Integrand size = 21, antiderivative size = 201 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=6 \sqrt [6]{x}+x-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}+4 \sqrt [6]{x}}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}+4 \sqrt [6]{x}\right )\right )+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right ) \] Output:
6*x^(1/6)+x-3/5*(10+2*5^(1/2))^(1/2)*arctan((1-5^(1/2)+4*x^(1/6))/(10+2*5^ (1/2))^(1/2))-3/5*(10-2*5^(1/2))^(1/2)*arctan(1/20*(50+10*5^(1/2))^(1/2)*( 1+5^(1/2)+4*x^(1/6)))+6/5*ln(1-x^(1/6))-3/10*(-5^(1/2)+1)*ln(2+x^(1/6)-5^( 1/2)*x^(1/6)+2*x^(1/3))-3/10*(5^(1/2)+1)*ln(2+x^(1/6)+5^(1/2)*x^(1/6)+2*x^ (1/3))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=6 \sqrt [6]{x}+x+\frac {6}{5} \log \left (-1+\sqrt [6]{x}\right )-\frac {6}{5} \text {RootSum}\left [1+\text {$\#$1}+\text {$\#$1}^2+\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {4 \log \left (\sqrt [6]{x}-\text {$\#$1}\right )+3 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{1+2 \text {$\#$1}+3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[Sqrt[x]/(-x^(-1/3) + Sqrt[x]),x]
Output:
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
Time = 0.74 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {10, 25, 864, 831, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {x}}{\sqrt {x}-\frac {1}{\sqrt [3]{x}}} \, dx\) |
| \(\Big \downarrow \) 10 |
| \(\displaystyle \int -\frac {x^{5/6}}{1-x^{5/6}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {x^{5/6}}{1-x^{5/6}}dx\) |
| \(\Big \downarrow \) 864 |
| \(\displaystyle -6 \int \frac {x^{5/3}}{1-x^{5/6}}d\sqrt [6]{x}\) |
| \(\Big \downarrow \) 831 |
| \(\displaystyle -6 \int \left (-x^{5/6}+\frac {1}{1-x^{5/6}}-1\right )d\sqrt [6]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 \left (\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 \sqrt [6]{x}-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (4 \sqrt [6]{x}+\sqrt {5}+1\right )\right )-\frac {x}{6}-\sqrt [6]{x}-\frac {1}{5} \log \left (1-\sqrt [6]{x}\right )+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (\sqrt [3]{x}+\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [6]{x}+1\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (\sqrt [3]{x}+\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [6]{x}+1\right )\right )\) |
Input:
Int[Sqrt[x]/(-x^(-1/3) + Sqrt[x]),x]
Output:
-6*(-x^(1/6) - x/6 + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6 ))/Sqrt[2*(5 + Sqrt[5])]])/5 + (Sqrt[(5 - Sqrt[5])/2]*ArcTan[(Sqrt[(5 + Sq rt[5])/10]*(1 + Sqrt[5] + 4*x^(1/6)))/2])/5 - Log[1 - x^(1/6)]/5 + ((1 - S qrt[5])*Log[1 + ((1 - Sqrt[5])*x^(1/6))/2 + x^(1/3)])/20 + ((1 + Sqrt[5])* Log[1 + ((1 + Sqrt[5])*x^(1/6))/2 + x^(1/3)])/20)
Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x _Symbol] :> Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ [e, 0]) && PosQ[s - r]
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] && Gt Q[m, 2*n - 1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[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]
