3.25.94 \(\int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9)}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx\) [2494]

Optimal. Leaf size=35 \[ e^x+\frac {1}{4} \left (1-\frac {2 \left (-3-\frac {4}{x}+2 x\right )}{x \left (-3-x+x^3\right )}\right ) \]

[Out]

1/4-1/2/(x^3-x-3)/x*(2*x-4/x-3)+exp(x)

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(9556\) vs. \(2(35)=70\).
time = 107.83, antiderivative size = 9556, normalized size of antiderivative = 273.03, number of steps used = 109, number of rules used = 22, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.234, Rules used = {6873, 12, 6820, 6874, 2225, 2090, 754, 814, 648, 632, 210, 642, 2104, 907, 652, 836, 1660, 6, 27, 21, 1600, 1642} \begin {gather*} \text {Too large to display} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(24 + 21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + E^x*(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9))/(18
*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9),x]

[Out]

E^x + 93312/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3))^3*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))
^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6^(1/3)*(2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(
1/3)) - 6*x)) + (13608*6^(1/3))/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3))^2*(6 + 6*3^(1/3)*
(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6^(1/3)*(2*(3/(27 + Sqrt[717]))^(1/3) + (2
*(27 + Sqrt[717]))^(1/3)) - 6*x)) + (648*6^(2/3))/((2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3))
*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6^(1/3)*(2*(3/(27 + Sqrt[7
17]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x)) + 124416/((12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/
3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))
^2*(6^(1/3)*(2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x)) - (2592*(58079*2^(2/3)*3^(5/6)
*Sqrt[239]*(27 + Sqrt[717])^(1/3) - 6480*2^(1/3)*3^(1/6)*Sqrt[239]*(27 + Sqrt[717])^(2/3) + 239*(348474 + 1301
4*Sqrt[717] + 6507*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) - 242*2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))))/((27 + Sqrt[
717])^(7/3)*(12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2
/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(3*(27 + Sqrt[717])^(5/3) + 2^(1/3)*((484*3^(2/
3) + 54*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(1/3) + 2*6^(1/3)*(3267 + 122*Sqrt[717])))*(6^(1/3)*(2*(3/(27 + Sq
rt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3)) - 6*x)) - 2/(3*x^2) - 7/(6*x) + (8*(27 + Sqrt[717])^3)/(27*Sqrt[
3]*(9*Sqrt[3] + Sqrt[239])^3*x) + (648*6^(2/3)*(54 + 2^(1/3)*3^(2/3)*(2*2^(1/3)*(3/(27 + Sqrt[717]))^(2/3) + (
27 + Sqrt[717])^(2/3))*x))/((12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(
6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(2*(3/(27 + Sqrt[717]))^(1/3) +
