3.11.22 \(\int \frac {e^{2 x} (-4 x^3-2 x^4-2 x^5+2 x^6)+e^x (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8)}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx\)

Optimal. Leaf size=18 \[ \left (47+\frac {e^x}{\left (1-\frac {1}{x}+x\right )^2}\right )^2 \]

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Rubi [C]  time = 37.39, antiderivative size = 2487, normalized size of antiderivative = 138.17, number of steps used = 908, number of rules used = 6, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6688, 12, 6742, 2177, 2178, 2268}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(E^(2*x)*(-4*x^3 - 2*x^4 - 2*x^5 + 2*x^6) + E^x*(-188*x + 282*x^2 + 282*x^3 - 470*x^5 - 376*x^6 + 94*x^7 +
94*x^8))/(-1 + 5*x - 5*x^2 - 10*x^3 + 15*x^4 + 11*x^5 - 15*x^6 - 10*x^7 + 5*x^8 + 5*x^9 + x^10),x]

[Out]

(-64*E^(2*x))/(25*Sqrt[5]*(1 - Sqrt[5] + 2*x)^4) + (56*(5 - Sqrt[5])*E^(2*x))/(125*(1 - Sqrt[5] + 2*x)^4) - (3
2*E^(2*x))/(25*(1 - Sqrt[5] + 2*x)^3) - (64*E^(2*x))/(75*Sqrt[5]*(1 - Sqrt[5] + 2*x)^3) + (112*(5 - 3*Sqrt[5])
*E^(2*x))/(375*(1 - Sqrt[5] + 2*x)^3) - (8*(1 - Sqrt[5])*E^(2*x))/(5*(1 - Sqrt[5] + 2*x)^3) + (56*(5 - Sqrt[5]
)*E^(2*x))/(375*(1 - Sqrt[5] + 2*x)^3) - (376*E^x)/(5*Sqrt[5]*(1 - Sqrt[5] + 2*x)^2) + (564*(5 - Sqrt[5])*E^x)
/(25*(1 - Sqrt[5] + 2*x)^2) - (16*E^(2*x))/(25*(1 - Sqrt[5] + 2*x)^2) - (68*E^(2*x))/(75*Sqrt[5]*(1 - Sqrt[5]
+ 2*x)^2) + (28*(5 - 3*Sqrt[5])*E^(2*x))/(75*(1 - Sqrt[5] + 2*x)^2) - (12*(5 - 2*Sqrt[5])*E^(2*x))/(25*(1 - Sq
rt[5] + 2*x)^2) - (4*(1 - Sqrt[5])*E^(2*x))/(5*(1 - Sqrt[5] + 2*x)^2) + (148*(5 - Sqrt[5])*E^(2*x))/(375*(1 -
Sqrt[5] + 2*x)^2) - (1692*E^x)/(25*(1 - Sqrt[5] + 2*x)) - (188*E^x)/(5*Sqrt[5]*(1 - Sqrt[5] + 2*x)) - (282*(1
- Sqrt[5])*E^x)/(5*(1 - Sqrt[5] + 2*x)) + (564*(3 - Sqrt[5])*E^x)/(25*(1 - Sqrt[5] + 2*x)) + (282*(5 - Sqrt[5]
)*E^x)/(25*(1 - Sqrt[5] + 2*x)) - (96*E^(2*x))/(125*(1 - Sqrt[5] + 2*x)) - (68*E^(2*x))/(75*Sqrt[5]*(1 - Sqrt[
