Integrand size = 81, antiderivative size = 76 \[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}} \]
arctan(x/d^(1/4)/(a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/4)-arctanh(x/d^(1/4)/( a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/4)
Time = 10.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [4]{d} \sqrt {x (-a+x) (-b+x)}}\right )-\text {arctanh}\left (\frac {x}{\sqrt [4]{d} \sqrt {x (-a+x) (-b+x)}}\right )}{\sqrt [4]{d}} \]
Integrate[(-(a*b*x) + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2*d - 2*a*b*( a + b)*d*x + (-1 + a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*x^4) ),x]
(ArcTan[x/(d^(1/4)*Sqrt[x*(-a + x)*(-b + x)])] - ArcTanh[x/(d^(1/4)*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.
| \(\displaystyle \int \frac {x^3-a b x}{\sqrt {x (x-a) (x-b)} \left (x^2 \left (a^2 d+4 a b d+b^2 d-1\right )+a^2 b^2 d-2 d x^3 (a+b)-2 a b d x (a+b)+d x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x \left (x^2-a b\right )}{\sqrt {x (x-a) (x-b)} \left (x^2 \left (a^2 d+4 a b d+b^2 d-1\right )+a^2 b^2 d-2 d x^3 (a+b)-2 a b d x (a+b)+d x^4\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {\sqrt {x} \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 (a+b) d x^3-\left (-d a^2-4 b d a-b^2 d+1\right ) x^2-2 a b (a+b) d x+a^2 b^2 d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {\sqrt {x} \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 (a+b) d x^3-\left (-d a^2-4 b d a-b^2 d+1\right ) x^2-2 a b (a+b) d x+a^2 b^2 d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 (a+b) d x^3-\left (-d a^2-4 b d a-b^2 d+1\right ) x^2-2 a b (a+b) d x+a^2 b^2 d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (-d x^4+2 a \left (\frac {b}{a}+1\right ) d x^3+\left (1-\left (a^2+4 b a+b^2\right ) d\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) d x-a^2 b^2 d\right )}+\frac {a b x}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 a \left (\frac {b}{a}+1\right ) d x^3-\left (1-\left (a^2+4 b a+b^2\right ) d\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) d x+a^2 b^2 d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (\int \frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (-d x^4+2 a \left (\frac {b}{a}+1\right ) d x^3+\left (1-\left (a^2+4 b a+b^2\right ) d\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) d x-a^2 b^2 d\right )}d\sqrt {x}+a b \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (d x^4-2 a \left (\frac {b}{a}+1\right ) d x^3-\left (1-\left (a^2+4 b a+b^2\right ) d\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) d x+a^2 b^2 d\right )}d\sqrt {x}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Int[(-(a*b*x) + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2*d - 2*a*b*(a + b) *d*x + (-1 + a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*x^4)),x]
3.11.3.3.1 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 = 0.72 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(-\frac {\left (\frac {1}{d}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-\left (\frac {1}{d}\right )^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )+2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right )\right )}{2}\) | \(84\) |
| pseudoelliptic | \(-\frac {\left (\frac {1}{d}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-\left (\frac {1}{d}\right )^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )+2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right )\right )}{2}\) | \(84\) |
| elliptic | \(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 d b \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +d \,b^{2}-1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-a b \right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a b d +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a b d +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +3 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha a b d -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +a^{2} b d +a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b}\) | \(303\) |
int((-a*b*x+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+(a^2*d +4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x,method=_RETURNVERBOSE)
-1/2*(1/d)^(1/4)*(ln(((1/d)^(1/4)*x+(x*(a-x)*(b-x))^(1/2))/(-(1/d)^(1/4)*x +(x*(a-x)*(b-x))^(1/2)))+2*arctan((x*(a-x)*(b-x))^(1/2)/x/(1/d)^(1/4)))
Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 787, normalized size of antiderivative = 10.36 \[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=-\frac {\log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d^{\frac {1}{4}} x + \frac {a b d - {\left (a + b\right )} d x + d x^{2}}{d^{\frac {1}{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} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d^{\frac {1}{4}} x + \frac {a b d - {\left (a + b\right )} d x + d x^{2}}{d^{\frac {1}{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} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (i \, d^{\frac {1}{4}} x + \frac {-i \, a b d + i \, {\left (a + b\right )} d x - i \, d x^{2}}{d^{\frac {1}{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} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (-i \, d^{\frac {1}{4}} x + \frac {i \, a b d - i \, {\left (a + b\right )} d x + i \, d x^{2}}{d^{\frac {1}{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} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} \]
integrate((-a*b*x+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+ (a^2*d+4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x, algorithm="fricas")
-1/4*log((a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + (( a^2 + 4*a*b + b^2)*d + 1)*x^2 + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(1/4) *x + (a*b*d - (a + b)*d*x + d*x^2)/d^(1/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)* d*x + ((a^2 + 4*a*b + b^2)*d - 1)*x^2))/d^(1/4) + 1/4*log((a^2*b^2*d - 2*( a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2)*d + 1) *x^2 - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(1/4)*x + (a*b*d - (a + b)*d*x + d*x^2)/d^(1/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2 *d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2 )*d - 1)*x^2))/d^(1/4) - 1/4*I*log((a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2)*d + 1)*x^2 - 2*sqrt(a*b*x - ( a + b)*x^2 + x^3)*(I*d^(1/4)*x + (-I*a*b*d + I*(a + b)*d*x - I*d*x^2)/d^(1 /4)) - 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2*d - 2*(a + b) *d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2)*d - 1)*x^2)) /d^(1/4) + 1/4*I*log((a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b ^2)*d*x + ((a^2 + 4*a*b + b^2)*d + 1)*x^2 - 2*sqrt(a*b*x - (a + b)*x^2 + x ^3)*(-I*d^(1/4)*x + (I*a*b*d - I*(a + b)*d*x + I*d*x^2)/d^(1/4)) - 2*(a*b* d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2*d - 2*(a + b)*d*x^3 + d*x^4 - 2*(a^2*b + a*b^2)*d*x + ((a^2 + 4*a*b + b^2)*d - 1)*x^2))/d^(1/4)
Timed out. \[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\text {Timed out} \]
integrate((-a*b*x+x**3)/(x*(-a+x)*(-b+x))**(1/2)/(a**2*b**2*d-2*a*b*(a+b)* d*x+(a**2*d+4*a*b*d+b**2*d-1)*x**2-2*(a+b)*d*x**3+d*x**4),x)
\[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\int { -\frac {a b x - x^{3}}{{\left (a^{2} b^{2} d - 2 \, {\left (a + b\right )} a b d x - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} + {\left (a^{2} d + 4 \, a b d + b^{2} d - 1\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((-a*b*x+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+ (a^2*d+4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x, algorithm="maxima")
-integrate((a*b*x - x^3)/((a^2*b^2*d - 2*(a + b)*a*b*d*x - 2*(a + b)*d*x^3 + d*x^4 + (a^2*d + 4*a*b*d + b^2*d - 1)*x^2)*sqrt((a - x)*(b - x)*x)), x)
\[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\int { -\frac {a b x - x^{3}}{{\left (a^{2} b^{2} d - 2 \, {\left (a + b\right )} a b d x - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} + {\left (a^{2} d + 4 \, a b d + b^{2} d - 1\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((-a*b*x+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2*d-2*a*b*(a+b)*d*x+ (a^2*d+4*a*b*d+b^2*d-1)*x^2-2*(a+b)*d*x^3+d*x^4),x, algorithm="giac")
integrate(-(a*b*x - x^3)/((a^2*b^2*d - 2*(a + b)*a*b*d*x - 2*(a + b)*d*x^3 + d*x^4 + (a^2*d + 4*a*b*d + b^2*d - 1)*x^2)*sqrt((a - x)*(b - x)*x)), x)
Time = 13.62 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.30 \[ \int \frac {-a b x+x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx=\frac {\ln \left (\frac {x-2\,d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}+\sqrt {d}\,x^2+a\,b\,\sqrt {d}-a\,\sqrt {d}\,x-b\,\sqrt {d}\,x}{x-\sqrt {d}\,x^2-a\,b\,\sqrt {d}+a\,\sqrt {d}\,x+b\,\sqrt {d}\,x}\right )}{2\,d^{1/4}}+\frac {\ln \left (\frac {x-\sqrt {d}\,x^2-a\,b\,\sqrt {d}+a\,\sqrt {d}\,x+b\,\sqrt {d}\,x+d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,2{}\mathrm {i}}{x+\sqrt {d}\,x^2+a\,b\,\sqrt {d}-a\,\sqrt {d}\,x-b\,\sqrt {d}\,x}\right )\,1{}\mathrm {i}}{2\,d^{1/4}} \]
int((x^3 - a*b*x)/((x*(a - x)*(b - x))^(1/2)*(x^2*(a^2*d + b^2*d + 4*a*b*d - 1) + d*x^4 + a^2*b^2*d - 2*d*x^3*(a + b) - 2*a*b*d*x*(a + b))),x)
log((x - 2*d^(1/4)*(x*(a - x)*(b - x))^(1/2) + d^(1/2)*x^2 + a*b*d^(1/2) - a*d^(1/2)*x - b*d^(1/2)*x)/(x - d^(1/2)*x^2 - a*b*d^(1/2) + a*d^(1/2)*x + b*d^(1/2)*x))/(2*d^(1/4)) + (log((x + d^(1/4)*(x*(a - x)*(b - x))^(1/2)*2 i - d^(1/2)*x^2 - a*b*d^(1/2) + a*d^(1/2)*x + b*d^(1/2)*x)/(x + d^(1/2)*x^ 2 + a*b*d^(1/2) - a*d^(1/2)*x - b*d^(1/2)*x))*1i)/(2*d^(1/4))