Integrand size = 11, antiderivative size = 104 \[ \int \frac {1}{(-1+x)^{2/3} x^5} \, dx=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}-\frac {110 \arctan \left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {55}{81} \log \left (1+\sqrt [3]{-1+x}\right )-\frac {55 \log (x)}{243} \] Output:
1/4*(-1+x)^(1/3)/x^4+11/36*(-1+x)^(1/3)/x^3+11/27*(-1+x)^(1/3)/x^2+55/81*( -1+x)^(1/3)/x+55/81*ln(1+(-1+x)^(1/3))-55/243*ln(x)-110/243*arctan(1/3*(1- 2*(-1+x)^(1/3))*3^(1/2))*3^(1/2)
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(-1+x)^{2/3} x^5} \, dx=\frac {1}{972} \left (\frac {3 \sqrt [3]{-1+x} \left (81+99 x+132 x^2+220 x^3\right )}{x^4}-440 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )+440 \log \left (1+\sqrt [3]{-1+x}\right )-220 \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )\right ) \] Input:
Integrate[1/((-1 + x)^(2/3)*x^5),x]
Output:
((3*(-1 + x)^(1/3)*(81 + 99*x + 132*x^2 + 220*x^3))/x^4 - 440*Sqrt[3]*ArcT an[(1 - 2*(-1 + x)^(1/3))/Sqrt[3]] + 440*Log[1 + (-1 + x)^(1/3)] - 220*Log [1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)])/972
Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {52, 52, 52, 52, 70, 16, 1083, 217}
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.
| \(\displaystyle \int \frac {1}{(x-1)^{2/3} x^5} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {11}{12} \int \frac {1}{(x-1)^{2/3} x^4}dx+\frac {\sqrt [3]{x-1}}{4 x^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {11}{12} \left (\frac {8}{9} \int \frac {1}{(x-1)^{2/3} x^3}dx+\frac {\sqrt [3]{x-1}}{3 x^3}\right )+\frac {\sqrt [3]{x-1}}{4 x^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \int \frac {1}{(x-1)^{2/3} x^2}dx+\frac {\sqrt [3]{x-1}}{2 x^2}\right )+\frac {\sqrt [3]{x-1}}{3 x^3}\right )+\frac {\sqrt [3]{x-1}}{4 x^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \int \frac {1}{(x-1)^{2/3} x}dx+\frac {\sqrt [3]{x-1}}{x}\right )+\frac {\sqrt [3]{x-1}}{2 x^2}\right )+\frac {\sqrt [3]{x-1}}{3 x^3}\right )+\frac {\sqrt [3]{x-1}}{4 x^4}\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x-1}+1}d\sqrt [3]{x-1}+\frac {3}{2} \int \frac {1}{(x-1)^{2/3}-\sqrt [3]{x-1}+1}d\sqrt [3]{x-1}-\frac {\log (x)}{2}\right )+\frac {\sqrt [3]{x-1}}{x}\right )+\frac {\sqrt [3]{x-1}}{2 x^2}\right )+\frac {\sqrt [3]{x-1}}{3 x^3}\right )+\frac {\sqrt [3]{x-1}}{4 x^4}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{(x-1)^{2/3}-\sqrt [3]{x-1}+1}d\sqrt [3]{x-1}+\frac {3}{2} \log \left (\sqrt [3]{x-1}+1\right )-\frac {\log (x)}{2}\right )+\frac {\sqrt [3]{x-1}}{x}\right )+\frac {\sqrt [3]{x-1}}{2 x^2}\right )+\frac {\sqrt [3]{x-1}}{3 x^3}\right )+\frac {\sqrt [3]{x-1}}{4 x^4}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (-3 \int \frac {1}{-(x-1)^{2/3}-3}d\left (2 \sqrt [3]{x-1}-1\right )+\frac {3}{2} \log \left (\sqrt [3]{x-1}+1\right )-\frac {\log (x)}{2}\right )+\frac {\sqrt [3]{x-1}}{x}\right )+\frac {\sqrt [3]{x-1}}{2 x^2}\right )+\frac {\sqrt [3]{x-1}}{3 x^3}\right )+\frac {\sqrt [3]{x-1}}{4 x^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x-1}-1}{\sqrt {3}}\right )+\frac {3}{2} \log \left (\sqrt [3]{x-1}+1\right )-\frac {\log (x)}{2}\right )+\frac {\sqrt [3]{x-1}}{x}\right )+\frac {\sqrt [3]{x-1}}{2 x^2}\right )+\frac {\sqrt [3]{x-1}}{3 x^3}\right )+\frac {\sqrt [3]{x-1}}{4 x^4}\) |