5] + 2*x)) + (28*(5 - 3*Sqrt[5])*E^(2*x))/(75*(1 - Sqrt[5] + 2*x)) - (12*(5 - 2*Sqrt[5])*E^(2*x))/(25*(1 - Sqr
t[5] + 2*x)) - (4*(1 - Sqrt[5])*E^(2*x))/(5*(1 - Sqrt[5] + 2*x)) + (8*(3 - Sqrt[5])*E^(2*x))/(25*(1 - Sqrt[5]
+ 2*x)) - (32*(5 - Sqrt[5])*E^(2*x))/(375*(1 - Sqrt[5] + 2*x)) + (28*(7 - Sqrt[5])*E^(2*x))/(125*(1 - Sqrt[5]
+ 2*x)) + (64*E^(2*x))/(25*Sqrt[5]*(1 + Sqrt[5] + 2*x)^4) + (56*(5 + Sqrt[5])*E^(2*x))/(125*(1 + Sqrt[5] + 2*x
)^4) - (32*E^(2*x))/(25*(1 + Sqrt[5] + 2*x)^3) + (64*E^(2*x))/(75*Sqrt[5]*(1 + Sqrt[5] + 2*x)^3) - (8*(1 + Sqr
t[5])*E^(2*x))/(5*(1 + Sqrt[5] + 2*x)^3) + (56*(5 + Sqrt[5])*E^(2*x))/(375*(1 + Sqrt[5] + 2*x)^3) + (112*(5 +
3*Sqrt[5])*E^(2*x))/(375*(1 + Sqrt[5] + 2*x)^3) + (376*E^x)/(5*Sqrt[5]*(1 + Sqrt[5] + 2*x)^2) + (564*(5 + Sqrt
[5])*E^x)/(25*(1 + Sqrt[5] + 2*x)^2) - (16*E^(2*x))/(25*(1 + Sqrt[5] + 2*x)^2) + (68*E^(2*x))/(75*Sqrt[5]*(1 +
Sqrt[5] + 2*x)^2) - (4*(1 + Sqrt[5])*E^(2*x))/(5*(1 + Sqrt[5] + 2*x)^2) + (148*(5 + Sqrt[5])*E^(2*x))/(375*(1
+ Sqrt[5] + 2*x)^2) - (12*(5 + 2*Sqrt[5])*E^(2*x))/(25*(1 + Sqrt[5] + 2*x)^2) + (28*(5 + 3*Sqrt[5])*E^(2*x))/
(75*(1 + Sqrt[5] + 2*x)^2) - (1692*E^x)/(25*(1 + Sqrt[5] + 2*x)) + (188*E^x)/(5*Sqrt[5]*(1 + Sqrt[5] + 2*x)) -
(282*(1 + Sqrt[5])*E^x)/(5*(1 + Sqrt[5] + 2*x)) + (564*(3 + Sqrt[5])*E^x)/(25*(1 + Sqrt[5] + 2*x)) + (282*(5
+ Sqrt[5])*E^x)/(25*(1 + Sqrt[5] + 2*x)) - (96*E^(2*x))/(125*(1 + Sqrt[5] + 2*x)) + (68*E^(2*x))/(75*Sqrt[5]*(
1 + Sqrt[5] + 2*x)) - (4*(1 + Sqrt[5])*E^(2*x))/(5*(1 + Sqrt[5] + 2*x)) + (8*(3 + Sqrt[5])*E^(2*x))/(25*(1 + S
qrt[5] + 2*x)) - (32*(5 + Sqrt[5])*E^(2*x))/(375*(1 + Sqrt[5] + 2*x)) + (28*(7 + Sqrt[5])*E^(2*x))/(125*(1 + S
qrt[5] + 2*x)) - (12*(5 + 2*Sqrt[5])*E^(2*x))/(25*(1 + Sqrt[5] + 2*x)) + (28*(5 + 3*Sqrt[5])*E^(2*x))/(75*(1 +
Sqrt[5] + 2*x)) + (846*E^((-1 + Sqrt[5])/2)*ExpIntegralEi[(1 - Sqrt[5] + 2*x)/2])/25 + (282*E^((-1 + Sqrt[5])
