3.22.12 \(\int \frac {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6)+e^{e^3+x} (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7)}{125-150 x+360 x^2-248 x^3+288 x^4-96 x^5+64 x^6} \, dx\)

Optimal. Leaf size=33 \[ \left (e^{e^3+x}-x-\frac {400}{\frac {1}{5} \left (-2+\frac {5}{x}-x\right )+x}\right )^2 \]

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Rubi [C]  time = 2.87, antiderivative size = 868, normalized size of antiderivative = 26.30, number of steps used = 70, number of rules used = 19, integrand size = 152, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6741, 6742, 2194, 638, 614, 618, 204, 738, 728, 722, 818, 773, 634, 628, 800, 2176, 2177, 2178, 2270} \begin {gather*} -\frac {32 (5-x) x^6}{19 \left (4 x^2-2 x+5\right )^2}+\frac {48 (5-x) x^5}{19 \left (4 x^2-2 x+5\right )^2}-\frac {12 (195-58 x) x^4}{361 \left (4 x^2-2 x+5\right )}-\frac {144 (5-x) x^4}{19 \left (4 x^2-2 x+5\right )^2}+\frac {12 (245-68 x) x^3}{361 \left (4 x^2-2 x+5\right )}+\frac {7955820 (1-4 x) x^3}{19 \left (4 x^2-2 x+5\right )^2}+\frac {8124 (5-x) x^3}{19 \left (4 x^2-2 x+5\right )^2}-\frac {212 x^3}{361}-\frac {18 (395-98 x) x^2}{361 \left (4 x^2-2 x+5\right )}+\frac {1272 x^2}{361}-2 e^{x+e^3} x+\frac {11994660 (5-x) x}{361 \left (4 x^2-2 x+5\right )}+\frac {30075 (5-x) x}{19 \left (4 x^2-2 x+5\right )^2}-\frac {750 x}{361}+e^{2 x+2 e^3}-\frac {500}{19} \left (19+3 i \sqrt {19}\right ) e^{\frac {1}{4}+\frac {i \sqrt {19}}{4}+e^3} \text {Ei}\left (\frac {1}{4} \left (4 x-i \sqrt {19}-1\right )\right )-\frac {500}{19} \left (1+i \sqrt {19}\right ) e^{\frac {1}{4}+\frac {i \sqrt {19}}{4}+e^3} \text {Ei}\left (\frac {1}{4} \left (4 x-i \sqrt {19}-1\right )\right )+\frac {2000 i e^{\frac {1}{4}+\frac {i \sqrt {19}}{4}+e^3} \text {Ei}\left (\frac {1}{4} \left (4 x-i \sqrt {19}-1\right )\right )}{\sqrt {19}}+\frac {10000}{19} e^{\frac {1}{4}+\frac {i \sqrt {19}}{4}+e^3} \text {Ei}\left (\frac {1}{4} \left (4 x-i \sqrt {19}-1\right )\right )-\frac {500}{19} \left (19-3 i \sqrt {19}\right ) e^{\frac {1}{4}-\frac {i \sqrt {19}}{4}+e^3} \text {Ei}\left (\frac {1}{4} \left (4 x+i \sqrt {19}-1\right )\right )-\frac {500}{19} \left (1-i \sqrt {19}\right ) e^{\frac {1}{4}-\frac {i \sqrt {19}}{4}+e^3} \text {Ei}\left (\frac {1}{4} \left (4 x+i \sqrt {19}-1\right )\right )-\frac {2000 i e^{\frac {1}{4}-\frac {i \sqrt {19}}{4}+e^3} \text {Ei}\left (\frac {1}{4} \left (4 x+i \sqrt {19}-1\right )\right )}{\sqrt {19}}+\frac {10000}{19} e^{\frac {1}{4}-\frac {i \sqrt {19}}{4}+e^3} \text {Ei}\left (\frac {1}{4} \left (4 x+i \sqrt {19}-1\right )\right )-\frac {2000 \left (1-i \sqrt {19}\right ) e^{x+e^3}}{19 \left (-4 x-i \sqrt {19}+1\right )}+\frac {40000 e^{x+e^3}}{19 \left (-4 x-i \sqrt {19}+1\right )}-\frac {2000 \left (1+i \sqrt {19}\right ) e^{x+e^3}}{19 \left (-4 x+i \sqrt {19}+1\right )}+\frac {40000 e^{x+e^3}}{19 \left (-4 x+i \sqrt {19}+1\right )}+\frac {30075 (15-22 x)}{722 \left (4 x^2-2 x+5\right )}-\frac {60300375 (1-4 x)}{1444 \left (4 x^2-2 x+5\right )}-\frac {20100125 (5-x)}{38 \left (4 x^2-2 x+5\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(40200250*x - 120300*x^2 - 31823280*x^3 - 32496*x^4 + 576*x^5 - 192*x^6 + 128*x^7 + E^(2*E^3 + 2*x)*(250 -
300*x + 720*x^2 - 496*x^3 + 576*x^4 - 192*x^5 + 128*x^6) + E^(E^3 + x)*(-100250 - 59950*x + 79580*x^2 - 20822
4*x^3 + 127920*x^4 - 64384*x^5 + 64*x^6 - 128*x^7))/(125 - 150*x + 360*x^2 - 248*x^3 + 288*x^4 - 96*x^5 + 64*x
^6),x]

