Integrand size = 70, antiderivative size = 84 \[ \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx=\frac {2 \sqrt {a b x-a x^2-b x^2+x^3}}{(b-x) x}+2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} x (-b+x)}\right ) \]
2*(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(b-x)/x+2*d^(1/2)*arctanh((a*b*x+(-a-b)*x^ 2+x^3)^(1/2)/d^(1/2)/x/(-b+x))
Time = 14.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.70 \[ \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx=\frac {2 (a-x)}{\sqrt {x (-a+x) (-b+x)}}+2 \sqrt {d} \text {arctanh}\left (\frac {-a+x}{\sqrt {d} \sqrt {x (-a+x) (-b+x)}}\right ) \]
Integrate[(a^2*b - a*(2*a + b)*x + 3*a*x^2 - x^3)/(x*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a + (-1 - b*d)*x + d*x^2)),x]
(2*(a - x))/Sqrt[x*(-a + x)*(-b + x)] + 2*Sqrt[d]*ArcTanh[(-a + x)/(Sqrt[d ]*Sqrt[x*(-a + x)*(-b + x)])]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.77 (sec) , antiderivative size = 1216, normalized size of antiderivative = 14.48, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2467, 25, 2035, 1395, 2019, 7279, 2009}
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.
| \(\displaystyle \int \frac {a^2 b-a x (2 a+b)+3 a x^2-x^3}{x (x-b) \sqrt {x (x-a) (x-b)} \left (a+x (-b d-1)+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {-x^3+3 a x^2-a (2 a+b) x+a^2 b}{(b-x) x^{3/2} \sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {-x^3+3 a x^2-a (2 a+b) x+a^2 b}{(b-x) x^{3/2} \sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {-x^3+3 a x^2-a (2 a+b) x+a^2 b}{(b-x) x \sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {-x^3+3 a x^2-a (2 a+b) x+a^2 b}{\sqrt {a-x} (b-x)^{3/2} x \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {\sqrt {a-x} \left (x^2-2 a x+a b\right )}{(b-x)^{3/2} x \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (\frac {\sqrt {a-x} b}{(b-x)^{3/2} x}+\frac {\sqrt {a-x} \left (d b^2+b-2 a+(1-b d) x\right )}{(b-x)^{3/2} \left (d x^2-(b d+1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {a-x} \sqrt {b-x} \sqrt {x} \left (-\frac {2 \sqrt {a} d \sqrt {b-x} \sqrt {1-\frac {x}{a}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\left (-b d-\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {a-x} \sqrt {1-\frac {x}{b}}}-\frac {2 \sqrt {a} d \sqrt {b-x} \sqrt {1-\frac {x}{a}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\left (-b d+\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {a-x} \sqrt {1-\frac {x}{b}}}+\frac {\left (b d-\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {a-x} \sqrt {1-\frac {x}{b}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{\sqrt {b} \left (-b d-\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {b-x} \sqrt {1-\frac {x}{a}}}+\frac {\left (b d+\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {a-x} \sqrt {1-\frac {x}{b}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{\sqrt {b} \left (-b d+\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {b-x} \sqrt {1-\frac {x}{a}}}-\frac {2 \sqrt {a-x} \sqrt {1-\frac {x}{b}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{\sqrt {b} \sqrt {b-x} \sqrt {1-\frac {x}{a}}}+\frac {\sqrt {b} \left (-2 a d+b d-\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right ),\frac {b}{a}\right )}{\left (-b d-\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {a-x} \sqrt {b-x}}+\frac {\sqrt {b} \left (-2 a d+b d+\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right ),\frac {b}{a}\right )}{\left (-b d+\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {a-x} \sqrt {b-x}}-\frac {2 a \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right ),\frac {b}{a}\right )}{\sqrt {b} \sqrt {a-x} \sqrt {b-x}}+\frac {(2 a-b) \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right ),\frac {b}{a}\right )}{\sqrt {b} \sqrt {a-x} \sqrt {b-x}}-\frac {\sqrt {a} \left (-b d-\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticPi}\left (\frac {2 a d}{b d-\sqrt {(b d+1)^2-4 a d}+1},\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{2 \sqrt {a-x} \sqrt {b-x}}-\frac {\sqrt {a} \left (-b d+\sqrt {(b d+1)^2-4 a d}+1\right ) \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticPi}\left (\frac {2 a d}{b d+\sqrt {(b d+1)^2-4 a d}+1},\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{2 \sqrt {a-x} \sqrt {b-x}}-\frac {2 \sqrt {a-x} \sqrt {x}}{b \sqrt {b-x}}-\frac {2 \sqrt {a-x} \sqrt {b-x}}{b \sqrt {x}}+\frac {\sqrt {a-x}}{\sqrt {b-x} \sqrt {x}}\right )}{\sqrt {(a-x) (b-x) x}}\) |
