3.11.75 \(\int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8)}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx\)

Optimal. Leaf size=24 \[ e^x+x^2-\frac {75}{x \left (-7-x-x^2\right )^2} \]

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Rubi [B]  time = 1.61, antiderivative size = 354, normalized size of antiderivative = 14.75, number of steps used = 63, number of rules used = 17, integrand size = 120, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.142, Rules used = {6741, 6742, 2194, 614, 618, 204, 740, 822, 800, 634, 628, 638, 738, 728, 722, 818, 773} \begin {gather*} \frac {50 x^3}{243}-\frac {8 (23 x+259) x^2}{81 \left (x^2+x+7\right )}+\frac {818 x^2}{243}-\frac {434 (x+14) x}{243 \left (x^2+x+7\right )}-\frac {49 (x+14) x}{9 \left (x^2+x+7\right )^2}+\frac {25 (227-32 x)}{2646 \left (x^2+x+7\right )}+\frac {32 (2 x+1)}{243 \left (x^2+x+7\right )}+\frac {98 (5 x+7)}{81 \left (x^2+x+7\right )}+\frac {25 (13-x)}{42 \left (x^2+x+7\right )^2}-\frac {343 (x+14)}{27 \left (x^2+x+7\right )^2}+\frac {125 (2 x+1)}{18 \left (x^2+x+7\right )^2}+\frac {25 (281-41 x)}{1134 \left (x^2+x+7\right ) x}+\frac {25 (13-x)}{18 \left (x^2+x+7\right )^2 x}-\frac {(x+14) x^6}{27 \left (x^2+x+7\right )^2}-\frac {(x+14) x^5}{9 \left (x^2+x+7\right )^2}-\frac {(41 x+385) x^4}{243 \left (x^2+x+7\right )}-\frac {8 (x+14) x^4}{9 \left (x^2+x+7\right )^2}-\frac {2 (16 x+161) x^3}{81 \left (x^2+x+7\right )}-\frac {43 (x+14) x^3}{27 \left (x^2+x+7\right )^2}+\frac {56 (2 x+1) x^3}{9 \left (x^2+x+7\right )^2}+\frac {980 x}{243}+e^x-\frac {11050}{3969 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(525 + 225*x + 375*x^2 + 686*x^3 + 294*x^4 + 336*x^5 + 86*x^6 + 48*x^7 + 6*x^8 + 2*x^9 + E^x*(343*x^2 + 14
7*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8))/(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8
),x]

[Out]

E^x - 11050/(3969*x) + (980*x)/243 + (818*x^2)/243 + (50*x^3)/243 + (25*(13 - x))/(42*(7 + x + x^2)^2) + (25*(
13 - x))/(18*x*(7 + x + x^2)^2) - (343*(14 + x))/(27*(7 + x + x^2)^2) - (49*x*(14 + x))/(9*(7 + x + x^2)^2) -
(43*x^3*(14 + x))/(27*(7 + x + x^2)^2) - (8*x^4*(14 + x))/(9*(7 + x + x^2)^2) - (x^5*(14 + x))/(9*(7 + x + x^2
)^2) - (x^6*(14 + x))/(27*(7 + x + x^2)^2) + (125*(1 + 2*x))/(18*(7 + x + x^2)^2) + (56*x^3*(1 + 2*x))/(9*(7 +
x + x^2)^2) + (25*(227 - 32*x))/(2646*(7 + x + x^2)) + (25*(281 - 41*x))/(1134*x*(7 + x + x^2)) - (434*x*(14
+ x))/(243*(7 + x + x^2)) + (32*(1 + 2*x))/(243*(7 + x + x^2)) + (98*(7 + 5*x))/(81*(7 + x + x^2)) - (2*x^3*(1
61 + 16*x))/(81*(7 + x + x^2)) - (8*x^2*(259 + 23*x))/(81*(7 + x + x^2)) - (x^4*(385 + 41*x))/(243*(7 + x + x^
2))

Rule 204

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

Rule 614

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

Rule 618

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 628

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 634

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 638

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)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), 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 722

