Integrand size = 51, antiderivative size = 857 \[ \int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (\frac {3 a \sqrt [3]{b-x} (-a+x)^{2/3}}{2 (a-b)^2 (-b+x)}-\frac {3 b \sqrt [3]{b-x} (-a+x)^{2/3}}{2 (a-b)^2 (-b+x)}-\frac {\sqrt {3} a (-1+d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} b c (-1+d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} (a-b d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} c (a-b d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {a (-1+d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {b c (-1+d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {c (a-b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}+\frac {(-a+b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}+\frac {a (-1+d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {b c (-1+d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {(a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {c (a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}\right )}{\sqrt [3]{(b-x)^2 (-a+x)}} \]
(b-x)^(2/3)*(-a+x)^(1/3)*(3/2*a*(b-x)^(1/3)*(-a+x)^(2/3)/(a-b)^2/(-b+x)-3/ 2*b*(b-x)^(1/3)*(-a+x)^(2/3)/(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^(1/3)-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^(1/3)-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^(1/3)-3^(1/2)*c*(-b*d+a)*arcta n(3^(1/2)*(-a+x)^(1/3)/(-2*d^(1/3)*(b-x)^(1/3)+(-a+x)^(1/3)))/(a-b)^2/d^(1 /3)-a*(-1+d)*ln(d^(1/3)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b)^2/d^(1/3)-b*c*(-1+ d)*ln(d^(1/3)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b)^2/d^(1/3)-c*(-b*d+a)*ln(d^(1 /3)*(b-x)^(1/3)+(-a+x)^(1/3))/(a-b)^2/d^(1/3)+(b*d-a)*ln(d^(1/3)*(b-x)^(1/ 3)+(-a+x)^(1/3))/(a-b)^2/d^(1/3)+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^(1/3)+1/2*b*c*(-1+d)*l n(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^(1/3)+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^(1/3)+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^(1/3))/((b-x)^2*( -a+x))^(1/3)
Time = 0.30 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.33 \[ \int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\frac {3 a \sqrt [3]{d}-3 \sqrt [3]{d} x+2 \sqrt {3} (c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )+(c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (1+\frac {d^{2/3} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 c (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 d (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )}{2 (a-b) \sqrt [3]{d} \sqrt [3]{(b-x)^2 (-a+x)}} \]
Integrate[(-a - b*c + (1 + c)*x)/((-b + x)*((-a + x)*(-b + x)^2)^(1/3)*(a - b*d + (-1 + d)*x)),x]
(3*a*d^(1/3) - 3*d^(1/3)*x + 2*Sqrt[3]*(c + d)*(b - x)^(2/3)*(-a + x)^(1/3 )*ArcTan[(1 - (2*d^(1/3)*(b - x)^(1/3))/(-a + x)^(1/3))/Sqrt[3]] + (c + d) *(b - x)^(2/3)*(-a + x)^(1/3)*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*c*(b - x)^(2/3)*(-a + x)^ (1/3)*Log[1 + (d^(1/3)*(b - x)^(1/3))/(-a + x)^(1/3)] - 2*d*(b - x)^(2/3)* (-a + x)^(1/3)*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 = 1.40 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, 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.
