Integrand size = 44, antiderivative size = 849 \[ \int \frac {-a-b c+(1+c) x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\frac {(b-x)^{4/3} (-a+x)^{2/3} \left (-\frac {3 a (b-x)^{2/3} \sqrt [3]{-a+x}}{(a-b)^2 (-b+x)}+\frac {3 b (b-x)^{2/3} \sqrt [3]{-a+x}}{(a-b)^2 (-b+x)}+\frac {\sqrt {3} a (-1+d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 d^{2/3}}+\frac {\sqrt {3} b c (-1+d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 d^{2/3}}+\frac {\sqrt {3} (a-b d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 d^{2/3}}+\frac {\sqrt {3} c (a-b d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 d^{2/3}}-\frac {a (-1+d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 d^{2/3}}-\frac {b c (-1+d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 d^{2/3}}-\frac {c (a-b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 d^{2/3}}+\frac {(-a+b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 d^{2/3}}+\frac {a (-1+d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b c (-1+d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 d^{2/3}}+\frac {(a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 d^{2/3}}+\frac {c (a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 d^{2/3}}\right )}{\left ((b-x)^2 (-a+x)\right )^{2/3}} \]
(b-x)^(4/3)*(-a+x)^(2/3)*(-3*a*(b-x)^(2/3)*(-a+x)^(1/3)/(a-b)^2/(-b+x)+3*b *(b-x)^(2/3)*(-a+x)^(1/3)/(a-b)^2/(-b+x)+3^(1/2)*a*(-1+d)*arctan(3^(1/2)*( -a+x)^(1/3)/(-2*d^(1/3)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)^2/d^(2/3)+3^(1/2) *b*c*(-1+d)*arctan(3^(1/2)*(-a+x)^(1/3)/(-2*d^(1/3)*(b-x)^(1/3)+(-a+x)^(1/ 3)))/(a-b)^2/d^(2/3)+3^(1/2)*(-b*d+a)*arctan(3^(1/2)*(-a+x)^(1/3)/(-2*d^(1 /3)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)^2/d^(2/3)+3^(1/2)*c*(-b*d+a)*arctan(3 ^(1/2)*(-a+x)^(1/3)/(-2*d^(1/3)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)^2/d^(2/3) -a*(-1+d)*ln(d^(1/3)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b)^2/d^(2/3)-b*c*(-1+d)* ln(d^(1/3)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b)^2/d^(2/3)-c*(-b*d+a)*ln(d^(1/3) *(b-x)^(1/3)+(-a+x)^(1/3))/(a-b)^2/d^(2/3)+(b*d-a)*ln(d^(1/3)*(b-x)^(1/3)+ (-a+x)^(1/3))/(a-b)^2/d^(2/3)+1/2*a*(-1+d)*ln(d^(2/3)*(b-x)^(2/3)-d^(1/3)* (b-x)^(1/3)*(-a+x)^(1/3)+(-a+x)^(2/3))/(a-b)^2/d^(2/3)+1/2*b*c*(-1+d)*ln(d ^(2/3)*(b-x)^(2/3)-d^(1/3)*(b-x)^(1/3)*(-a+x)^(1/3)+(-a+x)^(2/3))/(a-b)^2/ d^(2/3)+1/2*(-b*d+a)*ln(d^(2/3)*(b-x)^(2/3)-d^(1/3)*(b-x)^(1/3)*(-a+x)^(1/ 3)+(-a+x)^(2/3))/(a-b)^2/d^(2/3)+1/2*c*(-b*d+a)*ln(d^(2/3)*(b-x)^(2/3)-d^( 1/3)*(b-x)^(1/3)*(-a+x)^(1/3)+(-a+x)^(2/3))/(a-b)^2/d^(2/3))/((b-x)^2*(-a+ x))^(2/3)
Time = 0.33 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.28 \[ \int \frac {-a-b c+(1+c) x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\frac {-6 d^{2/3} (-a+x) (-b+x)+2 \sqrt {3} (c+d) (b-x)^{4/3} (-a+x)^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-a+x}}{\sqrt [3]{d} \sqrt [3]{b-x}}}{\sqrt {3}}\right )-2 (c+d) (b-x)^{4/3} (-a+x)^{2/3} \log \left (\sqrt [3]{d}+\frac {\sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )+(c+d) (b-x)^{4/3} (-a+x)^{2/3} \log \left (d^{2/3}-\frac {\sqrt [3]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {(-a+x)^{2/3}}{(b-x)^{2/3}}\right )}{2 (a-b) d^{2/3} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]
(-6*d^(2/3)*(-a + x)*(-b + x) + 2*Sqrt[3]*(c + d)*(b - x)^(4/3)*(-a + x)^( 2/3)*ArcTan[(1 - (2*(-a + x)^(1/3))/(d^(1/3)*(b - x)^(1/3)))/Sqrt[3]] - 2* (c + d)*(b - x)^(4/3)*(-a + x)^(2/3)*Log[d^(1/3) + (-a + x)^(1/3)/(b - x)^ (1/3)] + (c + d)*(b - x)^(4/3)*(-a + x)^(2/3)*Log[d^(2/3) - (d^(1/3)*(-a + x)^(1/3))/(b - x)^(1/3) + (-a + x)^(2/3)/(b - x)^(2/3)])/(2*(a - b)*d^(2/ 3)*((b - x)^2*(-a + x))^(2/3))
Time = 0.84 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, 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.