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

Rule 728

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

Rule 738

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

Rule 740

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 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

Rule 800

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 818

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

Rule 822

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 2194

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 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{x^2 \left (7+x+x^2\right )^3} \, dx\\ &=\int \left (e^x+\frac {375}{\left (7+x+x^2\right )^3}+\frac {525}{x^2 \left (7+x+x^2\right )^3}+\frac {225}{x \left (7+x+x^2\right )^3}+\frac {686 x}{\left (7+x+x^2\right )^3}+\frac {294 x^2}{\left (7+x+x^2\right )^3}+\frac {336 x^3}{\left (7+x+x^2\right )^3}+\frac {86 x^4}{\left (7+x+x^2\right )^3}+\frac {48 x^5}{\left (7+x+x^2\right )^3}+\frac {6 x^6}{\left (7+x+x^2\right )^3}+\frac {2 x^7}{\left (7+x+x^2\right )^3}\right ) \, dx\\ &=2 \int \frac {x^7}{\left (7+x+x^2\right )^3} \, dx+6 \int \frac {x^6}{\left (7+x+x^2\right )^3} \, dx+48 \int \frac {x^5}{\left (7+x+x^2\right )^3} \, dx+86 \int \frac {x^4}{\left (7+x+x^2\right )^3} \, dx+225 \int \frac {1}{x \left (7+x+x^2\right )^3} \, dx+294 \int \frac {x^2}{\left (7+x+x^2\right )^3} \, dx+336 \int \frac {x^3}{\left (7+x+x^2\right )^3} \, dx+375 \int \frac {1}{\left (7+x+x^2\right )^3} \, dx+525 \int \frac {1}{x^2 \left (7+x+x^2\right )^3} \, dx+686 \int \frac {x}{\left (7+x+x^2\right )^3} \, dx+\int e^x \, dx\\ &=e^x+\frac {25 (13-x)}{42 \left (7+x+x^2\right )^2}+\frac {25 (13-x)}{18 x \left (7+x+x^2\right )^2}-\frac {343 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {49 x (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {43 x^3 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {8 x^4 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^5 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^6 (14+x)}{27 \left (7+x+x^2\right )^2}+\frac {125 (1+2 x)}{18 \left (7+x+x^2\right )^2}+\frac {56 x^3 (1+2 x)}{9 \left (7+x+x^2\right )^2}+\frac {1}{27} \int \frac {x^5 (84+3 x)}{\left (7+x+x^2\right )^2} \, dx+\frac {1}{9} \int \frac {x^4 (70+2 x)}{\left (7+x+x^2\right )^2} \, dx+\frac {25}{42} \int \frac {54-3 x}{x \left (7+x+x^2\right )^2} \, dx+\frac {8}{9} \int \frac {x^3 (56+x)}{\left (7+x+x^2\right )^2} \, dx+\frac {25}{18} \int \frac {67-4 x}{x^2 \left (7+x+x^2\right )^2} \, dx+\frac {49}{9} \int \frac {14-2 x}{\left (7+x+x^2\right )^2} \, dx-\frac {56}{3} \int \frac {x^2}{\left (7+x+x^2\right )^2} \, dx-\frac {343}{9} \int \frac {1}{\left (7+x+x^2\right )^2} \, dx+\frac {125}{3} \int \frac {1}{\left (7+x+x^2\right )^2} \, dx+\frac {602}{9} \int \frac {x^2}{\left (7+x+x^2\right )^2} \, dx\\ &=e^x+\frac {25 (13-x)}{42 \left (7+x+x^2\right )^2}+\frac {25 (13-x)}{18 x \left (7+x+x^2\right )^2}-\frac {343 