∫ −2+x______ (1+x)√ ________

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

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

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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 )\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}

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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])

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IntegrateAlgebraic [A] time = 0.13, size = 40, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {3-a} x}{\sqrt {a x^2+x^3+3 x+1}}\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]

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fricas [B] time = 0.45, 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)]

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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 = 0.75, size = 3006, normalized size = 75.15 \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+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))

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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 [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)

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