3.6.17 \(\int \frac {-2+x}{(1+x) \sqrt {1+3 x+a x^2+x^3}} \, dx\)

Optimal. Leaf size=40 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {3-a} x}{\sqrt {a x^2+x^3+3 x+1}}\right )}{\sqrt {3-a}} \]

________________________________________________________________________________________

Rubi [C]  time = 131.81, antiderivative size = 5437, normalized size of antiderivative = 135.92, number of steps used = 13, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {6742, 2067, 2066, 718, 419, 2081, 2080, 934, 169, 538, 537}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(-2 + x)/((1 + x)*Sqrt[1 + 3*x + a*x^2 + x^3]),x]

[Out]

(2^(1/3)*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^
2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 -
27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(4/3)]*Sqrt[(-18*2^(1/3) + 2*2^(1/3)*a^2 + (54 - 54*a +
4*a^3 - 6*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) + 2*a*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15
+ 4*a)])^(1/3) + 6*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3)*x)/(-54 + 6*a^2 + 3*2^(1
/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36
*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(2
7 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[
(-3 + a)^2*(15 + 4*a)])^(4/3)])]*Sqrt[-((2*(9 - a^2) + 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(
15 + 4*a)])^(2/3) + (2*(9 - a^2)^2)/((-27*a)/2 + a^3 + (3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)]))/2)^(2/3) +
18*(a/3 + x)^2 + (2^(1/3)*(18 - 2*a^2 - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2
/3))*(a + 3*x))/(-27*a + 2*a^3 + 3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)]))^(1/3))/((18 - 2*a^2 - 2^(1/3)*(27
- 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3))^2/(18*2^(1/3)*(-27*a + 2*a^3 + 3*(9 - Sqrt[3]*S
qrt[(3 - a)^2*(15 + 4*a)]))^(2/3)) - (2*(2*(9 - a^2) + 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(
15 + 4*a)])^(2/3) + (2*(9 - a^2)^2)/((-27*a)/2 + a^3 + (3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)]))/2)^(2/3)))
/9))]*EllipticF[ArcSin[Sqrt[(18*2^(1/3) - 2*2^(1/3)*a^2 - (54 - 54*a + 4*a^3 - 6*Sqrt[3]*Sqrt[(-3 + a)^2*(15 +
4*a)])^(2/3) + 4*a*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3) + Sqrt[6]*Sqrt[-162*2^(2
/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) +
4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[
3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(4/3)] + 12*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3)*
x)/Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15
+ 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a
+ 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(4/3)]]/(2^(3/4)*3^(1/4))], (2*2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2
/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) + 4
*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]
*Sqrt[(3 - a)^2*(15 + 4*a)])^(4/3)])/(54 - 6*a^2 - 3*2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15
+ 4*a)])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 -
3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)]
)^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(4/3)])])/(3*Sqrt[3]*(27 - 27*a +
2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3)*Sqrt[1 + 3*x + a*x^2 + x^3]) - (6*2^(1/6)*Sqrt[3]*Sqrt[-
54 + 6*a^2 + 3*2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt
[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])
^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 -
3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(4/3)]]*Sqrt[a/3 + (-18 + 2*a^2 + 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3
]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3))/(3*2^(2/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(
1/3)) + x]*Sqrt[-((Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt
[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/
3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(4/3)])/(27 - 27*a + 2*a^3 - 3*Sqrt[3
]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3)) - (-18 + 2*a^2 + 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*
(15 + 4*a)])^(2/3))/((-27*a)/2 + a^3 - (3*(-9 + Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)]))/2)^(1/3) + 4*(a + 3*x)]*
