Integrand size = 68, antiderivative size = 92 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt [4]{d} (a-x)}\right )}{\sqrt [4]{d}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right )}{\sqrt [4]{d}} \]
-arctan((a*b*x+(-a-b)*x^2+x^3)^(1/2)/d^(1/4)/(a-x))/d^(1/4)-arctanh(d^(1/4 )*(a*b*x+(-a-b)*x^2+x^3)^(1/2)/x/(-b+x))/d^(1/4)
Time = 15.84 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.72 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\frac {\arctan \left (\frac {x (-b+x)}{\sqrt [4]{d} \sqrt {x (-a+x) (-b+x)}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} (-a+x)}{\sqrt {x (-a+x) (-b+x)}}\right )}{\sqrt [4]{d}} \]
Integrate[(x*(-b + x)*(a*b - 2*a*x + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(-(a ^2*d) + 2*a*d*x + (b^2 - d)*x^2 - 2*b*x^3 + x^4)),x]
(ArcTan[(x*(-b + x))/(d^(1/4)*Sqrt[x*(-a + x)*(-b + x)])] - ArcTanh[(d^(1/ 4)*(-a + x))/Sqrt[x*(-a + x)*(-b + x)]])/d^(1/4)
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.
| \(\displaystyle \int \frac {x (x-b) \left (a b-2 a x+x^2\right )}{\sqrt {x (x-a) (x-b)} \left (-a^2 d+2 a d x+x^2 \left (b^2-d\right )-2 b x^3+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(b-x) \sqrt {x} \left (x^2-2 a x+a b\right )}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-\left (b^2-d\right ) x^2-2 a d x+a^2 d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(b-x) x \left (x^2-2 a x+a b\right )}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-\left (b^2-d\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{\sqrt {x^2-(a+b) x+a b}}-\frac {-\left ((2 a-b) x^3\right )+\left (-b^2+3 a b+d\right ) x^2-a \left (b^2+2 d\right ) x+a^2 d}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-\left (b^2-d\right ) x^2-2 a d x+a^2 d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (a^2 (-d) \int \frac {1}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-b^2 \left (1-\frac {d}{b^2}\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}+a \left (b^2+2 d\right ) \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-b^2 \left (1-\frac {d}{b^2}\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}-\left (3 a b-b^2+d\right ) \int \frac {x^2}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-b^2 \left (1-\frac {d}{b^2}\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}+(2 a-b) \int \frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-b^2 \left (1-\frac {d}{b^2}\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}+\frac {\left (\sqrt {a} \sqrt {b}+x\right ) \sqrt {\frac {-x (a+b)+a b+x^2}{\left (\sqrt {a} \sqrt {b}+x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-x (a+b)+a b+x^2}}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Int[(x*(-b + x)*(a*b - 2*a*x + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^2*d) + 2*a*d*x + (b^2 - d)*x^2 - 2*b*x^3 + x^4)),x]
3.13.68.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.37 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.88
| method | result | size |
| default | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3} a +\underline {\hspace {1.25 ex}}\alpha ^{3} b +3 \underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\underline {\hspace {1.25 ex}}\alpha a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d +a^{2} d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\) | \(357\) |
| elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a -\underline {\hspace {1.25 ex}}\alpha ^{3} b -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a b +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha a d -a^{2} d \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\) | \(357\) |
int(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2*d+2*a*d*x+(b^2- d)*x^2-2*b*x^3+x^4),x,method=_RETURNVERBOSE)
-2*b*(-(-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^ 2+x^3)^(1/2)*EllipticF((-(-b+x)/b)^(1/2),(b/(-a+b))^(1/2))-b/d*sum((-2*_al pha^3*a+_alpha^3*b+3*_alpha^2*a*b-_alpha^2*b^2-_alpha*a*b^2+_alpha^2*d-2*_ alpha*a*d+a^2*d)/(-2*_alpha^3+3*_alpha^2*b-_alpha*b^2+_alpha*d-a*d)*(_alph a^3-_alpha^2*b-_alpha*d+2*a*d-b*d)/(a^2-2*a*b+b^2)*(-(-b+x)/b)^(1/2)*((-a+ x)/(-a+b))^(1/2)*(x/b)^(1/2)/(x*(a*b-a*x-b*x+x^2))^(1/2)*EllipticPi((-(-b+ x)/b)^(1/2),-(_alpha^3-_alpha^2*b-_alpha*d+2*a*d-b*d)*b/d/(a^2-2*a*b+b^2), (b/(-a+b))^(1/2)),_alpha=RootOf(_Z^4-2*b*_Z^3+(b^2-d)*_Z^2+2*a*d*_Z-a^2*d) )
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 601, normalized size of antiderivative = 6.53 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=-\frac {\log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (\frac {a d - d x}{d^{\frac {1}{4}}} + \frac {b d x - d x^{2}}{d^{\frac {3}{4}}}\right )} - \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} + \frac {\log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (\frac {a d - d x}{d^{\frac {1}{4}}} + \frac {b d x - d x^{2}}{d^{\frac {3}{4}}}\right )} - \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (\frac {i \, a d - i \, d x}{d^{\frac {1}{4}}} + \frac {-i \, b d x + i \, d x^{2}}{d^{\frac {3}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {2 \, b x^{3} - x^{4} - a^{2} d + 2 \, a d x - {\left (b^{2} + d\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (\frac {-i \, a d + i \, d x}{d^{\frac {1}{4}}} + \frac {i \, b d x - i \, d x^{2}}{d^{\frac {3}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} \]
integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2*d+2*a*d*x +(b^2-d)*x^2-2*b*x^3+x^4),x, algorithm="fricas")
-1/4*log((2*b*x^3 - x^4 - a^2*d + 2*a*d*x - (b^2 + d)*x^2 + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*((a*d - d*x)/d^(1/4) + (b*d*x - d*x^2)/d^(3/4)) - 2*(a *b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2))/d^(1/4) + 1/4*log((2*b*x^3 - x^4 - a^2*d + 2*a*d*x - (b^ 2 + d)*x^2 - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*((a*d - d*x)/d^(1/4) + (b*d *x - d*x^2)/d^(3/4)) - 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(2*b*x ^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2))/d^(1/4) - 1/4*I*log((2*b*x^3 - x^4 - a^2*d + 2*a*d*x - (b^2 + d)*x^2 + 2*sqrt(a*b*x - (a + b)*x^2 + x^3 )*((I*a*d - I*d*x)/d^(1/4) + (-I*b*d*x + I*d*x^2)/d^(3/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2))/d^(1/4) + 1/4*I*log((2*b*x^3 - x^4 - a^2*d + 2*a*d*x - (b^2 + d)* x^2 + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*((-I*a*d + I*d*x)/d^(1/4) + (I*b*d *x - I*d*x^2)/d^(3/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(2*b *x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2))/d^(1/4)
Timed out. \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate(x*(-b+x)*(a*b-2*a*x+x**2)/(x*(-a+x)*(-b+x))**(1/2)/(-a**2*d+2*a* d*x+(b**2-d)*x**2-2*b*x**3+x**4),x)
\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int { \frac {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2*d+2*a*d*x +(b^2-d)*x^2-2*b*x^3+x^4),x, algorithm="maxima")
integrate((a*b - 2*a*x + x^2)*(b - x)*x/((2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2)*sqrt((a - x)*(b - x)*x)), x)
\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int { \frac {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2*d+2*a*d*x +(b^2-d)*x^2-2*b*x^3+x^4),x, algorithm="giac")
integrate((a*b - 2*a*x + x^2)*(b - x)*x/((2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2)*sqrt((a - x)*(b - x)*x)), x)
Time = 6.31 (sec) , antiderivative size = 705, normalized size of antiderivative = 7.66 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\left (\sum _{k=1}^4\left (-\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (-d\,a^2+a\,b^2\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-3\,a\,b\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2+2\,a\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^3+2\,d\,a\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )+b^2\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2-b\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^3-d\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2\right )}{\left (\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (2\,b^2\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-6\,b\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2+4\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^3-2\,d\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )+2\,a\,d\right )}\right )\right )-\frac {2\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
int((x*(b - x)*(a*b - 2*a*x + x^2))/((x*(a - x)*(b - x))^(1/2)*(x^2*(d - b ^2) + a^2*d + 2*b*x^3 - x^4 - 2*a*d*x)),x)
symsum(-(2*b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipti cPi(-b/(root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k) - b), asin(((b - x)/b)^(1/2)), -b/(a - b))*(2*a*root(z^4 - 2*b*z^3 - z^2*(d - b^ 2) + 2*a*d*z - a^2*d, z, k)^3 - b*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a *d*z - a^2*d, z, k)^3 - d*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a ^2*d, z, k)^2 - a^2*d + b^2*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k)^2 + 2*a*d*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2 *d, z, k) - 3*a*b*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k)^2 + a*b^2*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k)) )/((root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k) - b)*(x*(a - x)*(b - x))^(1/2)*(2*a*d - 2*d*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a *d*z - a^2*d, z, k) - 6*b*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a ^2*d, z, k)^2 + 2*b^2*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d , z, k) + 4*root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k)^3) ), k, 1, 4) - (2*b*ellipticF(asin(((b - x)/b)^(1/2)), -b/(a - b))*(x/b)^(1 /2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2))/(x^3 - x^2*(a + b) + a*b*x) ^(1/2)