Integrand size = 84, antiderivative size = 44 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \]
Time = 10.80 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x)}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \]
Integrate[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)* (-b + x)]*(-(a^3*d) + (b + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]
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.
| \(\displaystyle \int \frac {a^2 b-x^2 (2 b-a)-a x (2 a-b)+x^3}{\sqrt {x (x-a) (x-b)} \left (a^3 (-d)+x \left (3 a^2 d+b\right )-x^2 (3 a d+1)+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {x^3+(a-2 b) x^2-a (2 a-b) x+a^2 b}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (d a^3-d x^3+(3 a d+1) x^2-\left (3 d a^2+b\right ) x\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^3+(a-2 b) x^2-a (2 a-b) x+a^2 b}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (d a^3-d x^3+(3 a d+1) x^2-\left (3 d a^2+b\right ) x\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^3+(a-2 b) x^2-a (2 a-b) x+a^2 b}{\sqrt {x^2-(a+b) x+a b} \left (d a^3-d x^3+(3 a d+1) x^2-\left (3 d a^2+b\right ) x\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(a-x) \left (-x^2-2 (a-b) x+a b\right )}{\sqrt {x^2-(a+b) x+a b} \left (d a^3-d x^3+(3 a d+1) x^2-\left (3 d a^2+b\right ) x\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {(a+b) d a^2+(4 a d-2 b d+1) x^2-\left (5 d a^2+b (1-a d)\right ) x}{d \sqrt {x^2-(a+b) x+a b} \left (d a^3-d x^3+(3 a d+1) x^2-\left (3 d a^2+b\right ) x\right )}-\frac {1}{d \sqrt {x^2-(a+b) x+a b}}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (a^2 (a+b) \int \frac {1}{\sqrt {x^2-(a+b) x+a b} \left (d a^3-d x^3+(3 a d+1) x^2-b \left (\frac {3 d a^2}{b}+1\right ) x\right )}d\sqrt {x}-\frac {\left (5 a^2 d-a b d+b\right ) \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (d a^3-d x^3+(3 a d+1) x^2-b \left (\frac {3 d a^2}{b}+1\right ) x\right )}d\sqrt {x}}{d}+\frac {(4 a d-2 b d+1) \int \frac {x^2}{\sqrt {x^2-(a+b) x+a b} \left (d a^3-d x^3+(3 a d+1) x^2-b \left (\frac {3 d a^2}{b}+1\right ) x\right )}d\sqrt {x}}{d}-\frac {\left (\sqrt {a} \sqrt {b}+x\right ) \sqrt {\frac {-x (a+b)+a b+x^2}{\left (\sqrt {a} \sqrt {b}+x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt {-x (a+b)+a b+x^2}}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Int[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^3*d) + (b + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]
3.6.70.3.1 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 = 1.14 (sec) , antiderivative size = 399, normalized size of antiderivative = 9.07
| method | result | size |
| default | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b \right ) \textit {\_Z} -d \,a^{3}\right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d +\underline {\hspace {1.25 ex}}\alpha a b d +d \,a^{3}+a^{2} b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha \right ) b}{d \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +6 \underline {\hspace {1.25 ex}}\alpha a d -3 a^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -b \right ) \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2}}\) | \(399\) |
| elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b \right ) \textit {\_Z} -d \,a^{3}\right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d -\underline {\hspace {1.25 ex}}\alpha a b d -d \,a^{3}-a^{2} b d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha \right ) b}{d \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -6 \underline {\hspace {1.25 ex}}\alpha a d +3 a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha +b \right ) \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2}}\) | \(401\) |
int((b*a^2-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-d*a^3+( 3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x^3),x,method=_RETURNVERBOSE)
-2/d*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))-2/d^2*b*sum(( 4*_alpha^2*a*d-2*_alpha^2*b*d-5*_alpha*a^2*d+_alpha*a*b*d+a^3*d+a^2*b*d+_a lpha^2-_alpha*b)/(-3*_alpha^2*d+6*_alpha*a*d-3*a^2*d+2*_alpha-b)*(_alpha^2 *d-3*_alpha*a*d+_alpha*b*d+3*a^2*d-3*a*b*d+b^2*d-_alpha)/(a^3-3*a^2*b+3*a* b^2-b^3)*(-(-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^2*d-3*_alpha*a*d+_alp ha*b*d+3*a^2*d-3*a*b*d+b^2*d-_alpha)*b/d/(a^3-3*a^2*b+3*a*b^2-b^3),(b/(-a+ b))^(1/2)),_alpha=RootOf(d*_Z^3+(-3*a*d-1)*_Z^2+(3*a^2*d+b)*_Z-d*a^3))
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (35) = 70\).
