Integrand size = 31, antiderivative size = 543 \[ \int \frac {1}{\sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\frac {\sqrt {-3-3 i \sqrt {3}} \sqrt [3]{d} \arctan \left (\frac {\sqrt {3} (a-b)^{2/3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{2 \sqrt [3]{-1} b (-a d+b d)^{2/3}-2 \sqrt [3]{-1} (-a d+b d)^{2/3} x+(a-b)^{2/3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{\sqrt {2} \sqrt [3]{a-b} (-((a-b) d))^{2/3}}+\frac {\left (\sqrt [3]{d}-i \sqrt {3} \sqrt [3]{d}\right ) \log \left (\sqrt [3]{-1} (-a d+b d)^{2/3} (b-x)-(a-b)^{2/3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{2 \sqrt [3]{a-b} (-((a-b) d))^{2/3}}+\frac {i \left (i \sqrt [3]{d}+\sqrt {3} \sqrt [3]{d}\right ) \log \left ((-1)^{2/3} d \sqrt [3]{-a d+b d} \left (a b^2-b^3-2 a b x+2 b^2 x+a x^2-b x^2\right )+\sqrt [3]{-1} (a-b)^{2/3} \sqrt [3]{d} (-a d+b d)^{2/3} (-b+x) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}-(a-b)^{4/3} d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{4 \sqrt [3]{a-b} (-((a-b) d))^{2/3}} \]
1/2*(-3-3*I*3^(1/2))^(1/2)*d^(1/3)*arctan(3^(1/2)*(a-b)^(2/3)*d^(1/3)*(-a* b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/3)/(2*(-1)^(1/3)*b*(-a*d+b*d)^(2/3) -2*(-1)^(1/3)*(-a*d+b*d)^(2/3)*x+(a-b)^(2/3)*d^(1/3)*(-a*b^2+(2*a*b+b^2)*x +(-a-2*b)*x^2+x^3)^(1/3)))*2^(1/2)/(a-b)^(1/3)/(-(a-b)*d)^(2/3)+1/2*(d^(1/ 3)-I*3^(1/2)*d^(1/3))*ln((-1)^(1/3)*(-a*d+b*d)^(2/3)*(b-x)-(a-b)^(2/3)*d^( 1/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/3))/(a-b)^(1/3)/(-(a-b)*d) ^(2/3)+1/4*I*(I*d^(1/3)+3^(1/2)*d^(1/3))*ln((-1)^(2/3)*d*(-a*d+b*d)^(1/3)* (a*b^2-2*a*b*x+a*x^2-b^3+2*b^2*x-b*x^2)+(-1)^(1/3)*(a-b)^(2/3)*d^(1/3)*(-a *d+b*d)^(2/3)*(-b+x)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/3)-(a-b)^( 4/3)*d^(2/3)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(2/3))/(a-b)^(1/3)/(- (a-b)*d)^(2/3)
Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )+\log \left (1+\frac {d^{2/3} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )\right )}{2 (a-b) \sqrt [3]{d} \sqrt [3]{(b-x)^2 (-a+x)}} \]
((b - x)^(2/3)*(-a + x)^(1/3)*(2*Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(b - x)^(1 /3))/(-a + x)^(1/3))/Sqrt[3]] + Log[1 + (d^(2/3)*(b - x)^(2/3))/(-a + x)^( 2/3) - (d^(1/3)*(b - x)^(1/3))/(-a + x)^(1/3)] - 2*Log[1 + (d^(1/3)*(b - x )^(1/3))/(-a + x)^(1/3)]))/(2*(a - b)*d^(1/3)*((b - x)^2*(-a + x))^(1/3))
Time = 0.58 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.71, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2490, 2483, 27, 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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{(x-a) (x-b)^2} (a-b d+(d-1) x)} \, dx\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \int \frac {1}{\sqrt [3]{-\frac {1}{3} (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )+\left (\frac {1}{3} (-a-2 b)+x\right )^3-\frac {2}{27} (a-b)^3} \left ((d-1) \left (\frac {1}{3} (-a-2 b)+x\right )+\frac {1}{3} (3 (a-b d)-(d-1) (-a-2 b))\right )}d\left (\frac {1}{3} (-a-2 b)+x\right )\) |
| \(\Big \downarrow \) 2483 |
| \(\displaystyle \frac {2^{2/3} \left (-3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-(a-b)^3\right )^{2/3} \sqrt [3]{3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-2 (a-b)^3} \int \frac {27}{2^{2/3} \left (-(a-b)^3-3 \left (\frac {1}{3} (-a-2 b)+x\right ) (a-b)^2\right )^{2/3} \sqrt [3]{3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-2 (a-b)^3} \left ((a-b) (d+2)-3 (1-d) \left (\frac {1}{3} (-a-2 b)+x\right )\right )}d\left (\frac {1}{3} (-a-2 b)+x\right )}{3 \sqrt [3]{-9 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )+27 \left (\frac {1}{3} (-a-2 b)+x\right )^3-2 (a-b)^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9 \left (-3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-(a-b)^3\right )^{2/3} \sqrt [3]{3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-2 (a-b)^3} \int \frac {1}{\left (-(a-b)^3-3 \left (\frac {1}{3} (-a-2 b)+x\right ) (a-b)^2\right )^{2/3} \sqrt [3]{3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-2 (a-b)^3} \left ((a-b) (d+2)-3 (1-d) \left (\frac {1}{3} (-a-2 b)+x\right )\right )}d\left (\frac {1}{3} (-a-2 b)+x\right )}{\sqrt [3]{-9 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )+27 \left (\frac {1}{3} (-a-2 b)+x\right )^3-2 (a-b)^3}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {9 \left (-3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-(a-b)^3\right )^{2/3} \sqrt [3]{3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-2 (a-b)^3} \left (-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-2 (a-b)^3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-(a-b)^3}}\right )}{3 \sqrt {3} \sqrt [3]{d} (a-b)^3}+\frac {\log \left ((d+2) (a-b)-3 (1-d) \left (\frac {1}{3} (-a-2 b)+x\right )\right )}{18 \sqrt [3]{d} (a-b)^3}-\frac {\log \left (-\frac {\sqrt [3]{3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-2 (a-b)^3}}{\sqrt [3]{d}}-\sqrt [3]{-3 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )-(a-b)^3}\right )}{6 \sqrt [3]{d} (a-b)^3}\right )}{\sqrt [3]{-9 (a-b)^2 \left (\frac {1}{3} (-a-2 b)+x\right )+27 \left (\frac {1}{3} (-a-2 b)+x\right )^3-2 (a-b)^3}}\) |
