Integrand size = 72, antiderivative size = 369 \[ \int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx=\frac {\sqrt {\frac {1}{2} \left (-3+3 i \sqrt {3}\right )} \arctan \left (\frac {3 \sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-3 i a+\sqrt {3} a+3 i x-\sqrt {3} x+\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{d^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) \log \left ((-1)^{2/3} (a-x)+\sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )}{2 d^{2/3}}-\frac {i \left (-i+\sqrt {3}\right ) \log \left (\sqrt [3]{-1} \left (a^2-2 a x+x^2\right )+(-1)^{2/3} \sqrt [3]{d} (a-x) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}-d^{2/3} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]
1/2*(-6+6*I*3^(1/2))^(1/2)*arctan(3*d^(1/3)*(-a*b*c+(a*b+a*c+b*c)*x+(-a-b- c)*x^2+x^3)^(1/3)/(-3*I*a+3^(1/2)*a+3*I*x-x*3^(1/2)+3^(1/2)*d^(1/3)*(-a*b* c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/3)))/d^(2/3)+1/2*(1+I*3^(1/2))*ln(( -1)^(2/3)*(a-x)+d^(1/3)*(-a*b*c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/3))/d ^(2/3)-1/4*I*(-I+3^(1/2))*ln((-1)^(1/3)*(a^2-2*a*x+x^2)+(-1)^(2/3)*d^(1/3) *(a-x)*(-a*b*c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/3)-d^(2/3)*(-a*b*c+(a* b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(2/3))/d^(2/3)
\[ \int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx=\int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx \]
Integrate[(a*b + a*c - 2*b*c + (-2*a + b + c)*x)/(((-a + x)*(-b + x)*(-c + x))^(1/3)*(a^2 - b*c*d + (-2*a + b*d + c*d)*x + (1 - d)*x^2)),x]
Integrate[(a*b + a*c - 2*b*c + (-2*a + b + c)*x)/(((-a + x)*(-b + x)*(-c + x))^(1/3)*(a^2 - b*c*d + (-2*a + b*d + c*d)*x + (1 - d)*x^2)), x]
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.
| \(\displaystyle \int \frac {x (-2 a+b+c)+a b+a c-2 b c}{\sqrt [3]{(x-a) (x-b) (x-c)} \left (a^2+x (-2 a+b d+c d)-b c d+(1-d) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle \frac {\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \int -\frac {2 b c-a (b+c)+(2 a-b-c) x}{\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \left (a^2+(1-d) x^2-b c d-(2 a-(b+c) d) x\right )}dx}{\sqrt [3]{-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \int \frac {2 b c-a (b+c)+(2 a-b-c) x}{\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \left (a^2+(1-d) x^2-b c d-(2 a-(b+c) d) x\right )}dx}{\sqrt [3]{-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \int \left (\frac {2 a-b-c-\frac {\sqrt {4 a^2-4 b a-4 c a+4 b c+b^2 d+c^2 d-2 b c d}}{\sqrt {d}}}{\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \left (-2 a+(b+c) d+2 (1-d) x+\sqrt {d} \sqrt {4 a^2-4 b a-4 c a+4 b c+b^2 d+c^2 d-2 b c d}\right )}+\frac {2 a-b-c+\frac {\sqrt {4 a^2-4 b a-4 c a+4 b c+b^2 d+c^2 d-2 b c d}}{\sqrt {d}}}{\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \left (-2 a+(b+c) d+2 (1-d) x-\sqrt {d} \sqrt {4 a^2-4 b a-4 c a+4 b c+b^2 d+c^2 d-2 b c d}\right )}\right )dx}{\sqrt [3]{-((a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \left (\left (\frac {\sqrt {4 a^2-4 a (b+c)+b^2 d+2 b c (2-d)+c^2 d}}{\sqrt {d}}+2 a-b-c\right ) \int \frac {1}{\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \left (-2 a+(b+c) d+2 (1-d) x-\sqrt {d} \sqrt {4 a^2-4 b a-4 c a+4 b c+b^2 d+c^2 d-2 b c d}\right )}dx+\left (-\frac {\sqrt {4 a^2-4 a (b+c)+b^2 d+2 b c (2-d)+c^2 d}}{\sqrt {d}}+2 a-b-c\right ) \int \frac {1}{\sqrt [3]{x-a} \sqrt [3]{x-b} \sqrt [3]{x-c} \left (-2 a+(b+c) d+2 (1-d) x+\sqrt {d} \sqrt {4 a^2-4 b a-4 c a+4 b c+b^2 d+c^2 d-2 b c d}\right )}dx\right )}{\sqrt [3]{-((a-x) (b-x) (c-x))}}\) |
