Integrand size = 81, antiderivative size = 44 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3+\left (3 a^2+b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}{(a-x)^2}\right )}{\sqrt {d}} \]
Time = 10.84 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3+\left (3 a^2+b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x (-a+x) (-b+x)}}{(a-x)^2}\right )}{\sqrt {d}} \]
Integrate[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)* (-b + x)]*(-a^3 + (3*a^2 + b*d)*x - (3*a + d)*x^2 + 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.
| \(\displaystyle \int \frac {a^2 b-x^2 (2 b-a)-a x (2 a-b)+x^3}{\sqrt {x (x-a) (x-b)} \left (-a^3+x \left (3 a^2+b d\right )-x^2 (3 a+d)+x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {x^3+(a-2 b) x^2-a (2 a-b) x+a^2 b}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (a^3-x^3+(3 a+d) x^2-\left (3 a^2+b d\right ) x\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^3+(a-2 b) x^2-a (2 a-b) x+a^2 b}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (a^3-x^3+(3 a+d) x^2-\left (3 a^2+b d\right ) x\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^3+(a-2 b) x^2-a (2 a-b) x+a^2 b}{\sqrt {x^2-(a+b) x+a b} \left (a^3-x^3+(3 a+d) x^2-\left (3 a^2+b d\right ) x\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(a-x) \left (-x^2-2 (a-b) x+a b\right )}{\sqrt {x^2-(a+b) x+a b} \left (a^3-x^3+(3 a+d) x^2-\left (3 a^2+b d\right ) x\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {(a+b) a^2+(4 a-2 b+d) x^2-\left (5 a^2-b a+b d\right ) x}{\sqrt {x^2-(a+b) x+a b} \left (a^3-x^3+(3 a+d) x^2-\left (3 a^2+b d\right ) x\right )}-\frac {1}{\sqrt {x^2-(a+b) x+a b}}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (a^2 (a+b) \int \frac {1}{\sqrt {x^2-(a+b) x+a b} \left (a^3-3 \left (\frac {b d}{3 a^2}+1\right ) x a^2+3 \left (\frac {d}{3 a}+1\right ) x^2 a-x^3\right )}d\sqrt {x}-\left (5 a^2-a b+b d\right ) \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (a^3-3 \left (\frac {b d}{3 a^2}+1\right ) x a^2+3 \left (\frac {d}{3 a}+1\right ) x^2 a-x^3\right )}d\sqrt {x}+(4 a-2 b+d) \int \frac {x^2}{\sqrt {x^2-(a+b) x+a b} \left (a^3-3 \left (\frac {b d}{3 a^2}+1\right ) x a^2+3 \left (\frac {d}{3 a}+1\right ) x^2 a-x^3\right )}d\sqrt {x}-\frac {\left (\sqrt {a} \sqrt {b}+x\right ) \sqrt {\frac {-x (a+b)+a b+x^2}{\left (\sqrt {a} \sqrt {b}+x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-x (a+b)+a b+x^2}}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Int[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-a^3 + (3*a^2 + b*d)*x - (3*a + d)*x^2 + x^3)),x]
3.6.69.3.1 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 = 1.08 (sec) , antiderivative size = 370, normalized size of antiderivative = 8.41
| method | result | size |
| default | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -d \right ) \textit {\_Z}^{2}+\left (3 a^{2}+d b \right ) \textit {\_Z} -a^{3}\right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b +\underline {\hspace {1.25 ex}}\alpha ^{2} d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2}+\underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha b d +a^{3}+b \,a^{2}\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a b +b^{2}\right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a b +b^{2}\right ) b}{a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+6 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha d -3 a^{2}-d b \right ) \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )\) | \(370\) |
| elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -d \right ) \textit {\_Z}^{2}+\left (3 a^{2}+d b \right ) \textit {\_Z} -a^{3}\right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha ^{2} d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha b d -a^{3}-b \,a^{2}\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a b +b^{2}\right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3 \underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha d +3 a^{2}-3 a b +b^{2}\right ) b}{a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+6 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha d -3 a^{2}-d b \right ) \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )\) | \(374\) |
int((b*a^2-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3+(3* a^2+b*d)*x-(3*a+d)*x^2+x^3),x,method=_RETURNVERBOSE)
-2*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*b*sum((4*_alp ha^2*a-2*_alpha^2*b+_alpha^2*d-5*_alpha*a^2+_alpha*a*b-_alpha*b*d+a^3+a^2* b)/(-3*_alpha^2+6*_alpha*a+2*_alpha*d-3*a^2-b*d)*(_alpha^2-3*_alpha*a+_alp ha*b-_alpha*d+3*a^2-3*a*b+b^2)/(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^2-3*_alpha*a+_alpha*b-_alpha*d+3*a^2-3*a*b+b^2) *b/(a^3-3*a^2*b+3*a*b^2-b^3),(b/(-a+b))^(1/2)),_alpha=RootOf(_Z^3+(-3*a-d) *_Z^2+(3*a^2+b*d)*_Z-a^3))
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (35) = 70\).