[Out]

E^(2*E^3 + 2*x) + (40000*E^(E^3 + x))/(19*(1 - I*Sqrt[19] - 4*x)) - (2000*(1 - I*Sqrt[19])*E^(E^3 + x))/(19*(1
- I*Sqrt[19] - 4*x)) + (40000*E^(E^3 + x))/(19*(1 + I*Sqrt[19] - 4*x)) - (2000*(1 + I*Sqrt[19])*E^(E^3 + x))/
(19*(1 + I*Sqrt[19] - 4*x)) - (750*x)/361 - 2*E^(E^3 + x)*x + (1272*x^2)/361 - (212*x^3)/361 - (20100125*(5 -
x))/(38*(5 - 2*x + 4*x^2)^2) + (30075*(5 - x)*x)/(19*(5 - 2*x + 4*x^2)^2) + (7955820*(1 - 4*x)*x^3)/(19*(5 - 2
*x + 4*x^2)^2) + (8124*(5 - x)*x^3)/(19*(5 - 2*x + 4*x^2)^2) - (144*(5 - x)*x^4)/(19*(5 - 2*x + 4*x^2)^2) + (4
8*(5 - x)*x^5)/(19*(5 - 2*x + 4*x^2)^2) - (32*(5 - x)*x^6)/(19*(5 - 2*x + 4*x^2)^2) + (30075*(15 - 22*x))/(722
*(5 - 2*x + 4*x^2)) - (60300375*(1 - 4*x))/(1444*(5 - 2*x + 4*x^2)) + (11994660*(5 - x)*x)/(361*(5 - 2*x + 4*x
^2)) - (18*(395 - 98*x)*x^2)/(361*(5 - 2*x + 4*x^2)) + (12*(245 - 68*x)*x^3)/(361*(5 - 2*x + 4*x^2)) - (12*(19
5 - 58*x)*x^4)/(361*(5 - 2*x + 4*x^2)) + (10000*E^(1/4 + (I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 - I*Sqrt[19]
+ 4*x)/4])/19 + ((2000*I)*E^(1/4 + (I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 - I*Sqrt[19] + 4*x)/4])/Sqrt[19] -
(500*(1 + I*Sqrt[19])*E^(1/4 + (I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 - I*Sqrt[19] + 4*x)/4])/19 - (500*(19 +
(3*I)*Sqrt[19])*E^(1/4 + (I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 - I*Sqrt[19] + 4*x)/4])/19 + (10000*E^(1/4 -
(I/4)*Sqrt[19] + E^3)*ExpIntegralEi[(-1 + I*Sqrt[19] + 4*x)/4])/19 - ((2000*I)*E^(1/4 - (I/4)*Sqrt[19] + E^3)
*ExpIntegralEi[(-1 + I*Sqrt[19] + 4*x)/4])/Sqrt[19] - (500*(1 - I*Sqrt[19])*E^(1/4 - (I/4)*Sqrt[19] + E^3)*Exp
IntegralEi[(-1 + I*Sqrt[19] + 4*x)/4])/19 - (500*(19 - (3*I)*Sqrt[19])*E^(1/4 - (I/4)*Sqrt[19] + E^3)*ExpInteg
ralEi[(-1 + I*Sqrt[19] + 4*x)/4])/19