Time = 0.70 (sec) , antiderivative size = 441, normalized size of antiderivative = 10.02 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\left [\frac {\log \left (\frac {a^{6} d^{2} + d^{2} x^{6} - 6 \, {\left (a d^{2} - d\right )} x^{5} + {\left (15 \, a^{2} d^{2} - 6 \, {\left (3 \, a + b\right )} d + 1\right )} x^{4} - 2 \, {\left (10 \, a^{3} d^{2} - 9 \, {\left (a^{2} + a b\right )} d + b\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 4 \, {\left (a^{4} d + d x^{4} - {\left (4 \, a d - 1\right )} x^{3} + {\left (6 \, a^{2} d - a - b\right )} x^{2} - {\left (4 \, a^{3} d - a b\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 6 \, {\left (a^{5} d^{2} - a^{3} b d\right )} x}{a^{6} d^{2} + d^{2} x^{6} - 2 \, {\left (3 \, a d^{2} + d\right )} x^{5} + {\left (15 \, a^{2} d^{2} + 2 \, {\left (3 \, a + b\right )} d + 1\right )} x^{4} - 2 \, {\left (10 \, a^{3} d^{2} + 3 \, {\left (a^{2} + a b\right )} d + b\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} d^{2} + a^{3} b d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a^{3} d - d x^{3} + {\left (3 \, a d - 1\right )} x^{2} - {\left (3 \, a^{2} d - b\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a^{2} b d x + {\left (2 \, a + b\right )} d x^{3} - d x^{4} - {\left (a^{2} + 2 \, a b\right )} d x^{2}\right )}}\right )}{d}\right ] \]
integrate((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a ^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x^3),x, algorithm="fricas")
[1/2*log((a^6*d^2 + d^2*x^6 - 6*(a*d^2 - d)*x^5 + (15*a^2*d^2 - 6*(3*a + b )*d + 1)*x^4 - 2*(10*a^3*d^2 - 9*(a^2 + a*b)*d + b)*x^3 + (15*a^4*d^2 + b^ 2 - 6*(a^3 + 3*a^2*b)*d)*x^2 - 4*(a^4*d + d*x^4 - (4*a*d - 1)*x^3 + (6*a^2 *d - a - b)*x^2 - (4*a^3*d - a*b)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt( d) - 6*(a^5*d^2 - a^3*b*d)*x)/(a^6*d^2 + d^2*x^6 - 2*(3*a*d^2 + d)*x^5 + ( 15*a^2*d^2 + 2*(3*a + b)*d + 1)*x^4 - 2*(10*a^3*d^2 + 3*(a^2 + a*b)*d + b) *x^3 + (15*a^4*d^2 + b^2 + 2*(a^3 + 3*a^2*b)*d)*x^2 - 2*(3*a^5*d^2 + a^3*b *d)*x))/sqrt(d), sqrt(-d)*arctan(1/2*(a^3*d - d*x^3 + (3*a*d - 1)*x^2 - (3 *a^2*d - b)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(-d)/(a^2*b*d*x + (2*a + b)*d*x^3 - d*x^4 - (a^2 + 2*a*b)*d*x^2))/d]
Timed out. \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Timed out} \]
integrate((a**2*b-a*(2*a-b)*x-(-a+2*b)*x**2+x**3)/(x*(-a+x)*(-b+x))**(1/2) /(-a**3*d+(3*a**2*d+b)*x-(3*a*d+1)*x**2+d*x**3),x)