Time = 0.50 (sec) , antiderivative size = 407, normalized size of antiderivative = 9.25 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3+\left (3 a^2+b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\left [\frac {\log \left (\frac {a^{6} - 6 \, {\left (a - d\right )} x^{5} + x^{6} + {\left (15 \, a^{2} - 6 \, {\left (3 \, a + b\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (10 \, a^{3} + b d^{2} - 9 \, {\left (a^{2} + a b\right )} d\right )} x^{3} + {\left (15 \, a^{4} + b^{2} d^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 4 \, {\left (a^{4} - {\left (4 \, a - d\right )} x^{3} + x^{4} + {\left (6 \, a^{2} - {\left (a + b\right )} d\right )} x^{2} - {\left (4 \, a^{3} - a b d\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 6 \, {\left (a^{5} - a^{3} b d\right )} x}{a^{6} - 2 \, {\left (3 \, a + d\right )} x^{5} + x^{6} + {\left (15 \, a^{2} + 2 \, {\left (3 \, a + b\right )} d + d^{2}\right )} x^{4} - 2 \, {\left (10 \, a^{3} + b d^{2} + 3 \, {\left (a^{2} + a b\right )} d\right )} x^{3} + {\left (15 \, a^{4} + b^{2} d^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} + a^{3} b d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a^{3} + {\left (3 \, a - d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} - b d\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a^{2} b d x + {\left (2 \, a + b\right )} d x^{3} - d x^{4} - {\left (a^{2} + 2 \, a b\right )} d x^{2}\right )}}\right )}{d}\right ] \]
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+(3*a^2+b*d)*x-(3*a+d)*x^2+x^3),x, algorithm="fricas")
[1/2*log((a^6 - 6*(a - d)*x^5 + x^6 + (15*a^2 - 6*(3*a + b)*d + d^2)*x^4 - 2*(10*a^3 + b*d^2 - 9*(a^2 + a*b)*d)*x^3 + (15*a^4 + b^2*d^2 - 6*(a^3 + 3 *a^2*b)*d)*x^2 - 4*(a^4 - (4*a - d)*x^3 + x^4 + (6*a^2 - (a + b)*d)*x^2 - (4*a^3 - a*b*d)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(d) - 6*(a^5 - a^3* b*d)*x)/(a^6 - 2*(3*a + d)*x^5 + x^6 + (15*a^2 + 2*(3*a + b)*d + d^2)*x^4 - 2*(10*a^3 + b*d^2 + 3*(a^2 + a*b)*d)*x^3 + (15*a^4 + b^2*d^2 + 2*(a^3 + 3*a^2*b)*d)*x^2 - 2*(3*a^5 + a^3*b*d)*x))/sqrt(d), sqrt(-d)*arctan(1/2*(a^ 3 + (3*a - d)*x^2 - x^3 - (3*a^2 - b*d)*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. \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3+\left (3 a^2+b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\text {Timed out} \]
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+(3*a**2+b*d)*x-(3*a+d)*x**2+x**3),x)
\[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3+\left (3 a^2+b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int { -\frac {a^{2} b - {\left (2 \, a - b\right )} a x + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
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+(3*a^2+b*d)*x-(3*a+d)*x^2+x^3),x, algorithm="maxima")
-integrate((a^2*b - (2*a - b)*a*x + (a - 2*b)*x^2 + x^3)/((a^3 + (3*a + d) *x^2 - x^3 - (3*a^2 + b*d)*x)*sqrt((a - x)*(b - x)*x)), x)
\[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3+\left (3 a^2+b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\int { -\frac {a^{2} b - {\left (2 \, a - b\right )} a x + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} + {\left (3 \, a + d\right )} x^{2} - x^{3} - {\left (3 \, a^{2} + b d\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]
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+(3*a^2+b*d)*x-(3*a+d)*x^2+x^3),x, algorithm="giac")
integrate(-(a^2*b - (2*a - b)*a*x + (a - 2*b)*x^2 + x^3)/((a^3 + (3*a + d) *x^2 - x^3 - (3*a^2 + b*d)*x)*sqrt((a - x)*(b - x)*x)), x)
Time = 5.56 (sec) , antiderivative size = 589, normalized size of antiderivative = 13.39 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3+\left (3 a^2+b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx=\left (\sum _{k=1}^3\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )-b};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a^2\,b+a^3+4\,a\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )}^2-5\,a^2\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )-2\,b\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )}^2+d\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )}^2+a\,b\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )-b\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )\right )}{\left (\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )-b\right )\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (3\,a^2-6\,a\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )+3\,{\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )}^2-2\,d\,\mathrm {root}\left (z^3-z^2\,\left (3\,a+d\right )+z\,\left (b\,d+3\,a^2\right )-a^3,z,k\right )+b\,d\right )}\right )-\frac {2\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
int((a^2*b + x^2*(a - 2*b) + x^3 - a*x*(2*a - b))/((x*(a - x)*(b - x))^(1/ 2)*(x*(b*d + 3*a^2) - x^2*(3*a + d) - a^3 + x^3)),x)
symsum((2*b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*elliptic Pi(-b/(root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k) - b), asin( ((b - x)/b)^(1/2)), -b/(a - b))*(a^2*b + a^3 + 4*a*root(z^3 - z^2*(3*a + d ) + z*(b*d + 3*a^2) - a^3, z, k)^2 - 5*a^2*root(z^3 - z^2*(3*a + d) + z*(b *d + 3*a^2) - a^3, z, k) - 2*b*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k)^2 + d*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k) ^2 + a*b*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k) - b*d*roo t(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k)))/((root(z^3 - z^2*(3 *a + d) + z*(b*d + 3*a^2) - a^3, z, k) - b)*(x*(a - x)*(b - x))^(1/2)*(b*d + 3*root(z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k)^2 - 6*a*root( z^3 - z^2*(3*a + d) + z*(b*d + 3*a^2) - a^3, z, k) - 2*d*root(z^3 - z^2*(3 *a + d) + z*(b*d + 3*a^2) - a^3, z, k) + 3*a^2)), k, 1, 3) - (2*b*elliptic F(asin(((b - x)/b)^(1/2)), -b/(a - b))*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2))/(x^3 - x^2*(a + b) + a*b*x)^(1/2)