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

Optimal. Leaf size=54 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a-3} x \sqrt {-a x^2+x^3+3 x-1}}{a x^2-x^3-3 x+1}\right )}{\sqrt {a-3}} \]

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Rubi [C]  time = 137.71, antiderivative size = 5514, normalized size of antiderivative = 102.11, number of steps used = 13, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, 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 + 2*a*(27 - 27
*a + 2*a^3 + 3*Sqrt[3]*Sqrt[(-3 + a)^2*(15 + 4*a)])^(1/3) + (54 - 54*a + 4*a^3 + 6*Sqrt[3]*Sqrt[(-3 + a)^2*(15
+ 4*a)])^(2/3) - 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*(-1/3*a + 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]*Sqrt[(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 - 4*a*(27 - 27*a + 2*a^3 + 3*Sqrt[3]*Sqrt[(-3 + a
)^2*(15 + 4*a)])^(1/3) + (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)] + 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[-16
2*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*S
qrt[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]) - (3*2^(2/3)*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[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)])]*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)])]*Sqrt[-((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)) - 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[-((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)) + 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[-2*a + (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) + 6*x]*Ell
ipticPi[(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)*Sqr
t[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[-2*a + (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) + 6*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*Sq
rt[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*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))*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]*S
qrt[(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[-1 + 3*x - a*x^2 + x^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\\ &=3 \int \frac {1}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx+\int \frac {1}{\sqrt {-1+3 x-a x^2+x^3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (-3+a)+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 )+\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 )\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}

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Mathematica [C]  time = 1.05, size = 1148, normalized size = 21.26

result too large to display

Warning: Unable to verify antiderivative.

[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 & , 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])]*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] - 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 & , 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] - Root[3 - a + (6 - 2*a)*#1 + (3 - a)*#1^2 + #1^3 & , 3])]*(-1 + x - Root[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 & , 2])/(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])])/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])]))/Sqrt[-1 + 3*x - a*x^2 + x^3]

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IntegrateAlgebraic [A]  time = 0.13, size = 54, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {a-3} x \sqrt {-a x^2+x^3+3 x-1}}{a x^2-x^3-3 x+1}\right )}{\sqrt {a-3}} \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])/(1 - 3*x + a*x^2 - x^3)])/Sqrt[-3 + a]

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fricas [A]  time = 0.49, size = 234, normalized size = 4.33 \begin {gather*} \left [-\frac {\sqrt {-a + 3} \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 \, {\left (a - 3\right )}}, \frac {\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 )}{\sqrt {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*sqrt(-a + 3)*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))/(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))/sqrt(a - 3)]

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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)

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maple [C]  time = 0.70, size = 3008, normalized size = 55.70 \begin {gather*} \text {output too large to display} \end {gather*}

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)

[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
+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+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)*((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+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^2-162*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+1
08+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+40
5)^(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)*EllipticF(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^2-162*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),(-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+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)*((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+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^2-162*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)))*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)/(-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)*EllipticPi(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^2-162*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),-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))

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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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x+2}{\left (x-1\right )\,\sqrt {x^3-a\,x^2+3\,x-1}} \,d x \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]

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

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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)

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