Integrand size = 51, antiderivative size = 1387 \[ \int \frac {-b-a c+(1+c) x}{(-a+x) \sqrt [3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx=\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{a-x} (-b+x)^{2/3} \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right ) \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right ) \left (\frac {3 \left (a \sqrt [3]{-b+x}-i \sqrt {3} a \sqrt [3]{-b+x}\right )}{2 (a-b)^2 \sqrt [3]{a-x}}+\frac {3 i \left (i b \sqrt [3]{-b+x}+\sqrt {3} b \sqrt [3]{-b+x}\right )}{2 (a-b)^2 \sqrt [3]{a-x}}+\frac {3 \left (a c \sqrt [3]{-b+x}-i \sqrt {3} a c \sqrt [3]{-b+x}\right )}{2 (a-b)^2 \sqrt [3]{a-x}}+\frac {3 i \left (i a c \sqrt [3]{-b+x}+\sqrt {3} a c \sqrt [3]{-b+x}\right )}{2 (a-b)^2 \sqrt [3]{a-x}}+\frac {\left (3 i b-\sqrt {3} b-3 i a d+\sqrt {3} a d\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-3 i b+\sqrt {3} b+3 i b d-\sqrt {3} b d\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-3 i a c+\sqrt {3} a c+3 i a c d-\sqrt {3} a c d\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (3 i b c-\sqrt {3} b c-3 i a c d+\sqrt {3} a c d\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (b-i \sqrt {3} b-a d+i \sqrt {3} a d\right ) \log \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-b+i \sqrt {3} b+b d-i \sqrt {3} b d\right ) \log \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-a c+i \sqrt {3} a c+a c d-i \sqrt {3} a c d\right ) \log \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (b c-i \sqrt {3} b c-a c d+i \sqrt {3} a c d\right ) \log \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-b+i \sqrt {3} b+a d-i \sqrt {3} a d\right ) \log \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (b-i \sqrt {3} b-b d+i \sqrt {3} b d\right ) \log \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (-b c+i \sqrt {3} b c+a c d-i \sqrt {3} a c d\right ) \log \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (a c-i \sqrt {3} a c-a c d+i \sqrt {3} a c d\right ) \log \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}\right )}{2 \sqrt [3]{-\left ((a-x) (b-x)^2\right )} (-b+a d+x-d x)} \]
1/2*(1+I*3^(1/2))*(a-x)^(1/3)*(-b+x)^(2/3)*(d^(1/3)*(a-x)^(1/3)+(-b+x)^(1/ 3))*(d^(2/3)*(a-x)^(2/3)-d^(1/3)*(a-x)^(1/3)*(-b+x)^(1/3)+(-b+x)^(2/3))*(3 /2*(a*(-b+x)^(1/3)-I*3^(1/2)*a*(-b+x)^(1/3))/(a-b)^2/(a-x)^(1/3)+3/2*I*(I* b*(-b+x)^(1/3)+3^(1/2)*b*(-b+x)^(1/3))/(a-b)^2/(a-x)^(1/3)+3/2*(a*c*(-b+x) ^(1/3)-I*3^(1/2)*a*c*(-b+x)^(1/3))/(a-b)^2/(a-x)^(1/3)+3/2*I*(I*a*c*(-b+x) ^(1/3)+3^(1/2)*a*c*(-b+x)^(1/3))/(a-b)^2/(a-x)^(1/3)+1/2*(3*I*b-3^(1/2)*b- 3*I*a*d+3^(1/2)*a*d)*arctan(3^(1/2)*(-b+x)^(1/3)/(-2*d^(1/3)*(a-x)^(1/3)+( -b+x)^(1/3)))/(a-b)^2/d^(2/3)+1/2*(-3*I*b+3^(1/2)*b+3*I*b*d-3^(1/2)*b*d)*a rctan(3^(1/2)*(-b+x)^(1/3)/(-2*d^(1/3)*(a-x)^(1/3)+(-b+x)^(1/3)))/(a-b)^2/ d^(2/3)+1/2*(-3*I*a*c+3^(1/2)*a*c+3*I*a*c*d-3^(1/2)*a*c*d)*arctan(3^(1/2)* (-b+x)^(1/3)/(-2*d^(1/3)*(a-x)^(1/3)+(-b+x)^(1/3)))/(a-b)^2/d^(2/3)+1/2*(3 *I*b*c-3^(1/2)*b*c-3*I*a*c*d+3^(1/2)*a*c*d)*arctan(3^(1/2)*(-b+x)^(1/3)/(- 2*d^(1/3)*(a-x)^(1/3)+(-b+x)^(1/3)))/(a-b)^2/d^(2/3)+1/2*(b-I*3^(1/2)*b-a* d+I*3^(1/2)*a*d)*ln(d^(1/3)*(a-x)^(1/3)+(-b+x)^(1/3))/(a-b)^2/d^(2/3)+1/2* (-b+I*3^(1/2)*b+b*d-I*3^(1/2)*b*d)*ln(d^(1/3)*(a-x)^(1/3)+(-b+x)^(1/3))/(a -b)^2/d^(2/3)+1/2*(-a*c+I*3^(1/2)*a*c+a*c*d-I*3^(1/2)*a*c*d)*ln(d^(1/3)*(a -x)^(1/3)+(-b+x)^(1/3))/(a-b)^2/d^(2/3)+1/2*(b*c-I*3^(1/2)*b*c-a*c*d+I*3^( 1/2)*a*c*d)*ln(d^(1/3)*(a-x)^(1/3)+(-b+x)^(1/3))/(a-b)^2/d^(2/3)+1/4*(-b+I *3^(1/2)*b+a*d-I*3^(1/2)*a*d)*ln(d^(2/3)*(a-x)^(2/3)-d^(1/3)*(a-x)^(1/3)*( -b+x)^(1/3)+(-b+x)^(2/3))/(a-b)^2/d^(2/3)+1/4*(b-I*3^(1/2)*b-b*d+I*3^(1...
