Integrand size = 59, antiderivative size = 77 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx=-\frac {2 \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}+2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} (a-x)}\right ) \]
-2*(a*b*x+(-a-b)*x^2+x^3)^(1/2)/(a-x)+2*d^(1/2)*arctanh((a*b*x+(-a-b)*x^2+ x^3)^(1/2)/d^(1/2)/(a-x))
Time = 15.56 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx=\frac {2 x (-b+x)}{\sqrt {x (-a+x) (-b+x)}}+2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x)}}{\sqrt {d} (a-x)}\right ) \]
Integrate[(x*(-b + x)*(a*b - 2*a*x + x^2))/((-a + x)*Sqrt[x*(-a + x)*(-b + x)]*(a*d + (-b - d)*x + x^2)),x]
(2*x*(-b + x))/Sqrt[x*(-a + x)*(-b + x)] + 2*Sqrt[d]*ArcTanh[Sqrt[x*(-a + x)*(-b + x)]/(Sqrt[d]*(a - x))]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 9.66 (sec) , antiderivative size = 2548, normalized size of antiderivative = 33.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2467, 2035, 1395, 7279, 6, 6, 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.
| \(\displaystyle \int \frac {x (x-b) \left (a b-2 a x+x^2\right )}{(x-a) \sqrt {x (x-a) (x-b)} \left (a d+x (-b-d)+x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(b-x) \sqrt {x} \left (x^2-2 a x+a b\right )}{(a-x) \sqrt {x^2-(a+b) x+a b} \left (x^2-(b+d) x+a d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(b-x) x \left (x^2-2 a x+a b\right )}{(a-x) \sqrt {x^2-(a+b) x+a b} \left (x^2-(b+d) x+a d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {\sqrt {b-x} x \left (x^2-2 a x+a b\right )}{(a-x)^{3/2} \left (x^2-(b+d) x+a d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {2 \sqrt {b-x} a}{(a-x)^{3/2}}+\frac {\sqrt {b-x} x}{(a-x)^{3/2}}+\frac {b \sqrt {b-x}}{(a-x)^{3/2}}+\frac {d \sqrt {b-x}}{(a-x)^{3/2}}+\frac {\sqrt {b-x} \left (a (2 a-b-d) d+\left ((b+d)^2-a (b+3 d)\right ) x\right )}{(a-x)^{3/2} \left (x^2+(-b-d) x+a d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (\frac {\sqrt {b-x} (b-2 a)}{(a-x)^{3/2}}+\frac {\sqrt {b-x} x}{(a-x)^{3/2}}+\frac {d \sqrt {b-x}}{(a-x)^{3/2}}+\frac {\sqrt {b-x} \left (a (2 a-b-d) d+\left ((b+d)^2-a (b+3 d)\right ) x\right )}{(a-x)^{3/2} \left (x^2+(-b-d) x+a d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (\frac {\sqrt {b-x} (-2 a+b+d)}{(a-x)^{3/2}}+\frac {\sqrt {b-x} x}{(a-x)^{3/2}}+\frac {\sqrt {b-x} \left (a (2 a-b-d) d+\left ((b+d)^2-a (b+3 d)\right ) x\right )}{(a-x)^{3/2} \left (x^2+(-b-d) x+a d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {a-x} \sqrt {b-x} \sqrt {x} \left (\frac {\sqrt {b} \sqrt {a-x} \sqrt {1-\frac {x}{b}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right ) (2 a-b-d)}{a \sqrt {b-x} \sqrt {1-\frac {x}{a}}}-\frac {\sqrt {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 ) (2 a-b-d)}{\sqrt {a-x} \sqrt {b-x}}-\frac {\sqrt {b-x} \sqrt {x} (2 a-b-d)}{a \sqrt {a-x}}-\frac {2 \sqrt {a} \left ((b+d)^2-a (b+3 d)-\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\right ) \sqrt {b-x} \sqrt {1-\frac {x}{a}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\left (2 a-b-d+\sqrt {b^2+2 d b+d^2-4 a d}\right )^2 \sqrt {a-x} \sqrt {1-\frac {x}{b}}}-\frac {2 \sqrt {a} \left ((b+d)^2-a (b+3 d)+\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\right ) \sqrt {b-x} \sqrt {1-\frac {x}{a}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\left (2 a-b-d-\sqrt {b^2+2 d b+d^2-4 a d}\right )^2 \sqrt {a-x} \sqrt {1-\frac {x}{b}}}+\frac {\sqrt {b} \left (b+d-\sqrt {b^2+2 d b+d^2-4 a d}\right ) \left ((b+d)^2-a (b+3 d)-\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\right ) \sqrt {a-x} \sqrt {1-\frac {x}{b}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \left (2 a-b-d+\sqrt {b^2+2 d b+d^2-4 a d}\right )^2 \sqrt {b-x} \sqrt {1-\frac {x}{a}}}+\frac {\sqrt {b} \left (b+d+\sqrt {b^2+2 d b+d^2-4 a d}\right ) \left ((b+d)^2-a (b+3 d)+\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\right ) \sqrt {a-x} \sqrt {1-\frac {x}{b}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{a \left (2 a-b-d-\sqrt {b^2+2 d b+d^2-4 a d}\right )^2 \sqrt {b-x} \sqrt {1-\frac {x}{a}}}-\frac {2 \sqrt {b} \sqrt {a-x} \sqrt {1-\frac {x}{b}} E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {b}}\right )|\frac {b}{a}\right )}{\sqrt {b-x} \sqrt {1-\frac {x}{a}}}+\frac {\sqrt {b} \left (b-d+\sqrt {b^2+2 d b+d^2-4 a d}\right ) \left ((b+d)^2-a (b+3 d)-\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\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 (2 a-b-d+\sqrt {b^2+2 d b+d^2-4 a d}\right )^2 \sqrt {a-x} \sqrt {b-x}}-\frac {\sqrt {b} \left ((b+d)^2-a (b+3 d)-\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\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 (2 a-b-d+\sqrt {b^2+2 d b+d^2-4 a d}\right ) \sqrt {a-x} \sqrt {b-x}}+\frac {\sqrt {b} \left (b-d-\sqrt {b^2+2 d b+d^2-4 a d}\right ) \left ((b+d)^2-a (b+3 d)+\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\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 (2 a-b-d-\sqrt {b^2+2 d b+d^2-4 a d}\right )^2 \sqrt {a-x} \sqrt {b-x}}-\frac {\sqrt {b} \left ((b+d)^2-a (b+3 d)+\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\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 (2 a-b-d-\sqrt {b^2+2 d b+d^2-4 a d}\right ) \sqrt {a-x} \sqrt {b-x}}+\frac {(2 a-b) \sqrt {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 {a-x} \sqrt {b-x}}-\frac {\sqrt {a} \left (b-d+\sqrt {b^2+2 d b+d^2-4 a d}\right ) \left ((b+d)^2-a (b+3 d)-\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\right ) \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticPi}\left (\frac {2 a}{b+d-\sqrt {b^2+2 d b+d^2-4 a d}},\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\left (b+d-\sqrt {b^2+2 d b+d^2-4 a d}\right ) \left (2 a-b-d+\sqrt {b^2+2 d b+d^2-4 a d}\right ) \sqrt {a-x} \sqrt {b-x}}-\frac {\sqrt {a} \left (b-d-\sqrt {b^2+2 d b+d^2-4 a d}\right ) \left ((b+d)^2-a (b+3 d)+\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\right ) \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticPi}\left (\frac {2 a}{b+d+\sqrt {b^2+2 d b+d^2-4 a d}},\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\left (2 a-b-d-\sqrt {b^2+2 d b+d^2-4 a d}\right ) \left (b+d+\sqrt {b^2+2 d b+d^2-4 a d}\right ) \sqrt {a-x} \sqrt {b-x}}+\frac {\left ((b+d)^2-a (b+3 d)-\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\right ) \sqrt {b-x} \sqrt {x}}{a \left (2 a-b-d+\sqrt {b^2+2 d b+d^2-4 a d}\right ) \sqrt {a-x}}+\frac {\left ((b+d)^2-a (b+3 d)+\frac {(b+d)^3+4 a^2 d-a \left (b^2+6 d b+5 d^2\right )}{\sqrt {b^2+2 d b-(4 a-d) d}}\right ) \sqrt {b-x} \sqrt {x}}{a \left (2 a-b-d-\sqrt {b^2+2 d b+d^2-4 a d}\right ) \sqrt {a-x}}+\frac {\sqrt {b-x} \sqrt {x}}{\sqrt {a-x}}\right )}{\sqrt {(a-x) (b-x) x}}\) |
