Integrand size = 87, antiderivative size = 59 \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt {d} x^2}\right )}{\sqrt {d}} \]
Time = 10.57 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x) (-c+x)}}{\sqrt {d} x^2}\right )}{\sqrt {d}} \]
Integrate[(3*a*b*c*x - 2*(a*b + a*c + b*c)*x^2 + (a + b + c)*x^3)/(Sqrt[x* (-a + x)*(-b + x)*(-c + x)]*(a*b*c - (a*b + a*c + b*c)*x + (a + b + c)*x^2 + (-1 + d)*x^3)),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.
| \(\displaystyle \int \frac {x^3 (a+b+c)-2 x^2 (a b+a c+b c)+3 a b c x}{\sqrt {x (x-a) (x-b) (x-c)} \left (x^2 (a+b+c)-x (a b+a c+b c)+a b c+(d-1) x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \int \frac {x \left (x^2 (a+b+c)-2 x (a b+a c+b c)+3 a b c\right )}{\sqrt {x (x-a) (x-b) (x-c)} \left (x^2 (a+b+c)-x (a b+a c+b c)+a b c+(d-1) x^3\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {\sqrt {x} \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c} \left (-\left ((1-d) x^3\right )+(a+b+c) x^2-(b c+a (b+c)) x+a b c\right )}dx}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {x \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c} \left (-\left ((1-d) x^3\right )+(a+b+c) x^2-(b c+a (b+c)) x+a b c\right )}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {x \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {-((a-x) (x-b) (x-c))} \left (-\left ((1-d) x^3\right )+(a+b+c) x^2-(b c+a (b+c)) x+a b c\right )}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {x \left ((a+b+c) x^2-2 (b c+a (b+c)) x+3 a b c\right )}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (-\left ((1-d) x^3\right )+(a+b+c) x^2-(b c+a (b+c)) x+a b c\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (-\frac {a+b+c}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}-\frac {\left (a^2+2 (b+c) d a+b^2+c^2+2 b c d\right ) x^2-\left ((b+c) a^2+\left (b^2+3 c d b+c^2\right ) a+b c (b+c)\right ) x+a b c (a+b+c)}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (-\left ((1-d) x^3\right )+(a+b+c) x^2-(b c+a (b+c)) x+a b c\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (a^2+2 (b+c) d a+b^2+c^2+2 b c d\right ) x^2-\left ((b+c) a^2+\left (b^2+3 c d b+c^2\right ) a+b c (b+c)\right ) x+a b c (a+b+c)}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (-\left ((1-d) x^3\right )+a \left (\frac {b+c}{a}+1\right ) x^2-a b \left (\frac {(a+b) c}{a b}+1\right ) x+a b c\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {a+b+c}{(d-1) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}+\frac {\left (a^2+2 (b+c) d a+b^2+c^2+2 b c d\right ) x^2-\left ((b+c) a^2+\left (b^2+3 c d b+c^2\right ) a+b c (b+c)\right ) x+a b c (a+b+c)}{(1-d) \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (-\left ((1-d) x^3\right )+a \left (\frac {b+c}{a}+1\right ) x^2-a b \left (\frac {(a+b) c}{a b}+1\right ) x+a b c\right )}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
Int[(3*a*b*c*x - 2*(a*b + a*c + b*c)*x^2 + (a + b + c)*x^3)/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(a*b*c - (a*b + a*c + b*c)*x + (a + b + c)*x^2 + (-1 + d)*x^3)),x]
3.8.56.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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]
Time = 5.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {-x \left (a -x \right ) \left (b -x \right ) \left (c -x \right )}}{x^{2} \sqrt {d}}\right )}{\sqrt {d}}\) | \(34\) |
| default | \(-\frac {2 \left (a +b +c \right ) c \sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}\, \left (-a +x \right )^{2} \sqrt {\frac {a \left (-b +x \right )}{b \left (-a +x \right )}}\, \sqrt {\frac {a \left (-c +x \right )}{c \left (-a +x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{\left (d -1\right ) \left (c -a \right ) a \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}+\frac {2 c \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\left (d -1\right ) \textit {\_Z}^{3}+\left (a +b +c \right ) \textit {\_Z}^{2}+\left (-a b -a c -b c \right ) \textit {\_Z} +a b c \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} a c d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b c d +3 \underline {\hspace {1.25 ex}}\alpha a b c d -\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} c^{2}+\underline {\hspace {1.25 ex}}\alpha \,a^{2} b +\underline {\hspace {1.25 ex}}\alpha \,a^{2} c +\underline {\hspace {1.25 ex}}\alpha a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha a \,c^{2}+\underline {\hspace {1.25 ex}}\alpha \,b^{2} c +\underline {\hspace {1.25 ex}}\alpha b \,c^{2}-a^{2} b c -a \,b^{2} c -a b \,c^{2}\right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d +a^{2} d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b +\underline {\hspace {1.25 ex}}\alpha c -b c \right ) \sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}\, \sqrt {\frac {a \left (-b +x \right )}{b \left (-a +x \right )}}\, \sqrt {\frac {a \left (-c +x \right )}{c \left (-a +x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )-\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +a b +a c +b c \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \frac {-\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +a b +a c}{b \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{b c}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +3 \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha a -2 \underline {\hspace {1.25 ex}}\alpha b -2 \underline {\hspace {1.25 ex}}\alpha c +a b +a c +b c \right ) \left (c -a \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{\left (d -1\right ) a^{4} d}\) | \(625\) |