Time = 0.06 (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 | \(x +6 x^{\frac {1}{6}}+\frac {3 \left (-\sqrt {5}-1\right ) \ln \left (2+x^{\frac {1}{6}}+\sqrt {5}\, x^{\frac {1}{6}}+2 x^{\frac {1}{3}}\right )}{10}+\frac {12 \left (-4-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+\sqrt {5}+4 x^{\frac {1}{6}}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {3 \left (-\sqrt {5}+1\right ) \ln \left (2+x^{\frac {1}{6}}-\sqrt {5}\, x^{\frac {1}{6}}+2 x^{\frac {1}{3}}\right )}{10}-\frac {12 \left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {1-\sqrt {5}+4 x^{\frac {1}{6}}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {6 \ln \left (-1+x^{\frac {1}{6}}\right )}{5}\) | \(166\) |
| default | \(x +6 x^{\frac {1}{6}}+\frac {3 \left (-\sqrt {5}-1\right ) \ln \left (2+x^{\frac {1}{6}}+\sqrt {5}\, x^{\frac {1}{6}}+2 x^{\frac {1}{3}}\right )}{10}+\frac {12 \left (-4-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+\sqrt {5}+4 x^{\frac {1}{6}}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {3 \left (-\sqrt {5}+1\right ) \ln \left (2+x^{\frac {1}{6}}-\sqrt {5}\, x^{\frac {1}{6}}+2 x^{\frac {1}{3}}\right )}{10}-\frac {12 \left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {1-\sqrt {5}+4 x^{\frac {1}{6}}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {6 \ln \left (-1+x^{\frac {1}{6}}\right )}{5}\) | \(166\) |
Input:
int(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x,method=_RETURNVERBOSE)
Output:
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)*arc tan(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)))))
Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=\frac {6}{5} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 5} \arctan \left (\frac {1}{10} \, \sqrt {\frac {1}{2}} {\left (2 \, x^{\frac {1}{6}} {\left (\sqrt {5} - 5\right )} + 3 \, \sqrt {5} - 5\right )} \sqrt {\sqrt {5} + 5}\right ) - \frac {3}{10} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{\frac {1}{6}} {\left (\sqrt {5} + 1\right )} + 2 \, x^{\frac {1}{3}} + 2\right ) + \frac {3}{10} \, {\left (\sqrt {5} - 1\right )} \log \left (-x^{\frac {1}{6}} {\left (\sqrt {5} - 1\right )} + 2 \, x^{\frac {1}{3}} + 2\right ) - \frac {2}{5} \, \sqrt {-\frac {9}{2} \, \sqrt {5} + \frac {45}{2}} \arctan \left (\frac {1}{15} \, x^{\frac {1}{6}} {\left (\sqrt {5} + 5\right )} \sqrt {-\frac {9}{2} \, \sqrt {5} + \frac {45}{2}} + \frac {1}{30} \, {\left (3 \, \sqrt {5} + 5\right )} \sqrt {-\frac {9}{2} \, \sqrt {5} + \frac {45}{2}}\right ) + x + 6 \, x^{\frac {1}{6}} + \frac {6}{5} \, \log \left (x^{\frac {1}{6}} - 1\right ) \] Input:
integrate(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x, algorithm="fricas")
Output:
6/5*sqrt(1/2)*sqrt(sqrt(5) + 5)*arctan(1/10*sqrt(1/2)*(2*x^(1/6)*(sqrt(5) - 5) + 3*sqrt(5) - 5)*sqrt(sqrt(5) + 5)) - 3/10*(sqrt(5) + 1)*log(x^(1/6)* (sqrt(5) + 1) + 2*x^(1/3) + 2) + 3/10*(sqrt(5) - 1)*log(-x^(1/6)*(sqrt(5) - 1) + 2*x^(1/3) + 2) - 2/5*sqrt(-9/2*sqrt(5) + 45/2)*arctan(1/15*x^(1/6)* (sqrt(5) + 5)*sqrt(-9/2*sqrt(5) + 45/2) + 1/30*(3*sqrt(5) + 5)*sqrt(-9/2*s qrt(5) + 45/2)) + x + 6*x^(1/6) + 6/5*log(x^(1/6) - 1)
\[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=\int \frac {x^{\frac {5}{6}}}{\left (\sqrt [6]{x} - 1\right ) \left (\sqrt [6]{x} + x^{\frac {2}{3}} + \sqrt [3]{x} + \sqrt {x} + 1\right )}\, dx \] Input:
integrate(x**(1/2)/(-1/x**(1/3)+x**(1/2)),x)
Output:
Integral(x**(5/6)/((x**(1/6) - 1)*(x**(1/6) + x**(2/3) + x**(1/3) + sqrt(x ) + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (134) = 268\).
