Integrand size = 63, antiderivative size = 79 \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\frac {2 \sqrt {a b x-a x^2-b x^2+x^3}}{(-a+x) (-b+x)}-2 \sqrt {d} \text {arctanh}\left (\frac {x}{\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}\right ) \]
2*(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(-a+x)/(-b+x)-2*d^(1/2)*arctanh(x/d^(1/2)/ (a*b*x+(-a-b)*x^2+x^3)^(1/2))
Time = 15.39 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65 \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\frac {2 x}{\sqrt {x (-a+x) (-b+x)}}-2 \sqrt {d} \text {arctanh}\left (\frac {x}{\sqrt {d} \sqrt {x (-a+x) (-b+x)}}\right ) \]
Integrate[(-(a*b*x) + x^3)/((-a + x)*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a *b*d - (1 + a*d + b*d)*x + d*x^2)),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.
| \(\displaystyle \int \frac {x^3-a b x}{(x-a) (x-b) \sqrt {x (x-a) (x-b)} \left (-x (a d+b d+1)+a b d+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x \left (x^2-a b\right )}{(x-a) (x-b) \sqrt {x (x-a) (x-b)} \left (-x (a d+b d+1)+a b d+d x^2\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {\sqrt {x} \left (a b-x^2\right )}{(a-x) (b-x) \sqrt {x^2-(a+b) x+a b} \left (d x^2-(a d+b d+1) x+a b d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {\sqrt {x} \left (a b-x^2\right )}{(a-x) (b-x) \sqrt {x^2-(a+b) x+a b} \left (d x^2-(a d+b d+1) x+a b d\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x \left (a b-x^2\right )}{(a-x) (b-x) \sqrt {x^2-(a+b) x+a b} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \frac {x \left (a b-x^2\right )}{(a-x)^{3/2} (b-x)^{3/2} \left (d x^2-(a d+b d+1) x+a b d\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {b-x} \int \left (-\frac {a d+b d+1}{d^2 (a-x)^{3/2} (b-x)^{3/2}}-\frac {x}{d (a-x)^{3/2} (b-x)^{3/2}}+\frac {a b d (a d+b d+1)-\left (a^2 d^2+2 a d+(b d+1)^2\right ) x}{d^2 (a-x)^{3/2} (b-x)^{3/2} \left (d x^2+(-a d-b d-1) x+a b d\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
Int[(-(a*b*x) + x^3)/((-a + x)*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a*b*d - (1 + a*d + b*d)*x + d*x^2)),x]
3.11.45.3.1 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[(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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, 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]
Time = 3.45 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\frac {-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {d}\, \sqrt {x \left (a -x \right ) \left (b -x \right )}}{x}\right ) \sqrt {x \left (a -x \right ) \left (b -x \right )}+2 x}{\sqrt {x \left (a -x \right ) \left (b -x \right )}}\) | \(60\) |
| pseudoelliptic | \(\frac {-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {d}\, \sqrt {x \left (a -x \right ) \left (b -x \right )}}{x}\right ) \sqrt {x \left (a -x \right ) \left (b -x \right )}+2 x}{\sqrt {x \left (a -x \right ) \left (b -x \right )}}\) | \(60\) |
| elliptic | \(\text {Expression too large to display}\) | \(4026\) |
int((-a*b*x+x^3)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a*b*d-(a*d+b*d+1)* x+d*x^2),x,method=_RETURNVERBOSE)
2*(-d^(1/2)*arctanh(d^(1/2)*(x*(a-x)*(b-x))^(1/2)/x)*(x*(a-x)*(b-x))^(1/2) +x)/(x*(a-x)*(b-x))^(1/2)
Time = 0.48 (sec) , antiderivative size = 395, normalized size of antiderivative = 5.00 \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\left [\frac {{\left (a b - {\left (a + b\right )} x + x^{2}\right )} \sqrt {d} \log \left (\frac {a^{2} b^{2} d^{2} + d^{2} x^{4} - 2 \, {\left ({\left (a + b\right )} d^{2} - 3 \, d\right )} x^{3} + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} - 6 \, {\left (a + b\right )} d + 1\right )} x^{2} - 4 \, {\left (a b d + d x^{2} - {\left ({\left (a + b\right )} d - 1\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} + 2 \, {\left (3 \, a b d - {\left (a^{2} b + a b^{2}\right )} d^{2}\right )} x}{a^{2} b^{2} d^{2} + d^{2} x^{4} - 2 \, {\left ({\left (a + b\right )} d^{2} + d\right )} x^{3} + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d^{2} + 2 \, {\left (a + b\right )} d + 1\right )} x^{2} - 2 \, {\left (a b d + {\left (a^{2} b + a b^{2}\right )} d^{2}\right )} x}\right ) + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{2 \, {\left (a b - {\left (a + b\right )} x + x^{2}\right )}}, \frac {{\left (a b - {\left (a + b\right )} x + x^{2}\right )} \sqrt {-d} \arctan \left (\frac {{\left (a b d + d x^{2} - {\left ({\left (a + b\right )} d - 1\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}\right ) + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{a b - {\left (a + b\right )} x + x^{2}}\right ] \]
