Integrand size = 77, antiderivative size = 63 \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt {d} (a-x)}\right )}{\sqrt {d}} \]
Time = 10.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x) (-c+x)}}{\sqrt {d} (a-x)}\right )}{\sqrt {d}} \]
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*d + (b*c - d)*x - (b + c)*x^2 + 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.
| \(\displaystyle \int \frac {-x^2 (3 a+b+c)+2 a x (b+c)-a b c+2 x^3}{\sqrt {x (x-a) (x-b) (x-c)} \left (a d+x (b c-d)-x^2 (b+c)+x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int -\frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {x} \left (x^3-(b+c) x^2+(b c-d) x+a d\right ) \sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c}}dx}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {x} \left (x^3-(b+c) x^2+(b c-d) x+a d\right ) \sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c}}dx}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\left (x^3-(b+c) x^2+(b c-d) x+a d\right ) \sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c}}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {-((a-x) (x-b) (x-c))} \left (x^3-(b+c) x^2+(b c-d) x+a d\right )}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {-2 x^3+(3 a+b+c) x^2-2 a (b+c) x+a b c}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (x^3-(b+c) x^2+(b c-d) x+a d\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(3 a-b-c) x^2+2 (b c-a (b+c)-d) x+a (b c+2 d)}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (x^3-(b+c) x^2+(b c-d) x+a d\right )}-\frac {2}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(3 a-b-c) x^2+2 (b c-a (b+c)-d) x+a (b c+2 d)}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (x^3-b \left (\frac {c}{b}+1\right ) x^2+b c \left (1-\frac {d}{b c}\right ) x+a d\right )}-\frac {2}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(3 a-b-c) x^2+2 (b c-a (b+c)-d) x+a (b c+2 d)}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (x^3-b \left (\frac {c}{b}+1\right ) x^2+b c \left (1-\frac {d}{b c}\right ) x+a d\right )}-\frac {2}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\) |
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*d + (b*c - d)*x - (b + c)*x^2 + x^3)),x]
3.9.27.3.1 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 = 8.06 (sec) , antiderivative size = 509, normalized size of antiderivative = 8.08
| 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 )}{\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 (\textit {\_Z}^{3}+\left (-b -c \right ) \textit {\_Z}^{2}+\left (b c -d \right ) \textit {\_Z} +a d \right )}{\sum }\frac {\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b +\underline {\hspace {1.25 ex}}\alpha ^{2} c +2 \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 c -a b c +2 \underline {\hspace {1.25 ex}}\alpha d -2 a d \right ) \left (-a +x \right )^{2} \left (\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^{2}-a b -a c +b c -d \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}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c -d \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, -\frac {\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}{d \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{d}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha b +2 \underline {\hspace {1.25 ex}}\alpha c -b c +d \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 )}{a^{2}}\) | \(509\) |
| 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 )}{\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 (\textit {\_Z}^{3}+\left (-b -c \right ) \textit {\_Z}^{2}+\left (b c -d \right ) \textit {\_Z} +a d \right )}{\sum }\frac {\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha ^{2} c -2 \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 c +a b c -2 \underline {\hspace {1.25 ex}}\alpha d +2 a d \right ) \left (-a +x \right )^{2} \left (\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^{2}-a b -a c +b c -d \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}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c -d \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, -\frac {\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}{d \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{d}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha b +2 \underline {\hspace {1.25 ex}}\alpha c -b c +d \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 )}{a^{2}}\) | \(510\) |
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*d+(b*c-d)*x-(b+c)*x^2+x^3),x,method=_RETURNVERBOSE)
-4*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*c/a^2*sum((-3*_alpha^2*a+_alp ha^2*b+_alpha^2*c+2*_alpha*a*b+2*_alpha*a*c-2*_alpha*b*c-a*b*c+2*_alpha*d- 2*a*d)/(-3*_alpha^2+2*_alpha*b+2*_alpha*c-b*c+d)*(-a+x)^2/(c-a)*(_alpha^2+ _alpha*a-_alpha*b-_alpha*c+a^2-a*b-a*c+b*c-d)/(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-_alpha*b-_alpha*c+b*c-d)/d*EllipticPi(((c-a)*x/c/(-a +x))^(1/2),-(_alpha^2-_alpha*b-_alpha*c+b*c)*c/d/(-c+a),((a-b)*c/b/(-c+a)) ^(1/2))),_alpha=RootOf(_Z^3+(-b-c)*_Z^2+(b*c-d)*_Z+a*d))
Timed out. \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=\text {Timed out} \]
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*d+(b*c-d)*x-(b+c)*x^2+x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=\text {Timed out} \]
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*d+(b*c-d)*x-(b+c)*x**2+x**3),x)
\[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=\int { \frac {a b c - 2 \, a {\left (b + c\right )} x + {\left (3 \, a + b + c\right )} x^{2} - 2 \, x^{3}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )} x} {\left ({\left (b + c\right )} x^{2} - x^{3} - a d - {\left (b c - d\right )} x\right )}} \,d x } \]
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*d+(b*c-d)*x-(b+c)*x^2+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)*x^2 - x^3 - a*d - (b*c - d)*x)), x)
\[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=\int { \frac {a b c - 2 \, a {\left (b + c\right )} x + {\left (3 \, a + b + c\right )} x^{2} - 2 \, x^{3}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )} x} {\left ({\left (b + c\right )} x^{2} - x^{3} - a d - {\left (b c - d\right )} x\right )}} \,d x } \]
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*d+(b*c-d)*x-(b+c)*x^2+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)*x^2 - x^3 - a*d - (b*c - d)*x)), x)
Timed out. \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=-\int \frac {-2\,x^3+\left (3\,a+b+c\right )\,x^2-2\,a\,\left (b+c\right )\,x+a\,b\,c}{\left (x^3+\left (-b-c\right )\,x^2+\left (b\,c-d\right )\,x+a\,d\right )\,\sqrt {-x\,\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}} \,d x \]
int(-(x^2*(3*a + b + c) - 2*x^3 - 2*a*x*(b + c) + a*b*c)/((a*d - x^2*(b + c) - x*(d - b*c) + x^3)*(-x*(a - x)*(b - x)*(c - x))^(1/2)),x)