(2*(27 + Sqrt[717]))^(1/3) - 6^(2/3)*x)*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717
]))^(2/3) - 6*3^(1/3)*((6/(27 + Sqrt[717]))^(1/3) + ((27 + Sqrt[717])/2)^(1/3))*x - 18*x^2)) - (7128*6^(2/3)*(
2 + (3^(2/3)*(2/(27 + Sqrt[717]))^(1/3) + ((3*(27 + Sqrt[717]))/2)^(1/3))*x))/((12 - 6*3^(1/3)*(2/(27 + Sqrt[7
17]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 +
Sqrt[717]))^(2/3))*(2*(3/(27 + Sqrt[717]))^(1/3) + (2*(27 + Sqrt[717]))^(1/3) - 6^(2/3)*x)*(6 - 6*3^(1/3)*(2/
(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 6*3^(1/3)*((6/(27 + Sqrt[717]))^(1/3) + ((27 +
Sqrt[717])/2)^(1/3))*x - 18*x^2)) - (3*((27 + Sqrt[717])^(2/3)*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(
1/3)*(3*(27 + Sqrt[717]))^(2/3))*(6*3^(1/6)*(9*Sqrt[3] + Sqrt[239])*(2/(27 + Sqrt[717]))^(1/3) + (27 + Sqrt[71
7])*(108 + 2^(2/3)*(3*(27 + Sqrt[717]))^(1/3))) - 72*(2*(27 + Sqrt[717])^(5/3) + 6^(1/3)*((3*(27 + Sqrt[717]))
^(1/3) + 2^(1/3)*(3240 + 121*Sqrt[717])))*x))/((12 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + S
qrt[717]))^(2/3))*(3*(27 + Sqrt[717])^(5/3) + 2^(1/3)*((484*3^(2/3) + 54*3^(1/6)*Sqrt[239])*(27 + Sqrt[717])^(
1/3) + 2*6^(1/3)*(3267 + 122*Sqrt[717])))*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[71
7]))^(2/3) - 6*3^(1/3)*((6/(27 + Sqrt[717]))^(1/3) + ((27 + Sqrt[717])/2)^(1/3))*x - 18*x^2)) - (23328*(27 + S
qrt[717])^(2/3)*((270 + 10*Sqrt[717] - 4*3^(2/3)*(2/(27 + Sqrt[717]))^(1/3) - 9*2^(2/3)*3^(5/6)*Sqrt[239]*(27
+ Sqrt[717])^(1/3) - 241*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3))/(27 + Sqrt[717]) - (27 - 4*3^(2/3)*(2/(27 + Sqrt[
717]))^(1/3) - 2*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3))*x))/((2*2^(2/3)*3^(1/3) - 4*(27 + Sqrt[717])^(2/3) + 3^(1
/6)*(9*Sqrt[3] + Sqrt[239])*(2*(27 + Sqrt[717]))^(1/3))*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3
*(27 + Sqrt[717]))^(2/3))*(6 + 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6
- 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3) - 3*2^(2/3)*(6^(2/3)/(27 + Sqrt[7
17])^(1/3) + (3*(27 + Sqrt[717]))^(1/3))*x - 18*x^2)) - (181398528*((4*(273*(27 + Sqrt[717])^(1/3)*(58079 + 21
69*Sqrt[717]) - 2^(1/3)*3^(1/6)*(2103804*Sqrt[3] + 235704*Sqrt[239] + 2607075*2^(1/3)*3^(1/6)*(27 + Sqrt[717])
^(2/3) + 97363*2^(1/3)*Sqrt[239]*(3*(27 + Sqrt[717]))^(2/3))))/(27 + Sqrt[717]) - (27 + Sqrt[717])^(1/3)*(8164
80 + 30492*Sqrt[717] - 887*2^(2/3)*3^(5/6)*Sqrt[239]*(27 + Sqrt[717])^(1/3) + 2070*2^(1/3)*3^(1/6)*Sqrt[239]*(
27 + Sqrt[717])^(2/3) - 23751*2^(2/3)*(3*(27 + Sqrt[717]))^(1/3) + 18476*2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))*x
))/((27 + Sqrt[717])^(8/3)*(2*2^(2/3)*3^(1/3) - 4*(27 + Sqrt[717])^(2/3) + 3^(1/6)*(9*Sqrt[3] + Sqrt[239])*(2*
(27 + Sqrt[717]))^(1/3))*(6 - 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^3*(6
+ 6*3^(1/3)*(2/(27 + Sqrt[717]))^(2/3) + 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3))^2*(6 - 6*3^(1/3)*(2/(27 + Sqrt[71
7]))^(2/3) - 2^(1/3)*(3*(27 + Sqrt[717]))^(2/3)...