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {49 x (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {43 x^3 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {8 x^4 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^5 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^6 (14+x)}{27 \left (7+x+x^2\right )^2}+\frac {125 (1+2 x)}{18 \left (7+x+x^2\right )^2}+\frac {56 x^3 (1+2 x)}{9 \left (7+x+x^2\right )^2}+\frac {25 (227-32 x)}{2646 \left (7+x+x^2\right )}+\frac {25 (281-41 x)}{1134 x \left (7+x+x^2\right )}-\frac {434 x (14+x)}{243 \left (7+x+x^2\right )}+\frac {32 (1+2 x)}{243 \left (7+x+x^2\right )}+\frac {98 (7+5 x)}{81 \left (7+x+x^2\right )}-\frac {2 x^3 (161+16 x)}{81 \left (7+x+x^2\right )}-\frac {8 x^2 (259+23 x)}{81 \left (7+x+x^2\right )}-\frac {x^4 (385+41 x)}{243 \left (7+x+x^2\right )}-\frac {1}{729} \int \frac {(-4620-450 x) x^3}{7+x+x^2} \, dx+\frac {25 \int \frac {1458-96 x}{x \left (7+x+x^2\right )} \, dx}{7938}-\frac {1}{243} \int \frac {(-2898-246 x) x^2}{7+x+x^2} \, dx+\frac {25 \int \frac {2652-246 x}{x^2 \left (7+x+x^2\right )} \, dx}{3402}-\frac {8}{243} \int \frac {(-1554-96 x) x}{7+x+x^2} \, dx-\frac {686}{243} \int \frac {1}{7+x+x^2} \, dx+\frac {250}{81} \int \frac {1}{7+x+x^2} \, dx+\frac {490}{81} \int \frac {1}{7+x+x^2} \, dx-\frac {784}{81} \int \frac {1}{7+x+x^2} \, dx+\frac {8428}{243} \int \frac {1}{7+x+x^2} \, dx\\ &=e^x+\frac {256 x}{81}+\frac {25 (13-x)}{42 \left (7+x+x^2\right )^2}+\frac {25 (13-x)}{18 x \left (7+x+x^2\right )^2}-\frac {343 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {49 x (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {43 x^3 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {8 x^4 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^5 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^6 (14+x)}{27 \left (7+x+x^2\right )^2}+\frac {125 (1+2 x)}{18 \left (7+x+x^2\right )^2}+\frac {56 x^3 (1+2 x)}{9 \left (7+x+x^2\right )^2}+\frac {25 (227-32 x)}{2646 \left (7+x+x^2\right )}+\frac {25 (281-41 x)}{1134 x \left (7+x+x^2\right )}-\frac {434 x (14+x)}{243 \left (7+x+x^2\right )}+\frac {32 (1+2 x)}{243 \left (7+x+x^2\right )}+\frac {98 (7+5 x)}{81 \left (7+x+x^2\right )}-\frac {2 x^3 (161+16 x)}{81 \left (7+x+x^2\right )}-\frac {8 x^2 (259+23 x)}{81 \left (7+x+x^2\right )}-\frac {x^4 (385+41 x)}{243 \left (7+x+x^2\right )}-\frac {1}{729} \int \left (7320-4170 x-450 x^2-\frac {30 (1708-729 x)}{7+x+x^2}\right ) \, dx+\frac {25 \int \left (\frac {1458}{7 x}-\frac {6 (355+243 x)}{7 \left (7+x+x^2\right )}\right ) \, dx}{7938}-\frac {1}{243} \int \left (-2652-246 x+\frac {6 (3094+729 x)}{7+x+x^2}\right ) \, dx+\frac {25 \int \left (\frac {2652}{7 x^2}-\frac {4374}{49 x}+\frac {6 (-2365+729 x)}{49 \left (7+x+x^2\right )}\right ) \, dx}{3402}-\frac {8}{243} \int \frac {672-1458 x}{7+x+x^2} \, dx+\frac {1372}{243} \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )-\frac {500}{81} \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )-\frac {980}{81} \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )+\frac {1568}{81} \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )-\frac {16856}{243} \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )\\ &=e^x-\frac {11050}{3969 x}+\frac {980 x}{243}+\frac {818 x^2}{243}+\frac {50 x^3}{243}+\frac {25 (13-x)}{42 \left (7+x+x^2\right )^2}+\frac {25 (13-x)}{18 x \left (7+x+x^2\right )^2}-\frac {343 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {49 x (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {43 x^3 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {8 x^4 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^5 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^6 (14+x)}{27 \left (7+x+x^2\right )^2}+\frac {125 (1+2 x)}{18 \left (7+x+x^2\right )^2}+\frac {56 x^3 (1+2 x)}{9 \left (7+x+x^2\right )^2}+\frac {25 (227-32 x)}{2646 \left (7+x+x^2\right )}+\frac {25 (281-41 x)}{1134 x \left (7+x+x^2\right )}-\frac {434 x (14+x)}{243 \left (7+x+x^2\right )}+\frac {32 (1+2 x)}{243 \left (7+x+x^2\right )}+\frac {98 (7+5 x)}{81 \left (7+x+x^2\right )}-\frac {2 x^3 (161+16 x)}{81 \left (7+x+x^2\right )}-\frac {8 x^2 (259+23 x)}{81 \left (7+x+x^2\right )}-\frac {x^4 (385+41 x)}{243 \left (7+x+x^2\right )}+\frac {15220 \tan ^{-1}\left (\frac {1+2 x}{3 \sqrt {3}}\right )}{729 \sqrt {3}}+\frac {25 \int \frac {-2365+729 x}{7+x+x^2} \, dx}{27783}-\frac {25 \int \frac {355+243 x}{7+x+x^2} \, dx}{9261}-\frac {2}{81} \int \frac {3094+729 x}{7+x+x^2} \, dx+\frac {10}{243} \int \frac {1708-729 x}{7+x+x^2} \, dx+24 \int \frac {1+2 x}{7+x+x^2} \, dx-\frac {3736}{81} \int \frac {1}{7+x+x^2} \, dx\\ &=e^x-\frac {11050}{3969 x}+\frac {980 x}{243}+\frac {818 x^2}{243}+\frac {50 x^3}{243}+\frac {25 (13-x)}{42 \left (7+x+x^2\right )^2}+\frac {25 (13-x)}{18 x \left (7+x+x^2\right )^2}-\frac {343 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {49 x (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {43 x^3 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {8 x^4 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^5 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^6 (14+x)}{27 \left (7+x+x^2\right )^2}+\frac {125 (1+2 x)}{18 \left (7+x+x^2\right )^2}+\frac {56 x^3 (1+2 x)}{9 \left (7+x+x^2\right )^2}+\frac {25 (227-32 x)}{2646 \left (7+x+x^2\right )}+\frac {25 (281-41 x)}{1134 x \left (7+x+x^2\right )}-\frac {434 x (14+x)}{243 \left (7+x+x^2\right )}+\frac {32 (1+2 x)}{243 \left (7+x+x^2\right )}+\frac {98 (7+5 x)}{81 \left (7+x+x^2\right )}-\frac {2 x^3 (161+16 x)}{81 \left (7+x+x^2\right )}-\frac {8 x^2 (259+23 x)}{81 \left (7+x+x^2\right )}-\frac {x^4 (385+41 x)}{243 \left (7+x+x^2\right )}+\frac {15220 \tan ^{-1}\left (\frac {1+2 x}{3 \sqrt {3}}\right )}{729 \sqrt {3}}+24 \log \left (7+x+x^2\right )-\frac {11675 \int \frac {1}{7+x+x^2} \, dx}{18522}-\frac {136475 \int \frac {1}{7+x+x^2} \, dx}{55566}-9 \int \frac {1+2 x}{7+x+x^2} \, dx-15 \int \frac {1+2 x}{7+x+x^2} \, dx-\frac {5459}{81} \int \frac {1}{7+x+x^2} \, dx+\frac {20725}{243} \int \frac {1}{7+x+x^2} \, dx+\frac {7472}{81} \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )\\ &=e^x-\frac {11050}{3969 x}+\frac {980 x}{243}+\frac {818 x^2}{243}+\frac {50 x^3}{243}+\frac {25 (13-x)}{42 \left (7+x+x^2\right )^2}+\frac {25 (13-x)}{18 x \left (7+x+x^2\right )^2}-\frac {343 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {49 x (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {43 x^3 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {8 x^4 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^5 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^6 (14+x)}{27 \left (7+x+x^2\right )^2}+\frac {125 (1+2 x)}{18 \left (7+x+x^2\right )^2}+\frac {56 x^3 (1+2 x)}{9 \left (7+x+x^2\right )^2}+\frac {25 (227-32 x)}{2646 \left (7+x+x^2\right )}+\frac {25 (281-41 x)}{1134 x \left (7+x+x^2\right )}-\frac {434 x (14+x)}{243 \left (7+x+x^2\right )}+\frac {32 (1+2 x)}{243 \left (7+x+x^2\right )}+\frac {98 (7+5 x)}{81 \left (7+x+x^2\right )}-\frac {2 x^3 (161+16 x)}{81 \left (7+x+x^2\right )}-\frac {8 x^2 (259+23 x)}{81 \left (7+x+x^2\right )}-\frac {x^4 (385+41 x)}{243 \left (7+x+x^2\right )}-\frac {7196 \tan ^{-1}\left (\frac {1+2 x}{3 \sqrt {3}}\right )}{729 \sqrt {3}}+\frac {11675 \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )}{9261}+\frac {136475 \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )}{27783}+\frac {10918}{81} \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )-\frac {41450}{243} \operatorname {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,1+2 x\right )\\ &=e^x-\frac {11050}{3969 x}+\frac {980 x}{243}+\frac {818 x^2}{243}+\frac {50 x^3}{243}+\frac {25 (13-x)}{42 \left (7+x+x^2\right )^2}+\frac {25 (13-x)}{18 x \left (7+x+x^2\right )^2}-\frac {343 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {49 x (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {43 x^3 (14+x)}{27 \left (7+x+x^2\right )^2}-\frac {8 x^4 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^5 (14+x)}{9 \left (7+x+x^2\right )^2}-\frac {x^6 (14+x)}{27 \left (7+x+x^2\right )^2}+\frac {125 (1+2 x)}{18 \left (7+x+x^2\right )^2}+\frac {56 x^3 (1+2 x)}{9 \left (7+x+x^2\right )^2}+\frac {25 (227-32 x)}{2646 \left (7+x+x^2\right )}+\frac {25 (281-41 x)}{1134 x \left (7+x+x^2\right )}-\frac {434 x (14+x)}{243 \left (7+x+x^2\right )}+\frac {32 (1+2 x)}{243 \left (7+x+x^2\right )}+\frac {98 (7+5 x)}{81 \left (7+x+x^2\right )}-\frac {2 x^3 (161+16 x)}{81 \left (7+x+x^2\right )}-\frac {8 x^2 (259+23 x)}{81 \left (7+x+x^2\right )}-\frac {x^4 (385+41 x)}{243 \left (7+x+x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 41, normalized size = 1.71 \begin {gather*} e^x+\frac {-75+49 x^3+14 x^4+15 x^5+2 x^6+x^7}{x \left (7+x+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(525 + 225*x + 375*x^2 + 686*x^3 + 294*x^4 + 336*x^5 + 86*x^6 + 48*x^7 + 6*x^8 + 2*x^9 + E^x*(343*x^
2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8))/(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7
+ x^8),x]