Sqrt[(Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3
+ a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*
(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(4/3)])/(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 +
a)^2*(15 + 4*a)])^(1/3) - (-18 + 2*a^2 + 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(
2/3))/((-27*a)/2 + a^3 - (3*(-9 + Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)]))/2)^(1/3) + 4*(a + 3*x)]*Sqrt[1 - (2*(1
8 - 2*a^2 - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(2/3)*(27 - 27*a + 2
*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3)*(a + 3*x)))/(54 - 6*a^2 - 3*2^(1/3)*(27 - 27*a + 2*a^3 - 3
*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a
^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[
3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(4
/3)])]*Sqrt[1 + (2*(18 - 2*a^2 - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2
^(2/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3)*(a + 3*x)))/(-54 + 6*a^2 + 3*2^(1/3)*
(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(
2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 -
27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3
+ a)^2*(15 + 4*a)])^(4/3)])]*EllipticPi[(54 - 6*a^2 - 3*2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*
(15 + 4*a)])^(2/3) - 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^
3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*
a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(4/3)])/(2*(18 - 2*a^2 + 3*2^(
2/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(1/3) - 2^(2/3)*a*(27 - 27*a + 2*a^3 - 3*Sqrt[
3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(1/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3
))), ArcSin[(2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(1/6)*Sqrt[(-18 + 2*a^2 + 2^(1
/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3))/((-27*a)/2 + a^3 - (3*(-9 + Sqrt[3]*Sqr
t[(-3 + a)^2*(15 + 4*a)]))/2)^(1/3) + 2*(a + 3*x)])/Sqrt[-54 + 6*a^2 + 3*2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3
]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*
(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(
3 - a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(4/3)]]], (54
- 6*a^2 - 3*2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) - 2^(1/6)*Sqrt[3]*Sqrt[-
162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(
2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3
*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(4/3)])/(54 - 6*a^2 - 3*2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 -
a)^2*(15 + 4*a)])^(2/3) + 2^(1/6)*Sqrt[3]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a +
2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15
+ 4*a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(4/3)])])/((-27*a + 2*a^3
+ 3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)]))^(1/6)*(6 - 2*a + (18 - 2*a^2 - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*S
qrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)])^(2/3))/((-27*a)/2 + a^3 + (3*(9 - Sqrt[3]*Sqrt[(3 - a)^2*(15 + 4*a)]))/2)^(
1/3))*Sqrt[1 + 3*x + a*x^2 + x^3]*Sqrt[(18*2^(1/3) - 2*2^(1/3)*a^2 - (54 - 54*a + 4*a^3 - 6*Sqrt[3]*Sqrt[(-3 +
a)^2*(15 + 4*a)])^(2/3) - Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3
- 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*
a)])^(2/3) - 2^(1/3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(4/3)] + 4*(27 - 27*a + 2*a^3
- 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3)*(a + 3*x))/(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15
+ 4*a)])^(1/3)]*Sqrt[(18*2^(1/3) - 2*2^(1/3)*a^2 - (54 - 54*a + 4*a^3 - 6*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])
^(2/3) + Sqrt[6]*Sqrt[-162*2^(2/3) + 36*2^(2/3)*a^2 - 2*2^(2/3)*a^4 - 36*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(
-3 + a)^2*(15 + 4*a)])^(2/3) + 4*a^2*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(2/3) - 2^(1/
3)*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(4/3)] + 4*(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[
(-3 + a)^2*(15 + 4*a)])^(1/3)*(a + 3*x))/(27 - 27*a + 2*a^3 - 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3)])