integrate((-a*b*x+x^3)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a*b*d-(a*d+b *d+1)*x+d*x^2),x, algorithm="fricas")
[1/2*((a*b - (a + b)*x + x^2)*sqrt(d)*log((a^2*b^2*d^2 + d^2*x^4 - 2*((a + b)*d^2 - 3*d)*x^3 + ((a^2 + 4*a*b + b^2)*d^2 - 6*(a + b)*d + 1)*x^2 - 4*( a*b*d + d*x^2 - ((a + b)*d - 1)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(d) + 2*(3*a*b*d - (a^2*b + a*b^2)*d^2)*x)/(a^2*b^2*d^2 + d^2*x^4 - 2*((a + b )*d^2 + d)*x^3 + ((a^2 + 4*a*b + b^2)*d^2 + 2*(a + b)*d + 1)*x^2 - 2*(a*b* d + (a^2*b + a*b^2)*d^2)*x)) + 4*sqrt(a*b*x - (a + b)*x^2 + x^3))/(a*b - ( a + b)*x + x^2), ((a*b - (a + b)*x + x^2)*sqrt(-d)*arctan(1/2*(a*b*d + d*x ^2 - ((a + b)*d - 1)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*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))/(a*b - (a + b)*x + x^2)]
Timed out. \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\text {Timed out} \]
integrate((-a*b*x+x**3)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x))**(1/2)/(a*b*d-(a*d +b*d+1)*x+d*x**2),x)
\[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\int { -\frac {a b x - x^{3}}{{\left (a b d + d x^{2} - {\left (a d + b d + 1\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a - x\right )} {\left (b - x\right )}} \,d x } \]
integrate((-a*b*x+x^3)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a*b*d-(a*d+b *d+1)*x+d*x^2),x, algorithm="maxima")
-integrate((a*b*x - x^3)/((a*b*d + d*x^2 - (a*d + b*d + 1)*x)*sqrt((a - x) *(b - x)*x)*(a - x)*(b - x)), x)
\[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\int { -\frac {a b x - x^{3}}{{\left (a b d + d x^{2} - {\left (a d + b d + 1\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a - x\right )} {\left (b - x\right )}} \,d x } \]
integrate((-a*b*x+x^3)/(-a+x)/(-b+x)/(x*(-a+x)*(-b+x))^(1/2)/(a*b*d-(a*d+b *d+1)*x+d*x^2),x, algorithm="giac")
integrate(-(a*b*x - x^3)/((a*b*d + d*x^2 - (a*d + b*d + 1)*x)*sqrt((a - x) *(b - x)*x)*(a - x)*(b - x)), x)
Time = 6.05 (sec) , antiderivative size = 695, normalized size of antiderivative = 8.80 \[ \int \frac {-a b x+x^3}{(-a+x) (-b+x) \sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=-\frac {2\,a\,\sqrt {\frac {x}{a}}\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\middle |\frac {a}{b}\right )-\frac {a\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {x}{a}}\right )\right )}{2\,b\,\sqrt {1-\frac {x}{b}}}\right )\,\sqrt {\frac {a-x}{a}}\,\sqrt {\frac {b-x}{b}}}{\left (\frac {a}{b}-1\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1\right )}{d\,\left (b-\frac {a\,d+b\,d+\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1\right )}{d\,\left (b-\frac {a\,d+b\,d-\sqrt {a^2\,d^2-2\,a\,b\,d^2+2\,a\,d+b^2\,d^2+2\,b\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,a\,b\,\left (\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )+\frac {b\,\sin \left (2\,\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\right )}{2\,\sqrt {\frac {b-x}{a-b}+1}\,\left (a-b\right )}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\left (\frac {b}{a-b}+1\right )\,\left (a-b\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \]
int((x^3 - a*b*x)/((a - x)*(b - x)*(d*x^2 - x*(a*d + b*d + 1) + a*b*d)*(x* (a - x)*(b - x))^(1/2)),x)
- (2*a*(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))/((a /b - 1)*(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 - (a*d + b*d + (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^( 1/2)), -b/(a - b))*(a*d + b*d + (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b *d^2 + 1)^(1/2) + 1))/(d*(b - (a*d + b*d + (2*a*d + 2*b*d + a^2*d^2 + b^2* d^2 - 2*a*b*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 - (a*d + b*d - (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^(1/2)), -b/(a - b))*(a*d + b*d - (2*a*d + 2* b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 + 1)^(1/2) + 1))/(d*(b - (a*d + b*d - (2*a*d + 2*b*d + a^2*d^2 + b^2*d^2 - 2*a*b*d^2 + 1)^(1/2) + 1)/(2*d))*(x^3 - x^2*(a + b) + a*b*x)^(1/2)) - (2*a*b*(ellipticE(asin(((b - x)/b)^(1/2)) , -b/(a - b)) + (b*sin(2*asin(((b - x)/b)^(1/2))))/(2*((b - x)/(a - b) + 1 )^(1/2)*(a - b)))*(x/b)^(1/2)*((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))