Int[(a^2*b - a*(2*a + b)*x + 3*a*x^2 - x^3)/(x*(-b + x)*Sqrt[x*(-a + x)*(- b + x)]*(a + (-1 - b*d)*x + d*x^2)),x]
(-2*Sqrt[a - x]*Sqrt[b - x]*Sqrt[x]*(Sqrt[a - x]/(Sqrt[b - x]*Sqrt[x]) - ( 2*Sqrt[a - x]*Sqrt[b - x])/(b*Sqrt[x]) - (2*Sqrt[a - x]*Sqrt[x])/(b*Sqrt[b - x]) - (2*Sqrt[a]*d*Sqrt[b - x]*Sqrt[1 - x/a]*EllipticE[ArcSin[Sqrt[x]/S qrt[a]], a/b])/((1 - b*d - Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[a - x]*Sqrt[1 - x/b]) - (2*Sqrt[a]*d*Sqrt[b - x]*Sqrt[1 - x/a]*EllipticE[ArcSin[Sqrt[x]/ Sqrt[a]], a/b])/((1 - b*d + Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[a - x]*Sqrt[1 - x/b]) - (2*Sqrt[a - x]*Sqrt[1 - x/b]*EllipticE[ArcSin[Sqrt[x]/Sqrt[b]], b/a])/(Sqrt[b]*Sqrt[b - x]*Sqrt[1 - x/a]) + ((1 + b*d - Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[a - x]*Sqrt[1 - x/b]*EllipticE[ArcSin[Sqrt[x]/Sqrt[b]], b/ a])/(Sqrt[b]*(1 - b*d - Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[b - x]*Sqrt[1 - x /a]) + ((1 + b*d + Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[a - x]*Sqrt[1 - x/b]*E llipticE[ArcSin[Sqrt[x]/Sqrt[b]], b/a])/(Sqrt[b]*(1 - b*d + Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[b - x]*Sqrt[1 - x/a]) - (2*a*Sqrt[1 - x/a]*Sqrt[1 - x/b ]*EllipticF[ArcSin[Sqrt[x]/Sqrt[b]], b/a])/(Sqrt[b]*Sqrt[a - x]*Sqrt[b - x ]) + ((2*a - b)*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[ b]], b/a])/(Sqrt[b]*Sqrt[a - x]*Sqrt[b - x]) + (Sqrt[b]*(1 - 2*a*d + b*d - Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticF[ArcSin[ Sqrt[x]/Sqrt[b]], b/a])/((1 - b*d - Sqrt[-4*a*d + (1 + b*d)^2])*Sqrt[a - x ]*Sqrt[b - x]) + (Sqrt[b]*(1 - 2*a*d + b*d + Sqrt[-4*a*d + (1 + b*d)^2])*S qrt[1 - x/a]*Sqrt[1 - x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[b]], b/a])/((1...
3.12.25.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 2.16 (sec) , antiderivative size = 2553, normalized size of antiderivative = 30.39
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2553\) |
| risch | \(\text {Expression too large to display}\) | \(2839\) |
| default | \(\text {Expression too large to display}\) | \(2935\) |
int((a^2*b-a*(2*a+b)*x+3*a*x^2-x^3)/x/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a+(- b*d-1)*x+d*x^2),x,method=_RETURNVERBOSE)
2*(a*b-a*x-b*x+x^2)/b/(x*(a*b-a*x-b*x+x^2))^(1/2)-2*(-a*x+x^2)/b/((-b+x)*( -a*x+x^2))^(1/2)+2*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(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))+1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b^3*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a +b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d* (b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(d* b+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*d+2/(b^2*d^2-4*a*d+2 *b*d+1)^(1/2)*b^2*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2) /(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^ (1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(d*b+1+(b^2*d^2-4*a*d+2*b*d +1)^(1/2))),(b/(-a+b))^(1/2))-4/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1 /2)*(-1/(-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/ 2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b )^(1/2),b/(b-1/2/d*(d*b+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2) )*a+1/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(-1/(-a+b)*a+1/(-a+b)* x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2 *d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(d*b+1+ (b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))/d-2*b*(1-x/b)^(1/2)*(-1/ (-a+b)*a+1/(-a+b)*x)^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2* b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/...
