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

Optimal. Leaf size=44 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^2 (-a-b)+a b x+x^3}}{\sqrt {d} (a-x)}\right )}{\sqrt {d}} \]

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Rubi [C]  time = 6.25, antiderivative size = 377, normalized size of antiderivative = 8.57, number of steps used = 15, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {6718, 6728, 117, 116, 169, 538, 537} \begin {gather*} \frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \left (\sqrt {-4 a d+b^2+2 b d+d^2}+2 a-b-d\right ) \Pi \left (\frac {2 a}{b+d-\sqrt {b^2+2 d b+d^2-4 a d}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\left (-\sqrt {-4 a d+b^2+2 b d+d^2}+b+d\right ) \sqrt {x (a-x) (b-x)}}+\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \left (-\sqrt {-4 a d+b^2+2 b d+d^2}+2 a-b-d\right ) \Pi \left (\frac {2 a}{b+d+\sqrt {b^2+2 d b+d^2-4 a d}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\left (\sqrt {-4 a d+b^2+2 b d+d^2}+b+d\right ) \sqrt {x (a-x) (b-x)}}+\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a-x) (b-x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*b - 2*a*x + x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a*d - (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]*(2*a - b - d + Sqrt[b^2 - 4*a*d + 2*b*d + d^2])*Sqrt[x]*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticPi[
(2*a)/(b + d - Sqrt[b^2 - 4*a*d + 2*b*d + d^2]), ArcSin[Sqrt[x]/Sqrt[a]], a/b])/((b + d - Sqrt[b^2 - 4*a*d + 2
*b*d + d^2])*Sqrt[(a - x)*(b - x)*x]) + (2*Sqrt[a]*(2*a - b - d - Sqrt[b^2 - 4*a*d + 2*b*d + d^2])*Sqrt[x]*Sqr
t[1 - x/a]*Sqrt[1 - x/b]*EllipticPi[(2*a)/(b + d + Sqrt[b^2 - 4*a*d + 2*b*d + d^2]), ArcSin[Sqrt[x]/Sqrt[a]],
a/b])/((b + d + Sqrt[b^2 - 4*a*d + 2*b*d + d^2])*Sqrt[(a - x)*(b - x)*x])

Rule 116

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

Rule 117

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 169

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 538

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

Rule 6718

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 6728

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*} \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a d-(b+d) x+x^2\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {a b-2 a x+x^2}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a d-(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 {a (b-d)-(2 a-b-d) x}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a d+(-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 {a (b-d)-(2 a-b-d) x}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a d+(-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 {-2 a+b+d+\sqrt {b^2-4 a d+2 b d+d^2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-b-d-\sqrt {b^2-4 a d+2 b d+d^2}+2 x\right )}+\frac {-2 a+b+d-\sqrt {b^2-4 a d+2 b d+d^2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-b-d+\sqrt {b^2-4 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}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}+\frac {\left (\left (-2 a+b+d-\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-b-d+\sqrt {b^2-4 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-2 a+b+d+\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-b-d-\sqrt {b^2-4 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}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-2 a+b+d-\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+d-\sqrt {b^2-4 a d+2 b d+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 (-2 a+b+d+\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+d+\sqrt {b^2-4 a d+2 b d+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}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-2 a+b+d-\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {-b+x} \sqrt {1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+d-\sqrt {b^2-4 a d+2 b d+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 (-2 a+b+d+\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {-b+x} \sqrt {1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+d+\sqrt {b^2-4 a d+2 b d+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}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-2 a+b+d-\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+d-\sqrt {b^2-4 a d+2 b d+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 (-2 a+b+d+\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+d+\sqrt {b^2-4 a d+2 b d+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}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\sqrt {(a-x) (b-x) x}}+\frac {2 \sqrt {a} \left (2 a-b-d+\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \Pi \left (\frac {2 a}{b+d-\sqrt {b^2-4 a d+2 b d+d^2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\left (b+d-\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {(a-x) (b-x) x}}+\frac {2 \sqrt {a} \left (2 a-b-d-\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \Pi \left (\frac {2 a}{b+d+\sqrt {b^2-4 a d+2 b d+d^2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{\left (b+d+\sqrt {b^2-4 a d+2 b d+d^2}\right ) \sqrt {(a-x) (b-x) x}}\\ \end {align*}

