\(\int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} (a b-(a+b+d) x+x^2)} \, dx\) [471]

Optimal result

Integrand size = 42, antiderivative size = 38 \[ \int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt {d}} \]

[Out]

2*arctanh(d^(1/2)*x/(a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 4.35 (sec) , antiderivative size = 273, normalized size of antiderivative = 7.18, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6850, 6860, 118, 117, 175, 552, 551} \[ \int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b+d-\sqrt {a^2-2 (b-d) a+(b+d)^2}},\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {x (a-x) (b-x)}}+\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b+d+\sqrt {a^2-2 (b-d) a+(b+d)^2}},\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {x (a-x) (b-x)}}-\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {x (a-x) (b-x)}} \]

[In]

Int[(a*b - x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a*b - (a + b + d)*x + x^2)),x]

[Out]

(-2*Sqrt[a]*Sqrt[x]*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[a]], a/b])/Sqrt[(a - x)*(b - x)*
x] + (2*Sqrt[a]*Sqrt[x]*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticPi[(2*a)/(a + b + d - Sqrt[a^2 - 2*a*(b - d) + (b
+ d)^2]), ArcSin[Sqrt[x]/Sqrt[a]], a/b])/Sqrt[(a - x)*(b - x)*x] + (2*Sqrt[a]*Sqrt[x]*Sqrt[1 - x/a]*Sqrt[1 - x
/b]*EllipticPi[(2*a)/(a + b + d + Sqrt[a^2 - 2*a*(b - d) + (b + d)^2]), ArcSin[Sqrt[x]/Sqrt[a]], a/b])/Sqrt[(a
- x)*(b - x)*x]

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])

Rule 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[1 + d*(x/c)]*
(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x])), Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 175

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 6850

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n*z^q)^FracP
art[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p]))), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !FreeQ[w, x] &&  !FreeQ[z, x]

Rule 6860

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] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {a b-x^2}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a b-(a+b+d) x+x^2\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \left (-\frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}}+\frac {2 a b-(a+b+d) x}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a b+(-a-b-d) x+x^2\right )}\right ) \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}} \, dx}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {2 a b-(a+b+d) x}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a b+(-a-b-d) x+x^2\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \left (\frac {-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )}+\frac {-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )}\right ) \, dx}{\sqrt {x (-a+x) (-b+x)}}-\frac {\left (\sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}+\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b+d+\sqrt {a^2-2 a (b-d)+(b+d)^2}-2 x^2\right ) \sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 \left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b+d-\sqrt {a^2-2 a (b-d)+(b+d)^2}-2 x^2\right ) \sqrt {-a+x^2} \sqrt {-b+x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {x} \sqrt {-b+x} \sqrt {1-\frac {x}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b+d+\sqrt {a^2-2 a (b-d)+(b+d)^2}-2 x^2\right ) \sqrt {-b+x^2} \sqrt {1-\frac {x^2}{a}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 \left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {x} \sqrt {-b+x} \sqrt {1-\frac {x}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b+d-\sqrt {a^2-2 a (b-d)+(b+d)^2}-2 x^2\right ) \sqrt {-b+x^2} \sqrt {1-\frac {x^2}{a}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b+d+\sqrt {a^2-2 a (b-d)+(b+d)^2}-2 x^2\right ) \sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 \left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b+d-\sqrt {a^2-2 a (b-d)+(b+d)^2}-2 x^2\right ) \sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = -\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}+\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b+d-\sqrt {a^2-2 a (b-d)+(b+d)^2}},\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}+\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b+d+\sqrt {a^2-2 a (b-d)+(b+d)^2}},\arcsin \left (\frac {\sqrt {x}}{\sqrt {a}}\right ),\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {x (-a+x) (-b+x)}}\right )}{\sqrt {d}} \]

[In]

Integrate[(a*b - x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a*b - (a + b + d)*x + x^2)),x]

[Out]

(2*ArcTanh[(Sqrt[d]*x)/Sqrt[x*(-a + x)*(-b + x)]])/Sqrt[d]

Maple [A] (verified)

Time = 1.94 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74

method	result	size
default	\(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \sqrt {d}}\right )}{\sqrt {d}}\)	\(28\)
pseudoelliptic	\(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \sqrt {d}}\right )}{\sqrt {d}}\)	\(28\)
elliptic	\(\text {Expression too large to display}\)	\(3397\)

[In]

int((a*b-x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a*b-(a+b+d)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

2/d^(1/2)*arctanh((x*(a-x)*(b-x))^(1/2)/x/d^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (29) = 58\).