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 419

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

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 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 934

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 2066

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27
*a^2*d^2], 3]}, Dist[(a + b*x + d*x^3)^p/(Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (1
2^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p), I
nt[Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1
/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, p}, x] && NeQ[
4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2067

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x,
x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rule 2080

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[(a + b*x + d*x^3)^p/(Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d
*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1
/3))*x + d^2*x^2, x]^p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12
^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x],
x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {-2+x}{(1+x) \sqrt {1+3 x+a x^2+x^3}} \, dx &=\int \left (\frac {1}{\sqrt {1+3 x+a x^2+x^3}}-\frac {3}{(1+x) \sqrt {1+3 x+a x^2+x^3}}\right ) \, dx\\ &=-\left (3 \int \frac {1}{(1+x) \sqrt {1+3 x+a x^2+x^3}} \, dx\right )+\int \frac {1}{\sqrt {1+3 x+a x^2+x^3}} \, dx\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{\left (\frac {3-a}{3}+x\right ) \sqrt {\frac {1}{27} \left (27-27 a+2 a^3\right )+\frac {1}{3} \left (9-a^2\right ) x+x^3}} \, dx,x,\frac {a}{3}+x\right )\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{27} \left (27-27 a+2 a^3\right )+\frac {1}{3} \left (9-a^2\right ) x+x^3}} \, dx,x,\frac {a}{3}+x\right )\\ &=\frac {\left (\sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x} \sqrt {2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+18 \left (\frac {a}{3}+x\right )^2+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+x} \sqrt {\frac {1}{18} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}\right )+\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) x}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+x^2}} \, dx,x,\frac {a}{3}+x\right )}{3 \sqrt {2} \sqrt {1+3 x+a x^2+x^3}}-\frac {\left (\sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x} \sqrt {2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+18 \left (\frac {a}{3}+x\right )^2+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {3-a}{3}+x\right ) \sqrt {-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+x} \sqrt {\frac {1}{18} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}\right )+\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) x}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+x^2}} \, dx,x,\frac {a}{3}+x\right )}{\sqrt {2} \sqrt {1+3 x+a x^2+x^3}}\\ &=-\frac {\left (\sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x} \sqrt {-\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)} \sqrt {\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {3-a}{3}+x\right ) \sqrt {-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+x} \sqrt {\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}-\frac {\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt {6} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+2 x} \sqrt {\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+\frac {\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt {6} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+2 x}} \, dx,x,\frac {a}{3}+x\right )}{2 \sqrt {1+3 x+a x^2+x^3}}+\frac {\left (\sqrt [3]{2} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {-18 \sqrt [3]{2}+2 \sqrt [3]{2} a^2+\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+2 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+6 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{-54+6 a^2+3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {-\frac {2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+18 \left (\frac {a}{3}+x\right )^2+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}}{\frac {\left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right )^2}{18 \sqrt [3]{2} \left (-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}-\frac {2}{9} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}\right )}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {\sqrt {\frac {2}{3}} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}} x^2}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} \left (-\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right )}{3 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}-\frac {\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt {6} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}\right )}}} \, dx,x,\frac {\sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )}{3 \sqrt {3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} \sqrt {1+3 x+a x^2+x^3}}\\ &=\frac {2^{5/6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-2 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-6 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {\frac {\left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+2 (a+3 x)^2\right )}{162\ 2^{2/3}-36\ 2^{2/3} a^2+2\ 2^{2/3} a^4+36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{3 \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \sqrt {1+3 x+a x^2+x^3}}+\frac {\left (\sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x} \sqrt {-\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)} \sqrt {\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (-2 (3-a)-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}\right )-x^2\right ) \sqrt {\frac {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+2 x^2} \sqrt {\frac {54 \sqrt [3]{2}-6 \sqrt [3]{2} a^2-3 \left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{6 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+2 x^2}} \, dx,x,\sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x}\right )}{\sqrt {1+3 x+a x^2+x^3}}\\ &=\frac {2^{5/6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-2 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-6 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {\frac {\left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+2 (a+3 x)^2\right )}{162\ 2^{2/3}-36\ 2^{2/3} a^2+2\ 2^{2/3} a^4+36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{3 \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \sqrt {1+3 x+a x^2+x^3}}+\frac {\left (\sqrt {6} \sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x} \sqrt {-\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)} \sqrt {\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)} \sqrt {1+\frac {2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (2 a+\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (-2 (3-a)-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}\right )-x^2\right ) \sqrt {\frac {54 \sqrt [3]{2}-6 \sqrt [3]{2} a^2-3 \left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{6 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+2 x^2} \sqrt {1+\frac {6\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x^2}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}} \, dx,x,\sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x}\right )}{\sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {1+3 x+a x^2+x^3}}\\ &=\frac {2^{5/6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-2 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-6 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {\frac {\left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+2 (a+3 x)^2\right )}{162\ 2^{2/3}-36\ 2^{2/3} a^2+2\ 2^{2/3} a^4+36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{3 \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \sqrt {1+3 x+a x^2+x^3}}+\frac {\left (6 \sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x} \sqrt {-\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)} \sqrt {\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)} \sqrt {1+\frac {2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (2 a+\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {1+\frac {2 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (2 a+\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+6 x\right )}{54 \sqrt [3]{2}-6 \sqrt [3]{2} a^2-3 \left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{6} \left (-2 (3-a)-\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}\right )-x^2\right ) \sqrt {1+\frac {6\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x^2}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {1+\frac {12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x^2}{54 \sqrt [3]{2}-6 \sqrt [3]{2} a^2-3 \left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}} \, dx,x,\sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x}\right )}{\sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {1+3 x+a x^2+x^3} \sqrt {\frac {54 \sqrt [3]{2}-6 \sqrt [3]{2} a^2-3 \left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+2 \left (2 a+\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+6 x\right )}}\\ &=\frac {2^{5/6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}} \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-2 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-6 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {\frac {\left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3} \left (2 \left (9-a^2\right )+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\frac {2 \left (9-a^2\right )^2}{\left (-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )\right )^{2/3}}+\frac {\sqrt [3]{2} \left (18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right ) (a+3 x)}{\sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}+2 (a+3 x)^2\right )}{162\ 2^{2/3}-36\ 2^{2/3} a^2+2\ 2^{2/3} a^4+36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} x}{\sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{3 \sqrt [3]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \sqrt {1+3 x+a x^2+x^3}}-\frac {6 \sqrt [6]{2} \sqrt {3} \sqrt {-54+6 a^2+3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}} \sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x} \sqrt {-\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)} \sqrt {\frac {\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}-\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3-\frac {3}{2} \left (-9+\sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )}}+4 (a+3 x)} \sqrt {1+\frac {2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (2 a+\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+6 x\right )}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \sqrt {1+\frac {2 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \left (2 a+\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+6 x\right )}{54 \sqrt [3]{2}-6 \sqrt [3]{2} a^2-3 \left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}} \Pi \left (\frac {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{2 \left (18-2 a^2+3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-2^{2/3} a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}\right )};\sin ^{-1}\left (\frac {2^{5/6} \sqrt {3} \sqrt [6]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}} \sqrt {\frac {a}{3}+\frac {-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}}{3\ 2^{2/3} \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+x}}{\sqrt {-54+6 a^2+3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}}\right )|\frac {54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{54-6 a^2-3 \sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt [6]{2} \sqrt {3} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}\right )}{\sqrt [6]{-27 a+2 a^3+3 \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )} \left (2 (3-a)+\frac {18-2 a^2-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}}{\sqrt [3]{-\frac {27 a}{2}+a^3+\frac {3}{2} \left (9-\sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )}}\right ) \sqrt {\frac {18 \sqrt [3]{2}-2 \sqrt [3]{2} a^2-\left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}-\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{4/3}}+12 \sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}} x}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}} \sqrt {1+3 x+a x^2+x^3} \sqrt {\frac {54 \sqrt [3]{2}-6 \sqrt [3]{2} a^2-3 \left (54-54 a+4 a^3-6 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+\sqrt {6} \sqrt {-162 2^{2/3}+36\ 2^{2/3} a^2-2\ 2^{2/3} a^4-36 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}+4 a^2 \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{2/3}-\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}\right )^{4/3}}}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(3-a)^2 (15+4 a)}}}+2 \left (2 a+\frac {\sqrt [3]{2} \left (-18+2 a^2+\sqrt [3]{2} \left (27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}\right )^{2/3}\right )}{\sqrt [3]{27-27 a+2 a^3-3 \sqrt {3} \sqrt {(-3+a)^2 (15+4 a)}}}+6 x\right )}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.76, size = 824, normalized size = 20.60 \begin {gather*} \frac {2 \sqrt {-\frac {\left (x-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,2\right ]+1\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]+1\right )}{\left (\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]\right )^2}} \left (\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]\right ) \sqrt {\frac {x-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,1\right ]+1}{\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,1\right ]}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {x-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]+1}{\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]}}\right )|\frac {\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]}\right ) \text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]-3 \Pi \left (1-\frac {\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,2\right ]}{\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]};\sin ^{-1}\left (\sqrt {\frac {x-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]+1}{\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]}}\right )|\frac {\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]}\right )\right )}{\sqrt {x^3+a x^2+3 x+1} \text {Root}\left [\text {$\#$1}^3+(a-3) \text {$\#$1}^2+(6-2 a) \text {$\#$1}+a-3\&,3\right ]} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + x)/((1 + x)*Sqrt[1 + 3*x + a*x^2 + x^3]),x]