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Mathematica [C]  time = 4.81, size = 346, normalized size = 7.86 \begin {gather*} -\frac {i a x^{3/2} \sqrt {1-\frac {a}{x}} \sqrt {1-\frac {b}{x}} \left (\left (-b \left (\sqrt {-4 a d+b^2+2 b d+d^2}-2 d\right )+d \left (\sqrt {-4 a d+b^2+2 b d+d^2}-4 a+d\right )+b^2\right ) \Pi \left (\frac {2 d}{b+d-\sqrt {b^2+2 d b+d^2-4 a d}};i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )-\left (b \left (\sqrt {-4 a d+b^2+2 b d+d^2}+2 d\right )+d \left (-\sqrt {-4 a d+b^2+2 b d+d^2}-4 a+d\right )+b^2\right ) \Pi \left (\frac {2 d}{b+d+\sqrt {b^2+2 d b+d^2-4 a d}};i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )+2 b \sqrt {d (d-4 a)+b^2+2 b d} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )\right )}{(-a)^{3/2} d \sqrt {d (d-4 a)+b^2+2 b d} \sqrt {x (x-a) (x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*a*Sqrt[1 - a/x]*Sqrt[1 - b/x]*x^(3/2)*(2*b*Sqrt[b^2 + 2*b*d + d*(-4*a + d)]*EllipticF[I*ArcSinh[Sqrt[-a]
/Sqrt[x]], b/a] + (b^2 - b*(-2*d + Sqrt[b^2 - 4*a*d + 2*b*d + d^2]) + d*(-4*a + d + Sqrt[b^2 - 4*a*d + 2*b*d +
d^2]))*EllipticPi[(2*d)/(b + d - Sqrt[b^2 - 4*a*d + 2*b*d + d^2]), I*ArcSinh[Sqrt[-a]/Sqrt[x]], b/a] - (b^2 +
d*(-4*a + d - Sqrt[b^2 - 4*a*d + 2*b*d + d^2]) + b*(2*d + Sqrt[b^2 - 4*a*d + 2*b*d + d^2]))*EllipticPi[(2*d)/
(b + d + Sqrt[b^2 - 4*a*d + 2*b*d + d^2]), I*ArcSinh[Sqrt[-a]/Sqrt[x]], b/a]))/((-a)^(3/2)*d*Sqrt[b^2 + 2*b*d
+ d*(-4*a + d)]*Sqrt[x*(-a + x)*(-b + x)])

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IntegrateAlgebraic [A]  time = 0.35, size = 44, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} (a-x)}\right )}{\sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(Sqrt[d]*(a - x))])/Sqrt[d]

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fricas [B]  time = 1.36, size = 224, normalized size = 5.09 \begin {gather*} \left [\frac {\log \left (\frac {a^{2} d^{2} - 2 \, {\left (b - 3 \, d\right )} x^{3} + x^{4} + {\left (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 d + {\left (b - d\right )} x - x^{2}\right )} \sqrt {d} + 2 \, {\left (3 \, a b d - a d^{2}\right )} x}{a^{2} d^{2} - 2 \, {\left (b + d\right )} x^{3} + x^{4} + {\left (b^{2} + 2 \, {\left (a + b\right )} d + d^{2}\right )} x^{2} - 2 \, {\left (a b d + a d^{2}\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 d + {\left (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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b - 2 \, a x + x^{2}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a d - {\left (b + d\right )} x + x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.24, size = 2321, normalized size = 52.75

