3.5.72 \(\int \frac {a b-x^2}{\sqrt {x (-a+x) (-b+x)} (a b d-(1+a d+b d) x+d x^2)} \, dx\)

Optimal. Leaf size=38 \[ \frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {d} \sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{\sqrt {d}} \]

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Rubi [C]  time = 8.70, antiderivative size = 308, normalized size of antiderivative = 8.11, number of steps used = 15, number of rules used = 7, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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}} \Pi \left (\frac {2 a d}{a d+b d-\sqrt {a^2 d^2+2 a (1-b d) d+(b d+1)^2}+1};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {x (a-x) (b-x)}}+\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \Pi \left (\frac {2 a d}{a d+b d+\sqrt {a^2 d^2+2 a (1-b d) d+(b d+1)^2}+1};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \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 )}{d \sqrt {x (a-x) (b-x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*b - x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a*b*d - (1 + a*d + b*d)*x + d*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])/(d*Sqrt[(a - x)*(b -
x)*x]) + (2*Sqrt[a]*Sqrt[x]*Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticPi[(2*a*d)/(1 + a*d + b*d - Sqrt[a^2*d^2 + 2*a
*d*(1 - b*d) + (1 + b*d)^2]), ArcSin[Sqrt[x]/Sqrt[a]], a/b])/(d*Sqrt[(a - x)*(b - x)*x]) + (2*Sqrt[a]*Sqrt[x]*
Sqrt[1 - x/a]*Sqrt[1 - x/b]*EllipticPi[(2*a*d)/(1 + a*d + b*d + Sqrt[a^2*d^2 + 2*a*d*(1 - b*d) + (1 + b*d)^2])
, ArcSin[Sqrt[x]/Sqrt[a]], a/b])/(d*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-x^2}{\sqrt {x (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx &=\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 d-(1+a d+b d) x+d 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}{d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}}+\frac {2 a b d-(1+a d+b d) x}{d \sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a b d+(-1-a d-b d) x+d 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}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {2 a b d-(1+a d+b d) x}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (a b d+(-1-a d-b d) x+d x^2\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \left (\frac {-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )}+\frac {-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )}\right ) \, dx}{d \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}{d \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 )}{d \sqrt {(a-x) (b-x) x}}+\frac {\left (\left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+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 (-1-a d-b d-\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \sqrt {x (-a+x) (-b+x)}}+\frac {\left (\left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+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 (-1-a d-b d+\sqrt {1+2 a d+2 b d+a^2 d^2-2 a b d^2+b^2 d^2}+2 d x\right )} \, dx}{d \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 )}{d \sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (1+a d+b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 \left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (1+a d+b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \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 )}{d \sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-b+x} \sqrt {1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+x^2} \sqrt {1-\frac {x^2}{a}} \left (1+a d+b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 \left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {-b+x} \sqrt {1-\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+x^2} \sqrt {1-\frac {x^2}{a}} \left (1+a d+b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \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 )}{d \sqrt {(a-x) (b-x) x}}-\frac {\left (2 \left (-1-a d-b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}} \left (1+a d+b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \sqrt {x (-a+x) (-b+x)}}-\frac {\left (2 \left (-1-a d-b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}\right ) \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}} \sqrt {1-\frac {x^2}{b}} \left (1+a d+b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}-2 d x^2\right )} \, dx,x,\sqrt {x}\right )}{d \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 )}{d \sqrt {(a-x) (b-x) x}}+\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \Pi \left (\frac {2 a d}{1+a d+b d-\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {(a-x) (b-x) x}}+\frac {2 \sqrt {a} \sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} \Pi \left (\frac {2 a d}{1+a d+b d+\sqrt {a^2 d^2+2 a d (1-b d)+(1+b d)^2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{d \sqrt {(a-x) (b-x) x}}\\ \end {align*}