Time = 0.48 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.73 \[ \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx=\left [\frac {{\left (b x - x^{2}\right )} \sqrt {d} \log \left (\frac {d^{2} x^{4} - 2 \, {\left (b d^{2} - 3 \, d\right )} x^{3} + {\left (b^{2} d^{2} - 6 \, {\left (a + b\right )} d + 1\right )} x^{2} + a^{2} + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d x^{2} - {\left (b d - 1\right )} x - a\right )} \sqrt {d} + 2 \, {\left (3 \, a b d - a\right )} x}{d^{2} x^{4} - 2 \, {\left (b d^{2} + d\right )} x^{3} + {\left (b^{2} d^{2} + 2 \, {\left (a + b\right )} d + 1\right )} x^{2} + a^{2} - 2 \, {\left (a b d + a\right )} x}\right ) + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{2 \, {\left (b x - x^{2}\right )}}, -\frac {{\left (b x - x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d x^{2} - {\left (b d - 1\right )} x - a\right )} \sqrt {-d}}{2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}\right ) - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{b x - x^{2}}\right ] \]
integrate((a^2*b-a*(2*a+b)*x+3*a*x^2-x^3)/x/(-b+x)/(x*(-a+x)*(-b+x))^(1/2) /(a+(-b*d-1)*x+d*x^2),x, algorithm="fricas")
[1/2*((b*x - x^2)*sqrt(d)*log((d^2*x^4 - 2*(b*d^2 - 3*d)*x^3 + (b^2*d^2 - 6*(a + b)*d + 1)*x^2 + a^2 + 4*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d*x^2 - (b *d - 1)*x - a)*sqrt(d) + 2*(3*a*b*d - a)*x)/(d^2*x^4 - 2*(b*d^2 + d)*x^3 + (b^2*d^2 + 2*(a + b)*d + 1)*x^2 + a^2 - 2*(a*b*d + a)*x)) + 4*sqrt(a*b*x - (a + b)*x^2 + x^3))/(b*x - x^2), -((b*x - x^2)*sqrt(-d)*arctan(1/2*sqrt( a*b*x - (a + b)*x^2 + x^3)*(d*x^2 - (b*d - 1)*x - a)*sqrt(-d)/(a*b*d*x - ( a + b)*d*x^2 + d*x^3)) - 2*sqrt(a*b*x - (a + b)*x^2 + x^3))/(b*x - x^2)]
Timed out. \[ \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx=\text {Timed out} \]
integrate((a**2*b-a*(2*a+b)*x+3*a*x**2-x**3)/x/(-b+x)/(x*(-a+x)*(-b+x))**( 1/2)/(a+(-b*d-1)*x+d*x**2),x)
\[ \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx=\int { -\frac {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (d x^{2} - {\left (b d + 1\right )} x + a\right )} {\left (b - x\right )} x} \,d x } \]
integrate((a^2*b-a*(2*a+b)*x+3*a*x^2-x^3)/x/(-b+x)/(x*(-a+x)*(-b+x))^(1/2) /(a+(-b*d-1)*x+d*x^2),x, algorithm="maxima")
-integrate((a^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/(sqrt((a - x)*(b - x)*x )*(d*x^2 - (b*d + 1)*x + a)*(b - x)*x), x)
\[ \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx=\int { -\frac {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (d x^{2} - {\left (b d + 1\right )} x + a\right )} {\left (b - x\right )} x} \,d x } \]
integrate((a^2*b-a*(2*a+b)*x+3*a*x^2-x^3)/x/(-b+x)/(x*(-a+x)*(-b+x))^(1/2) /(a+(-b*d-1)*x+d*x^2),x, algorithm="giac")
integrate(-(a^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/(sqrt((a - x)*(b - x)*x )*(d*x^2 - (b*d + 1)*x + a)*(b - x)*x), x)
Time = 0.19 (sec) , antiderivative size = 628, normalized size of antiderivative = 7.48 \[ \int \frac {a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a+(-1-b d) x+d x^2\right )} \, dx=\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (b\,d-2\,a\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1\right )}{d\,\left (b-\frac {b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {b\,d-\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (2\,a\,d-b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}-1\right )}{d\,\left (b-\frac {b\,d-\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,a\,\left (a-b\right )\,\sqrt {\frac {x}{a}}\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\middle |\frac {a}{b}\right )-\frac {a\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\right )}{2\,b\,\sqrt {1-\frac {x}{b}}}\right )\,\sqrt {\frac {a-x}{a}}\,\sqrt {\frac {b-x}{b}}}{b\,\left (\frac {a}{b}-1\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,a\,\left (\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\frac {\sqrt {\frac {b-x}{a-b}+1}\,\sqrt {\frac {b-x}{b}}}{\sqrt {1-\frac {b-x}{b}}}}{\frac {b}{a-b}+1}-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
int(-(a^2*b + 3*a*x^2 - x^3 - a*x*(2*a + b))/(x*(b - x)*(x*(a - x)*(b - x) )^(1/2)*(a - x*(b*d + 1) + d*x^2)),x)
(b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(b/(b - (b*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^ (1/2)), -b/(a - b))*(b*d - 2*a*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1 ))/(d*(b - (b*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d))*(x^3 - x ^2*(a + b) + a*b*x)^(1/2)) - (b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(b/(b - (b*d - (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^(1/2)), -b/(a - b))*(2*a*d - b*d + (2*b*d - 4 *a*d + b^2*d^2 + 1)^(1/2) - 1))/(d*(b - (b*d - (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d))*(x^3 - x^2*(a + b) + a*b*x)^(1/2)) - (2*a*(a - b)*(x/ a)^(1/2)*(ellipticE(asin((x/a)^(1/2)), a/b) - (a*sin(2*asin((x/a)^(1/2)))) /(2*b*(1 - x/b)^(1/2)))*((a - x)/a)^(1/2)*((b - x)/b)^(1/2))/(b*(a/b - 1)* (x^3 - x^2*(a + b) + a*b*x)^(1/2)) - (2*a*((ellipticE(asin(((b - x)/b)^(1/ 2)), -b/(a - b)) - (((b - x)/(a - b) + 1)^(1/2)*((b - x)/b)^(1/2))/(1 - (b - x)/b)^(1/2))/(b/(a - b) + 1) - ellipticF(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)