| \(\displaystyle \int \frac {-a-b c+(c+1) x}{(x-b) \sqrt [3]{(x-a) (x-b)^2} (a-b d+(d-1) x)} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [3]{x-a} (x-b)^{2/3} \int -\frac {a+b c-(c+1) x}{\sqrt [3]{x-a} (x-b)^{5/3} (a-b d-(1-d) x)}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \int \frac {a+b c-(c+1) x}{\sqrt [3]{x-a} (x-b)^{5/3} (a-b d-(1-d) x)}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (\frac {3 \int -\frac {2 (a-b)^2 (c+d)}{3 \sqrt [3]{x-a} (x-b)^{2/3} (a-b d-(1-d) x)}dx}{2 (a-b)^2}+\frac {3 (x-a)^{2/3}}{2 (a-b) (x-b)^{2/3}}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (\frac {3 (x-a)^{2/3}}{2 (a-b) (x-b)^{2/3}}-(c+d) \int \frac {1}{\sqrt [3]{x-a} (x-b)^{2/3} (a-b d-(1-d) x)}dx\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (\frac {3 (x-a)^{2/3}}{2 (a-b) (x-b)^{2/3}}-(c+d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{d} (a-b)}+\frac {\log (a-b d-(1-d) x)}{2 \sqrt [3]{d} (a-b)}-\frac {3 \log \left (\frac {\sqrt [3]{x-a}}{\sqrt [3]{d}}-\sqrt [3]{x-b}\right )}{2 \sqrt [3]{d} (a-b)}\right )\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
-(((-a + x)^(1/3)*(-b + x)^(2/3)*((3*(-a + x)^(2/3))/(2*(a - b)*(-b + x)^( 2/3)) - (c + d)*(-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(-a + x)^(1/3))/(Sqrt[3] *d^(1/3)*(-b + x)^(1/3))])/((a - b)*d^(1/3))) + Log[a - b*d - (1 - d)*x]/( 2*(a - b)*d^(1/3)) - (3*Log[(-a + x)^(1/3)/d^(1/3) - (-b + x)^(1/3)])/(2*( a - b)*d^(1/3)))))/(-((a - x)*(b - x)^2))^(1/3))
3.32.35.3.1 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]
\[\int \frac {-a -b c +\left (1+c \right ) x}{\left (-b +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (a -b d +\left (-1+d \right ) x \right )}d x\]
Time = 0.26 (sec) , antiderivative size = 976, normalized size of antiderivative = 1.14 \[ \int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\text {Too large to display} \]
integrate((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(a-b*d+(-1+d)*x) ,x, algorithm="fricas")
[-1/2*(sqrt(3)*(b^2*c*d + b^2*d^2 + (c*d + d^2)*x^2 - 2*(b*c*d + b*d^2)*x) *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)) - (b^2*c + b^2*d + (c + d)*x^2 - 2*(b*c + b*d)*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*(b^2*c + b^2*d + (c + d)*x^2 - 2*(b*c + b*d)*x)*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)) + 3*(-a*b^2 - (a + 2*b)*x ^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d)/((a - b)*d*x^2 - 2*(a*b - b^2)*d*x + (a*b^2 - b^3)*d), 1/2*((b^2*c + b^2*d + (c + d)*x^2 - 2*(b*c + b*d)*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*(b^2*c + b^2*d + (c + d)*x^ 2 - 2*(b*c + b*d)*x)*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)) - 2*sqrt(3)*(b^2*c*d + b^2*d^2 + (c*d + d^2)*x^2 - 2*(b*c*d + b*d^2)*x)*arctan(1/3*sqrt(3)*((b - x)*d...
\[ \int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\int \frac {- a - b c + c x + x}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (- b + x\right ) \left (a - b d + d x - x\right )}\, dx \]
Integral((-a - b*c + c*x + x)/(((-a + x)*(-b + x)**2)**(1/3)*(-b + x)*(a - b*d + d*x - x)), x)
\[ \int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\int { -\frac {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )} {\left (b - x\right )}} \,d x } \]
integrate((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(a-b*d+(-1+d)*x) ,x, algorithm="maxima")
-integrate((b*c - (c + 1)*x + a)/((-(a - x)*(b - x)^2)^(1/3)*(b*d - (d - 1 )*x - a)*(b - x)), x)
\[ \int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=\int { -\frac {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )} {\left (b - x\right )}} \,d x } \]
integrate((-a-b*c+(1+c)*x)/(-b+x)/((-a+x)*(-b+x)^2)^(1/3)/(a-b*d+(-1+d)*x) ,x, algorithm="giac")
integrate(-(b*c - (c + 1)*x + a)/((-(a - x)*(b - x)^2)^(1/3)*(b*d - (d - 1 )*x - a)*(b - x)), x)
Timed out. \[ \int \frac {-a-b c+(1+c) x}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx=-\int -\frac {a+b\,c-x\,\left (c+1\right )}{\left (b-x\right )\,{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (a-b\,d+x\,\left (d-1\right )\right )} \,d x \]