Input:
Int[1/((-1 + x)^(2/3)*x^5),x]
Output:
(-1 + x)^(1/3)/(4*x^4) + (11*((-1 + x)^(1/3)/(3*x^3) + (8*((-1 + x)^(1/3)/ (2*x^2) + (5*((-1 + x)^(1/3)/x + (2*(Sqrt[3]*ArcTan[(-1 + 2*(-1 + x)^(1/3) )/Sqrt[3]] + (3*Log[1 + (-1 + x)^(1/3)])/2 - Log[x]/2))/3))/6))/9))/12
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) , x] + (Simp[3/(2*b*q) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 /3)], x] + Simp[3/(2*b*q^2) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.42 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.82
| method | result | size |
| meijerg | \(\frac {\left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {2}{3}} \left (-\frac {\Gamma \left (\frac {2}{3}\right )}{4 x^{4}}-\frac {2 \Gamma \left (\frac {2}{3}\right )}{9 x^{3}}-\frac {5 \Gamma \left (\frac {2}{3}\right )}{18 x^{2}}-\frac {40 \Gamma \left (\frac {2}{3}\right )}{81 x}+\frac {110 \left (\frac {877}{1320}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+\ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{243}+\frac {308 \Gamma \left (\frac {2}{3}\right ) x \operatorname {hypergeom}\left (\left [1, 1, \frac {17}{3}\right ], \left [2, 6\right ], x\right )}{729}\right )}{\Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (-1+x \right )^{\frac {2}{3}}}\) | \(85\) |
| risch | \(\frac {220 x^{4}-88 x^{3}-33 x^{2}-18 x -81}{324 x^{4} \left (-1+x \right )^{\frac {2}{3}}}+\frac {110 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+\ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {2 \Gamma \left (\frac {2}{3}\right ) x \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x\right )}{3}\right )}{243 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (-1+x \right )^{\frac {2}{3}}}\) | \(87\) |
| derivativedivides | \(-\frac {-75 \left (-1+x \right )^{\frac {7}{3}}+190 \left (-1+x \right )^{2}-350 \left (-1+x \right )^{\frac {5}{3}}+\frac {1157 \left (-1+x \right )^{\frac {4}{3}}}{4}+\frac {149}{4}-138 x -116 \left (-1+x \right )^{\frac {2}{3}}+137 \left (-1+x \right )^{\frac {1}{3}}}{243 \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )^{4}}-\frac {55 \ln \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )}{243}+\frac {110 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{243}-\frac {1}{324 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{4}}-\frac {5}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{3}}-\frac {20}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{2}}-\frac {25}{81 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}+\frac {110 \ln \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}{243}\) | \(158\) |