| \(\displaystyle \int \frac {-a-b c+(c+1) x}{\left ((x-a) (x-b)^2\right )^{2/3} (a-b d+(d-1) x)} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{4/3} \int -\frac {a+b c-(c+1) x}{(x-a)^{2/3} (x-b)^{4/3} (a-b d-(1-d) x)}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \int \frac {a+b c-(c+1) x}{(x-a)^{2/3} (x-b)^{4/3} (a-b d-(1-d) x)}dx}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \left (\frac {3 \int -\frac {(a-b)^2 (c+d)}{3 (x-a)^{2/3} \sqrt [3]{x-b} (a-b d-(1-d) x)}dx}{(a-b)^2}+\frac {3 \sqrt [3]{x-a}}{(a-b) \sqrt [3]{x-b}}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \left (\frac {3 \sqrt [3]{x-a}}{(a-b) \sqrt [3]{x-b}}-(c+d) \int \frac {1}{(x-a)^{2/3} \sqrt [3]{x-b} (a-b d-(1-d) x)}dx\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{4/3} \left (\frac {3 \sqrt [3]{x-a}}{(a-b) \sqrt [3]{x-b}}-(c+d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{d^{2/3} (a-b)}+\frac {\log (a-b d-(1-d) x)}{2 d^{2/3} (a-b)}-\frac {3 \log \left (\sqrt [3]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{2 d^{2/3} (a-b)}\right )\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\) |
-(((-a + x)^(2/3)*(-b + x)^(4/3)*((3*(-a + x)^(1/3))/((a - b)*(-b + x)^(1/ 3)) - (c + d)*(-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(-b + x)^(1/3))/(S qrt[3]*(-a + x)^(1/3))])/((a - b)*d^(2/3))) + Log[a - b*d - (1 - d)*x]/(2* (a - b)*d^(2/3)) - (3*Log[-(-a + x)^(1/3) + d^(1/3)*(-b + x)^(1/3)])/(2*(a - b)*d^(2/3)))))/(-((a - x)*(b - x)^2))^(2/3))
3.32.34.3.1 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]
\[\int \frac {-a -b c +\left (1+c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (a -b d +\left (-1+d \right ) x \right )}d x\]
Time = 0.25 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.46 \[ \int \frac {-a-b c+(1+c) x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=-\frac {2 \, \sqrt {3} {\left (b c d + b d^{2} - {\left (c d + d^{2}\right )} x\right )} {\left (d^{2}\right )}^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b d - d x\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, {\left (b d^{2} - d^{2} x\right )}}\right ) + {\left (b c + b d - {\left (c + d\right )} x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} {\left (b - x\right )} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d - {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, {\left (b c + b d - {\left (c + d\right )} x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (d^{2}\right )}^{\frac {2}{3}} {\left (b - x\right )} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b - x}\right ) + 6 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d^{2}}{2 \, {\left ({\left (a - b\right )} d^{2} x - {\left (a b - b^{2}\right )} d^{2}\right )}} \]
-1/2*(2*sqrt(3)*(b*c*d + b*d^2 - (c*d + d^2)*x)*(d^2)^(1/6)*arctan(1/3*sqr t(3)*(d^2)^(1/6)*((b*d - d*x)*(d^2)^(1/3) - 2*(-a*b^2 - (a + 2*b)*x^2 + x^ 3 + (2*a*b + b^2)*x)^(1/3)*(d^2)^(2/3))/(b*d^2 - d^2*x)) + (b*c + b*d - (c + d)*x)*(d^2)^(2/3)*log(-((-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x )^(1/3)*(d^2)^(2/3)*(b - x) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2 )*x)^(2/3)*d - (b^2*d - 2*b*d*x + d*x^2)*(d^2)^(1/3))/(b^2 - 2*b*x + x^2)) - 2*(b*c + b*d - (c + d)*x)*(d^2)^(2/3)*log(-((d^2)^(2/3)*(b - x) + (-a*b ^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d)/(b - x)) + 6*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d^2)/((a - b)*d^2*x - (a*b - b^2)*d^2)
\[ \int \frac {-a-b c+(1+c) x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\int \frac {- a - b c + c x + x}{\left (\left (- a + x\right ) \left (- b + x\right )^{2}\right )^{\frac {2}{3}} \left (a - b d + d x - x\right )}\, dx \]
\[ \int \frac {-a-b c+(1+c) x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\int { \frac {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )}} \,d x } \]
\[ \int \frac {-a-b c+(1+c) x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\int { \frac {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )}} \,d x } \]
Timed out. \[ \int \frac {-a-b c+(1+c) x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\int -\frac {a+b\,c-x\,\left (c+1\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (a-b\,d+x\,\left (d-1\right )\right )} \,d x \]