x^(1/6) + 270)
*((3*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/10 - (3*5^(1/2))/10 + 3/10) + log(270*2^(1/2)*(- 5^(1/2) - 5)^(1/2) + 270*
5^(1/2) - 1080*x^(1/6) - 270)*((3*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/10 + (3*5^(1/2))/10 - 3/10) + 6*x^(1/6) - log
(270*5^(1/2) + 1080*x^(1/6) - 270*2^(1/2)*(5^(1/2) - 5)^(1/2) + 270)*((3*5^(1/2))/10 - (3*2^(1/2)*(5^(1/2) - 5
)^(1/2))/10 + 3/10) - log(270*5^(1/2) + 1080*x^(1/6) + 270*2^(1/2)*(5^(1/2) - 5)^(1/2) + 270)*((3*5^(1/2))/10
+ (3*2^(1/2)*(5^(1/2) - 5)^(1/2))/10 + 3/10)