[Out]

E^x + (-75 + 49*x^3 + 14*x^4 + 15*x^5 + 2*x^6 + x^7)/(x*(7 + x + x^2)^2)

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fricas [B]  time = 1.36, size = 75, normalized size = 3.12 \begin {gather*} \frac {x^{7} + 2 \, x^{6} + 15 \, x^{5} + 14 \, x^{4} + 49 \, x^{3} + {\left (x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x\right )} e^{x} - 75}{x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+
686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm="fricas")

[Out]

(x^7 + 2*x^6 + 15*x^5 + 14*x^4 + 49*x^3 + (x^5 + 2*x^4 + 15*x^3 + 14*x^2 + 49*x)*e^x - 75)/(x^5 + 2*x^4 + 15*x
^3 + 14*x^2 + 49*x)

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giac [B]  time = 0.29, size = 82, normalized size = 3.42 \begin {gather*} \frac {x^{7} + 2 \, x^{6} + x^{5} e^{x} + 15 \, x^{5} + 2 \, x^{4} e^{x} + 14 \, x^{4} + 15 \, x^{3} e^{x} + 49 \, x^{3} + 14 \, x^{2} e^{x} + 49 \, x e^{x} - 75}{x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+
686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm="giac")

[Out]

(x^7 + 2*x^6 + x^5*e^x + 15*x^5 + 2*x^4*e^x + 14*x^4 + 15*x^3*e^x + 49*x^3 + 14*x^2*e^x + 49*x*e^x - 75)/(x^5
+ 2*x^4 + 15*x^3 + 14*x^2 + 49*x)

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maple [A]  time = 0.15, size = 32, normalized size = 1.33

method	result	size
risch	\(x^{2}-\frac {75}{x \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+{\mathrm e}^{x}\)	\(32\)
norman	\(\frac {-75+x^{7}+x^{5} {\mathrm e}^{x}-343 x -98 x^{2}-56 x^{3}+8 x^{5}+2 x^{6}+49 \,{\mathrm e}^{x} x +14 \,{\mathrm e}^{x} x^{2}+15 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{x} x^{4}}{x \left (x^{2}+x +7\right )^{2}}\)	\(73\)
default	\({\mathrm e}^{x}+x^{2}-\frac {75}{49 x}-\frac {2 \left (\frac {5797}{243} x^{3}+\frac {11305}{162} x^{2}+\frac {14602}{81} x +\frac {138229}{486}\right )}{\left (x^{2}+x +7\right )^{2}}-\frac {6 \left (-\frac {1280}{243} x^{3}+\frac {899}{81} x^{2}-\frac {1232}{81} x +\frac {42385}{486}\right )}{\left (x^{2}+x +7\right )^{2}}+\frac {\frac {5984}{81} x^{3}+\frac {10120}{27} x^{2}+\frac {17024}{27} x +\frac {134848}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {12470}{243} x^{3}+\frac {731}{81} x^{2}-\frac {15050}{81} x +\frac {48461}{243}}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {784}{81} x^{3}-\frac {4928}{27} x^{2}-\frac {3136}{27} x -\frac {52136}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {\frac {490}{81} x^{3}+\frac {245}{27} x^{2}-\frac {686}{27} x +\frac {4802}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {343 x}{27}-\frac {4802}{27}}{\left (x^{2}+x +7\right )^{2}}+\frac {\frac {125 x}{9}+\frac {125}{18}}{\left (x^{2}+x +7\right )^{2}}+\frac {\frac {64 x}{243}+\frac {32}{243}}{x^{2}+x +7}-\frac {225 \left (\frac {112}{243} x^{3}-\frac {455}{162} x^{2}+\frac {70}{81} x -\frac {8428}{243}\right )}{343 \left (x^{2}+x +7\right )^{2}}+\frac {-\frac {4975}{3969} x^{3}-\frac {9025}{2646} x^{2}-\frac {17875}{1323} x -\frac {21625}{1134}}{\left (x^{2}+x +7\right )^{2}}+\frac {{\mathrm e}^{x} \left (1009 x^{3}+3237 x^{2}+7392 x +11123\right )}{162 x^{4}+324 x^{3}+2430 x^{2}+2268 x +7938}-\frac {43 \,{\mathrm e}^{x} \left (68 x^{3}+255 x^{2}+483 x +490\right )}{486 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {{\mathrm e}^{x} \left (1480 x^{3}-8301 x^{2}-1995 x -55174\right )}{486 x^{4}+972 x^{3}+7290 x^{2}+6804 x +23814}-\frac {4 \,{\mathrm e}^{x} \left (173 x^{3}-546 x^{2}-147 x -3773\right )}{81 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {343 \,{\mathrm e}^{x} \left (4 x^{3}+15 x^{2}+57 x +86\right )}{486 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {49 \,{\mathrm e}^{x} \left (7 x^{3}+6 x^{2}+39 x -133\right )}{162 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {28 \,{\mathrm e}^{x} \left (x^{3}-57 x^{2}-168 x -343\right )}{81 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}\)	\(528\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+686*x^
3+375*x^2+225*x+525)/(x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x,method=_RETURNVERBOSE)