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 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 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 2270

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

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 {40200250 x-120300 x^2-31823280 x^3-32496 x^4+576 x^5-192 x^6+128 x^7+e^{2 e^3+2 x} \left (250-300 x+720 x^2-496 x^3+576 x^4-192 x^5+128 x^6\right )+e^{e^3+x} \left (-100250-59950 x+79580 x^2-208224 x^3+127920 x^4-64384 x^5+64 x^6-128 x^7\right )}{\left (5-2 x+4 x^2\right )^3} \, dx\\ &=\int \left (2 e^{2 e^3+2 x}+\frac {40200250 x}{\left (5-2 x+4 x^2\right )^3}-\frac {120300 x^2}{\left (5-2 x+4 x^2\right )^3}-\frac {31823280 x^3}{\left (5-2 x+4 x^2\right )^3}-\frac {32496 x^4}{\left (5-2 x+4 x^2\right )^3}+\frac {576 x^5}{\left (5-2 x+4 x^2\right )^3}-\frac {192 x^6}{\left (5-2 x+4 x^2\right )^3}+\frac {128 x^7}{\left (5-2 x+4 x^2\right )^3}-\frac {2 e^{e^3+x} \left (10025+10005 x-11976 x^2+8028 x^3+16 x^5\right )}{\left (5-2 x+4 x^2\right )^2}\right ) \, dx\\ &=2 \int e^{2 e^3+2 x} \, dx-2 \int \frac {e^{e^3+x} \left (10025+10005 x-11976 x^2+8028 x^3+16 x^5\right )}{\left (5-2 x+4 x^2\right )^2} \, dx+128 \int \frac {x^7}{\left (5-2 x+4 x^2\right )^3} \, dx-192 \int \frac {x^6}{\left (5-2 x+4 x^2\right )^3} \, dx+576 \int \frac {x^5}{\left (5-2 x+4 x^2\right )^3} \, dx-32496 \int \frac {x^4}{\left (5-2 x+4 x^2\right )^3} \, dx-120300 \int \frac {x^2}{\left (5-2 x+4 x^2\right )^3} \, dx-31823280 \int \frac {x^3}{\left (5-2 x+4 x^2\right )^3} \, dx+40200250 \int \frac {x}{\left (5-2 x+4 x^2\right )^3} \, dx\\ &=e^{2 e^3+2 x}-\frac {20100125 (5-x)}{38 \left (5-2 x+4 x^2\right )^2}+\frac {30075 (5-x) x}{19 \left (5-2 x+4 x^2\right )^2}+\frac {7955820 (1-4 x) x^3}{19 \left (5-2 x+4 x^2\right )^2}+\frac {8124 (5-x) x^3}{19 \left (5-2 x+4 x^2\right )^2}-\frac {144 (5-x) x^4}{19 \left (5-2 x+4 x^2\right )^2}+\frac {48 (5-x) x^5}{19 \left (5-2 x+4 x^2\right )^2}-\frac {32 (5-x) x^6}{19 \left (5-2 x+4 x^2\right )^2}+\frac {16}{19} \int \frac {(60-6 x) x^5}{\left (5-2 x+4 x^2\right )^2} \, dx-\frac {24}{19} \int \frac {(50-4 x) x^4}{\left (5-2 x+4 x^2\right )^2} \, dx-2 \int \left (e^{e^3+x}+e^{e^3+x} x-\frac {4000 e^{e^3+x} (-5+x)}{\left (5-2 x+4 x^2\right )^2}+\frac {2000 e^{e^3+x} (-1+x)}{5-2 x+4 x^2}\right ) \, dx+\frac {72}{19} \int \frac {(40-2 x) x^3}{\left (5-2 x+4 x^2\right )^2} \, dx-\frac {30075}{38} \int \frac {10+4 x}{\left (5-2 x+4 x^2\right )^2} \, dx-\frac {121860}{19} \int \frac {x^2}{\left (5-2 x+4 x^2\right )^2} \, dx-\frac {23867460}{19} \int \frac {x^2}{\left (5-2 x+4 x^2\right )^2} \, dx+\frac {60300375}{38} \int \frac {1}{\left (5-2 x+4 x^2\right )^2} \, dx\\ &=e^{2 e^3+2 x}-\frac {20100125 (5-x)}{38 \left (5-2 x+4 x^2\right )^2}+\frac {30075 (5-x) x}{19 \left (5-2 x+4 x^2\right )^2}+\frac {7955820 (1-4 x) x^3}{19 \left (5-2 x+4 x^2\right )^2}+\frac {8124 (5-x) x^3}{19 \left (5-2 x+4 x^2\right )^2}-\frac {144 (5-x) x^4}{19 \left (5-2 x+4 x^2\right )^2}+\frac {48 (5-x) x^5}{19 \left (5-2 x+4 x^2\right )^2}-\frac {32 (5-x) x^6}{19 \left (5-2 x+4 x^2\right )^2}+\frac {30075 (15-22 x)}{722 \left (5-2 x+4 x^2\right )}-\frac {60300375 (1-4 x)}{1444 \left (5-2 x+4 x^2\right )}+\frac {11994660 (5-x) x}{361 \left (5-2 x+4 x^2\right )}-\frac {18 (395-98 x) x^2}{361 \left (5-2 x+4 x^2\right )}+\frac {12 (245-68 x) x^3}{361 \left (5-2 x+4 x^2\right )}-\frac {12 (195-58 x) x^4}{361 \left (5-2 x+4 x^2\right )}-\frac {1}{361} \int \frac {x^3 (-9360+2544 x)}{5-2 x+4 x^2} \, dx+\frac {3}{722} \int \frac {x^2 (-5880+1392 x)}{5-2 x+4 x^2} \, dx-\frac {9}{722} \int \frac {x (-3160+544 x)}{5-2 x+4 x^2} \, dx-2 \int e^{e^3+x} \, dx-2 \int e^{e^3+x} x \, dx-\frac {304650}{361} \int \frac {1}{5-2 x+4 x^2} \, dx-\frac {330825}{361} \int \frac {1}{5-2 x+4 x^2} \, dx-4000 \int \frac {e^{e^3+x} (-1+x)}{5-2 x+4 x^2} \, dx+8000 \int \frac {e^{e^3+x} (-5+x)}{\left (5-2 x+4 x^2\right )^2} \, dx-\frac {59668650}{361} \int \frac {1}{5-2 x+4 x^2} \, dx+\frac {60300375}{361} \int \frac {1}{5-2 x+4 x^2} \, dx\\ &=-2 e^{e^3+x}+e^{2 e^3+2 x}-\frac {612 x}{361}-2 e^{e^3+x} x-\frac {20100125 (5-x)}{38 \left (5-2 x+4 x^2\right )^2}+\frac {30075 (5-x) x}{19 \left (5-2 x+4 x^2\right )^2}+\frac {7955820 (1-4 x) x^3}{19 \left (5-2 x+4 x^2\right )^2}+\frac {8124 (5-x) x^3}{19 \left (5-2 x+4 x^2\right )^2}-\frac {144 (5-x) x^4}{19 \left (5-2 x+4 x^2\right )^2}+\frac {48 (5-x) x^5}{19 \left (5-2 x+4 x^2\right )^2}-\frac {32 (5-x) x^6}{19 \left (5-2 x+4 x^2\right )^2}+\frac {30075 (15-22 x)}{722 \left (5-2 x+4 x^2\right )}-\frac {60300375 (1-4 x)}{1444 \left (5-2 x+4 x^2\right )}+\frac {11994660 (5-x) x}{361 \left (5-2 x+4 x^2\right )}-\frac {18 (395-98 x) x^2}{361 \left (5-2 x+4 x^2\right )}+\frac {12 (245-68 x) x^3}{361 \left (5-2 x+4 x^2\right )}-\frac {12 (195-58 x) x^4}{361 \left (5-2 x+4 x^2\right )}-\frac {1}{361} \int \left (-1806-2022 x+636 x^2+\frac {6 (1505+1083 x)}{5-2 x+4 x^2}\right ) \, dx-\frac {9 \int \frac {-2720-11552 x}{5-2 x+4 x^2} \, dx}{2888}+\frac {3}{722} \int \left (-1296+348 x+\frac {12 (540-361 x)}{5-2 x+4 x^2}\right ) \, dx+2 \int e^{e^3+x} \, dx+\frac {609300}{361} \operatorname {Subst}\left (\int \frac {1}{-76-x^2} \, dx,x,-2+8 x\right )+\frac {661650}{361} \operatorname {Subst}\left (\int \frac {1}{-76-x^2} \, dx,x,-2+8 x\right )-4000 \int \left (\frac {\left (1+\frac {3 i}{\sqrt {19}}\right ) e^{e^3+x}}{-2-2 i \sqrt {19}+8 x}+\frac {\left (1-\frac {3 i}{\sqrt {19}}\right ) e^{e^3+x}}{-2+2 i \sqrt {19}+8 x}\right ) \, dx+8000 \int \left (-\frac {5 e^{e^3+x}}{\left (5-2 x+4 x^2\right )^2}+\frac {e^{e^3+x} x}{\left (5-2 x+4 x^2\right )^2}\right ) \, dx+\frac {119337300}{361} \operatorname {Subst}\left (\int \frac {1}{-76-x^2} \, dx,x,-2+8 x\right )-\frac {120600750}{361} \operatorname {Subst}\left (\int \frac {1}{-76-x^2} \, dx,x,-2+8 x\right )\\ &=e^{2 e^3+2 x}-\frac {750 x}{361}-2 e^{e^3+x} x+\frac {1272 x^2}{361}-\frac {212 x^3}{361}-\frac {20100125 (5-x)}{38 \left (5-2 x+4 x^2\right )^2}+\frac {30075 (5-x) x}{19 \left (5-2 x+4 x^2\right )^2}+\frac {7955820 (1-4 x) x^3}{19 \left (5-2 x+4 x^2\right )^2}+\frac {8124 (5-x) x^3}{19 \left (5-2 x+4 x^2\right )^2}-\frac {144 (5-x) x^4}{19 \left (5-2 x+4 x^2\right )^2}+\frac {48 (5-x) x^5}{19 \left (5-2 x+4 x^2\right )^2}-\frac {32 (5-x) x^6}{19 \left (5-2 x+4 x^2\right )^2}+\frac {30075 (15-22 x)}{722 \left (5-2 x+4 x^2\right )}-\frac {60300375 (1-4 x)}{1444 \left (5-2 x+4 x^2\right )}+\frac {11994660 (5-x) x}{361 \left (5-2 x+4 x^2\right )}-\frac {18 (395-98 x) x^2}{361 \left (5-2 x+4 x^2\right )}+\frac {12 (245-68 x) x^3}{361 \left (5-2 x+4 x^2\right )}-\frac {12 (195-58 x) x^4}{361 \left (5-2 x+4 x^2\right )}+\frac {3750 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {19}}\right )}{361 \sqrt {19}}-\frac {6}{361} \int \frac {1505+1083 x}{5-2 x+4 x^2} \, dx+\frac {18}{361} \int \frac {540-361 x}{5-2 x+4 x^2} \, dx+\frac {9}{2} \int \frac {-2+8 x}{5-2 x+4 x^2} \, dx+\frac {6309}{361} \int \frac {1}{5-2 x+4 x^2} \, dx+8000 \int \frac {e^{e^3+x} x}{\left (5-2 x+4 x^2\right )^2} \, dx-40000 \int \frac {e^{e^3+x}}{\left (5-2 x+4 x^2\right )^2} \, dx-\frac {1}{19} \left (4000 \left (19-3 i \sqrt {19}\right )\right ) \int \frac {e^{e^3+x}}{-2+2 i \sqrt {19}+8 x} \, dx-\frac {1}{19} \left (4000 \left (19+3 i \sqrt {19}\right )\right ) \int \frac {e^{e^3+x}}{-2-2 i \sqrt {19}+8 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.12, size = 92, normalized size = 2.79 \begin {gather*} -2 \left (-\frac {1}{2} e^{2 \left (e^3+x\right )}-\frac {x^2}{2}-\frac {500000 (-5+2 x)}{\left (5-2 x+4 x^2\right )^2}-\frac {500 (995+2 x)}{5-2 x+4 x^2}+\frac {e^{e^3+x} x \left (2005-2 x+4 x^2\right )}{5-2 x+4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(40200250*x - 120300*x^2 - 31823280*x^3 - 32496*x^4 + 576*x^5 - 192*x^6 + 128*x^7 + E^(2*E^3 + 2*x)*
(250 - 300*x + 720*x^2 - 496*x^3 + 576*x^4 - 192*x^5 + 128*x^6) + E^(E^3 + x)*(-100250 - 59950*x + 79580*x^2 -
208224*x^3 + 127920*x^4 - 64384*x^5 + 64*x^6 - 128*x^7))/(125 - 150*x + 360*x^2 - 248*x^3 + 288*x^4 - 96*x^5
+ 64*x^6),x]