Int[(x*(-b + x)*(a*b - 2*a*x + x^2))/((-a + x)*Sqrt[x*(-a + x)*(-b + x)]*( a*d + (-b - d)*x + x^2)),x]
(2*Sqrt[a - x]*Sqrt[b - x]*Sqrt[x]*((Sqrt[b - x]*Sqrt[x])/Sqrt[a - x] - (( 2*a - b - d)*Sqrt[b - x]*Sqrt[x])/(a*Sqrt[a - x]) + (((b + d)^2 - a*(b + 3 *d) - (4*a^2*d + (b + d)^3 - a*(b^2 + 6*b*d + 5*d^2))/Sqrt[b^2 + 2*b*d - ( 4*a - d)*d])*Sqrt[b - x]*Sqrt[x])/(a*(2*a - b - d + Sqrt[b^2 - 4*a*d + 2*b *d + d^2])*Sqrt[a - x]) + (((b + d)^2 - a*(b + 3*d) + (4*a^2*d + (b + d)^3 - a*(b^2 + 6*b*d + 5*d^2))/Sqrt[b^2 + 2*b*d - (4*a - d)*d])*Sqrt[b - x]*S qrt[x])/(a*(2*a - b - d - Sqrt[b^2 - 4*a*d + 2*b*d + d^2])*Sqrt[a - x]) - (2*Sqrt[a]*((b + d)^2 - a*(b + 3*d) - (4*a^2*d + (b + d)^3 - a*(b^2 + 6*b* d + 5*d^2))/Sqrt[b^2 + 2*b*d - (4*a - d)*d])*Sqrt[b - x]*Sqrt[1 - x/a]*Ell ipticE[ArcSin[Sqrt[x]/Sqrt[a]], a/b])/((2*a - b - d + Sqrt[b^2 - 4*a*d + 2 *b*d + d^2])^2*Sqrt[a - x]*Sqrt[1 - x/b]) - (2*Sqrt[a]*((b + d)^2 - a*(b + 3*d) + (4*a^2*d + (b + d)^3 - a*(b^2 + 6*b*d + 5*d^2))/Sqrt[b^2 + 2*b*d - (4*a - d)*d])*Sqrt[b - x]*Sqrt[1 - x/a]*EllipticE[ArcSin[Sqrt[x]/Sqrt[a]] , a/b])/((2*a - b - d - Sqrt[b^2 - 4*a*d + 2*b*d + d^2])^2*Sqrt[a - x]*Sqr t[1 - x/b]) - (2*Sqrt[b]*Sqrt[a - x]*Sqrt[1 - x/b]*EllipticE[ArcSin[Sqrt[x ]/Sqrt[b]], b/a])/(Sqrt[b - x]*Sqrt[1 - x/a]) + (Sqrt[b]*(2*a - b - d)*Sqr t[a - x]*Sqrt[1 - x/b]*EllipticE[ArcSin[Sqrt[x]/Sqrt[b]], b/a])/(a*Sqrt[b - x]*Sqrt[1 - x/a]) + (Sqrt[b]*(b + d - Sqrt[b^2 - 4*a*d + 2*b*d + d^2])*( (b + d)^2 - a*(b + 3*d) - (4*a^2*d + (b + d)^3 - a*(b^2 + 6*b*d + 5*d^2))/ Sqrt[b^2 + 2*b*d - (4*a - d)*d])*Sqrt[a - x]*Sqrt[1 - x/b]*EllipticE[Ar...
3.11.9.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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[(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 = 1.91 (sec) , antiderivative size = 2341, normalized size of antiderivative = 30.40
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2341\) |
| default | \(\text {Expression too large to display}\) | \(2765\) |
int(x*(-b+x)*(a*b-2*a*x+x^2)/(-a+x)/(x*(-a+x)*(-b+x))^(1/2)/(a*d+(-b-d)*x+ x^2),x,method=_RETURNVERBOSE)
2*(-b*x+x^2)/((-a+x)*(-b*x+x^2))^(1/2)-2*d*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))+4/(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-b+x)/b)^(1 /2),b/(1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(b/(-a+b))^(1/2))*a*d ^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)/(1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*Ellipti cPi((-(-b+x)/b)^(1/2),b/(1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(b/ (-a+b))^(1/2))*d*a-1/(-4*a*d+b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(1/ 2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(b/(-a+b))^(1/2))*d-2/(-4*a*d+ b^2+2*b*d+d^2)^(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*(-4*a*d+b^2+2*b*d+d ^2)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b *d+d^2)^(1/2)),(b/(-a+b))^(1/2))*d^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*(-4 *a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(1/2*b-1/2*d-1/2 *(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(b/(-a+b))^(1/2))*d-1/(-4*a*d+b^2+2*b*d+...