Time = 0.35 (sec) , antiderivative size = 250, normalized size of antiderivative = 6.58 \[ \int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} b^{2} - 2 \, {\left (a + b - 3 \, d\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - 6 \, {\left (a + b\right )} d + d^{2}\right )} x^{2} + 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (a b - {\left (a + b - d\right )} x + x^{2}\right )} \sqrt {d} - 2 \, {\left (a^{2} b + a b^{2} - 3 \, a b d\right )} x}{a^{2} b^{2} - 2 \, {\left (a + b + d\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} + 2 \, {\left (a + b\right )} d + d^{2}\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2} + a b d\right )} x}\right )}{2 \, \sqrt {d}}, -\frac {\sqrt {-d} \arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (a b - {\left (a + b - d\right )} x + x^{2}\right )} \sqrt {-d}}{2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}\right )}{d}\right ] \]

[In]

integrate((a*b-x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a*b-(a+b+d)*x+x^2),x, algorithm="fricas")

[Out]

[1/2*log((a^2*b^2 - 2*(a + b - 3*d)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - 6*(a + b)*d + d^2)*x^2 + 4*sqrt(a*b*x - (
a + b)*x^2 + x^3)*(a*b - (a + b - d)*x + x^2)*sqrt(d) - 2*(a^2*b + a*b^2 - 3*a*b*d)*x)/(a^2*b^2 - 2*(a + b + d
)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + 2*(a + b)*d + d^2)*x^2 - 2*(a^2*b + a*b^2 + a*b*d)*x))/sqrt(d), -sqrt(-d)*a
rctan(1/2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(a*b - (a + b - d)*x + x^2)*sqrt(-d)/(a*b*d*x - (a + b)*d*x^2 + d*x^
3))/d]

Sympy [F(-1)]

Timed out. \[ \int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*b-x**2)/(x*(-a+x)*(-b+x))**(1/2)/(a*b-(a+b+d)*x+x**2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a*b-x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a*b-(a+b+d)*x+x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((d+b+a)^2-4*a*b>0)', see `assu
me?` for mor

Giac [F]

\[ \int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\int { \frac {a b - x^{2}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a b - {\left (a + b + d\right )} x + x^{2}\right )}} \,d x } \]

[In]

integrate((a*b-x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a*b-(a+b+d)*x+x^2),x, algorithm="giac")

[Out]

integrate((a*b - x^2)/(sqrt((a - x)*(b - x)*x)*(a*b - (a + b + d)*x + x^2)), x)

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 532, normalized size of antiderivative = 14.00 \[ \int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {2\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\left (2\,a\,b-\left (a+b+d\right )\,\left (\frac {a}{2}+\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )\right )\,\Pi \left (-\frac {b}{\frac {a}{2}-\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {a}{2}-\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )\,\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}+\frac {2\,b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\left (2\,a\,b-\left (a+b+d\right )\,\left (\frac {a}{2}+\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )\right )\,\Pi \left (-\frac {b}{\frac {a}{2}-\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {a}{2}-\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )\,\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}} \]

[In]

int((a*b - x^2)/((x*(a - x)*(b - x))^(1/2)*(a*b - x*(a + b + d) + x^2)),x)

[Out]

(2*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*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*(2*a*b - (a + b +
d)*(a/2 + b/2 + d/2 - (2*a*d - 2*a*b + 2*b*d + a^2 + b^2 + d^2)^(1/2)/2))*ellipticPi(-b/(a/2 - b/2 + d/2 - (2*
a*d - 2*a*b + 2*b*d + a^2 + b^2 + d^2)^(1/2)/2), asin(((b - x)/b)^(1/2)), -b/(a - b)))/((x^3 - x^2*(a + b) + a
*b*x)^(1/2)*(a/2 - b/2 + d/2 - (2*a*d - 2*a*b + 2*b*d + a^2 + b^2 + d^2)^(1/2)/2)*(2*a*d - 2*a*b + 2*b*d + a^2
+ b^2 + d^2)^(1/2)) + (2*b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*(2*a*b - (a + b + d)*(a/2 +
b/2 + d/2 + (2*a*d - 2*a*b + 2*b*d + a^2 + b^2 + d^2)^(1/2)/2))*ellipticPi(-b/(a/2 - b/2 + d/2 + (2*a*d - 2*a*
b + 2*b*d + a^2 + b^2 + d^2)^(1/2)/2), asin(((b - x)/b)^(1/2)), -b/(a - b)))/((x^3 - x^2*(a + b) + a*b*x)^(1/2
)*(a/2 - b/2 + d/2 + (2*a*d - 2*a*b + 2*b*d + a^2 + b^2 + d^2)^(1/2)/2)*(2*a*d - 2*a*b + 2*b*d + a^2 + b^2 + d
^2)^(1/2))