Integrand size = 85, antiderivative size = 296 \[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{a \sqrt [3]{d}-\sqrt [3]{d} x-2 \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{d^{2/3}}-\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )}{d^{2/3}}+\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]
-3^(1/2)*arctan((3^(1/2)*a*d^(1/3)-3^(1/2)*d^(1/3)*x)/(a*d^(1/3)-d^(1/3)*x -2*(-a*b*c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/3)))/d^(2/3)-ln(a*d^(1/3)- d^(1/3)*x+(-a*b*c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/3))/d^(2/3)+1/2*ln( a^2*d^(2/3)-2*a*d^(2/3)*x+d^(2/3)*x^2+(-a*d^(1/3)+d^(1/3)*x)*(-a*b*c+(a*b+ a*c+b*c)*x+(-a-b-c)*x^2+x^3)^(1/3)+(-a*b*c+(a*b+a*c+b*c)*x+(-a-b-c)*x^2+x^ 3)^(2/3))/d^(2/3)
\[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx \]
Integrate[(-(a*(a*b + a*c - 2*b*c)) + 2*(a^2 - b*c)*x + (-2*a + b + c)*x^2 )/(((-a + x)*(-b + x)*(-c + x))^(2/3)*(-(b*c) + a^2*d + (b + c - 2*a*d)*x + (-1 + d)*x^2)),x]
Integrate[(-(a*(a*b + a*c - 2*b*c)) + 2*(a^2 - b*c)*x + (-2*a + b + c)*x^2 )/(((-a + x)*(-b + x)*(-c + x))^(2/3)*(-(b*c) + a^2*d + (b + c - 2*a*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.
| \(\displaystyle \int \frac {2 x \left (a^2-b c\right )+x^2 (-2 a+b+c)-a (a b+a c-2 b c)}{((x-a) (x-b) (x-c))^{2/3} \left (a^2 d+x (-2 a d+b+c)-b c+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle \frac {(x-a)^{2/3} (x-b)^{2/3} (x-c)^{2/3} \int -\frac {-\left ((2 a-b-c) x^2\right )+2 \left (a^2-b c\right ) x+a (2 b c-a (b+c))}{(x-a)^{2/3} (x-b)^{2/3} (x-c)^{2/3} \left (-d a^2+(1-d) x^2+b c-(b+c-2 a d) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{2/3} (x-c)^{2/3} \int \frac {-\left ((2 a-b-c) x^2\right )+2 \left (a^2-b c\right ) x+a (2 b c-a (b+c))}{(x-a)^{2/3} (x-b)^{2/3} (x-c)^{2/3} \left (-d a^2+(1-d) x^2+b c-(b+c-2 a d) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{2/3}}\) |
| \(\Big \downarrow \) 2004 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{2/3} (x-c)^{2/3} \int \frac {\sqrt [3]{x-a} (-2 b c+a (b+c)+(-2 a+b+c) x)}{(x-b)^{2/3} (x-c)^{2/3} \left (-d a^2+(1-d) x^2+b c-(b+c-2 a d) x\right )}dx}{(-((a-x) (b-x) (c-x)))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{2/3} (x-c)^{2/3} \int \left (\frac {\sqrt [3]{x-a} \left (-2 a+b+c-\sqrt {4 d a^2-4 b d a-4 c d a+b^2+c^2-2 b c+4 b c d}\right )}{(x-b)^{2/3} (x-c)^{2/3} \left (-b-c+2 a d+2 (1-d) x+\sqrt {4 d a^2-4 b d a-4 c d a+b^2+c^2-2 b c+4 b c d}\right )}+\frac {\left (-2 a+b+c+\sqrt {4 d a^2-4 b d a-4 c d a+b^2+c^2-2 b c+4 b c d}\right ) \sqrt [3]{x-a}}{(x-b)^{2/3} (x-c)^{2/3} \left (-b-c+2 a d+2 (1-d) x-\sqrt {4 d a^2-4 b d a-4 c d a+b^2+c^2-2 b c+4 b c d}\right )}\right )dx}{(-((a-x) (b-x) (c-x)))^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(x-a)^{2/3} (x-b)^{2/3} (x-c)^{2/3} \left (-\left (\left (-\sqrt {4 a^2 d-2 b (2 a d-2 c d+c)-4 a c d+b^2+c^2}+2 a-b-c\right ) \int \frac {\sqrt [3]{x-a}}{(x-b)^{2/3} (x-c)^{2/3} \left (-b-c+2 a d+2 (1-d) x-\sqrt {4 d a^2-4 b d a-4 c d a+b^2+c^2-2 b c+4 b c d}\right )}dx\right )-\left (\sqrt {4 a^2 d-2 b (2 a d-2 c d+c)-4 a c d+b^2+c^2}+2 a-b-c\right ) \int \frac {\sqrt [3]{x-a}}{(x-b)^{2/3} (x-c)^{2/3} \left (-b-c+2 a d+2 (1-d) x+\sqrt {4 d a^2-4 b d a-4 c d a+b^2+c^2-2 b c+4 b c d}\right )}dx\right )}{(-((a-x) (b-x) (c-x)))^{2/3}}\) |