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1660

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 2090

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[1/d^(2*p), Int[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}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]

Rule 2104

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[1/d^(2*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}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6820

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

Rule 6873

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

Rule 6874

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

Rubi steps

Aborted

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Mathematica [A]
time = 1.79, size = 34, normalized size = 0.97 \begin {gather*} \frac {1}{2} \left (2 e^x+\frac {4+3 x-2 x^2}{x^2 \left (-3-x+x^3\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24 + 21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + E^x*(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9
))/(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9),x]

[Out]

(2*E^x + (4 + 3*x - 2*x^2)/(x^2*(-3 - x + x^3)))/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 736, normalized size = 21.03

method	result	size
risch	\(\frac {-x^{2}+\frac {3}{2} x +2}{x^{2} \left (x^{3}-x -3\right )}+{\mathrm e}^{x}\)	\(28\)
norman	\(\frac {2+x^{5} {\mathrm e}^{x}-x^{2}+\frac {3 x}{2}-3 \,{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x^{3}}{x^{2} \left (x^{3}-x -3\right )}\)	\(45\)
default	\(\text {Expression too large to display}\)	\(736\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*x^6+2*
x^5+12*x^4+18*x^3),x,method=_RETURNVERBOSE)

[Out]

-2/3/x^2-5/18/x+exp(x)-4/9*(-131/239*x^2+231/239*x+167/239)/(x^3-x-3)-6*(-27/239*x^2+2/239*x+18/239)/(x^3-x-3)
-4/2151*sum((239*_R^2-609*_R+940)/(3*_R^2-1)*ln(x-_R),_R=RootOf(_Z^3-_Z-3))+7/2151*sum((239*_R^2-474*_R-275)/(
3*_R^2-1)*ln(x-_R),_R=RootOf(_Z^3-_Z-3))+3*(2/239*x^2-9/239*x-81/239)/(x^3-x-3)-60/239*sum((_R-9)/(3*_R^2-1)*l
n(x-_R),_R=RootOf(_Z^3-_Z-3))+7/18*(-231/239*x^2-36/239*x+393/239)/(x^3-x-3)-6/239*sum((27*_R1^2-29*_R1-14)/(3
*_R1^2-1)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3-_Z-3))-6/239*exp(x)*(27*x^2-2*x-18)/(x^3-x-3)-11*(6/239*x^2-27
/239*x-4/239)/(x^3-x-3)-1/3*(12/239*x^2-54/239*x+231/239)/(x^3-x-3)+9/239*sum((6*_R1^2-33*_R1+50)/(3*_R1^2-1)*
exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3-_Z-3))+6/239*sum((9*_R1^2+70*_R1+75)/(3*_R1^2-1)*exp(_R1)*Ei(1,-x+_R1),_
R1=RootOf(_Z^3-_Z-3))+6/239*exp(x)*(9*x^2+79*x-6)/(x^3-x-3)-1/239*sum((106*_R1^2+612*_R1+963)/(3*_R1^2-1)*exp(
_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3-_Z-3))+2/239*sum((79*_R1^2+163*_R1+21)/(3*_R1^2-1)*exp(_R1)*Ei(1,-x+_R1),_R1
=RootOf(_Z^3-_Z-3))+9/239*exp(x)*(6*x^2-27*x-4)/(x^3-x-3)-1/239*exp(x)*(106*x^2+240*x+9)/(x^3-x-3)+2/239*exp(x
)*(79*x^2+3*x+27)/(x^3-x-3)-1/717*sum((239*_R^2+12*_R-347)/(3*_R^2-1)*ln(x-_R),_R=RootOf(_Z^3-_Z-3))+1/239*exp
(x)*(2*x^2-9*x-81)/(x^3-x-3)-6/239*sum((-27*_R+4)/(3*_R^2-1)*ln(x-_R),_R=RootOf(_Z^3-_Z-3))+1/239*sum((2*_R1^2
-11*_R1-63)/(3*_R1^2-1)*exp(_R1)*Ei(1,-x+_R1),_R1=RootOf(_Z^3-_Z-3))

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Maxima [A]
time = 0.27, size = 46, normalized size = 1.31 \begin {gather*} -\frac {2 \, x^{2} - 2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )} e^{x} - 3 \, x - 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*
x^6+2*x^5+12*x^4+18*x^3),x, algorithm="maxima")

[Out]

-1/2*(2*x^2 - 2*(x^5 - x^3 - 3*x^2)*e^x - 3*x - 4)/(x^5 - x^3 - 3*x^2)

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Fricas [A]
time = 0.35, size = 46, normalized size = 1.31 \begin {gather*} -\frac {2 \, x^{2} - 2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )} e^{x} - 3 \, x - 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*
x^6+2*x^5+12*x^4+18*x^3),x, algorithm="fricas")

[Out]

-1/2*(2*x^2 - 2*(x^5 - x^3 - 3*x^2)*e^x - 3*x - 4)/(x^5 - x^3 - 3*x^2)

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Sympy [A]
time = 0.10, size = 27, normalized size = 0.77 \begin {gather*} \frac {- 2 x^{2} + 3 x + 4}{2 x^{5} - 2 x^{3} - 6 x^{2}} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**9-4*x**7-12*x**6+2*x**5+12*x**4+18*x**3)*exp(x)+6*x**5-12*x**4-22*x**3+6*x**2+21*x+24)/(2*x**
9-4*x**7-12*x**6+2*x**5+12*x**4+18*x**3),x)

[Out]

(-2*x**2 + 3*x + 4)/(2*x**5 - 2*x**3 - 6*x**2) + exp(x)

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Giac [A]
time = 0.41, size = 49, normalized size = 1.40 \begin {gather*} \frac {2 \, x^{5} e^{x} - 2 \, x^{3} e^{x} - 6 \, x^{2} e^{x} - 2 \, x^{2} + 3 \, x + 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*
x^6+2*x^5+12*x^4+18*x^3),x, algorithm="giac")

[Out]

1/2*(2*x^5*e^x - 2*x^3*e^x - 6*x^2*e^x - 2*x^2 + 3*x + 4)/(x^5 - x^3 - 3*x^2)

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Mupad [B]
time = 1.64, size = 28, normalized size = 0.80 \begin {gather*} {\mathrm {e}}^x-\frac {-x^2+\frac {3\,x}{2}+2}{x^2\,\left (-x^3+x+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + exp(x)*(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9) + 24)/
(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9),x)

[Out]

exp(x) - ((3*x)/2 - x^2 + 2)/(x^2*(x - x^3 + 3))

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