Integrand size = 82, antiderivative size = 77 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}} \]
-arctan(d^(1/4)*x/(a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/4)-arctanh(d^(1/4)*x/ (a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/4)
Time = 9.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.73 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{d} x}{\sqrt {x (-a+x) (-b+x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt {x (-a+x) (-b+x)}}\right )}{\sqrt [4]{d}} \]
Integrate[((-a + x)*(-b + x)*(-(a*b) + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(a ^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x ^4)),x]
-((ArcTan[(d^(1/4)*x)/Sqrt[x*(-a + x)*(-b + x)]] + ArcTanh[(d^(1/4)*x)/Sqr t[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.
\(\displaystyle \int \frac {(x-a) (x-b) \left (x^2-a b\right )}{\sqrt {x (x-a) (x-b)} \left (x^2 \left (a^2+4 a b+b^2-d\right )+a^2 b^2-2 x^3 (a+b)-2 a b x (a+b)+x^4\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {(a-x) (b-x) \left (a b-x^2\right )}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(a-x) (b-x) \left (a b-x^2\right )}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(a-x) (b-x) \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {-\left ((a+b) x^3\right )+\left (a^2+4 b a+b^2-d\right ) x^2-3 a b (a+b) x+2 a^2 b^2}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}-\frac {1}{\sqrt {x^2-(a+b) x+a b}}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (2 a^2 b^2 \int \frac {1}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}-3 a b (a+b) \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}+\left (a^2+4 a b+b^2-d\right ) \int \frac {x^2}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}-(a+b) \int \frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}-\frac {\left (\sqrt {a} \sqrt {b}+x\right ) \sqrt {\frac {-x (a+b)+a b+x^2}{\left (\sqrt {a} \sqrt {b}+x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-x (a+b)+a b+x^2}}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Int[((-a + x)*(-b + x)*(-(a*b) + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x^4)),x ]
3.11.15.3.1 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]
Time = 1.54 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {-2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \,d^{\frac {1}{4}}}\right )+\ln \left (\frac {d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )}{2 d^{\frac {1}{4}}}\) | \(76\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \,d^{\frac {1}{4}}}\right )-\ln \left (\frac {d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )}{2 d^{\frac {1}{4}}}\) | \(78\) |
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 {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\left (-2 a -2 b \right ) \textit {\_Z}^{3}+\left (a^{2}+4 a b +b^{2}-d \right ) \textit {\_Z}^{2}+\left (-2 a^{2} b -2 a \,b^{2}\right ) \textit {\_Z} +a^{2} b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a -\underline {\hspace {1.25 ex}}\alpha ^{3} b +\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+4 \underline {\hspace {1.25 ex}}\alpha ^{2} a b +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+2 a^{2} b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} d \right ) \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 a b \underline {\hspace {1.25 ex}}\alpha +a^{2} b +\underline {\hspace {1.25 ex}}\alpha 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 {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 a b \underline {\hspace {1.25 ex}}\alpha +a^{2} b +\underline {\hspace {1.25 ex}}\alpha d +d b}{d b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-4 a b \underline {\hspace {1.25 ex}}\alpha -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+a^{2} b +a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b d}\) | \(427\) |
int((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b*(a+b)* x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x,method=_RETURNVERBOSE)
-1/2*(-2*arctan((x*(a-x)*(b-x))^(1/2)/x/d^(1/4))+ln((d^(1/4)*x+(x*(a-x)*(b -x))^(1/2))/(-d^(1/4)*x+(x*(a-x)*(b-x))^(1/2))))/d^(1/4)
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 731, normalized size of antiderivative = 9.49 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=-\frac {\log \left (\frac {a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} + d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d^{\frac {3}{4}} x + \frac {a b d - {\left (a + b\right )} 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}}}{a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right )}{4 \, d^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} + d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d^{\frac {3}{4}} x + \frac {a b d - {\left (a + b\right )} 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}}}{a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right )}{4 \, d^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} + d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (i \, d^{\frac {3}{4}} x + \frac {-i \, a b d + i \, {\left (a + b\right )} 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}}}{a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right )}{4 \, d^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} + d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (-i \, d^{\frac {3}{4}} x + \frac {i \, a b d - i \, {\left (a + b\right )} 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}}}{a^{2} b^{2} - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right )}{4 \, d^{\frac {1}{4}}} \]