/2)*ExpIntegralEi[(1 - Sqrt[5] + 2*x)/2])/(5*Sqrt[5]) + (141*(1 - Sqrt[5])*E^((-1 + Sqrt[5])/2)*ExpIntegralEi[
(1 - Sqrt[5] + 2*x)/2])/5 - (282*(3 - Sqrt[5])*E^((-1 + Sqrt[5])/2)*ExpIntegralEi[(1 - Sqrt[5] + 2*x)/2])/25 -
(141*(5 - Sqrt[5])*E^((-1 + Sqrt[5])/2)*ExpIntegralEi[(1 - Sqrt[5] + 2*x)/2])/25 + (96*E^(-1 + Sqrt[5])*ExpIn
tegralEi[1 - Sqrt[5] + 2*x])/125 + (68*E^(-1 + Sqrt[5])*ExpIntegralEi[1 - Sqrt[5] + 2*x])/(75*Sqrt[5]) - (28*(
5 - 3*Sqrt[5])*E^(-1 + Sqrt[5])*ExpIntegralEi[1 - Sqrt[5] + 2*x])/75 + (12*(5 - 2*Sqrt[5])*E^(-1 + Sqrt[5])*Ex
pIntegralEi[1 - Sqrt[5] + 2*x])/25 + (4*(1 - Sqrt[5])*E^(-1 + Sqrt[5])*ExpIntegralEi[1 - Sqrt[5] + 2*x])/5 - (
8*(3 - Sqrt[5])*E^(-1 + Sqrt[5])*ExpIntegralEi[1 - Sqrt[5] + 2*x])/25 + (32*(5 - Sqrt[5])*E^(-1 + Sqrt[5])*Exp
IntegralEi[1 - Sqrt[5] + 2*x])/375 - (28*(7 - Sqrt[5])*E^(-1 + Sqrt[5])*ExpIntegralEi[1 - Sqrt[5] + 2*x])/125
+ (846*E^(-1/2 - Sqrt[5]/2)*ExpIntegralEi[(1 + Sqrt[5] + 2*x)/2])/25 - (282*E^(-1/2 - Sqrt[5]/2)*ExpIntegralEi
[(1 + Sqrt[5] + 2*x)/2])/(5*Sqrt[5]) + (141*(1 + Sqrt[5])*E^(-1/2 - Sqrt[5]/2)*ExpIntegralEi[(1 + Sqrt[5] + 2*
x)/2])/5 - (282*(3 + Sqrt[5])*E^(-1/2 - Sqrt[5]/2)*ExpIntegralEi[(1 + Sqrt[5] + 2*x)/2])/25 - (141*(5 + Sqrt[5
])*E^(-1/2 - Sqrt[5]/2)*ExpIntegralEi[(1 + Sqrt[5] + 2*x)/2])/25 + (96*E^(-1 - Sqrt[5])*ExpIntegralEi[1 + Sqrt
[5] + 2*x])/125 - (68*E^(-1 - Sqrt[5])*ExpIntegralEi[1 + Sqrt[5] + 2*x])/(75*Sqrt[5]) + (4*(1 + Sqrt[5])*E^(-1
- Sqrt[5])*ExpIntegralEi[1 + Sqrt[5] + 2*x])/5 - (8*(3 + Sqrt[5])*E^(-1 - Sqrt[5])*ExpIntegralEi[1 + Sqrt[5]
+ 2*x])/25 + (32*(5 + Sqrt[5])*E^(-1 - Sqrt[5])*ExpIntegralEi[1 + Sqrt[5] + 2*x])/375 - (28*(7 + Sqrt[5])*E^(-
1 - Sqrt[5])*ExpIntegralEi[1 + Sqrt[5] + 2*x])/125 + (12*(5 + 2*Sqrt[5])*E^(-1 - Sqrt[5])*ExpIntegralEi[1 + Sq
rt[5] + 2*x])/25 - (28*(5 + 3*Sqrt[5])*E^(-1 - Sqrt[5])*ExpIntegralEi[1 + Sqrt[5] + 2*x])/75

