Integrand size = 77, antiderivative size = 63 \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{a-x}\right )}{\sqrt {d}} \]
Time = 10.52 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x (-a+x) (-b+x) (-c+x)}}{a-x}\right )}{\sqrt {d}} \]
Integrate[(-(a*b*c) + 2*a*(b + c)*x - (3*a + b + c)*x^2 + 2*x^3)/(Sqrt[x*( -a + x)*(-b + x)*(-c + x)]*(a + (-1 + b*c*d)*x - (b + c)*d*x^2 + 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.
| \(\displaystyle \int \frac {-x^2 (3 a+b+c)+2 a x (b+c)-a b c+2 x^3}{\sqrt {x (x-a) (x-b) (x-c)} \left (a-d x^2 (b+c)+x (b c d-1)+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int -\frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {x} \sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c} \left (d x^3-(b+c) d x^2-(1-b c d) x+a\right )}dx}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {x} \sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c} \left (d x^3-(b+c) d x^2-(1-b c d) x+a\right )}dx}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c} \left (d x^3-(b+c) d x^2-(1-b c d) x+a\right )}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {-((a-x) (x-b) (x-c))} \left (d x^3-(b+c) d x^2-(1-b c d) x+a\right )}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (d x^3-(b+c) d x^2-(1-b c d) x+a\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(3 a-b-c) d x^2-2 (-b c d+a (b+c) d+1) x+a (b c d+2)}{d \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (d x^3-(b+c) d x^2-(1-b c d) x+a\right )}-\frac {2}{d \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(3 a-b-c) d x^2-2 (-b c d+a (b+c) d+1) x+a (b c d+2)}{d \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (d x^3-b \left (\frac {c}{b}+1\right ) d x^2-(1-b c d) x+a\right )}-\frac {2}{d \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(3 a-b-c) d x^2-2 (-b c d+a (b+c) d+1) x+a (b c d+2)}{d \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (d x^3-b \left (\frac {c}{b}+1\right ) d x^2-(1-b c d) x+a\right )}-\frac {2}{d \sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
Int[(-(a*b*c) + 2*a*(b + c)*x - (3*a + b + c)*x^2 + 2*x^3)/(Sqrt[x*(-a + x )*(-b + x)*(-c + x)]*(a + (-1 + b*c*d)*x - (b + c)*d*x^2 + d*x^3)),x]
3.9.30.3.1 Defintions of rubi rules used
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 8.15 (sec) , antiderivative size = 531, normalized size of antiderivative = 8.43
| method | result | size |
| default | \(-\frac {4 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 )}{d \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 (d \,\textit {\_Z}^{3}+\left (-d b -c d \right ) \textit {\_Z}^{2}+\left (b c d -1\right ) \textit {\_Z} +a \right )}{\sum }\frac {\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +\underline {\hspace {1.25 ex}}\alpha ^{2} b d +\underline {\hspace {1.25 ex}}\alpha ^{2} c d +2 \underline {\hspace {1.25 ex}}\alpha a b d +2 \underline {\hspace {1.25 ex}}\alpha a c d -2 \underline {\hspace {1.25 ex}}\alpha b c d -a b c d +2 \underline {\hspace {1.25 ex}}\alpha -2 a \right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +a^{2} d -a b d -c a d +b c d -1\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 )+\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +b c d -1\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, -\frac {d \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c \right ) c}{-c +a}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha b d +2 \underline {\hspace {1.25 ex}}\alpha c d -b c d +1\right ) \left (c -a \right ) \left (a^{2}-a b -a c +b c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{d^{2} a^{2}}\) | \(531\) |