Int[(a*b + a*c - 2*b*c + (-2*a + b + c)*x)/(((-a + x)*(-b + x)*(-c + x))^( 1/3)*(a^2 - b*c*d + (-2*a + b*d + c*d)*x + (1 - d)*x^2)),x]
3.30.68.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Simp[ a^IntPart[p]*((a*v^m*w^n*z^q)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[ p])*z^(q*FracPart[p]))) Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; FreeQ[{a , m, n, p, q}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x] && !F reeQ[z, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
\[\int \frac {a b +a c -2 b c +\left (-2 a +b +c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {1}{3}} \left (a^{2}-b c d +\left (b d +c d -2 a \right ) x +\left (1-d \right ) x^{2}\right )}d x\]
int((a*b+a*c-2*b*c+(-2*a+b+c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(a^2-b*c*d+( b*d+c*d-2*a)*x+(1-d)*x^2),x)
int((a*b+a*c-2*b*c+(-2*a+b+c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(a^2-b*c*d+( b*d+c*d-2*a)*x+(1-d)*x^2),x)
Timed out. \[ \int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx=\text {Timed out} \]
integrate((a*b+a*c-2*b*c+(-2*a+b+c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(a^2-b *c*d+(b*d+c*d-2*a)*x+(1-d)*x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx=\text {Timed out} \]
integrate((a*b+a*c-2*b*c+(-2*a+b+c)*x)/((-a+x)*(-b+x)*(-c+x))**(1/3)/(a**2 -b*c*d+(b*d+c*d-2*a)*x+(1-d)*x**2),x)
\[ \int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx=\int { -\frac {a b + a c - 2 \, b c - {\left (2 \, a - b - c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{3}} {\left (b c d + {\left (d - 1\right )} x^{2} - a^{2} - {\left (b d + c d - 2 \, a\right )} x\right )}} \,d x } \]
integrate((a*b+a*c-2*b*c+(-2*a+b+c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(a^2-b *c*d+(b*d+c*d-2*a)*x+(1-d)*x^2),x, algorithm="maxima")
-integrate((a*b + a*c - 2*b*c - (2*a - b - c)*x)/((-(a - x)*(b - x)*(c - x ))^(1/3)*(b*c*d + (d - 1)*x^2 - a^2 - (b*d + c*d - 2*a)*x)), x)
\[ \int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx=\int { -\frac {a b + a c - 2 \, b c - {\left (2 \, a - b - c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{3}} {\left (b c d + {\left (d - 1\right )} x^{2} - a^{2} - {\left (b d + c d - 2 \, a\right )} x\right )}} \,d x } \]
integrate((a*b+a*c-2*b*c+(-2*a+b+c)*x)/((-a+x)*(-b+x)*(-c+x))^(1/3)/(a^2-b *c*d+(b*d+c*d-2*a)*x+(1-d)*x^2),x, algorithm="giac")
integrate(-(a*b + a*c - 2*b*c - (2*a - b - c)*x)/((-(a - x)*(b - x)*(c - x ))^(1/3)*(b*c*d + (d - 1)*x^2 - a^2 - (b*d + c*d - 2*a)*x)), x)
Timed out. \[ \int \frac {a b+a c-2 b c+(-2 a+b+c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx=\int \frac {a\,b+a\,c-2\,b\,c+x\,\left (b-2\,a+c\right )}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{1/3}\,\left (x\,\left (b\,d-2\,a+c\,d\right )+a^2-x^2\,\left (d-1\right )-b\,c\,d\right )} \,d x \]
int((a*b + a*c - 2*b*c + x*(b - 2*a + c))/((-(a - x)*(b - x)*(c - x))^(1/3 )*(x*(b*d - 2*a + c*d) + a^2 - x^2*(d - 1) - b*c*d)),x)