Rule 12

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

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 2268

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]

Rule 6688

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

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 {2 e^x x \left (2+x+x^2-x^3\right ) \left (47-94 x+\left (-47+e^x\right ) x^2+94 x^3+47 x^4\right )}{\left (1-x-x^2\right )^5} \, dx\\ &=2 \int \frac {e^x x \left (2+x+x^2-x^3\right ) \left (47-94 x+\left (-47+e^x\right ) x^2+94 x^3+47 x^4\right )}{\left (1-x-x^2\right )^5} \, dx\\ &=2 \int \left (\frac {47 e^x (-2+x) x \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5}-\frac {94 e^x (-2+x) x^2 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5}-\frac {47 e^x (-2+x) x^3 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5}+\frac {e^{2 x} (-2+x) x^3 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5}+\frac {94 e^x (-2+x) x^4 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5}+\frac {47 e^x (-2+x) x^5 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5}\right ) \, dx\\ &=2 \int \frac {e^{2 x} (-2+x) x^3 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5} \, dx+94 \int \frac {e^x (-2+x) x \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5} \, dx-94 \int \frac {e^x (-2+x) x^3 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5} \, dx+94 \int \frac {e^x (-2+x) x^5 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5} \, dx-188 \int \frac {e^x (-2+x) x^2 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5} \, dx+188 \int \frac {e^x (-2+x) x^4 \left (1+x+x^2\right )}{\left (-1+x+x^2\right )^5} \, dx\\ &=2 \int \left (-\frac {2 e^{2 x} (-4+7 x)}{\left (-1+x+x^2\right )^5}+\frac {e^{2 x} (14-15 x)}{\left (-1+x+x^2\right )^4}+\frac {e^{2 x} (7-4 x)}{\left (-1+x+x^2\right )^3}+\frac {e^{2 x}}{\left (-1+x+x^2\right )^2}\right ) \, dx+94 \int \left (-\frac {2 e^x (-1+3 x)}{\left (-1+x+x^2\right )^5}-\frac {3 e^x (-1+x)}{\left (-1+x+x^2\right )^4}+\frac {e^x}{\left (-1+x+x^2\right )^3}\right ) \, dx-94 \int \left (-\frac {2 e^x (-4+7 x)}{\left (-1+x+x^2\right )^5}+\frac {e^x (14-15 x)}{\left (-1+x+x^2\right )^4}+\frac {e^x (7-4 x)}{\left (-1+x+x^2\right )^3}+\frac {e^x}{\left (-1+x+x^2\right )^2}\right ) \, dx+94 \int \left (-\frac {2 e^x (-11+18 x)}{\left (-1+x+x^2\right )^5}+\frac {e^x (51-58 x)}{\left (-1+x+x^2\right )^4}-\frac {10 e^x (-4+3 x)}{\left (-1+x+x^2\right )^3}+\frac {e^x (12-5 x)}{\left (-1+x+x^2\right )^2}+\frac {e^x}{-1+x+x^2}\right ) \, dx-188 \int \left (\frac {2 e^x (-3+4 x)}{\left (-1+x+x^2\right )^5}+\frac {3 e^x (-3+2 x)}{\left (-1+x+x^2\right )^4}+\frac {e^x (-3+x)}{\left (-1+x+x^2\right )^3}\right ) \, dx+188 \int \left (\frac {2 e^x (-7+11 x)}{\left (-1+x+x^2\right )^5}+\frac {29 e^x (-1+x)}{\left (-1+x+x^2\right )^4}+\frac {e^x (-19+11 x)}{\left (-1+x+x^2\right )^3}+\frac {e^x (-4+x)}{\left (-1+x+x^2\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{2 x} (14-15 x)}{\left (-1+x+x^2\right )^4} \, dx+2 \int \frac {e^{2 x} (7-4 x)}{\left (-1+x+x^2\right )^3} \, dx+2 \int \frac {e^{2 x}}{\left (-1+x+x^2\right )^2} \, dx-4 \int \frac {e^{2 x} (-4+7 x)}{\left (-1+x+x^2\right )^5} \, dx+94 \int \frac {e^x (51-58 x)}{\left (-1+x+x^2\right )^4} \, dx-94 \int \frac {e^x (14-15 x)}{\left (-1+x+x^2\right )^4} \, dx+94 \int \frac {e^x}{\left (-1+x+x^2\right )^3} \, dx-94 \int \frac {e^x (7-4 x)}{\left (-1+x+x^2\right )^3} \, dx-94 \int \frac {e^x}{\left (-1+x+x^2\right )^2} \, dx+94 \int \frac {e^x (12-5 x)}{\left (-1+x+x^2\right )^2} \, dx+94 \int \frac {e^x}{-1+x+x^2} \, dx-188 \int \frac {e^x (-1+3 x)}{\left (-1+x+x^2\right )^5} \, dx+188 \int \frac {e^x (-4+7 x)}{\left (-1+x+x^2\right )^5} \, dx-188 \int \frac {e^x (-11+18 x)}{\left (-1+x+x^2\right )^5} \, dx-188 \int \frac {e^x (-3+x)}{\left (-1+x+x^2\right )^3} \, dx+188 \int \frac {e^x (-19+11 x)}{\left (-1+x+x^2\right )^3} \, dx+188 \int \frac {e^x (-4+x)}{\left (-1+x+x^2\right )^2} \, dx-282 \int \frac {e^x (-1+x)}{\left (-1+x+x^2\right )^4} \, dx-376 \int \frac {e^x (-3+4 x)}{\left (-1+x+x^2\right )^5} \, dx+376 \int \frac {e^x (-7+11 x)}{\left (-1+x+x^2\right )^5} \, dx-564 \int \frac {e^x (-3+2 x)}{\left (-1+x+x^2\right )^4} \, dx-940 \int \frac {e^x (-4+3 x)}{\left (-1+x+x^2\right )^3} \, dx+5452 \int \frac {e^x (-1+x)}{\left (-1+x+x^2\right )^4} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 5.04, size = 39, normalized size = 2.17 \begin {gather*} \frac {e^x x^2 \left (94-188 x+\left (-94+e^x\right ) x^2+188 x^3+94 x^4\right )}{\left (-1+x+x^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(-4*x^3 - 2*x^4 - 2*x^5 + 2*x^6) + E^x*(-188*x + 282*x^2 + 282*x^3 - 470*x^5 - 376*x^6 + 94
*x^7 + 94*x^8))/(-1 + 5*x - 5*x^2 - 10*x^3 + 15*x^4 + 11*x^5 - 15*x^6 - 10*x^7 + 5*x^8 + 5*x^9 + x^10),x]