Time = 0.11 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3 \, \sqrt {5} \left (-1\right )^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {2 \, \sqrt {5} - 10}} - \frac {3 \, \sqrt {5} \left (-1\right )^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} - 10}} - \frac {6}{5} \, \left (-1\right )^{\frac {1}{5}} \log \left (\left (-1\right )^{\frac {1}{5}} + x^{\frac {1}{6}}\right ) + x - \frac {3 \, {\left (\sqrt {5} + 3\right )} \log \left (-x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {4}{5}} + \left (-1\right )^{\frac {4}{5}}\right )}} - \frac {3 \, {\left (\sqrt {5} - 3\right )} \log \left (x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {4}{5}} - \left (-1\right )^{\frac {4}{5}}\right )}} + 6 \, x^{\frac {1}{6}} \] Input:
integrate(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x, algorithm="maxima")
Output:
-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*sqrt(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)
Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {3}{5} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {\sqrt {5} - 4 \, x^{\frac {1}{6}} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) - \frac {3}{5} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {\sqrt {5} + 4 \, x^{\frac {1}{6}} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {3}{10} \, \sqrt {5} \log \left (\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} + 1\right )} + x^{\frac {1}{3}} + 1\right ) + \frac {3}{10} \, \sqrt {5} \log \left (-\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} - 1\right )} + x^{\frac {1}{3}} + 1\right ) + x + 6 \, x^{\frac {1}{6}} - \frac {3}{10} \, \log \left (x^{\frac {2}{3}} + \sqrt {x} + x^{\frac {1}{3}} + x^{\frac {1}{6}} + 1\right ) + \frac {6}{5} \, \log \left ({\left | x^{\frac {1}{6}} - 1 \right |}\right ) \] Input:
integrate(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x, algorithm="giac")
Output:
-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))
Time = 0.12 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=x+\frac {6\,\ln \left (1296\,x^{1/6}-1296\right )}{5}-\ln \left (270\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}-270\,\sqrt {5}+1080\,x^{1/6}+270\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}-\frac {3\,\sqrt {5}}{10}+\frac {3}{10}\right )+\ln \left (270\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}+270\,\sqrt {5}-1080\,x^{1/6}-270\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}+\frac {3\,\sqrt {5}}{10}-\frac {3}{10}\right )+6\,x^{1/6}-\ln \left (270\,\sqrt {5}+1080\,x^{1/6}-270\,\sqrt {2}\,\sqrt {\sqrt {5}-5}+270\right )\,\left (\frac {3\,\sqrt {5}}{10}-\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )-\ln \left (270\,\sqrt {5}+1080\,x^{1/6}+270\,\sqrt {2}\,\sqrt {\sqrt {5}-5}+270\right )\,\left (\frac {3\,\sqrt {5}}{10}+\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right ) \] Input:
int(x^(1/2)/(x^(1/2) - 1/x^(1/3)),x)
Output:
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 + 3/1 0)
\[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=6 x^{\frac {1}{6}}-\left (\int \frac {x^{\frac {1}{3}}}{x^{\frac {1}{3}}-x^{2}}d x \right )-\left (\int \frac {1}{x^{\frac {5}{6}}-\sqrt {x}\, x^{2}}d x \right )+x \] Input:
int(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x)
Output:
6*x**(1/6) - int(x**(1/3)/(x**(1/3) - x**2),x) - int(1/(x**(5/6) - sqrt(x) *x**2),x) + x