Timed out. \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx=\text {Timed out} \]
integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(-a+x)/(x*(-a+x)*(-b+x))^(1/2)/(a*d+(-b -d)*x+x^2),x, algorithm="fricas")
Timed out. \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx=\int { \frac {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a d - {\left (b + d\right )} x + x^{2}\right )} {\left (a - x\right )}} \,d x } \]
integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(-a+x)/(x*(-a+x)*(-b+x))^(1/2)/(a*d+(-b -d)*x+x^2),x, algorithm="maxima")
integrate((a*b - 2*a*x + x^2)*(b - x)*x/(sqrt((a - x)*(b - x)*x)*(a*d - (b + d)*x + x^2)*(a - x)), x)
\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx=\int { \frac {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a d - {\left (b + d\right )} x + x^{2}\right )} {\left (a - x\right )}} \,d x } \]
integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(-a+x)/(x*(-a+x)*(-b+x))^(1/2)/(a*d+(-b -d)*x+x^2),x, algorithm="giac")
integrate((a*b - 2*a*x + x^2)*(b - x)*x/(sqrt((a - x)*(b - x)*x)*(a*d - (b + d)*x + x^2)*(a - x)), x)
Time = 0.14 (sec) , antiderivative size = 754, normalized size of antiderivative = 9.79 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{(-a+x) \sqrt {x (-a+x) (-b+x)} \left (a d+(-b-d) x+x^2\right )} \, dx=\frac {2\,a\,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}}-\frac {2\,b\,\left (a\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\left (a-b\right )\,\mathrm {E}\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}}-\frac {2\,b\,d\,\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}}+\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a\,d-\frac {b\,d}{2}+\frac {d\,\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}-\frac {d^2}{2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}\right )}+\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\frac {d}{2}-\frac {b}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (\frac {b\,d}{2}-a\,d+\frac {d\,\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}+\frac {d^2}{2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {d}{2}-\frac {b}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}\right )}+\frac {2\,b\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )+\frac {b\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\right )}{2\,\sqrt {\frac {b-x}{a-b}+1}\,\left (a-b\right )}\right )\,\sqrt {\frac {x}{b}}\,\left (a\,b-a^2\right )\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\left (\frac {b}{a-b}+1\right )\,\left (a-b\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
int((x*(b - x)*(a*b - 2*a*x + x^2))/((a - x)*(x*(a - x)*(b - x))^(1/2)*(a* d + x^2 - x*(b + d))),x)
(2*a*b*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) - (2* b*(a*ellipticF(asin(((b - x)/b)^(1/2)), -b/(a - b)) - (a - b)*ellipticE(as in(((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) - (2*b*d*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) + (2*b*(x/b)^(1/2)*((b - x )/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(b/(b/2 - d/2 + (2*b*d - 4*a* d + b^2 + d^2)^(1/2)/2), asin(((b - x)/b)^(1/2)), -b/(a - b))*(a*d - (b*d) /2 + (d*(2*b*d - 4*a*d + b^2 + d^2)^(1/2))/2 - d^2/2))/((x^3 - x^2*(a + b) + a*b*x)^(1/2)*(b/2 - d/2 + (2*b*d - 4*a*d + b^2 + d^2)^(1/2)/2)) + (2*b* (x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(-b/(d/2 - b/2 + (2*b*d - 4*a*d + b^2 + d^2)^(1/2)/2), asin(((b - x)/b)^(1/2)), -b/( a - b))*((b*d)/2 - a*d + (d*(2*b*d - 4*a*d + b^2 + d^2)^(1/2))/2 + d^2/2)) /((x^3 - x^2*(a + b) + a*b*x)^(1/2)*(d/2 - b/2 + (2*b*d - 4*a*d + b^2 + d^ 2)^(1/2)/2)) + (2*b*(ellipticE(asin(((b - x)/b)^(1/2)), -b/(a - b)) + (b*s in(2*asin(((b - x)/b)^(1/2))))/(2*((b - x)/(a - b) + 1)^(1/2)*(a - b)))*(x /b)^(1/2)*(a*b - a^2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2))/((b/(a - b) + 1)*(a - b)*(x^3 - x^2*(a + b) + a*b*x)^(1/2))