[Out]

(2*Sqrt[-(((1 + x - Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 2])*(1 + x - Root[-3 + a + (6 - 2*a)
*#1 + (-3 + a)*#1^2 + #1^3 & , 3]))/(Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 2] - Root[-3 + a +
(6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3])^2)]*(Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 2] - Ro
ot[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3])*Sqrt[(1 + x - Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1
^2 + #1^3 & , 1])/(-Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 1] + Root[-3 + a + (6 - 2*a)*#1 + (-
3 + a)*#1^2 + #1^3 & , 3])]*(-3*EllipticPi[1 - Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 2]/Root[-
3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3], ArcSin[Sqrt[(1 + x - Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)
*#1^2 + #1^3 & , 3])/(Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 2] - Root[-3 + a + (6 - 2*a)*#1 +
(-3 + a)*#1^2 + #1^3 & , 3])]], (Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 2] - Root[-3 + a + (6 -
2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3])/(Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 1] - Root[-3 +
a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3])] + EllipticF[ArcSin[Sqrt[(1 + x - Root[-3 + a + (6 - 2*a)*#1 +
(-3 + a)*#1^2 + #1^3 & , 3])/(Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 2] - Root[-3 + a + (6 - 2
*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3])]], (Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 2] - Root[-3 +
a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3])/(Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 1] - R
oot[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3])]*Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 &
, 3]))/(Sqrt[1 + 3*x + a*x^2 + x^3]*Root[-3 + a + (6 - 2*a)*#1 + (-3 + a)*#1^2 + #1^3 & , 3])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.11, size = 40, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {3-a} x}{\sqrt {1+3 x+a x^2+x^3}}\right )}{\sqrt {3-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-2 + x)/((1 + x)*Sqrt[1 + 3*x + a*x^2 + x^3]),x]

[Out]

(-2*ArcTan[(Sqrt[3 - a]*x)/Sqrt[1 + 3*x + a*x^2 + x^3]])/Sqrt[3 - a]

________________________________________________________________________________________

fricas [B]  time = 0.53, size = 221, normalized size = 5.52 \begin {gather*} \left [\frac {\log \left (\frac {2 \, {\left (4 \, a - 9\right )} x^{5} + x^{6} + {\left (8 \, a^{2} - 24 \, a + 15\right )} x^{4} + 4 \, {\left (6 \, a - 13\right )} x^{3} + {\left (8 \, a - 9\right )} x^{2} - 4 \, {\left ({\left (2 \, a - 3\right )} x^{3} + x^{4} + 3 \, x^{2} + x\right )} \sqrt {a x^{2} + x^{3} + 3 \, x + 1} \sqrt {a - 3} + 6 \, x + 1}{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1}\right )}{2 \, \sqrt {a - 3}}, \frac {\sqrt {-a + 3} \arctan \left (\frac {{\left ({\left (2 \, a - 3\right )} x^{2} + x^{3} + 3 \, x + 1\right )} \sqrt {a x^{2} + x^{3} + 3 \, x + 1} \sqrt {-a + 3}}{2 \, {\left ({\left (a - 3\right )} x^{4} + {\left (a^{2} - 3 \, a\right )} x^{3} + 3 \, {\left (a - 3\right )} x^{2} + {\left (a - 3\right )} x\right )}}\right )}{a - 3}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+x)/(1+x)/(a*x^2+x^3+3*x+1)^(1/2),x, algorithm="fricas")

[Out]

[1/2*log((2*(4*a - 9)*x^5 + x^6 + (8*a^2 - 24*a + 15)*x^4 + 4*(6*a - 13)*x^3 + (8*a - 9)*x^2 - 4*((2*a - 3)*x^
3 + x^4 + 3*x^2 + x)*sqrt(a*x^2 + x^3 + 3*x + 1)*sqrt(a - 3) + 6*x + 1)/(x^6 + 6*x^5 + 15*x^4 + 20*x^3 + 15*x^
2 + 6*x + 1))/sqrt(a - 3), sqrt(-a + 3)*arctan(1/2*((2*a - 3)*x^2 + x^3 + 3*x + 1)*sqrt(a*x^2 + x^3 + 3*x + 1)
*sqrt(-a + 3)/((a - 3)*x^4 + (a^2 - 3*a)*x^3 + 3*(a - 3)*x^2 + (a - 3)*x))/(a - 3)]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 2}{\sqrt {a x^{2} + x^{3} + 3 \, x + 1} {\left (x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+x)/(1+x)/(a*x^2+x^3+3*x+1)^(1/2),x, algorithm="giac")

[Out]

integrate((x - 2)/(sqrt(a*x^2 + x^3 + 3*x + 1)*(x + 1)), x)

