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

Optimal. Leaf size=60 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}}{a-x}\right )}{\sqrt {d}} \]

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Rubi [C]  time = 19.38, antiderivative size = 455, normalized size of antiderivative = 7.58, number of steps used = 16, number of rules used = 7, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6718, 6728, 121, 120, 169, 538, 537} \begin {gather*} -\frac {2 \sqrt {b-a} \sqrt {x-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \Pi \left (-\frac {2 (a-b) d}{-2 a d+b d+c d-\sqrt {b^2 d^2-4 a d+2 b (1-c d) d+(c d+1)^2}+1};\sin ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right )|\frac {a-b}{a-c}\right )}{d \sqrt {-((a-x) (b-x) (c-x))}}-\frac {2 \sqrt {b-a} \sqrt {x-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} \Pi \left (-\frac {2 (a-b) d}{-2 a d+b d+c d+\sqrt {b^2 d^2-4 a d+2 b (1-c d) d+(c d+1)^2}+1};\sin ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right )|\frac {a-b}{a-c}\right )}{d \sqrt {-((a-x) (b-x) (c-x))}}+\frac {2 \sqrt {b-a} \sqrt {x-a} \sqrt {-\frac {b-x}{a-b}} \sqrt {-\frac {c-x}{a-c}} F\left (\sin ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {b-a}}\right )|\frac {a-b}{a-c}\right )}{d \sqrt {-((a-x) (b-x) (c-x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-a + b]*Sqrt[-((b - x)/(a - b))]*Sqrt[-((c - x)/(a - c))]*Sqrt[-a + x]*EllipticF[ArcSin[Sqrt[-a + x]/S
qrt[-a + b]], (a - b)/(a - c)])/(d*Sqrt[-((a - x)*(b - x)*(c - x))]) - (2*Sqrt[-a + b]*Sqrt[-((b - x)/(a - b))
]*Sqrt[-((c - x)/(a - c))]*Sqrt[-a + x]*EllipticPi[(-2*(a - b)*d)/(1 - 2*a*d + b*d + c*d - Sqrt[-4*a*d + b^2*d
^2 + 2*b*d*(1 - c*d) + (1 + c*d)^2]), ArcSin[Sqrt[-a + x]/Sqrt[-a + b]], (a - b)/(a - c)])/(d*Sqrt[-((a - x)*(
b - x)*(c - x))]) - (2*Sqrt[-a + b]*Sqrt[-((b - x)/(a - b))]*Sqrt[-((c - x)/(a - c))]*Sqrt[-a + x]*EllipticPi[
(-2*(a - b)*d)/(1 - 2*a*d + b*d + c*d + Sqrt[-4*a*d + b^2*d^2 + 2*b*d*(1 - c*d) + (1 + c*d)^2]), ArcSin[Sqrt[-
a + x]/Sqrt[-a + b]], (a - b)/(a - c)])/(d*Sqrt[-((a - x)*(b - x)*(c - x))])

Rule 120

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

Rule 121

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

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

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Mathematica [C]  time = 8.96, size = 308, normalized size = 5.13 \begin {gather*} -\frac {2 i (x-a)^{3/2} \sqrt {\frac {b-x}{a-x}} \sqrt {\frac {c-x}{a-x}} \left (-\Pi \left (\frac {2 (a-c) d}{2 a d-b d-c d+\sqrt {b^2 d^2-4 a d-2 b (c d-1) d+(c d+1)^2}-1};i \sinh ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {x-a}}\right )|\frac {a-c}{a-b}\right )-\Pi \left (-\frac {2 (a-c) d}{-2 a d+b d+c d+\sqrt {b^2 d^2-4 a d-2 b (c d-1) d+(c d+1)^2}+1};i \sinh ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {x-a}}\right )|\frac {a-c}{a-b}\right )+F\left (i \sinh ^{-1}\left (\frac {\sqrt {a-b}}{\sqrt {x-a}}\right )|\frac {a-c}{a-b}\right )\right )}{d \sqrt {a-b} \sqrt {(x-a) (x-b) (x-c)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.81, size = 60, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right )}{\sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 29.89, size = 379, normalized size = 6.32 \begin {gather*} \left [\frac {\log \left (\frac {b^{2} c^{2} d^{2} + d^{2} x^{4} - 6 \, a b c d - 2 \, {\left ({\left (b + c\right )} d^{2} - 3 \, d\right )} x^{3} + {\left ({\left (b^{2} + 4 \, b c + c^{2}\right )} d^{2} - 6 \, {\left (a + b + c\right )} d + 1\right )} x^{2} + a^{2} - 4 \, \sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} {\left (b c d + d x^{2} - {\left ({\left (b + c\right )} d - 1\right )} x - a\right )} \sqrt {d} - 2 \, {\left ({\left (b^{2} c + b c^{2}\right )} d^{2} - 3 \, {\left (a b + {\left (a + b\right )} c\right )} d + a\right )} x}{b^{2} c^{2} d^{2} + d^{2} x^{4} + 2 \, a b c d - 2 \, {\left ({\left (b + c\right )} d^{2} + d\right )} x^{3} + {\left ({\left (b^{2} + 4 \, b c + c^{2}\right )} d^{2} + 2 \, {\left (a + b + c\right )} d + 1\right )} x^{2} + a^{2} - 2 \, {\left ({\left (b^{2} c + b c^{2}\right )} d^{2} + {\left (a b + {\left (a + b\right )} c\right )} d + a\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (-\frac {\sqrt {-a b c - {\left (a + b + c\right )} x^{2} + x^{3} + {\left (a b + {\left (a + b\right )} c\right )} x} {\left (b c d + d x^{2} - {\left ({\left (b + c\right )} d - 1\right )} x - a\right )} \sqrt {-d}}{2 \, {\left (a b c d + {\left (a + b + c\right )} d x^{2} - d x^{3} - {\left (a b + {\left (a + b\right )} c\right )} d x\right )}}\right )}{d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b + a c - b c - 2 \, a x + x^{2}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}} {\left (b c d + d x^{2} - {\left (b d + c d + 1\right )} x + a\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.34, size = 9198, normalized size = 153.30

