Integrand size = 17, antiderivative size = 150 \[ \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx=-\frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x}-\frac {\arctan \left (\frac {1-\frac {2 (-1+x)}{\sqrt [3]{(-1+x)^2 (1+x)}}}{\sqrt {3}}\right )}{\sqrt {3}}-\sqrt {3} \arctan \left (\frac {1+\frac {2 (-1+x)}{\sqrt [3]{(-1+x)^2 (1+x)}}}{\sqrt {3}}\right )+\frac {\log (x)}{6}-\frac {2}{3} \log (1+x)-\frac {3}{2} \log \left (1-\frac {-1+x}{\sqrt [3]{(-1+x)^2 (1+x)}}\right )-\frac {1}{2} \log \left (1+\frac {-1+x}{\sqrt [3]{(-1+x)^2 (1+x)}}\right ) \] Output:
-((-1+x)^2*(1+x))^(1/3)/x+1/6*ln(x)-2/3*ln(1+x)-3/2*ln(1+(1-x)/((-1+x)^2*( 1+x))^(1/3))-1/2*ln(1+(-1+x)/((-1+x)^2*(1+x))^(1/3))-1/3*arctan(1/3*(1-2*( -1+x)/((-1+x)^2*(1+x))^(1/3))*3^(1/2))*3^(1/2)-arctan(1/3*(1+2*(-1+x)/((-1 +x)^2*(1+x))^(1/3))*3^(1/2))*3^(1/2)
Time = 0.34 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx=-\frac {(-1+x)^{4/3} (1+x)^{2/3} \left (18 (-1+x)^{2/3} \sqrt [3]{1+x}-6 \sqrt {3} x \arctan \left (\frac {1-\frac {2}{\sqrt [3]{\frac {-1+x}{1+x}}}}{\sqrt {3}}\right )-18 \sqrt {3} x \arctan \left (\frac {1+\frac {2}{\sqrt [3]{\frac {-1+x}{1+x}}}}{\sqrt {3}}\right )-10 x \log \left (\frac {2}{-1+x}\right )-3 x \log \left (1+\frac {1}{\left (\frac {-1+x}{1+x}\right )^{2/3}}-\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )+28 x \log \left (-1+\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )+6 x \log \left (1+\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )+x \log \left (1+\frac {1}{\left (\frac {-1+x}{1+x}\right )^{2/3}}+\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )\right )}{18 x \left ((-1+x)^2 (1+x)\right )^{2/3}} \] Input:
Integrate[((-1 + x)^2*(1 + x))^(1/3)/x^2,x]
Output:
-1/18*((-1 + x)^(4/3)*(1 + x)^(2/3)*(18*(-1 + x)^(2/3)*(1 + x)^(1/3) - 6*S qrt[3]*x*ArcTan[(1 - 2/((-1 + x)/(1 + x))^(1/3))/Sqrt[3]] - 18*Sqrt[3]*x*A rcTan[(1 + 2/((-1 + x)/(1 + x))^(1/3))/Sqrt[3]] - 10*x*Log[2/(-1 + x)] - 3 *x*Log[1 + ((-1 + x)/(1 + x))^(-2/3) - ((-1 + x)/(1 + x))^(-1/3)] + 28*x*L og[-1 + ((-1 + x)/(1 + x))^(-1/3)] + 6*x*Log[1 + ((-1 + x)/(1 + x))^(-1/3) ] + x*Log[1 + ((-1 + x)/(1 + x))^(-2/3) + ((-1 + x)/(1 + x))^(-1/3)]))/(x* ((-1 + x)^2*(1 + x))^(2/3))
Time = 0.33 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2490, 2483, 27, 108, 27, 175, 72, 102}
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.