[Out]

-2*(-1/2*E^(2*(E^3 + x)) - x^2/2 - (500000*(-5 + 2*x))/(5 - 2*x + 4*x^2)^2 - (500*(995 + 2*x))/(5 - 2*x + 4*x^
2) + (E^(E^3 + x)*x*(2005 - 2*x + 4*x^2))/(5 - 2*x + 4*x^2))

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fricas [B]  time = 0.46, size = 114, normalized size = 3.45 \begin {gather*} \frac {16 \, x^{6} - 16 \, x^{5} + 44 \, x^{4} + 7980 \, x^{3} + 3976025 \, x^{2} + {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )} e^{\left (2 \, x + 2 \, e^{3}\right )} - 2 \, {\left (16 \, x^{5} - 16 \, x^{4} + 8044 \, x^{3} - 4020 \, x^{2} + 10025 \, x\right )} e^{\left (x + e^{3}\right )} + 20000 \, x - 25000}{16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+x)^2+(-128*x^7+64*x^6-64384*x^5+1279
20*x^4-208224*x^3+79580*x^2-59950*x-100250)*exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-12030
0*x^2+40200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x, algorithm="fricas")

[Out]

(16*x^6 - 16*x^5 + 44*x^4 + 7980*x^3 + 3976025*x^2 + (16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25)*e^(2*x + 2*e^3) -
2*(16*x^5 - 16*x^4 + 8044*x^3 - 4020*x^2 + 10025*x)*e^(x + e^3) + 20000*x - 25000)/(16*x^4 - 16*x^3 + 44*x^2 -
20*x + 25)

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giac [B]  time = 0.35, size = 166, normalized size = 5.03 \begin {gather*} \frac {16 \, x^{6} - 32 \, x^{5} e^{\left (x + e^{3}\right )} - 16 \, x^{5} + 16 \, x^{4} e^{\left (2 \, x + 2 \, e^{3}\right )} + 32 \, x^{4} e^{\left (x + e^{3}\right )} + 44 \, x^{4} - 16 \, x^{3} e^{\left (2 \, x + 2 \, e^{3}\right )} - 16088 \, x^{3} e^{\left (x + e^{3}\right )} + 7980 \, x^{3} + 44 \, x^{2} e^{\left (2 \, x + 2 \, e^{3}\right )} + 8040 \, x^{2} e^{\left (x + e^{3}\right )} + 3976025 \, x^{2} - 20 \, x e^{\left (2 \, x + 2 \, e^{3}\right )} - 20050 \, x e^{\left (x + e^{3}\right )} + 20000 \, x + 25 \, e^{\left (2 \, x + 2 \, e^{3}\right )} - 25000}{16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+x)^2+(-128*x^7+64*x^6-64384*x^5+1279
20*x^4-208224*x^3+79580*x^2-59950*x-100250)*exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-12030
0*x^2+40200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x, algorithm="giac")

[Out]

(16*x^6 - 32*x^5*e^(x + e^3) - 16*x^5 + 16*x^4*e^(2*x + 2*e^3) + 32*x^4*e^(x + e^3) + 44*x^4 - 16*x^3*e^(2*x +
2*e^3) - 16088*x^3*e^(x + e^3) + 7980*x^3 + 44*x^2*e^(2*x + 2*e^3) + 8040*x^2*e^(x + e^3) + 3976025*x^2 - 20*
x*e^(2*x + 2*e^3) - 20050*x*e^(x + e^3) + 20000*x + 25*e^(2*x + 2*e^3) - 25000)/(16*x^4 - 16*x^3 + 44*x^2 - 20
*x + 25)