Time = 0.34 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.22 \[ \int \frac {-b-a c+(1+c) x}{(-a+x) \sqrt [3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx=\frac {-6 b d^{2/3}+6 d^{2/3} x-2 \sqrt {3} (c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b-x}}{\sqrt [3]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )+2 (c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (\sqrt [3]{d}+\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-c (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (d^{2/3}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-d (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (d^{2/3}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )}{2 (a-b) d^{2/3} \sqrt [3]{(b-x)^2 (-a+x)}} \]
Integrate[(-b - a*c + (1 + c)*x)/((-a + x)*((-a + x)*(-b + x)^2)^(1/3)*(b - a*d + (-1 + d)*x)),x]
(-6*b*d^(2/3) + 6*d^(2/3)*x - 2*Sqrt[3]*(c + d)*(b - x)^(2/3)*(-a + x)^(1/ 3)*ArcTan[(1 - (2*(b - x)^(1/3))/(d^(1/3)*(-a + x)^(1/3)))/Sqrt[3]] + 2*(c + d)*(b - x)^(2/3)*(-a + x)^(1/3)*Log[d^(1/3) + (b - x)^(1/3)/(-a + x)^(1 /3)] - c*(b - x)^(2/3)*(-a + x)^(1/3)*Log[d^(2/3) + (b - x)^(2/3)/(-a + x) ^(2/3) - (d^(1/3)*(b - x)^(1/3))/(-a + x)^(1/3)] - d*(b - x)^(2/3)*(-a + x )^(1/3)*Log[d^(2/3) + (b - x)^(2/3)/(-a + x)^(2/3) - (d^(1/3)*(b - x)^(1/3 ))/(-a + x)^(1/3)])/(2*(a - b)*d^(2/3)*((b - x)^2*(-a + x))^(1/3))
Time = 1.39 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {7270, 25, 172, 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.
| \(\displaystyle \int \frac {-a c-b+(c+1) x}{(x-a) \sqrt [3]{(x-a) (x-b)^2} (-a d+b+(d-1) x)} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [3]{x-a} (x-b)^{2/3} \int -\frac {b+a c-(c+1) x}{(x-a)^{4/3} (x-b)^{2/3} (b-a d-(1-d) x)}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \int \frac {b+a c-(c+1) x}{(x-a)^{4/3} (x-b)^{2/3} (b-a d-(1-d) x)}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (\frac {3 \int -\frac {(a-b)^2 (c+d)}{3 \sqrt [3]{x-a} (x-b)^{2/3} (b-a d-(1-d) x)}dx}{(a-b)^2}-\frac {3 \sqrt [3]{x-b}}{(a-b) \sqrt [3]{x-a}}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (-\left ((c+d) \int \frac {1}{\sqrt [3]{x-a} (x-b)^{2/3} (b-a d-(1-d) x)}dx\right )-\frac {3 \sqrt [3]{x-b}}{(a-b) \sqrt [3]{x-a}}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (-\left ((c+d) \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{d^{2/3} (a-b)}-\frac {\log (-a d+b-(1-d) x)}{2 d^{2/3} (a-b)}+\frac {3 \log \left (\sqrt [3]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{2 d^{2/3} (a-b)}\right )\right )-\frac {3 \sqrt [3]{x-b}}{(a-b) \sqrt [3]{x-a}}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
-(((-a + x)^(1/3)*(-b + x)^(2/3)*((-3*(-b + x)^(1/3))/((a - b)*(-a + x)^(1 /3)) - (c + d)*((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(-a + x)^(1/3))/(Sq rt[3]*(-b + x)^(1/3))])/((a - b)*d^(2/3)) - Log[b - a*d - (1 - d)*x]/(2*(a - b)*d^(2/3)) + (3*Log[d^(1/3)*(-a + x)^(1/3) - (-b + x)^(1/3)])/(2*(a - b)*d^(2/3)))))/(-((a - x)*(b - x)^2))^(1/3))
3.32.47.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ (b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1) *(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f )*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int \frac {-b -a c +\left (1+c \right ) x}{\left (-a +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (b -a d +\left (-1+d \right ) x \right )}d x\]