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Mathematica [C]  time = 5.77, size = 210, normalized size = 5.53 \begin {gather*} \frac {2 i x^{3/2} \sqrt {1-\frac {a}{x}} \sqrt {1-\frac {b}{x}} \left (-\Pi \left (\frac {2 b d}{a d+b d-\sqrt {(a d+b d+1)^2-4 a b d^2}+1};i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )-\Pi \left (\frac {2 b d}{a d+b d+\sqrt {(a d+b d+1)^2-4 a b d^2}+1};i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )+F\left (i \sinh ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )\right )}{\sqrt {-a} d \sqrt {x (x-a) (x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.19, size = 38, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.86, size = 299, normalized size = 7.87 \begin {gather*} \left [\frac {\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 )}{2 \, \sqrt {d}}, -\frac {\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 )}{d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*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))/sqrt(d), -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))/d]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.32, size = 4000, normalized size = 105.26

method	result	size
default	\(\text {Expression too large to display}\)	\(4000\)
elliptic	\(\text {Expression too large to display}\)	\(4018\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*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))+1/d*(1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)*a^3*(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)/(1/2*a-1/2*b-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*
d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+
2*b*d+1)^(1/2))),(a/(a-b))^(1/2))*d-2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(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)/(1/2*a-1/2*b-1/2/d-1/2/d*(a^2*d^2-2*a*b*d
^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2
+2*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))*b*d+2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(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)/(1/2*a-1/2*b-1/2/d-1/2/d*(a^2*d^
2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^
2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))+1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(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)/(1/2*a-1/2*b-1/2/d-1/2/d*(a^2
*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b
*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))*b^2*d+2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(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)/(1/2*a-1/2*b-1/2/d-1
/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(a*d+b*d+1+(a^2*
d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))*b+1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(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)/(1/2*a-1/2*b-1/
2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(a*d+b*d+1+
(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))/d+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)/(1/2*a-1/2*b-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+
2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(
1/2))),(a/(a-b))^(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)/(1/2*a-1/2*b-1/2/d-1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/
(a-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))*b+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)/(1/2*a-1/2*b-1/2/d-1/2/d*(a^2*d^2-2*a*b*
d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-a+x)/a)^(1/2),a/(a-1/2/d*(a*d+b*d+1+(a^2*d^2-2*a*b*d^2+b^2*d^
2+2*a*d+2*b*d+1)^(1/2))),(a/(a-b))^(1/2))/d-1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)*a^3*(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)/(1/2*a-1/2*b+1/2/d*(a^2*d^2-2*a*b
*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-a*d-b*d+(a^2*d^2-2*a*b*d^2+b^
2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))*d+2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(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)/(1/2*a-1/2*b+1/2/d*(a^2*d
^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-a*d-b*d+(a^2*d^2-2*a*
b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))*b*d-2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(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)/(1/2*a-1/2*b+1/
2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-a*d-b*d+(a^
2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))-1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1
)^(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)/(1/2*a-1/2*
b+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-a*d-b*d
+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))*b^2*d-2/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a
*d+2*b*d+1)^(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)/(
1/2*a-1/2*b+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2
*(-a*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))*b+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)/(1/2*a-1/2*b+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d
^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-a*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*
d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(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)/(1/2*a-1/2*b+1/2/d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-
a+x)/a)^(1/2),a/(a+1/2*(-a*d-b*d+(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))*b+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)/(1/2*a-1/2*b+1/2/d*(a^2
*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-a*d-b*d+(a^2*d^2-2*
a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(a/(a-b))^(1/2))/d-1/(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(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)/(1/2*a-1/2*b+1/2/
d*(a^2*d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1/2/d)*EllipticPi((-(-a+x)/a)^(1/2),a/(a+1/2*(-a*d-b*d+(a^2*
d^2-2*a*b*d^2+b^2*d^2+2*a*d+2*b*d+1)^(1/2)-1)/d),(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-x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a*b*d-(a*d+b*d+1)*x+d*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((b*d+a*d+1)^2>0)', see `assume
?` for more details)Is (b*d+a*d+1)^2    -4*a*b*d^2 positive, negative or zero?

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mupad [B]  time = 0.35, size = 531, normalized size = 13.97 \begin {gather*} \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}}}{d\,\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^2\,\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^2\,\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

[Out]

Timed out

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