\[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { -\frac {a^{2} b - {\left (2 \, a - b\right )} a x + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} - {\left (3 \, a^{2} d + b\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a ^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x^3),x, algorithm="maxima")
-integrate((a^2*b - (2*a - b)*a*x + (a - 2*b)*x^2 + x^3)/((a^3*d - d*x^3 + (3*a*d + 1)*x^2 - (3*a^2*d + b)*x)*sqrt((a - x)*(b - x)*x)), x)
\[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { -\frac {a^{2} b - {\left (2 \, a - b\right )} a x + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} - {\left (3 \, a^{2} d + b\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
integrate((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a ^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x^3),x, algorithm="giac")
integrate(-(a^2*b - (2*a - b)*a*x + (a - 2*b)*x^2 + x^3)/((a^3*d - d*x^3 + (3*a*d + 1)*x^2 - (3*a^2*d + b)*x)*sqrt((a - x)*(b - x)*x)), x)
Time = 11.50 (sec) , antiderivative size = 368, normalized size of antiderivative = 8.36 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\frac {\ln \left (\frac {\left (a-b+x+a^2\,d-2\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}+d\,x^2-2\,a\,d\,x\right )\,\left (a\,x^2-a^4\,d-2\,b\,x^2+b^2\,x-2\,d\,x^4+x^3-a^5\,d^2+d^2\,x^5+2\,a^2\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-3\,a^2\,d\,x^2-5\,a\,d^2\,x^4+5\,a^4\,d^2\,x-a\,b\,x+10\,a^2\,d^2\,x^3-10\,a^3\,d^2\,x^2+a^3\,b\,d+4\,a\,d\,x^3+2\,a^3\,d\,x+2\,b\,d\,x^3-2\,a\,b\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-3\,a\,b\,d\,x^2\right )}{\left (-d\,a^3+3\,d\,a^2\,x-3\,d\,a\,x^2+d\,x^3-x^2+b\,x\right )\,\left (a^4\,d^2-4\,a^3\,d^2\,x+2\,a^3\,d-2\,a^2\,b\,d+6\,a^2\,d^2\,x^2-2\,a^2\,d\,x+a^2-2\,a\,b-4\,a\,d^2\,x^3+2\,a\,d\,x^2+2\,a\,x+b^2+2\,b\,d\,x^2-2\,b\,x+d^2\,x^4-2\,d\,x^3+x^2\right )}\right )}{\sqrt {d}} \]
int((a^2*b + x^2*(a - 2*b) + x^3 - a*x*(2*a - b))/((x*(a - x)*(b - x))^(1/ 2)*(x*(b + 3*a^2*d) - a^3*d + d*x^3 - x^2*(3*a*d + 1))),x)
log(((a - b + x + a^2*d - 2*d^(1/2)*(x*(a - x)*(b - x))^(1/2) + d*x^2 - 2* a*d*x)*(a*x^2 - a^4*d - 2*b*x^2 + b^2*x - 2*d*x^4 + x^3 - a^5*d^2 + d^2*x^ 5 + 2*a^2*d^(1/2)*(x*(a - x)*(b - x))^(1/2) - 3*a^2*d*x^2 - 5*a*d^2*x^4 + 5*a^4*d^2*x - a*b*x + 10*a^2*d^2*x^3 - 10*a^3*d^2*x^2 + a^3*b*d + 4*a*d*x^ 3 + 2*a^3*d*x + 2*b*d*x^3 - 2*a*b*d^(1/2)*(x*(a - x)*(b - x))^(1/2) - 3*a* b*d*x^2))/((b*x - a^3*d + d*x^3 - x^2 - 3*a*d*x^2 + 3*a^2*d*x)*(2*a*x - 2* a*b - 2*b*x + 2*a^3*d - 2*d*x^3 + a^2 + b^2 + x^2 + a^4*d^2 + d^2*x^4 - 4* a*d^2*x^3 - 4*a^3*d^2*x + 6*a^2*d^2*x^2 - 2*a^2*b*d + 2*a*d*x^2 - 2*a^2*d* x + 2*b*d*x^2)))/d^(1/2)