Time = 0.26 (sec) , antiderivative size = 473, normalized size of antiderivative = 0.34 \[ \int \frac {-b-a c+(1+c) x}{(-a+x) \sqrt [3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx=\frac {2 \, \sqrt {3} {\left (a b c d + a b d^{2} + {\left (c d + d^{2}\right )} x^{2} - {\left ({\left (a + b\right )} c d + {\left (a + b\right )} d^{2}\right )} x\right )} \sqrt {-\left (-d^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} {\left (\left (-d^{2}\right )^{\frac {1}{3}} {\left (b - x\right )} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d\right )} \sqrt {-\left (-d^{2}\right )^{\frac {1}{3}}}}{3 \, {\left (b d - d x\right )}}\right ) + 6 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d^{2} - {\left (a b c + a b d + {\left (c + d\right )} x^{2} - {\left ({\left (a + b\right )} c + {\left (a + b\right )} d\right )} x\right )} \left (-d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d^{2} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b d - d x\right )} \left (-d^{2}\right )^{\frac {1}{3}} + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d^{2}\right )^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) + 2 \, {\left (a b c + a b d + {\left (c + d\right )} x^{2} - {\left ({\left (a + b\right )} c + {\left (a + b\right )} d\right )} x\right )} \left (-d^{2}\right )^{\frac {2}{3}} \log \left (\frac {\left (-d^{2}\right )^{\frac {1}{3}} {\left (b - x\right )} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b - x}\right )}{2 \, {\left ({\left (a - b\right )} d^{2} x^{2} - {\left (a^{2} - b^{2}\right )} d^{2} x + {\left (a^{2} b - a b^{2}\right )} d^{2}\right )}} \]
integrate((-b-a*c+(1+c)*x)/(-a+x)/((-a+x)*(-b+x)^2)^(1/3)/(b-a*d+(-1+d)*x) ,x, algorithm="fricas")
1/2*(2*sqrt(3)*(a*b*c*d + a*b*d^2 + (c*d + d^2)*x^2 - ((a + b)*c*d + (a + b)*d^2)*x)*sqrt(-(-d^2)^(1/3))*arctan(-1/3*sqrt(3)*((-d^2)^(1/3)*(b - x) + 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d)*sqrt(-(-d^2)^ (1/3))/(b*d - d*x)) + 6*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^( 2/3)*d^2 - (a*b*c + a*b*d + (c + d)*x^2 - ((a + b)*c + (a + b)*d)*x)*(-d^2 )^(2/3)*log(((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d^2 + (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b*d - d*x)*(-d^2)^ (1/3) + (b^2 - 2*b*x + x^2)*(-d^2)^(2/3))/(b^2 - 2*b*x + x^2)) + 2*(a*b*c + a*b*d + (c + d)*x^2 - ((a + b)*c + (a + b)*d)*x)*(-d^2)^(2/3)*log(((-d^2 )^(1/3)*(b - x) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d )/(b - x)))/((a - b)*d^2*x^2 - (a^2 - b^2)*d^2*x + (a^2*b - a*b^2)*d^2)
\[ \int \frac {-b-a c+(1+c) x}{(-a+x) \sqrt [3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx=\int \frac {- a c - b + c x + x}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (- a + x\right ) \left (- a d + b + d x - x\right )}\, dx \]
Integral((-a*c - b + c*x + x)/(((-a + x)*(-b + x)**2)**(1/3)*(-a + x)*(-a* d + b + d*x - x)), x)
\[ \int \frac {-b-a c+(1+c) x}{(-a+x) \sqrt [3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx=\int { -\frac {a c - {\left (c + 1\right )} x + b}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (a d - {\left (d - 1\right )} x - b\right )} {\left (a - x\right )}} \,d x } \]
integrate((-b-a*c+(1+c)*x)/(-a+x)/((-a+x)*(-b+x)^2)^(1/3)/(b-a*d+(-1+d)*x) ,x, algorithm="maxima")
-integrate((a*c - (c + 1)*x + b)/((-(a - x)*(b - x)^2)^(1/3)*(a*d - (d - 1 )*x - b)*(a - x)), x)
\[ \int \frac {-b-a c+(1+c) x}{(-a+x) \sqrt [3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx=\int { -\frac {a c - {\left (c + 1\right )} x + b}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (a d - {\left (d - 1\right )} x - b\right )} {\left (a - x\right )}} \,d x } \]
integrate((-b-a*c+(1+c)*x)/(-a+x)/((-a+x)*(-b+x)^2)^(1/3)/(b-a*d+(-1+d)*x) ,x, algorithm="giac")
integrate(-(a*c - (c + 1)*x + b)/((-(a - x)*(b - x)^2)^(1/3)*(a*d - (d - 1 )*x - b)*(a - x)), x)
Timed out. \[ \int \frac {-b-a c+(1+c) x}{(-a+x) \sqrt [3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx=-\int -\frac {b+a\,c-x\,\left (c+1\right )}{\left (a-x\right )\,{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (b-a\,d+x\,\left (d-1\right )\right )} \,d x \]