method	result	size
default	\(-\frac {2 a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {4 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {2 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b^{2}}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b d}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d^{2}}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {4 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {2 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b^{2}}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b d}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d^{2}}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}\)	\(2321\)
elliptic	\(-\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticF \left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {4 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {2 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b^{2}}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b d}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d^{2}}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}-\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {4 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {2 a^{2} \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b^{2}}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {2 a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b d}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) b}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}+\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d^{2}}{\sqrt {-4 a d +b^{2}+2 b d +d^{2}}\, \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}-\frac {a \sqrt {1-\frac {x}{a}}\, \sqrt {-\frac {b}{a -b}+\frac {x}{a -b}}\, \sqrt {\frac {x}{a}}\, \EllipticPi \left (\sqrt {-\frac {-a +x}{a}}, \frac {a}{a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}}, \sqrt {\frac {a}{a -b}}\right ) d}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}\, \left (a -\frac {b}{2}-\frac {d}{2}+\frac {\sqrt {-4 a d +b^{2}+2 b d +d^{2}}}{2}\right )}\)	\(2326\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*(-(-a+x)/a)^(1/2)*((-b+x)/(a-b))^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)*EllipticF((-(-a+x)/a)^
(1/2),(a/(a-b))^(1/2))+4/(-4*a*d+b^2+2*b*d+d^2)^(1/2)*a^2*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)
^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(
1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d+2*a^2*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1
/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*E
llipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))-1/(-4*a*d+b^2+
2*b*d+d^2)^(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a
-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2
*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b^2-2/(-4*a*d+b^2+2*b*d+d^2)^(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^
(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi(
(-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b*d-a*(1-1/a*x)^(1/2)*(-
1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^
2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b-1
/(-4*a*d+b^2+2*b*d+d^2)^(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+
x^3)^(1/2)/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d-1/2*
(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d^2-a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2
)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),
a/(a-1/2*b-1/2*d-1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d-4/(-4*a*d+b^2+2*b*d+d^2)^(1/2)*a^2*(1-1/
a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d+1/2*(-4*a*d
+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b
))^(1/2))*d+2*a^2*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-
1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*
b*d+d^2)^(1/2)),(a/(a-b))^(1/2))+1/(-4*a*d+b^2+2*b*d+d^2)^(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)
*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a
+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b^2+2/(-4*a*d+b^2+2*b*d+d^2)^
(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*
d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(
1/2)),(a/(a-b))^(1/2))*b*d-a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3
)^(1/2)/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4
*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*b+1/(-4*a*d+b^2+2*b*d+d^2)^(1/2)*a*(1-1/a*x)^(1/2)*(-1/(a-b)*b+1/(
a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2))*Ell
ipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1/2))*d^2-a*(1-1/a*x)^
(1/2)*(-1/(a-b)*b+1/(a-b)*x)^(1/2)*(1/a*x)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+
2*b*d+d^2)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2*b-1/2*d+1/2*(-4*a*d+b^2+2*b*d+d^2)^(1/2)),(a/(a-b))^(1
/2))*d

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a*d-(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)^2-4*a*d>0)', see `assume
?` for more details)Is (d+b)^2-4*a*d    positive, negative or zero?

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mupad [B]  time = 0.37, size = 465, normalized size = 10.57 \begin {gather*} \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 (\left (\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}\right )\,\left (b-2\,a+d\right )+a\,b-a\,d\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 )\,\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}-\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}}\,\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 (\left (\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}}{2}\right )\,\left (b-2\,a+d\right )+a\,b-a\,d\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 )\,\sqrt {b^2+2\,b\,d+d^2-4\,a\,d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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))*((b/2 + d/2 - (2*b*d - 4*a*d + b^2 + d^2)^(1/2)/2)*(b - 2*a
+ d) + a*b - a*d))/((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*d
- 4*a*d + b^2 + d^2)^(1/2)) - (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)*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/2 + d/2 + (2*b*d - 4*a*d + b^2 + d^2)^(1/2)/2)*(b - 2*a + d) + a*b - a*d))/((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*d - 4*a*d + b^2 + d^2)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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