[Out]

(E^x*x^2*(94 - 188*x + (-94 + E^x)*x^2 + 188*x^3 + 94*x^4))/(-1 + x + x^2)^4

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fricas [B]  time = 0.65, size = 76, normalized size = 4.22 \begin {gather*} \frac {x^{4} e^{\left (2 \, x\right )} + 94 \, {\left (x^{6} + 2 \, x^{5} - x^{4} - 2 \, x^{3} + x^{2}\right )} e^{x}}{x^{8} + 4 \, x^{7} + 2 \, x^{6} - 8 \, x^{5} - 5 \, x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x^5+282*x^3+282*x^2-188*x)*exp(x))/(x
^10+5*x^9+5*x^8-10*x^7-15*x^6+11*x^5+15*x^4-10*x^3-5*x^2+5*x-1),x, algorithm="fricas")

[Out]

(x^4*e^(2*x) + 94*(x^6 + 2*x^5 - x^4 - 2*x^3 + x^2)*e^x)/(x^8 + 4*x^7 + 2*x^6 - 8*x^5 - 5*x^4 + 8*x^3 + 2*x^2
- 4*x + 1)

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giac [B]  time = 0.30, size = 85, normalized size = 4.72 \begin {gather*} \frac {94 \, x^{6} e^{x} + 188 \, x^{5} e^{x} + x^{4} e^{\left (2 \, x\right )} - 94 \, x^{4} e^{x} - 188 \, x^{3} e^{x} + 94 \, x^{2} e^{x}}{x^{8} + 4 \, x^{7} + 2 \, x^{6} - 8 \, x^{5} - 5 \, x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x^5+282*x^3+282*x^2-188*x)*exp(x))/(x
^10+5*x^9+5*x^8-10*x^7-15*x^6+11*x^5+15*x^4-10*x^3-5*x^2+5*x-1),x, algorithm="giac")