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maple [B]  time = 0.93, size = 80, normalized size = 2.42

method	result	size
risch	\(x^{2}+\frac {500 x^{3}+248500 x^{2}+1250 x -\frac {3125}{2}}{x^{4}-x^{3}+\frac {11}{4} x^{2}-\frac {5}{4} x +\frac {25}{16}}+{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x}-\frac {2 x \left (4 x^{2}-2 x +2005\right ) {\mathrm e}^{{\mathrm e}^{3}+x}}{4 x^{2}-2 x +5}\)	\(80\)
norman	\(\frac {8024 x^{4}+10025 x +3997970 x^{2}-16 x^{5}+16 x^{6}+25 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x}-20050 x \,{\mathrm e}^{{\mathrm e}^{3}+x}+8040 x^{2} {\mathrm e}^{{\mathrm e}^{3}+x}+16 x^{4} {\mathrm e}^{2 \,{\mathrm e}^{3}+2 x}-16088 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{3}+32 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{4}-32 \,{\mathrm e}^{{\mathrm e}^{3}+x} x^{5}-20 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x +44 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x^{2}-16 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2 x} x^{3}-\frac {50125}{4}}{\left (4 x^{2}-2 x +5\right )^{2}}\)	\(142\)
derivativedivides	\(\text {Expression too large to display}\)	\(94585\)
default	\(\text {Expression too large to display}\)	\(94585\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+x)^2+(-128*x^7+64*x^6-64384*x^5+127920*x^4
-208224*x^3+79580*x^2-59950*x-100250)*exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-120300*x^2+
40200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x,method=_RETURNVERBOSE)

[Out]

x^2+(500*x^3+248500*x^2+1250*x-3125/2)/(x^4-x^3+11/4*x^2-5/4*x+25/16)+exp(2*exp(3)+2*x)-2*x*(4*x^2-2*x+2005)/(
4*x^2-2*x+5)*exp(exp(3)+x)

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maxima [B]  time = 0.61, size = 349, normalized size = 10.58 \begin {gather*} x^{2} + \frac {92336 \, x^{3} - 98132 \, x^{2} + 120160 \, x - 59825}{1444 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} - \frac {3 \, {\left (17472 \, x^{3} + 24440 \, x^{2} + 3200 \, x + 31875\right )}}{1444 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} - \frac {9 \, {\left (4416 \, x^{3} - 8366 \, x^{2} + 6680 \, x - 6225\right )}}{361 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} + \frac {2031 \, {\left (3376 \, x^{3} + 356 \, x^{2} + 2020 \, x + 1225\right )}}{1444 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} - \frac {1988955 \, {\left (240 \, x^{3} - 1624 \, x^{2} + 520 \, x - 1025\right )}}{722 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} - \frac {30075 \, {\left (88 \, x^{3} - 66 \, x^{2} - 50 \, x - 75\right )}}{722 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} + \frac {20100125 \, {\left (48 \, x^{3} - 36 \, x^{2} + 104 \, x - 205\right )}}{1444 \, {\left (16 \, x^{4} - 16 \, x^{3} + 44 \, x^{2} - 20 \, x + 25\right )}} + \frac {{\left (4 \, x^{2} e^{\left (2 \, e^{3}\right )} - 2 \, x e^{\left (2 \, e^{3}\right )} + 5 \, e^{\left (2 \, e^{3}\right )}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (4 \, x^{3} e^{\left (e^{3}\right )} - 2 \, x^{2} e^{\left (e^{3}\right )} + 2005 \, x e^{\left (e^{3}\right )}\right )} e^{x}}{4 \, x^{2} - 2 \, x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^6-192*x^5+576*x^4-496*x^3+720*x^2-300*x+250)*exp(exp(3)+x)^2+(-128*x^7+64*x^6-64384*x^5+1279
20*x^4-208224*x^3+79580*x^2-59950*x-100250)*exp(exp(3)+x)+128*x^7-192*x^6+576*x^5-32496*x^4-31823280*x^3-12030
0*x^2+40200250*x)/(64*x^6-96*x^5+288*x^4-248*x^3+360*x^2-150*x+125),x, algorithm="maxima")