integrate((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b* (a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="fricas")
-1/4*log((a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2* (a^2*b + a*b^2)*x + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(3/4)*x + (a*b*d - (a + b)*d*x + d*x^2)/d^(3/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt (d))/(a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a^2 *b + a*b^2)*x))/d^(1/4) + 1/4*log((a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2*(a^2*b + a*b^2)*x - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(3/4)*x + (a*b*d - (a + b)*d*x + d*x^2)/d^(3/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a* b + b^2 - d)*x^2 - 2*(a^2*b + a*b^2)*x))/d^(1/4) + 1/4*I*log((a^2*b^2 - 2* (a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2*(a^2*b + a*b^2)*x - 2* sqrt(a*b*x - (a + b)*x^2 + x^3)*(I*d^(3/4)*x + (-I*a*b*d + I*(a + 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))/(a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a^2*b + a*b^2)*x ))/d^(1/4) - 1/4*I*log((a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2*(a^2*b + a*b^2)*x - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(-I*d^ (3/4)*x + (I*a*b*d - I*(a + 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))/(a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a^2*b + a*b^2)*x))/d^(1/4)
Timed out. \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate((-a+x)*(-b+x)*(-a*b+x**2)/(x*(-a+x)*(-b+x))**(1/2)/(a**2*b**2-2* a*b*(a+b)*x+(a**2+4*a*b+b**2-d)*x**2-2*(a+b)*x**3+x**4),x)
\[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b* (a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="maxima")
-integrate((a*b - x^2)*(a - x)*(b - x)/((a^2*b^2 - 2*(a + b)*a*b*x - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2)*sqrt((a - x)*(b - x)*x)), x)
\[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b* (a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="giac")
integrate(-(a*b - x^2)*(a - x)*(b - x)/((a^2*b^2 - 2*(a + b)*a*b*x - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2)*sqrt((a - x)*(b - x)*x)), x)
Time = 6.89 (sec) , antiderivative size = 1150, normalized size of antiderivative = 14.94 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\left (\sum _{k=1}^4\left (-\frac {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-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (-2\,a^2\,b^2+a^2\,b\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )\,3-a^2\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2+3\,a\,b^2\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )-4\,a\,b\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2+a\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^3-b^2\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2+b\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^3+d\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2\right )}{\left (\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (a^2\,b-a^2\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )+a\,b^2-4\,a\,b\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )+3\,a\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2-b^2\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )+3\,b\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^2-2\,{\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )}^3+d\,\mathrm {root}\left (z^4-z^3\,\left (2\,a+2\,b\right )+z^2\,\left (-d+4\,a\,b+a^2+b^2\right )-2\,a\,b\,z\,\left (a+b\right )+a^2\,b^2,z,k\right )\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(-((a - x)*(b - x)*(a*b - x^2))/((x*(a - x)*(b - x))^(1/2)*(x^4 - 2*x^3 *(a + b) + a^2*b^2 + x^2*(4*a*b - d + a^2 + b^2) - 2*a*b*x*(a + b))),x)
symsum(-(b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticP i(-b/(root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z *(a + b) + a^2*b^2, z, k) - b), asin(((b - x)/b)^(1/2)), -b/(a - b))*(a*ro ot(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^3 + b*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^ 2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^3 + d*root(z^4 - z^3*(2*a + 2* b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^2 - a^2*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*( a + b) + a^2*b^2, z, k)^2 - b^2*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4* a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^2 - 2*a^2*b^2 - 4*a*b* root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^2 + 3*a*b^2*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4* a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k) + 3*b*a^2*root(z^4 - z ^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2 , z, k)))/((root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2 *a*b*z*(a + b) + a^2*b^2, z, k) - b)*(x*(a - x)*(b - x))^(1/2)*(3*a*root(z ^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a ^2*b^2, z, k)^2 - a^2*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k) + 3*b*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^2 -...