[Out]

(94*x^6*e^x + 188*x^5*e^x + x^4*e^(2*x) - 94*x^4*e^x - 188*x^3*e^x + 94*x^2*e^x)/(x^8 + 4*x^7 + 2*x^6 - 8*x^5
- 5*x^4 + 8*x^3 + 2*x^2 - 4*x + 1)

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maple [A]  time = 0.16, size = 33, normalized size = 1.83

method	result	size
risch	\(\frac {x^{4} {\mathrm e}^{2 x}}{\left (x^{2}+x -1\right )^{4}}+\frac {94 x^{2} {\mathrm e}^{x}}{\left (x^{2}+x -1\right )^{2}}\)	\(33\)
norman	\(\frac {{\mathrm e}^{2 x} x^{4}+188 x^{5} {\mathrm e}^{x}+94 x^{6} {\mathrm e}^{x}+94 \,{\mathrm e}^{x} x^{2}-188 \,{\mathrm e}^{x} x^{3}-94 \,{\mathrm e}^{x} x^{4}}{\left (x^{2}+x -1\right )^{4}}\)	\(54\)
default	\(-\frac {47 \,{\mathrm e}^{x} \left (7 x^{7}-53 x^{6}-377 x^{5}-870 x^{4}+685 x^{3}+446 x^{2}-514 x +119\right )}{300 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {{\mathrm e}^{2 x} \left (12 x^{7}+2 x^{6}-142 x^{5}-205 x^{4}+240 x^{3}+96 x^{2}-154 x +39\right )}{250 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {47 \,{\mathrm e}^{x} \left (4 x^{7}-241 x^{6}-1269 x^{5}+185 x^{4}+1145 x^{3}-263 x^{2}-333 x +118\right )}{375 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {{\mathrm e}^{2 x} \left (128 x^{7}+538 x^{6}+552 x^{5}-870 x^{4}-340 x^{3}+574 x^{2}-126 x -9\right )}{750 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {47 \,{\mathrm e}^{x} \left (437 x^{7}+1952 x^{6}-232 x^{5}-2445 x^{4}+560 x^{3}+1111 x^{2}-599 x +79\right )}{1500 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {47 \,{\mathrm e}^{x} \left (399 x^{7}-971 x^{6}+761 x^{5}+2960 x^{4}-2405 x^{3}-1828 x^{2}+2152 x -517\right )}{1500 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {47 \,{\mathrm e}^{x} \left (253 x^{7}+788 x^{6}-108 x^{5}-1705 x^{4}+140 x^{3}+1459 x^{2}-531 x +151\right )}{750 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {47 \,{\mathrm e}^{x} \left (109 x^{7}+339 x^{6}-49 x^{5}-740 x^{4}+45 x^{3}+552 x^{2}-668 x +153\right )}{500 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {47 \,{\mathrm e}^{x} \left (18 x^{7}+53 x^{6}-23 x^{5}-155 x^{4}-85 x^{3}-371 x^{2}+239 x -44\right )}{500 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {{\mathrm e}^{2 x} \left (56 x^{7}+176 x^{6}+4 x^{5}-290 x^{4}+120 x^{3}+448 x^{2}-302 x +57\right )}{375 x^{8}+1500 x^{7}+750 x^{6}-3000 x^{5}-1875 x^{4}+3000 x^{3}+750 x^{2}-1500 x +375}+\frac {{\mathrm e}^{2 x} \left (52 x^{7}+192 x^{6}+118 x^{5}-155 x^{4}+140 x^{3}-34 x^{2}+16 x -6\right )}{750 x^{8}+3000 x^{7}+1500 x^{6}-6000 x^{5}-3750 x^{4}+6000 x^{3}+1500 x^{2}-3000 x +750}\)	\(879\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x^5+282*x^3+282*x^2-188*x)*exp(x))/(x^10+5*
x^9+5*x^8-10*x^7-15*x^6+11*x^5+15*x^4-10*x^3-5*x^2+5*x-1),x,method=_RETURNVERBOSE)