Int[(-(a*(a*b + a*c - 2*b*c)) + 2*(a^2 - b*c)*x + (-2*a + b + c)*x^2)/(((- a + x)*(-b + x)*(-c + x))^(2/3)*(-(b*c) + a^2*d + (b + c - 2*a*d)*x + (-1 + d)*x^2)),x]
3.29.53.3.1 Defintions of rubi rules used
Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) , x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b , c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
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]
\[\int \frac {-a \left (a b +a c -2 b c \right )+2 \left (a^{2}-b c \right ) x +\left (-2 a +b +c \right ) x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {2}{3}} \left (-b c +a^{2} d +\left (-2 a d +b +c \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
int((-a*(a*b+a*c-2*b*c)+2*(a^2-b*c)*x+(-2*a+b+c)*x^2)/((-a+x)*(-b+x)*(-c+x ))^(2/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x)
int((-a*(a*b+a*c-2*b*c)+2*(a^2-b*c)*x+(-2*a+b+c)*x^2)/((-a+x)*(-b+x)*(-c+x ))^(2/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x)
Timed out. \[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-a*(a*b+a*c-2*b*c)+2*(a^2-b*c)*x+(-2*a+b+c)*x^2)/((-a+x)*(-b+x) *(-c+x))^(2/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x, algorithm="fricas ")
Timed out. \[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
integrate((-a*(a*b+a*c-2*b*c)+2*(a**2-b*c)*x+(-2*a+b+c)*x**2)/((-a+x)*(-b+ x)*(-c+x))**(2/3)/(-b*c+a**2*d+(-2*a*d+b+c)*x+(-1+d)*x**2),x)
\[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (2 \, a - b - c\right )} x^{2} + {\left (a b + a c - 2 \, b c\right )} a - 2 \, {\left (a^{2} - b c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {2}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b c - {\left (2 \, a d - b - c\right )} x\right )}} \,d x } \]
integrate((-a*(a*b+a*c-2*b*c)+2*(a^2-b*c)*x+(-2*a+b+c)*x^2)/((-a+x)*(-b+x) *(-c+x))^(2/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x, algorithm="maxima ")
-integrate(((2*a - b - c)*x^2 + (a*b + a*c - 2*b*c)*a - 2*(a^2 - b*c)*x)/( (-(a - x)*(b - x)*(c - x))^(2/3)*(a^2*d + (d - 1)*x^2 - b*c - (2*a*d - b - c)*x)), x)
\[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (2 \, a - b - c\right )} x^{2} + {\left (a b + a c - 2 \, b c\right )} a - 2 \, {\left (a^{2} - b c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {2}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b c - {\left (2 \, a d - b - c\right )} x\right )}} \,d x } \]
integrate((-a*(a*b+a*c-2*b*c)+2*(a^2-b*c)*x+(-2*a+b+c)*x^2)/((-a+x)*(-b+x) *(-c+x))^(2/3)/(-b*c+a^2*d+(-2*a*d+b+c)*x+(-1+d)*x^2),x, algorithm="giac")
integrate(-((2*a - b - c)*x^2 + (a*b + a*c - 2*b*c)*a - 2*(a^2 - b*c)*x)/( (-(a - x)*(b - x)*(c - x))^(2/3)*(a^2*d + (d - 1)*x^2 - b*c - (2*a*d - b - c)*x)), x)
Timed out. \[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=-\int \frac {2\,x\,\left (b\,c-a^2\right )-x^2\,\left (b-2\,a+c\right )+a\,\left (a\,b+a\,c-2\,b\,c\right )}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{2/3}\,\left (x\,\left (b+c-2\,a\,d\right )-b\,c+a^2\,d+x^2\,\left (d-1\right )\right )} \,d x \]
int(-(2*x*(b*c - a^2) - x^2*(b - 2*a + c) + a*(a*b + a*c - 2*b*c))/((-(a - x)*(b - x)*(c - x))^(2/3)*(x*(b + c - 2*a*d) - b*c + a^2*d + x^2*(d - 1)) ),x)