| \(\displaystyle \int \frac {\sqrt [3]{(x-1)^2 (x+1)}}{x^2} \, dx\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \int \frac {\sqrt [3]{\left (x-\frac {1}{3}\right )^3-\frac {4}{3} \left (x-\frac {1}{3}\right )+\frac {16}{27}}}{x^2}d\left (x-\frac {1}{3}\right )\) |
| \(\Big \downarrow \) 2483 |
| \(\displaystyle \frac {3 \sqrt [3]{27 \left (x-\frac {1}{3}\right )^3-36 \left (x-\frac {1}{3}\right )+16} \int \frac {4\ 2^{2/3} \left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}{\left (3 \left (x-\frac {1}{3}\right )+1\right )^2}d\left (x-\frac {1}{3}\right )}{4\ 2^{2/3} \left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt [3]{27 \left (x-\frac {1}{3}\right )^3-36 \left (x-\frac {1}{3}\right )+16} \int \frac {\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}{\left (3 \left (x-\frac {1}{3}\right )+1\right )^2}d\left (x-\frac {1}{3}\right )}{\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {3 \sqrt [3]{27 \left (x-\frac {1}{3}\right )^3-36 \left (x-\frac {1}{3}\right )+16} \left (\frac {1}{3} \int -\frac {3 \left (3 \left (x-\frac {1}{3}\right )+2\right )}{\sqrt [3]{2-3 \left (x-\frac {1}{3}\right )} \left (3 \left (x-\frac {1}{3}\right )+1\right ) \left (3 \left (x-\frac {1}{3}\right )+4\right )^{2/3}}d\left (x-\frac {1}{3}\right )-\frac {\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}{3 \left (3 \left (x-\frac {1}{3}\right )+1\right )}\right )}{\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt [3]{27 \left (x-\frac {1}{3}\right )^3-36 \left (x-\frac {1}{3}\right )+16} \left (-\int \frac {3 \left (x-\frac {1}{3}\right )+2}{\sqrt [3]{2-3 \left (x-\frac {1}{3}\right )} \left (3 \left (x-\frac {1}{3}\right )+1\right ) \left (3 \left (x-\frac {1}{3}\right )+4\right )^{2/3}}d\left (x-\frac {1}{3}\right )-\frac {\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}{3 \left (3 \left (x-\frac {1}{3}\right )+1\right )}\right )}{\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {3 \sqrt [3]{27 \left (x-\frac {1}{3}\right )^3-36 \left (x-\frac {1}{3}\right )+16} \left (-\int \frac {1}{\sqrt [3]{2-3 \left (x-\frac {1}{3}\right )} \left (3 \left (x-\frac {1}{3}\right )+4\right )^{2/3}}d\left (x-\frac {1}{3}\right )-\int \frac {1}{\sqrt [3]{2-3 \left (x-\frac {1}{3}\right )} \left (3 \left (x-\frac {1}{3}\right )+1\right ) \left (3 \left (x-\frac {1}{3}\right )+4\right )^{2/3}}d\left (x-\frac {1}{3}\right )-\frac {\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}{3 \left (3 \left (x-\frac {1}{3}\right )+1\right )}\right )}{\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\) |
| \(\Big \downarrow \) 72 |
| \(\displaystyle \frac {3 \sqrt [3]{27 \left (x-\frac {1}{3}\right )^3-36 \left (x-\frac {1}{3}\right )+16} \left (-\int \frac {1}{\sqrt [3]{2-3 \left (x-\frac {1}{3}\right )} \left (3 \left (x-\frac {1}{3}\right )+1\right ) \left (3 \left (x-\frac {1}{3}\right )+4\right )^{2/3}}d\left (x-\frac {1}{3}\right )-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{2-3 \left (x-\frac {1}{3}\right )}}{\sqrt {3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\right )}{\sqrt {3}}-\frac {\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}{3 \left (3 \left (x-\frac {1}{3}\right )+1\right )}-\frac {1}{2} \log \left (\frac {\sqrt [3]{2-3 \left (x-\frac {1}{3}\right )}}{\sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}+1\right )-\frac {1}{6} \log \left (3 \left (x-\frac {1}{3}\right )+4\right )\right )}{\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {3 \sqrt [3]{27 \left (x-\frac {1}{3}\right )^3-36 \left (x-\frac {1}{3}\right )+16} \left (-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{2-3 \left (x-\frac {1}{3}\right )}}{\sqrt {3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {2 \sqrt [3]{2-3 \left (x-\frac {1}{3}\right )}}{\sqrt {3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}{3 \left (3 \left (x-\frac {1}{3}\right )+1\right )}-\frac {1}{2} \log \left (\frac {\sqrt [3]{2-3 \left (x-\frac {1}{3}\right )}}{\sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}+1\right )-\frac {1}{6} \log \left (\sqrt [3]{2-3 \left (x-\frac {1}{3}\right )}-\sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}\right )+\frac {1}{18} \log \left (3 \left (x-\frac {1}{3}\right )+1\right )-\frac {1}{6} \log \left (3 \left (x-\frac {1}{3}\right )+4\right )\right )}{\left (2-3 \left (x-\frac {1}{3}\right )\right )^{2/3} \sqrt [3]{3 \left (x-\frac {1}{3}\right )+4}}\) |