[Out]

x^4/(x^2+x-1)^4*exp(2*x)+94*x^2/(x^2+x-1)^2*exp(x)

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maxima [B]  time = 0.45, size = 76, normalized size = 4.22 \begin {gather*} \frac {x^{4} e^{\left (2 \, x\right )} + 94 \, {\left (x^{6} + 2 \, x^{5} - x^{4} - 2 \, x^{3} + x^{2}\right )} e^{x}}{x^{8} + 4 \, x^{7} + 2 \, x^{6} - 8 \, x^{5} - 5 \, x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x^5+282*x^3+282*x^2-188*x)*exp(x))/(x
^10+5*x^9+5*x^8-10*x^7-15*x^6+11*x^5+15*x^4-10*x^3-5*x^2+5*x-1),x, algorithm="maxima")

[Out]

(x^4*e^(2*x) + 94*(x^6 + 2*x^5 - x^4 - 2*x^3 + x^2)*e^x)/(x^8 + 4*x^7 + 2*x^6 - 8*x^5 - 5*x^4 + 8*x^3 + 2*x^2
- 4*x + 1)

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mupad [B]  time = 0.35, size = 40, normalized size = 2.22 \begin {gather*} \frac {x^2\,{\mathrm {e}}^x\,\left (x^2\,{\mathrm {e}}^x-188\,x-94\,x^2+188\,x^3+94\,x^4+94\right )}{{\left (x^2+x-1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(282*x^2 - 188*x + 282*x^3 - 470*x^5 - 376*x^6 + 94*x^7 + 94*x^8) - exp(2*x)*(4*x^3 + 2*x^4 + 2*x^
5 - 2*x^6))/(5*x - 5*x^2 - 10*x^3 + 15*x^4 + 11*x^5 - 15*x^6 - 10*x^7 + 5*x^8 + 5*x^9 + x^10 - 1),x)

[Out]

(x^2*exp(x)*(x^2*exp(x) - 188*x - 94*x^2 + 188*x^3 + 94*x^4 + 94))/(x + x^2 - 1)^4

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sympy [B]  time = 0.19, size = 133, normalized size = 7.39 \begin {gather*} \frac {\left (x^{8} + 2 x^{7} - x^{6} - 2 x^{5} + x^{4}\right ) e^{2 x} + \left (94 x^{10} + 376 x^{9} + 188 x^{8} - 752 x^{7} - 470 x^{6} + 752 x^{5} + 188 x^{4} - 376 x^{3} + 94 x^{2}\right ) e^{x}}{x^{12} + 6 x^{11} + 9 x^{10} - 10 x^{9} - 30 x^{8} + 6 x^{7} + 41 x^{6} - 6 x^{5} - 30 x^{4} + 10 x^{3} + 9 x^{2} - 6 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**6-2*x**5-2*x**4-4*x**3)*exp(x)**2+(94*x**8+94*x**7-376*x**6-470*x**5+282*x**3+282*x**2-188*x)
*exp(x))/(x**10+5*x**9+5*x**8-10*x**7-15*x**6+11*x**5+15*x**4-10*x**3-5*x**2+5*x-1),x)

[Out]

((x**8 + 2*x**7 - x**6 - 2*x**5 + x**4)*exp(2*x) + (94*x**10 + 376*x**9 + 188*x**8 - 752*x**7 - 470*x**6 + 752
*x**5 + 188*x**4 - 376*x**3 + 94*x**2)*exp(x))/(x**12 + 6*x**11 + 9*x**10 - 10*x**9 - 30*x**8 + 6*x**7 + 41*x*
*6 - 6*x**5 - 30*x**4 + 10*x**3 + 9*x**2 - 6*x + 1)

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