Integrand size = 75, antiderivative size = 94 \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right )}{d^{3/4}} \]
arctan(d^(1/4)*(a*b*x+(-a-b)*x^2+x^3)^(1/2)/x/(-b+x))/d^(3/4)-arctanh(d^(1 /4)*(a*b*x+(-a-b)*x^2+x^3)^(1/2)/x/(-b+x))/d^(3/4)
Time = 11.97 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.70 \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=-\frac {-\arctan \left (\frac {\sqrt [4]{d} (-a+x)}{\sqrt {x (-a+x) (-b+x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} (-a+x)}{\sqrt {x (-a+x) (-b+x)}}\right )}{d^{3/4}} \]
Integrate[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(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[(d^(1/4)*(-a + x))/Sqrt[x*(-a + x)*(-b + x)]] + ArcTanh[(d^(1/4 )*(-a + 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.
| \(\displaystyle \int \frac {-a^2 b+a x (2 a+b)-3 a x^2+x^3}{\sqrt {x (x-a) (x-b)} \left (-a^2 d+2 a d x+x^2 \left (b^2-d\right )-2 b x^3+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {-x^3+3 a x^2-a (2 a+b) x+a^2 b}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-\left (b^2-d\right ) x^2-2 a d x+a^2 d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {-x^3+3 a x^2-a (2 a+b) x+a^2 b}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-\left (b^2-d\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(a-x) \left (x^2-2 a x+a b\right )}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-\left (b^2-d\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 b x^3+b^2 \left (1-\frac {d}{b^2}\right ) x^2+2 a d x-a^2 d\right )}+\frac {3 a x^2}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-b^2 \left (1-\frac {d}{b^2}\right ) x^2-2 a d x+a^2 d\right )}+\frac {2 a^2 \left (\frac {b}{2 a}+1\right ) x}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 b x^3+b^2 \left (1-\frac {d}{b^2}\right ) x^2+2 a d x-a^2 d\right )}+\frac {a^2 b}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-b^2 \left (1-\frac {d}{b^2}\right ) x^2-2 a d x+a^2 d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (a^2 b \int \frac {1}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-b^2 \left (1-\frac {d}{b^2}\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}+3 a \int \frac {x^2}{\sqrt {x^2-(a+b) x+a b} \left (-x^4+2 b x^3-b^2 \left (1-\frac {d}{b^2}\right ) x^2-2 a d x+a^2 d\right )}d\sqrt {x}+a (2 a+b) \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 b x^3+b^2 \left (1-\frac {d}{b^2}\right ) x^2+2 a d x-a^2 d\right )}d\sqrt {x}+\int \frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 b x^3+b^2 \left (1-\frac {d}{b^2}\right ) x^2+2 a d x-a^2 d\right )}d\sqrt {x}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Int[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(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]
3.14.1.3.1 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 = 0.70 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.67
| method | result | size |
| default | \(\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-a b \underline {\hspace {1.25 ex}}\alpha +a^{2} b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\) | \(251\) |
| elliptic | \(\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (b^{2}-d \right ) \textit {\_Z}^{2}+2 a d \textit {\_Z} -a^{2} d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-a b \underline {\hspace {1.25 ex}}\alpha +a^{2} b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha d +2 a d -d b \right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d -a d \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\) | \(251\) |
int((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/(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)
b/d*sum((-_alpha^3+3*_alpha^2*a-2*_alpha*a^2-_alpha*a*b+a^2*b)/(-2*_alpha^ 3+3*_alpha^2*b-_alpha*b^2+_alpha*d-a*d)*(_alpha^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-_alph a^2*b-_alpha*d+2*a*d-b*d)*b/d/(a^2-2*a*b+b^2),(b/(-a+b))^(1/2)),_alpha=Roo tOf(_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 = 0.67 (sec) , antiderivative size = 673, normalized size of antiderivative = 7.16 \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\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 {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \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 ({\left (a d^{3} - d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (b d x - d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 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}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) + \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \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 ({\left (a d^{3} - d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (b d x - d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 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}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) + \frac {1}{4} i \, \frac {1}{d^{3}}^{\frac {1}{4}} \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 ({\left (i \, a d^{3} - i \, d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (-i \, b d x + i \, d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} + 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}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) - \frac {1}{4} i \, \frac {1}{d^{3}}^{\frac {1}{4}} \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 ({\left (-i \, a d^{3} + i \, d^{3} x\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (i \, b d x - i \, d x^{2}\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} + 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}}}}{2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}}\right ) \]
integrate((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/(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*(d^(-3))^(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^3 - d^3*x)*(d^(-3))^(3/4) + (b*d* x - d*x^2)*(d^(-3))^(1/4)) - 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqr t(d^(-3)))/(2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2)) + 1/4*(d^(-3 ))^(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^3 - d^3*x)*(d^(-3))^(3/4) + (b*d*x - d*x^2)* (d^(-3))^(1/4)) - 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/ (2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2)) + 1/4*I*(d^(-3))^(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)*((I*a*d^3 - I*d^3*x)*(d^(-3))^(3/4) + (-I*b*d*x + I*d*x^2)* (d^(-3))^(1/4)) + 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/ (2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2)) - 1/4*I*(d^(-3))^(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)*((-I*a*d^3 + I*d^3*x)*(d^(-3))^(3/4) + (I*b*d*x - I*d*x^2)* (d^(-3))^(1/4)) + 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/ (2*b*x^3 - x^4 + a^2*d - 2*a*d*x - (b^2 - d)*x^2))
Timed out. \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate((-a**2*b+a*(2*a+b)*x-3*a*x**2+x**3)/(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)
\[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int { \frac {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/(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^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/((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)
\[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\int { \frac {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{{\left (2 \, b x^{3} - x^{4} + a^{2} d - 2 \, a d x - {\left (b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/(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^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/((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 = 6.52 (sec) , antiderivative size = 453, normalized size of antiderivative = 4.82 \[ \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2 d+2 a d x+\left (b^2-d\right ) x^2-2 b x^3+x^4\right )} \, dx=\sum _{k=1}^4\frac {2\,b\,\left (\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-a\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left ({\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2-2\,a\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )+a\,b\right )}{\left (\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (2\,b^2\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )-6\,b\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^2+4\,{\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )}^3-2\,d\,\mathrm {root}\left (z^4-2\,b\,z^3-z^2\,\left (d-b^2\right )+2\,a\,d\,z-a^2\,d,z,k\right )+2\,a\,d\right )} \]
int((a^2*b + 3*a*x^2 - x^3 - a*x*(2*a + b))/((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*(root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k) - a)*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(-b/(r oot(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))*(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) + root(z^4 - 2*b*z^3 - z^2*(d - b^2) + 2*a*d*z - a^2*d, z, k)^2))/((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)