________________________________________________________________________________________

maple [C]  time = 0.70, size = 3006, normalized size = 75.15

method	result	size
default	\(\text {Expression too large to display}\)	\(3006\)
elliptic	\(\text {Expression too large to display}\)	\(3006\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2+x)/(1+x)/(a*x^2+x^3+3*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+1
2*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))*(-I*(x+1/12*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/
3)-3*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+1/3*a+1/2*I*3^(1/2)*(1/6*(108*a-10
8-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^
(1/2))^(1/3)))*3^(1/2)/(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-10
8-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)))^(1/2)*((x-1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405
)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+1/3*a)/(-1/4*(108*a-10
8-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+9*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^
(1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*
a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))))^(1/2)*(I*(x+1/12*(108*a-108-8*a^3+12*(12*a^3-27*a^2-1
62*a+405)^(1/2))^(1/3)-3*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+1/3*a-1/2*I*3^
(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^
3-27*a^2-162*a+405)^(1/2))^(1/3)))*3^(1/2)/(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(
1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)))^(1/2)/(a*x^2+x^3+3*x+1)^(1/2)*Elliptic
F(1/3*3^(1/2)*(-I*(x+1/12*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)-3*(1-1/9*a^2)/(108*a-108-
8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+1/3*a+1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-16
2*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)))*3^(1/2)/(1/6*
(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-16
2*a+405)^(1/2))^(1/3)))^(1/2),(-I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1
-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))/(-1/4*(108*a-108-8*a^3+12*(12*a^3-27*a^2
-162*a+405)^(1/2))^(1/3)+9*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)-1/2*I*3^(1/2
)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27
*a^2-162*a+405)^(1/2))^(1/3))))^(1/2))-2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(
1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))*(-I*(x+1/12*(108*a-108-8*a^3+12
*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)-3*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/
3)+1/3*a+1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-10
8-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)))*3^(1/2)/(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)
^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)))^(1/2)*((x-1/6*(108*a-
108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405
)^(1/2))^(1/3)+1/3*a)/(-1/4*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+9*(1-1/9*a^2)/(108*a-10
8-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+
405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))))^(1/2)*(I*(x+1/12
*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)-3*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-1
62*a+405)^(1/2))^(1/3)+1/3*a-1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(
1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)))*3^(1/2)/(1/6*(108*a-108-8*a^3+12*(12*a
^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)))^(
1/2)/(a*x^2+x^3+3*x+1)^(1/2)/(-1/12*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+3*(1-1/9*a^2)/(
108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)-1/3*a-1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3
-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))+1)*E
llipticPi(1/3*3^(1/2)*(-I*(x+1/12*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)-3*(1-1/9*a^2)/(10
8*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+1/3*a+1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-2
7*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)))*3^(1/
2)/(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-2
7*a^2-162*a+405)^(1/2))^(1/3)))^(1/2),-I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/
3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))/(-1/12*(108*a-108-8*a^3+12*(12*a^
3-27*a^2-162*a+405)^(1/2))^(1/3)+3*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)-1/3*
a-1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3
+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))+1),(-I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^
(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))/(-1/4*(108*a-108-8*a^3+
12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+9*(1-1/9*a^2)/(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(
1/3)-1/2*I*3^(1/2)*(1/6*(108*a-108-8*a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3)+6*(1-1/9*a^2)/(108*a-108-8*
a^3+12*(12*a^3-27*a^2-162*a+405)^(1/2))^(1/3))))^(1/2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 2}{\sqrt {a x^{2} + x^{3} + 3 \, x + 1} {\left (x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+x)/(1+x)/(a*x^2+x^3+3*x+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((x - 2)/(sqrt(a*x^2 + x^3 + 3*x + 1)*(x + 1)), x)

________________________________________________________________________________________

mupad [B]  time = 0.69, size = 62, normalized size = 1.55 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {x^3+a\,x^2+3\,x+1}+x\,\sqrt {a-3}\right )\,{\left (\sqrt {x^3+a\,x^2+3\,x+1}-x\,\sqrt {a-3}\right )}^3}{{\left (x+1\right )}^6}\right )}{\sqrt {a-3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 2)/((x + 1)*(3*x + a*x^2 + x^3 + 1)^(1/2)),x)

[Out]

log((((3*x + a*x^2 + x^3 + 1)^(1/2) + x*(a - 3)^(1/2))*((3*x + a*x^2 + x^3 + 1)^(1/2) - x*(a - 3)^(1/2))^3)/(x
+ 1)^6)/(a - 3)^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 2}{\left (x + 1\right ) \sqrt {a x^{2} + x^{3} + 3 x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+x)/(1+x)/(a*x**2+x**3+3*x+1)**(1/2),x)

[Out]

Integral((x - 2)/((x + 1)*sqrt(a*x**2 + x**3 + 3*x + 1)), x)

________________________________________________________________________________________