Integrand size = 73, antiderivative size = 76 \[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}} \]
arctan(d^(1/4)*x/(a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(3/4)-arctanh(d^(1/4)*x/( a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(3/4)
Time = 12.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d} x}{\sqrt {x (-a+x) (-b+x)}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt {x (-a+x) (-b+x)}}\right )}{d^{3/4}} \]
Integrate[(-(a*b*x) + x^3)/(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)/Sqrt[ x*(-a + x)*(-b + x)]])/d^(3/4)
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.
| \(\displaystyle \int \frac {x^3-a b x}{\sqrt {x (x-a) (x-b)} \left (x^2 \left (a^2+4 a b+b^2-d\right )+a^2 b^2-2 x^3 (a+b)-2 a b x (a+b)+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x \left (x^2-a b\right )}{\sqrt {x (x-a) (x-b)} \left (x^2 \left (a^2+4 a b+b^2-d\right )+a^2 b^2-2 x^3 (a+b)-2 a b x (a+b)+x^4\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {\sqrt {x} \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {\sqrt {x} \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 a \left (\frac {b}{a}+1\right ) x^3-a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) x-a^2 b^2\right )}+\frac {a b x}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (\int \frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 a \left (\frac {b}{a}+1\right ) x^3-a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) x-a^2 b^2\right )}d\sqrt {x}+a b \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Int[(-(a*b*x) + x^3)/(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]
3.10.100.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[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]
Time = 3.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {\ln \left (\frac {d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )+2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \,d^{\frac {1}{4}}}\right )}{2 d^{\frac {3}{4}}}\) | \(76\) |
| pseudoelliptic | \(-\frac {\ln \left (\frac {d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )+2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \,d^{\frac {1}{4}}}\right )}{2 d^{\frac {3}{4}}}\) | \(76\) |
| elliptic | \(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\left (-2 a -2 b \right ) \textit {\_Z}^{3}+\left (a^{2}+4 a b +b^{2}-d \right ) \textit {\_Z}^{2}+\left (-2 a^{2} b -2 a \,b^{2}\right ) \textit {\_Z} +a^{2} b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-a b \right ) \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 a b \underline {\hspace {1.25 ex}}\alpha +a^{2} b +\underline {\hspace {1.25 ex}}\alpha d +d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 a b \underline {\hspace {1.25 ex}}\alpha +a^{2} b +\underline {\hspace {1.25 ex}}\alpha d +d b}{d b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-4 a b \underline {\hspace {1.25 ex}}\alpha -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+a^{2} b +a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b d}\) | \(289\) |
int((-a*b*x+x^3)/(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/d^(3/4)*(ln((d^(1/4)*x+(x*(a-x)*(b-x))^(1/2))/(-d^(1/4)*x+(x*(a-x)*(b -x))^(1/2)))+2*arctan((x*(a-x)*(b-x))^(1/2)/x/d^(1/4)))
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 799, normalized size of antiderivative = 10.51 \[ \int \frac {-a b x+x^3}{\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 {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \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 \, {\left (d^{3} \frac {1}{d^{3}}^{\frac {3}{4}} x + {\left (a b d - {\left (a + b\right )} d x + d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} + 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{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 ) + \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \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 \, {\left (d^{3} \frac {1}{d^{3}}^{\frac {3}{4}} x + {\left (a b d - {\left (a + b\right )} d x + d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} + 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{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 ) - \frac {1}{4} i \, \frac {1}{d^{3}}^{\frac {1}{4}} \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 \, {\left (i \, d^{3} \frac {1}{d^{3}}^{\frac {3}{4}} x + {\left (-i \, a b d + i \, {\left (a + b\right )} d x - i \, d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{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 ) + \frac {1}{4} i \, \frac {1}{d^{3}}^{\frac {1}{4}} \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 \, {\left (-i \, d^{3} \frac {1}{d^{3}}^{\frac {3}{4}} x + {\left (i \, a b d - i \, {\left (a + b\right )} d x + i \, d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{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 ) \]
integrate((-a*b*x+x^3)/(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*(d^(-3))^(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*(d^3*(d^(-3))^(3/4)*x + (a*b*d - (a + b)*d*x + d*x^2)*(d^(-3))^(1/4))*sqrt(a*b*x - (a + b)*x^2 + x^3) + 2*(a*b* d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(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)) + 1/4*(d^(-3)) ^(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*(d^3*(d^(-3))^(3/4)*x + (a*b*d - (a + b)*d*x + d*x ^2)*(d^(-3))^(1/4))*sqrt(a*b*x - (a + b)*x^2 + x^3) + 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(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)) - 1/4*I*(d^(-3))^(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*(I*d^3*(d^(-3))^(3/4)*x + (-I*a*b*d + I*(a + b)*d*x - I*d*x^2 )*(d^(-3))^(1/4))*sqrt(a*b*x - (a + b)*x^2 + x^3) - 2*(a*b*d^2*x - (a + b) *d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(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)) + 1/4*I*(d^(-3))^(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*(-I*d^3*(d^(-3))^(3/4)*x + (I*a*b*d - I*(a + b)*d*x + I*d*x^2)* (d^(-3))^(1/4))*sqrt(a*b*x - (a + b)*x^2 + x^3) - 2*(a*b*d^2*x - (a + b)*d ^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(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))
Timed out. \[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate((-a*b*x+x**3)/(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)
\[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {a b x - x^{3}}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((-a*b*x+x^3)/(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 - x^3)/((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)
\[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {a b x - x^{3}}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((-a*b*x+x^3)/(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 - x^3)/((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 = 7.32 (sec) , antiderivative size = 714, normalized size of antiderivative = 9.39 \[ \int \frac {-a b x+x^3}{\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=\sum _{k=1}^4\left (-\frac {b\,\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 )}^3-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 )\right )\,\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 (\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 ) \]
int((x^3 - a*b*x)/((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*(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 - 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))*(x/b)^(1/2)*( (b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(-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)))/((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 - 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) + a*b^2 + a^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)^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) - 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))), k, 1, 4)