[Out]

x^2-75/x/(x^4+2*x^3+15*x^2+14*x+49)+exp(x)

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maxima [B]  time = 0.57, size = 385, normalized size = 16.04 \begin {gather*} x^{2} - \frac {25 \, {\left (884 \, x^{4} + 2055 \, x^{3} + 11580 \, x^{2} + 12859 \, x + 23814\right )}}{7938 \, {\left (x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x\right )}} - \frac {11594 \, x^{3} + 33915 \, x^{2} + 87612 \, x + 138229}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {2560 \, x^{3} - 5394 \, x^{2} + 7392 \, x - 42385}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {8 \, {\left (748 \, x^{3} + 3795 \, x^{2} + 6384 \, x + 16856\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {43 \, {\left (290 \, x^{3} - 51 \, x^{2} + 1050 \, x - 1127\right )}}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {25 \, {\left (32 \, x^{3} - 195 \, x^{2} + 60 \, x - 2408\right )}}{2646 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {56 \, {\left (14 \, x^{3} + 264 \, x^{2} + 168 \, x + 931\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {49 \, {\left (10 \, x^{3} + 15 \, x^{2} - 42 \, x + 98\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {125 \, {\left (4 \, x^{3} + 6 \, x^{2} + 48 \, x + 23\right )}}{162 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {343 \, {\left (2 \, x^{3} + 3 \, x^{2} + 24 \, x + 133\right )}}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+
686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm="maxima")

[Out]

x^2 - 25/7938*(884*x^4 + 2055*x^3 + 11580*x^2 + 12859*x + 23814)/(x^5 + 2*x^4 + 15*x^3 + 14*x^2 + 49*x) - 1/24
3*(11594*x^3 + 33915*x^2 + 87612*x + 138229)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 1/81*(2560*x^3 - 5394*x^2 +
7392*x - 42385)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 8/81*(748*x^3 + 3795*x^2 + 6384*x + 16856)/(x^4 + 2*x^3 +
15*x^2 + 14*x + 49) - 43/243*(290*x^3 - 51*x^2 + 1050*x - 1127)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 25/2646*
(32*x^3 - 195*x^2 + 60*x - 2408)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 56/81*(14*x^3 + 264*x^2 + 168*x + 931)/(
x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 49/81*(10*x^3 + 15*x^2 - 42*x + 98)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 1
25/162*(4*x^3 + 6*x^2 + 48*x + 23)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 343/243*(2*x^3 + 3*x^2 + 24*x + 133)/(
x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + e^x

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mupad [B]  time = 0.98, size = 32, normalized size = 1.33 \begin {gather*} {\mathrm {e}}^x-\frac {75}{x^5+2\,x^4+15\,x^3+14\,x^2+49\,x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((225*x + exp(x)*(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8) + 375*x^2 + 686*x^3 + 294*x^
4 + 336*x^5 + 86*x^6 + 48*x^7 + 6*x^8 + 2*x^9 + 525)/(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 +
x^8),x)

[Out]

exp(x) - 75/(49*x + 14*x^2 + 15*x^3 + 2*x^4 + x^5) + x^2

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sympy [A]  time = 0.22, size = 29, normalized size = 1.21 \begin {gather*} x^{2} + e^{x} - \frac {75}{x^{5} + 2 x^{4} + 15 x^{3} + 14 x^{2} + 49 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**8+3*x**7+24*x**6+43*x**5+168*x**4+147*x**3+343*x**2)*exp(x)+2*x**9+6*x**8+48*x**7+86*x**6+336*x
**5+294*x**4+686*x**3+375*x**2+225*x+525)/(x**8+3*x**7+24*x**6+43*x**5+168*x**4+147*x**3+343*x**2),x)

[Out]

x**2 + exp(x) - 75/(x**5 + 2*x**4 + 15*x**3 + 14*x**2 + 49*x)

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