| default | \(-\frac {-75 \left (-1+x \right )^{\frac {7}{3}}+190 \left (-1+x \right )^{2}-350 \left (-1+x \right )^{\frac {5}{3}}+\frac {1157 \left (-1+x \right )^{\frac {4}{3}}}{4}+\frac {149}{4}-138 x -116 \left (-1+x \right )^{\frac {2}{3}}+137 \left (-1+x \right )^{\frac {1}{3}}}{243 \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )^{4}}-\frac {55 \ln \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )}{243}+\frac {110 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{243}-\frac {1}{324 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{4}}-\frac {5}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{3}}-\frac {20}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{2}}-\frac {25}{81 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}+\frac {110 \ln \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}{243}\) | \(158\) |
| trager | \(\frac {\left (220 x^{3}+132 x^{2}+99 x +81\right ) \left (-1+x \right )^{\frac {1}{3}}}{324 x^{4}}-\frac {110 \ln \left (\frac {72 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {2}{3}}-1152 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x -72 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {1}{3}}+2304 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}-72 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x +120 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )-x +1}{x}\right )}{243}-\frac {1760 \ln \left (\frac {72 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {2}{3}}-1152 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x -72 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {1}{3}}+2304 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}-72 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x +120 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )-x +1}{x}\right ) \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )}{81}+\frac {1760 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \ln \left (-\frac {144 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {2}{3}}+2304 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x +3 \left (-1+x \right )^{\frac {2}{3}}-144 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {1}{3}}-4608 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}-48 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x -3 \left (-1+x \right )^{\frac {1}{3}}+48 \operatorname {RootOf}\left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )+1}{x}\right )}{81}\) | \(379\) |
Input:
int(1/(-1+x)^(2/3)/x^5,x,method=_RETURNVERBOSE)
Output:
1/GAMMA(2/3)/signum(-1+x)^(2/3)*(-signum(-1+x))^(2/3)*(-1/4*GAMMA(2/3)/x^4 -2/9*GAMMA(2/3)/x^3-5/18*GAMMA(2/3)/x^2-40/81*GAMMA(2/3)/x+110/243*(877/13 20+1/6*Pi*3^(1/2)-3/2*ln(3)+ln(x)+I*Pi)*GAMMA(2/3)+308/729*GAMMA(2/3)*x*hy pergeom([1,1,17/3],[2,6],x))
Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(-1+x)^{2/3} x^5} \, dx=\frac {440 \, \sqrt {3} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 220 \, x^{4} \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + 440 \, x^{4} \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (220 \, x^{3} + 132 \, x^{2} + 99 \, x + 81\right )} {\left (x - 1\right )}^{\frac {1}{3}}}{972 \, x^{4}} \] Input:
integrate(1/(-1+x)^(2/3)/x^5,x, algorithm="fricas")
Output:
1/972*(440*sqrt(3)*x^4*arctan(2/3*sqrt(3)*(x - 1)^(1/3) - 1/3*sqrt(3)) - 2 20*x^4*log((x - 1)^(2/3) - (x - 1)^(1/3) + 1) + 440*x^4*log((x - 1)^(1/3) + 1) + 3*(220*x^3 + 132*x^2 + 99*x + 81)*(x - 1)^(1/3))/x^4
Result contains complex when optimal does not.
Time = 36.14 (sec) , antiderivative size = 12993, normalized size of antiderivative = 124.93 \[ \int \frac {1}{(-1+x)^{2/3} x^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(-1+x)**(2/3)/x**5,x)
Output:
-440*(x - 1)**(35/3)*log(-(x - 1)**(1/3)*exp_polar(I*pi/3) + 1)*gamma(1/3) /(2916*(x - 1)**(35/3)*exp(I*pi/3)*gamma(4/3) + 32076*(x - 1)**(32/3)*exp( I*pi/3)*gamma(4/3) + 160380*(x - 1)**(29/3)*exp(I*pi/3)*gamma(4/3) + 48114 0*(x - 1)**(26/3)*exp(I*pi/3)*gamma(4/3) + 962280*(x - 1)**(23/3)*exp(I*pi /3)*gamma(4/3) + 1347192*(x - 1)**(20/3)*exp(I*pi/3)*gamma(4/3) + 1347192* (x - 1)**(17/3)*exp(I*pi/3)*gamma(4/3) + 962280*(x - 1)**(14/3)*exp(I*pi/3 )*gamma(4/3) + 481140*(x - 1)**(11/3)*exp(I*pi/3)*gamma(4/3) + 160380*(x - 1)**(8/3)*exp(I*pi/3)*gamma(4/3) + 32076*(x - 1)**(5/3)*exp(I*pi/3)*gamma (4/3) + 2916*(x - 1)**(2/3)*exp(I*pi/3)*gamma(4/3)) + 440*(x - 1)**(35/3)* exp(I*pi/3)*log(-(x - 1)**(1/3)*exp_polar(I*pi) + 1)*gamma(1/3)/(2916*(x - 1)**(35/3)*exp(I*pi/3)*gamma(4/3) + 32076*(x - 1)**(32/3)*exp(I*pi/3)*gam ma(4/3) + 160380*(x - 1)**(29/3)*exp(I*pi/3)*gamma(4/3) + 481140*(x - 1)** (26/3)*exp(I*pi/3)*gamma(4/3) + 962280*(x - 1)**(23/3)*exp(I*pi/3)*gamma(4 /3) + 1347192*(x - 1)**(20/3)*exp(I*pi/3)*gamma(4/3) + 1347192*(x - 1)**(1 7/3)*exp(I*pi/3)*gamma(4/3) + 962280*(x - 1)**(14/3)*exp(I*pi/3)*gamma(4/3 ) + 481140*(x - 1)**(11/3)*exp(I*pi/3)*gamma(4/3) + 160380*(x - 1)**(8/3)* exp(I*pi/3)*gamma(4/3) + 32076*(x - 1)**(5/3)*exp(I*pi/3)*gamma(4/3) + 291 6*(x - 1)**(2/3)*exp(I*pi/3)*gamma(4/3)) - 440*(x - 1)**(35/3)*exp(2*I*pi/ 3)*log(-(x - 1)**(1/3)*exp_polar(5*I*pi/3) + 1)*gamma(1/3)/(2916*(x - 1)** (35/3)*exp(I*pi/3)*gamma(4/3) + 32076*(x - 1)**(32/3)*exp(I*pi/3)*gamma...
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(-1+x)^{2/3} x^5} \, dx=\frac {110}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {220 \, {\left (x - 1\right )}^{\frac {10}{3}} + 792 \, {\left (x - 1\right )}^{\frac {7}{3}} + 1023 \, {\left (x - 1\right )}^{\frac {4}{3}} + 532 \, {\left (x - 1\right )}^{\frac {1}{3}}}{324 \, {\left ({\left (x - 1\right )}^{4} + 4 \, {\left (x - 1\right )}^{3} + 6 \, {\left (x - 1\right )}^{2} + 4 \, x - 3\right )}} - \frac {55}{243} \, \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {110}{243} \, \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:
integrate(1/(-1+x)^(2/3)/x^5,x, algorithm="maxima")
Output:
110/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x - 1)^(1/3) - 1)) + 1/324*(220*(x - 1)^(10/3) + 792*(x - 1)^(7/3) + 1023*(x - 1)^(4/3) + 532*(x - 1)^(1/3))/ ((x - 1)^4 + 4*(x - 1)^3 + 6*(x - 1)^2 + 4*x - 3) - 55/243*log((x - 1)^(2/ 3) - (x - 1)^(1/3) + 1) + 110/243*log((x - 1)^(1/3) + 1)