| elliptic | \(-\frac {4 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 )}{d \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 (d \,\textit {\_Z}^{3}+\left (-d b -c d \right ) \textit {\_Z}^{2}+\left (b c d -1\right ) \textit {\_Z} +a \right )}{\sum }\frac {\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha ^{2} c d -2 \underline {\hspace {1.25 ex}}\alpha a b d -2 \underline {\hspace {1.25 ex}}\alpha a c d +2 \underline {\hspace {1.25 ex}}\alpha b c d +a b c d -2 \underline {\hspace {1.25 ex}}\alpha +2 a \right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +a^{2} d -a b d -c a d +b c d -1\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 )+\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +b c d -1\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, -\frac {d \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c \right ) c}{-c +a}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -2 \underline {\hspace {1.25 ex}}\alpha b d -2 \underline {\hspace {1.25 ex}}\alpha c d +b c d -1\right ) \left (c -a \right ) \left (a^{2}-a b -a c +b c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{d^{2} a^{2}}\) | \(531\) |
int((-a*b*c+2*a*(b+c)*x-(3*a+b+c)*x^2+2*x^3)/(x*(-a+x)*(-b+x)*(-c+x))^(1/2 )/(a+(b*c*d-1)*x-(b+c)*d*x^2+d*x^3),x,method=_RETURNVERBOSE)
-4/d*c*((c-a)*x/c/(-a+x))^(1/2)*(-a+x)^2*(a*(-b+x)/b/(-a+x))^(1/2)*(a*(-c+ x)/c/(-a+x))^(1/2)/(c-a)/a/(x*(-a+x)*(-b+x)*(-c+x))^(1/2)*EllipticF(((c-a) *x/c/(-a+x))^(1/2),((a-b)*c/b/(-c+a))^(1/2))+2/d^2*c/a^2*sum((-3*_alpha^2* a*d+_alpha^2*b*d+_alpha^2*c*d+2*_alpha*a*b*d+2*_alpha*a*c*d-2*_alpha*b*c*d -a*b*c*d+2*_alpha-2*a)/(-3*_alpha^2*d+2*_alpha*b*d+2*_alpha*c*d-b*c*d+1)*( -a+x)^2/(c-a)*(_alpha^2*d+_alpha*a*d-_alpha*b*d-_alpha*c*d+a^2*d-a*b*d-a*c *d+b*c*d-1)/(a^2-a*b-a*c+b*c)*((c-a)*x/c/(-a+x))^(1/2)*(a*(-b+x)/b/(-a+x)) ^(1/2)*(a*(-c+x)/c/(-a+x))^(1/2)/(x*(-a+x)*(-b+x)*(-c+x))^(1/2)*(EllipticF (((c-a)*x/c/(-a+x))^(1/2),((a-b)*c/b/(-c+a))^(1/2))+(_alpha^2*d-_alpha*b*d -_alpha*c*d+b*c*d-1)*EllipticPi(((c-a)*x/c/(-a+x))^(1/2),-d*(_alpha^2-_alp ha*b-_alpha*c+b*c)*c/(-c+a),((a-b)*c/b/(-c+a))^(1/2))),_alpha=RootOf(d*_Z^ 3+(-b*d-c*d)*_Z^2+(b*c*d-1)*_Z+a))
Timed out. \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((-a*b*c+2*a*(b+c)*x-(3*a+b+c)*x^2+2*x^3)/(x*(-a+x)*(-b+x)*(-c+x) )^(1/2)/(a+(b*c*d-1)*x-(b+c)*d*x^2+d*x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((-a*b*c+2*a*(b+c)*x-(3*a+b+c)*x**2+2*x**3)/(x*(-a+x)*(-b+x)*(-c+ x))**(1/2)/(a+(b*c*d-1)*x-(b+c)*d*x**2+d*x**3),x)
\[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=\int { \frac {a b c - 2 \, a {\left (b + c\right )} x + {\left (3 \, a + b + c\right )} x^{2} - 2 \, x^{3}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )} x} {\left ({\left (b + c\right )} d x^{2} - d x^{3} - {\left (b c d - 1\right )} x - a\right )}} \,d x } \]
integrate((-a*b*c+2*a*(b+c)*x-(3*a+b+c)*x^2+2*x^3)/(x*(-a+x)*(-b+x)*(-c+x) )^(1/2)/(a+(b*c*d-1)*x-(b+c)*d*x^2+d*x^3),x, algorithm="maxima")
integrate((a*b*c - 2*a*(b + c)*x + (3*a + b + c)*x^2 - 2*x^3)/(sqrt(-(a - x)*(b - x)*(c - x)*x)*((b + c)*d*x^2 - d*x^3 - (b*c*d - 1)*x - a)), x)
\[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=\int { \frac {a b c - 2 \, a {\left (b + c\right )} x + {\left (3 \, a + b + c\right )} x^{2} - 2 \, x^{3}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )} x} {\left ({\left (b + c\right )} d x^{2} - d x^{3} - {\left (b c d - 1\right )} x - a\right )}} \,d x } \]
integrate((-a*b*c+2*a*(b+c)*x-(3*a+b+c)*x^2+2*x^3)/(x*(-a+x)*(-b+x)*(-c+x) )^(1/2)/(a+(b*c*d-1)*x-(b+c)*d*x^2+d*x^3),x, algorithm="giac")
integrate((a*b*c - 2*a*(b + c)*x + (3*a + b + c)*x^2 - 2*x^3)/(sqrt(-(a - x)*(b - x)*(c - x)*x)*((b + c)*d*x^2 - d*x^3 - (b*c*d - 1)*x - a)), x)
Timed out. \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=-\int \frac {-2\,x^3+\left (3\,a+b+c\right )\,x^2-2\,a\,\left (b+c\right )\,x+a\,b\,c}{\sqrt {-x\,\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}\,\left (d\,x^3-d\,\left (b+c\right )\,x^2+\left (b\,c\,d-1\right )\,x+a\right )} \,d x \]
int(-(x^2*(3*a + b + c) - 2*x^3 - 2*a*x*(b + c) + a*b*c)/((-x*(a - x)*(b - x)*(c - x))^(1/2)*(a + d*x^3 + x*(b*c*d - 1) - d*x^2*(b + c))),x)