Input:
Int[((-1 + x)^2*(1 + x))^(1/3)/x^2,x]
Output:
(3*(16 - 36*(-1/3 + x) + 27*(-1/3 + x)^3)^(1/3)*(-1/3*((2 - 3*(-1/3 + x))^ (2/3)*(4 + 3*(-1/3 + x))^(1/3))/(1 + 3*(-1/3 + x)) - ArcTan[1/Sqrt[3] - (2 *(2 - 3*(-1/3 + x))^(1/3))/(Sqrt[3]*(4 + 3*(-1/3 + x))^(1/3))]/Sqrt[3] - A rcTan[1/Sqrt[3] + (2*(2 - 3*(-1/3 + x))^(1/3))/(Sqrt[3]*(4 + 3*(-1/3 + x)) ^(1/3))]/(3*Sqrt[3]) - Log[1 + (2 - 3*(-1/3 + x))^(1/3)/(4 + 3*(-1/3 + x)) ^(1/3)]/2 - Log[(2 - 3*(-1/3 + x))^(1/3) - (4 + 3*(-1/3 + x))^(1/3)]/6 + L og[1 + 3*(-1/3 + x)]/18 - Log[4 + 3*(-1/3 + x)]/6))/((2 - 3*(-1/3 + x))^(2 /3)*(4 + 3*(-1/3 + x))^(1/3))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-d/b, 3]}, Simp[Sqrt[3]*(q/d)*ArcTan[1/Sqrt[3] - 2*q*((a + b* x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))], x] + (Simp[3*(q/(2*d))*Log[q*((a + b* x)^(1/3)/(c + d*x)^(1/3)) + 1], x] + Simp[(q/(2*d))*Log[c + d*x], x])] /; F reeQ[{a, b, c, d}, x] && NegQ[d/b]
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) *(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q *(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S ymbol] :> Simp[(a + b*x + d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p)) In t[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && !IntegerQ[p]
Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 , x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 *d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c , 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.61 (sec) , antiderivative size = 1246, normalized size of antiderivative = 8.31
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1246\) |
| trager | \(\text {Expression too large to display}\) | \(1374\) |
Input:
int(((-1+x)^2*(1+x))^(1/3)/x^2,x,method=_RETURNVERBOSE)
Output:
-((-1+x)^2*(1+x))^(1/3)/x+(-1/3*ln((157880368143-288529720857*x-4262769939 861*x^4+2395436537574*x^2+2841846626574*x^3-4262769939861*x^5+175558962851 1*(x^3+x^2-x-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^3-1412122229127*(x^3+x^2-x-1)^ (1/3)*RootOf(_Z^2-3*_Z+9)*x^4+585196542837*(x^3+x^2-x-1)^(2/3)*RootOf(_Z^2 -3*_Z+9)*x^2-941414819418*(x^3+x^2-x-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^3-1950 65514279*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(2/3)*x+627609879612*(x^3+x^2-x -1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^2+104601646602*RootOf(_Z^2-3*_Z+9)*(x^3+x^ 2-x-1)^(1/3)*x+108655360*RootOf(_Z^2-3*_Z+9)^2+195065514279*(x^3+x^2-x-1)^ (1/3)+38163044376*(x^3+x^2-x-1)^(2/3)-1030402198152*(x^3+x^2-x-1)^(2/3)*x^ 3+5266768885533*(x^3+x^2-x-1)^(1/3)*x^4-343467399384*(x^3+x^2-x-1)^(2/3)*x ^2+3511179257022*(x^3+x^2-x-1)^(1/3)*x^3-65021838093*RootOf(_Z^2-3*_Z+9)*( x^3+x^2-x-1)^(2/3)+114489133128*(x^3+x^2-x-1)^(2/3)*x-2340786171348*(x^3+x ^2-x-1)^(1/3)*x^2-52300823301*RootOf(_Z^2-3*_Z+9)*(x^3+x^2-x-1)^(1/3)-3901 31028558*(x^3+x^2-x-1)^(1/3)*x-2933694720*RootOf(_Z^2-3*_Z+9)^2*x^5+223110 876816*RootOf(_Z^2-3*_Z+9)*x^3+477460395840*RootOf(_Z^2-3*_Z+9)*x^2+266744 567736*RootOf(_Z^2-3*_Z+9)*x+12395048712*RootOf(_Z^2-3*_Z+9)+1955796480*Ro otOf(_Z^2-3*_Z+9)^2*x^3+21459433600*RootOf(_Z^2-3*_Z+9)^2*x^2+19612292480* RootOf(_Z^2-3*_Z+9)^2*x-334666315224*RootOf(_Z^2-3*_Z+9)*x^5-2933694720*Ro otOf(_Z^2-3*_Z+9)^2*x^4-334666315224*RootOf(_Z^2-3*_Z+9)*x^4)/x/(1+x))+1/9 *RootOf(_Z^2-3*_Z+9)*ln(-(33401336760+117256110840*x-901836092520*x^4+6...