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(-1+x)^{2/3} x^5} \, dx=\frac {110}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {220 \, {\left (x - 1\right )}^{\frac {10}{3}} + 792 \, {\left (x - 1\right )}^{\frac {7}{3}} + 1023 \, {\left (x - 1\right )}^{\frac {4}{3}} + 532 \, {\left (x - 1\right )}^{\frac {1}{3}}}{324 \, x^{4}} - \frac {55}{243} \, \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {110}{243} \, \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:
integrate(1/(-1+x)^(2/3)/x^5,x, algorithm="giac")
Output:
110/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x - 1)^(1/3) - 1)) + 1/324*(220*(x - 1)^(10/3) + 792*(x - 1)^(7/3) + 1023*(x - 1)^(4/3) + 532*(x - 1)^(1/3))/ x^4 - 55/243*log((x - 1)^(2/3) - (x - 1)^(1/3) + 1) + 110/243*log((x - 1)^ (1/3) + 1)
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(-1+x)^{2/3} x^5} \, dx=\frac {110\,\ln \left (\frac {12100\,{\left (x-1\right )}^{1/3}}{6561}+\frac {12100}{6561}\right )}{243}+\frac {\frac {133\,{\left (x-1\right )}^{1/3}}{81}+\frac {341\,{\left (x-1\right )}^{4/3}}{108}+\frac {22\,{\left (x-1\right )}^{7/3}}{9}+\frac {55\,{\left (x-1\right )}^{10/3}}{81}}{4\,x+6\,{\left (x-1\right )}^2+4\,{\left (x-1\right )}^3+{\left (x-1\right )}^4-3}-\ln \left (\frac {55}{27}-\frac {110\,{\left (x-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,55{}\mathrm {i}}{27}\right )\,\left (\frac {55}{243}+\frac {\sqrt {3}\,55{}\mathrm {i}}{243}\right )+\ln \left (\frac {110\,{\left (x-1\right )}^{1/3}}{27}-\frac {55}{27}+\frac {\sqrt {3}\,55{}\mathrm {i}}{27}\right )\,\left (-\frac {55}{243}+\frac {\sqrt {3}\,55{}\mathrm {i}}{243}\right ) \] Input:
int(1/(x^5*(x - 1)^(2/3)),x)
Output:
(110*log((12100*(x - 1)^(1/3))/6561 + 12100/6561))/243 + ((133*(x - 1)^(1/ 3))/81 + (341*(x - 1)^(4/3))/108 + (22*(x - 1)^(7/3))/9 + (55*(x - 1)^(10/ 3))/81)/(4*x + 6*(x - 1)^2 + 4*(x - 1)^3 + (x - 1)^4 - 3) - log((3^(1/2)*5 5i)/27 - (110*(x - 1)^(1/3))/27 + 55/27)*((3^(1/2)*55i)/243 + 55/243) + lo g((110*(x - 1)^(1/3))/27 + (3^(1/2)*55i)/27 - 55/27)*((3^(1/2)*55i)/243 - 55/243)
Time = 0.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(-1+x)^{2/3} x^5} \, dx=\frac {440 \sqrt {3}\, \mathit {atan} \left (2 \left (x -1\right )^{\frac {1}{6}}-\sqrt {3}\right ) x^{4}-440 \sqrt {3}\, \mathit {atan} \left (2 \left (x -1\right )^{\frac {1}{6}}+\sqrt {3}\right ) x^{4}+660 \left (x -1\right )^{\frac {1}{3}} x^{3}+396 \left (x -1\right )^{\frac {1}{3}} x^{2}+297 \left (x -1\right )^{\frac {1}{3}} x +243 \left (x -1\right )^{\frac {1}{3}}+440 \,\mathrm {log}\left (\left (x -1\right )^{\frac {1}{3}}+1\right ) x^{4}-220 \,\mathrm {log}\left (-\left (x -1\right )^{\frac {1}{6}} \sqrt {3}+\left (x -1\right )^{\frac {1}{3}}+1\right ) x^{4}-220 \,\mathrm {log}\left (\left (x -1\right )^{\frac {1}{6}} \sqrt {3}+\left (x -1\right )^{\frac {1}{3}}+1\right ) x^{4}}{972 x^{4}} \] Input:
int(1/(-1+x)^(2/3)/x^5,x)
Output:
(440*sqrt(3)*atan(2*(x - 1)**(1/6) - sqrt(3))*x**4 - 440*sqrt(3)*atan(2*(x - 1)**(1/6) + sqrt(3))*x**4 + 660*(x - 1)**(1/3)*x**3 + 396*(x - 1)**(1/3 )*x**2 + 297*(x - 1)**(1/3)*x + 243*(x - 1)**(1/3) + 440*log((x - 1)**(1/3 ) + 1)*x**4 - 220*log( - (x - 1)**(1/6)*sqrt(3) + (x - 1)**(1/3) + 1)*x**4 - 220*log((x - 1)**(1/6)*sqrt(3) + (x - 1)**(1/3) + 1)*x**4)/(972*x**4)