| elliptic | \(-\frac {2 \left (a +b +c \right ) c \sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}\, \left (-a +x \right )^{2} \sqrt {\frac {a \left (-b +x \right )}{b \left (-a +x \right )}}\, \sqrt {\frac {a \left (-c +x \right )}{c \left (-a +x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{\left (d -1\right ) \left (c -a \right ) a \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}+\frac {2 c \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\left (d -1\right ) \textit {\_Z}^{3}+\left (a +b +c \right ) \textit {\_Z}^{2}+\left (-a b -a c -b c \right ) \textit {\_Z} +a b c \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} a c d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b c d -3 \underline {\hspace {1.25 ex}}\alpha a b c d +\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} c^{2}-\underline {\hspace {1.25 ex}}\alpha \,a^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} c -\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-\underline {\hspace {1.25 ex}}\alpha a \,c^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2} c -\underline {\hspace {1.25 ex}}\alpha b \,c^{2}+a^{2} b c +a \,b^{2} c +a b \,c^{2}\right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d +a^{2} d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b +\underline {\hspace {1.25 ex}}\alpha c -b c \right ) \sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}\, \sqrt {\frac {a \left (-b +x \right )}{b \left (-a +x \right )}}\, \sqrt {\frac {a \left (-c +x \right )}{c \left (-a +x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )-\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +a b +a c +b c \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \frac {-\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +a b +a c}{b \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{b c}\right )}{\left (d -1\right ) \left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha b +2 \underline {\hspace {1.25 ex}}\alpha c -a b -a c -b c \right ) \left (c -a \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{a^{4} d}\) | \(628\) |
int((3*x*a*b*c-2*(a*b+a*c+b*c)*x^2+(a+b+c)*x^3)/(x*(-a+x)*(-b+x)*(-c+x))^( 1/2)/(a*b*c-(a*b+a*c+b*c)*x+(a+b+c)*x^2+(d-1)*x^3),x,method=_RETURNVERBOSE )
Timed out. \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=\text {Timed out} \]
integrate((3*a*b*c*x-2*(a*b+a*c+b*c)*x^2+(a+b+c)*x^3)/(x*(-a+x)*(-b+x)*(-c +x))^(1/2)/(a*b*c-(a*b+a*c+b*c)*x+(a+b+c)*x^2+(-1+d)*x^3),x, algorithm="fr icas")
Timed out. \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=\text {Timed out} \]
integrate((3*a*b*c*x-2*(a*b+a*c+b*c)*x**2+(a+b+c)*x**3)/(x*(-a+x)*(-b+x)*( -c+x))**(1/2)/(a*b*c-(a*b+a*c+b*c)*x+(a+b+c)*x**2+(-1+d)*x**3),x)
\[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=\int { \frac {3 \, a b c x + {\left (a + b + c\right )} x^{3} - 2 \, {\left (a b + a c + b c\right )} x^{2}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )} x} {\left ({\left (d - 1\right )} x^{3} + a b c + {\left (a + b + c\right )} x^{2} - {\left (a b + a c + b c\right )} x\right )}} \,d x } \]
integrate((3*a*b*c*x-2*(a*b+a*c+b*c)*x^2+(a+b+c)*x^3)/(x*(-a+x)*(-b+x)*(-c +x))^(1/2)/(a*b*c-(a*b+a*c+b*c)*x+(a+b+c)*x^2+(-1+d)*x^3),x, algorithm="ma xima")
integrate((3*a*b*c*x + (a + b + c)*x^3 - 2*(a*b + a*c + b*c)*x^2)/(sqrt(-( a - x)*(b - x)*(c - x)*x)*((d - 1)*x^3 + a*b*c + (a + b + c)*x^2 - (a*b + a*c + b*c)*x)), x)
Time = 0.55 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=-\frac {2 \, \arctan \left (\frac {\sqrt {-\frac {a b c}{x^{3}} + \frac {a b}{x^{2}} + \frac {a c}{x^{2}} + \frac {b c}{x^{2}} - \frac {a}{x} - \frac {b}{x} - \frac {c}{x} + 1}}{\sqrt {-d}}\right )}{\sqrt {-d}} \]
integrate((3*a*b*c*x-2*(a*b+a*c+b*c)*x^2+(a+b+c)*x^3)/(x*(-a+x)*(-b+x)*(-c +x))^(1/2)/(a*b*c-(a*b+a*c+b*c)*x+(a+b+c)*x^2+(-1+d)*x^3),x, algorithm="gi ac")
-2*arctan(sqrt(-a*b*c/x^3 + a*b/x^2 + a*c/x^2 + b*c/x^2 - a/x - b/x - c/x + 1)/sqrt(-d))/sqrt(-d)
Timed out. \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=\int \frac {x^3\,\left (a+b+c\right )-2\,x^2\,\left (a\,b+a\,c+b\,c\right )+3\,a\,b\,c\,x}{\left (\left (d-1\right )\,x^3+\left (a+b+c\right )\,x^2+\left (-a\,b-a\,c-b\,c\right )\,x+a\,b\,c\right )\,\sqrt {-x\,\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}} \,d x \]
int((x^3*(a + b + c) - 2*x^2*(a*b + a*c + b*c) + 3*a*b*c*x)/((x^2*(a + b + c) - x*(a*b + a*c + b*c) + x^3*(d - 1) + a*b*c)*(-x*(a - x)*(b - x)*(c - x))^(1/2)),x)