method	result	size
default	\(\text {Expression too large to display}\)	\(9198\)
elliptic	\(\text {Expression too large to display}\)	\(9394\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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+a*c-2*a*x-b*c+x^2)/((-a+x)*(-b+x)*(-c+x))^(1/2)/(a+b*c*d-(b*d+c*d+1)*x+d*x^2),x, algorithm="max
ima")

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

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mupad [B]  time = 0.79, size = 690, normalized size = 11.50 \begin {gather*} \frac {2\,\left (a-c\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}}{d\,\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}}+\frac {\left (a-c\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}\,\Pi \left (-\frac {a-c}{c-\frac {b\,d+c\,d+\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\left (b\,d-2\,a\,d+c\,d+\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1\right )}{d^2\,\left (c-\frac {b\,d+c\,d+\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}}+\frac {\left (a-c\right )\,\sqrt {\frac {a-x}{a-c}}\,\sqrt {-\frac {c-x}{a-c}}\,\sqrt {\frac {b-x}{b-c}}\,\Pi \left (-\frac {a-c}{c-\frac {b\,d+c\,d-\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {-\frac {c-x}{a-c}}\right )\middle |\frac {a-c}{b-c}\right )\,\left (b\,d-2\,a\,d+c\,d-\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1\right )}{d^2\,\left (c-\frac {b\,d+c\,d-\sqrt {b^2\,d^2-2\,b\,c\,d^2+2\,b\,d+c^2\,d^2+2\,c\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b-c\right )\,x^2+\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a - c)*ellipticF(asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c))*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c
))^(1/2)*((b - x)/(b - c))^(1/2))/(d*(x*(a*b + a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)^(1/2)) + ((a - c)*(
(a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^(1/2)*((b - x)/(b - c))^(1/2)*ellipticPi(-(a - c)/(c - (b*d + c*d +
(2*b*d - 4*a*d + 2*c*d + b^2*d^2 + c^2*d^2 - 2*b*c*d^2 + 1)^(1/2) + 1)/(2*d)), asin((-(c - x)/(a - c))^(1/2)),
(a - c)/(b - c))*(b*d - 2*a*d + c*d + (2*b*d - 4*a*d + 2*c*d + b^2*d^2 + c^2*d^2 - 2*b*c*d^2 + 1)^(1/2) + 1))
/(d^2*(c - (b*d + c*d + (2*b*d - 4*a*d + 2*c*d + b^2*d^2 + c^2*d^2 - 2*b*c*d^2 + 1)^(1/2) + 1)/(2*d))*(x*(a*b
+ a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)^(1/2)) + ((a - c)*((a - x)/(a - c))^(1/2)*(-(c - x)/(a - c))^(1/
2)*((b - x)/(b - c))^(1/2)*ellipticPi(-(a - c)/(c - (b*d + c*d - (2*b*d - 4*a*d + 2*c*d + b^2*d^2 + c^2*d^2 -
2*b*c*d^2 + 1)^(1/2) + 1)/(2*d)), asin((-(c - x)/(a - c))^(1/2)), (a - c)/(b - c))*(b*d - 2*a*d + c*d - (2*b*d
- 4*a*d + 2*c*d + b^2*d^2 + c^2*d^2 - 2*b*c*d^2 + 1)^(1/2) + 1))/(d^2*(c - (b*d + c*d - (2*b*d - 4*a*d + 2*c*
d + b^2*d^2 + c^2*d^2 - 2*b*c*d^2 + 1)^(1/2) + 1)/(2*d))*(x*(a*b + a*c + b*c) - x^2*(a + b + c) + x^3 - a*b*c)
^(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+a*c-2*a*x-b*c+x**2)/((-a+x)*(-b+x)*(-c+x))**(1/2)/(a+b*c*d-(b*d+c*d+1)*x+d*x**2),x)

[Out]

Timed out

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