(9*(-(a - b)^3 - 3*(a - b)^2*((-a - 2*b)/3 + x))^(2/3)*(-2*(a - b)^3 + 3*( a - b)^2*((-a - 2*b)/3 + x))^(1/3)*(-1/3*ArcTan[1/Sqrt[3] - (2*(-2*(a - b) ^3 + 3*(a - b)^2*((-a - 2*b)/3 + x))^(1/3))/(Sqrt[3]*d^(1/3)*(-(a - b)^3 - 3*(a - b)^2*((-a - 2*b)/3 + x))^(1/3))]/(Sqrt[3]*(a - b)^3*d^(1/3)) + Log [(a - b)*(2 + d) - 3*(1 - d)*((-a - 2*b)/3 + x)]/(18*(a - b)^3*d^(1/3)) - Log[-(-(a - b)^3 - 3*(a - b)^2*((-a - 2*b)/3 + x))^(1/3) - (-2*(a - b)^3 + 3*(a - b)^2*((-a - 2*b)/3 + x))^(1/3)/d^(1/3)]/(6*(a - b)^3*d^(1/3))))/(- 2*(a - b)^3 - 9*(a - b)^2*((-a - 2*b)/3 + x) + 27*((-a - 2*b)/3 + x)^3)^(1 /3)
3.31.96.3.1 Defintions of rubi rules used
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)*((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[((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]
\[\int \frac {1}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (a -b d +\left (-1+d \right ) x \right )}d x\]
Time = 0.27 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\left [-\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {b^{2} d + {\left (d + 2\right )} x^{2} + 2 \, a b + 3 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} d^{\frac {2}{3}} - 2 \, {\left (b d + a + b\right )} x + \sqrt {3} {\left ({\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b d - d x\right )} - {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} d^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}}}{b^{2} d + {\left (d - 1\right )} x^{2} - a b - {\left (2 \, b d - a - b\right )} x}\right ) - d^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} d^{\frac {1}{3}} - {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} d^{\frac {1}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}}}{b - x}\right )}{2 \, {\left (a - b\right )} d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (b - x\right )} d^{\frac {1}{3}} - 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (b - x\right )} d^{\frac {1}{3}}}\right ) - d^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b - x\right )} d^{\frac {1}{3}} - {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} d^{\frac {1}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}}}{b - x}\right )}{2 \, {\left (a - b\right )} d}\right ] \]
[-1/2*(sqrt(3)*d*sqrt(-1/d^(2/3))*log(-(b^2*d + (d + 2)*x^2 + 2*a*b + 3*(- a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b - x)*d^(2/3) - 2*( b*d + a + b)*x + sqrt(3)*((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x) ^(1/3)*(b*d - d*x) - (b^2*d - 2*b*d*x + d*x^2)*d^(1/3) + 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d^(2/3))*sqrt(-1/d^(2/3)))/(b^2*d + (d - 1)*x^2 - a*b - (2*b*d - a - b)*x)) - d^(2/3)*log(-((-a*b^2 - (a + 2 *b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b - x)*d^(1/3) - (b^2 - 2*b*x + x^ 2)*d^(2/3) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3))/(b^2 - 2*b*x + x^2)) + 2*d^(2/3)*log(-((b - x)*d^(1/3) + (-a*b^2 - (a + 2*b)*x^ 2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x)))/((a - b)*d), -1/2*(2*sqrt(3)*d ^(2/3)*arctan(1/3*sqrt(3)*((b - x)*d^(1/3) - 2*(-a*b^2 - (a + 2*b)*x^2 + x ^3 + (2*a*b + b^2)*x)^(1/3))/((b - x)*d^(1/3))) - d^(2/3)*log(-((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b - x)*d^(1/3) - (b^2 - 2*b* x + x^2)*d^(2/3) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)) /(b^2 - 2*b*x + x^2)) + 2*d^(2/3)*log(-((b - x)*d^(1/3) + (-a*b^2 - (a + 2 *b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3))/(b - x)))/((a - b)*d)]
\[ \int \frac {1}{\sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\int \frac {1}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (a - b d + d x - x\right )}\, dx \]
\[ \int \frac {1}{\sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\int { -\frac {1}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\int { -\frac {1}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\int \frac {1}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (a-b\,d+x\,\left (d-1\right )\right )} \,d x \]