[Out]

x^2 + 1/1444*(92336*x^3 - 98132*x^2 + 120160*x - 59825)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) - 3/1444*(17472
*x^3 + 24440*x^2 + 3200*x + 31875)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) - 9/361*(4416*x^3 - 8366*x^2 + 6680*
x - 6225)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) + 2031/1444*(3376*x^3 + 356*x^2 + 2020*x + 1225)/(16*x^4 - 16
*x^3 + 44*x^2 - 20*x + 25) - 1988955/722*(240*x^3 - 1624*x^2 + 520*x - 1025)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x
+ 25) - 30075/722*(88*x^3 - 66*x^2 - 50*x - 75)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) + 20100125/1444*(48*x^3
- 36*x^2 + 104*x - 205)/(16*x^4 - 16*x^3 + 44*x^2 - 20*x + 25) + ((4*x^2*e^(2*e^3) - 2*x*e^(2*e^3) + 5*e^(2*e
^3))*e^(2*x) - 2*(4*x^3*e^(e^3) - 2*x^2*e^(e^3) + 2005*x*e^(e^3))*e^x)/(4*x^2 - 2*x + 5)

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mupad [B]  time = 1.41, size = 80, normalized size = 2.42 \begin {gather*} {\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^3}+x^2+\frac {500\,x^3+248500\,x^2+1250\,x-\frac {3125}{2}}{x^4-x^3+\frac {11\,x^2}{4}-\frac {5\,x}{4}+\frac {25}{16}}-\frac {{\mathrm {e}}^{x+{\mathrm {e}}^3}\,\left (2\,x^3-x^2+\frac {2005\,x}{2}\right )}{x^2-\frac {x}{2}+\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x + exp(3))*(59950*x - 79580*x^2 + 208224*x^3 - 127920*x^4 + 64384*x^5 - 64*x^6 + 128*x^7 + 100250)
- exp(2*x + 2*exp(3))*(720*x^2 - 300*x - 496*x^3 + 576*x^4 - 192*x^5 + 128*x^6 + 250) - 40200250*x + 120300*x^
2 + 31823280*x^3 + 32496*x^4 - 576*x^5 + 192*x^6 - 128*x^7)/(360*x^2 - 150*x - 248*x^3 + 288*x^4 - 96*x^5 + 64
*x^6 + 125),x)

[Out]

exp(2*x + 2*exp(3)) + x^2 + (1250*x + 248500*x^2 + 500*x^3 - 3125/2)/((11*x^2)/4 - (5*x)/4 - x^3 + x^4 + 25/16
) - (exp(x + exp(3))*((2005*x)/2 - x^2 + 2*x^3))/(x^2 - x/2 + 5/4)

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sympy [B]  time = 0.26, size = 88, normalized size = 2.67 \begin {gather*} x^{2} + \frac {\left (4 x^{2} - 2 x + 5\right ) e^{2 x + 2 e^{3}} + \left (- 8 x^{3} + 4 x^{2} - 4010 x\right ) e^{x + e^{3}}}{4 x^{2} - 2 x + 5} + \frac {8000 x^{3} + 3976000 x^{2} + 20000 x - 25000}{16 x^{4} - 16 x^{3} + 44 x^{2} - 20 x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x**6-192*x**5+576*x**4-496*x**3+720*x**2-300*x+250)*exp(exp(3)+x)**2+(-128*x**7+64*x**6-64384*
x**5+127920*x**4-208224*x**3+79580*x**2-59950*x-100250)*exp(exp(3)+x)+128*x**7-192*x**6+576*x**5-32496*x**4-31
823280*x**3-120300*x**2+40200250*x)/(64*x**6-96*x**5+288*x**4-248*x**3+360*x**2-150*x+125),x)

[Out]

x**2 + ((4*x**2 - 2*x + 5)*exp(2*x + 2*exp(3)) + (-8*x**3 + 4*x**2 - 4010*x)*exp(x + exp(3)))/(4*x**2 - 2*x +
5) + (8000*x**3 + 3976000*x**2 + 20000*x - 25000)/(16*x**4 - 16*x**3 + 44*x**2 - 20*x + 25)

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