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (126) = 252\).
Time = 0.07 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.87 \[ \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx=\frac {6 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) - 2 \, \sqrt {3} x \arctan \left (-\frac {\sqrt {3} {\left (x - 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) + 3 \, x \log \left (\frac {x^{2} + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \, x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) + x \log \left (\frac {x^{2} - {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \, x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) - 2 \, x \log \left (\frac {x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} - 1}{x - 1}\right ) - 6 \, x \log \left (-\frac {x - {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} - 1}{x - 1}\right ) - 6 \, {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{6 \, x} \] Input:
integrate(((-1+x)^2*(1+x))^(1/3)/x^2,x, algorithm="fricas")
Output:
1/6*(6*sqrt(3)*x*arctan(1/3*(sqrt(3)*(x - 1) + 2*sqrt(3)*(x^3 - x^2 - x + 1)^(1/3))/(x - 1)) - 2*sqrt(3)*x*arctan(-1/3*(sqrt(3)*(x - 1) - 2*sqrt(3)* (x^3 - x^2 - x + 1)^(1/3))/(x - 1)) + 3*x*log((x^2 + (x^3 - x^2 - x + 1)^( 1/3)*(x - 1) - 2*x + (x^3 - x^2 - x + 1)^(2/3) + 1)/(x^2 - 2*x + 1)) + x*l og((x^2 - (x^3 - x^2 - x + 1)^(1/3)*(x - 1) - 2*x + (x^3 - x^2 - x + 1)^(2 /3) + 1)/(x^2 - 2*x + 1)) - 2*x*log((x + (x^3 - x^2 - x + 1)^(1/3) - 1)/(x - 1)) - 6*x*log(-(x - (x^3 - x^2 - x + 1)^(1/3) - 1)/(x - 1)) - 6*(x^3 - x^2 - x + 1)^(1/3))/x
\[ \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx=\int \frac {\sqrt [3]{\left (x - 1\right )^{2} \left (x + 1\right )}}{x^{2}}\, dx \] Input:
integrate(((-1+x)**2*(1+x))**(1/3)/x**2,x)
Output:
Integral(((x - 1)**2*(x + 1))**(1/3)/x**2, x)
\[ \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx=\int { \frac {\left ({\left (x + 1\right )} {\left (x - 1\right )}^{2}\right )^{\frac {1}{3}}}{x^{2}} \,d x } \] Input:
integrate(((-1+x)^2*(1+x))^(1/3)/x^2,x, algorithm="maxima")
Output:
integrate(((x + 1)*(x - 1)^2)^(1/3)/x^2, x)
\[ \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx=\int { \frac {\left ({\left (x + 1\right )} {\left (x - 1\right )}^{2}\right )^{\frac {1}{3}}}{x^{2}} \,d x } \] Input:
integrate(((-1+x)^2*(1+x))^(1/3)/x^2,x, algorithm="giac")
Output:
integrate(((x + 1)*(x - 1)^2)^(1/3)/x^2, x)
Timed out. \[ \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx=\int \frac {{\left ({\left (x-1\right )}^2\,\left (x+1\right )\right )}^{1/3}}{x^2} \,d x \] Input:
int(((x - 1)^2*(x + 1))^(1/3)/x^2,x)
Output:
int(((x - 1)^2*(x + 1))^(1/3)/x^2, x)
\[ \int \frac {\sqrt [3]{(-1+x)^2 (1+x)}}{x^2} \, dx=\frac {-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+\left (\int \frac {\left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}}{x^{3}-x}d x \right ) x +3 \left (\int \frac {\left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}}{x^{2}-1}d x \right ) x}{3 x} \] Input:
int(((-1+x)^2*(1+x))^(1/3)/x^2,x)
Output:
( - 3*(x**3 - x**2 - x + 1)**(1/3) + int((x**3 - x**2 - x + 1)**(1/3)/(x** 3 - x),x)*x + 3*int((x**3 - x**2 - x + 1)**(1/3)/(x**2 - 1),x)*x)/(3*x)