3.11.16 \(\int \frac {e^6 (450+360 x+72 x^3-18 x^4+e (-150 x+90 x^3-60 x^4)+e^2 (-50 x^3-50 x^4))+e^{6+x} (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e (540 x^3+120 x^4-60 x^5)+e^2 (-50 x^3-150 x^4-50 x^5))+e^{6+2 x} (-360 x^3+162 x^4-18 x^5+e (150 x^3+240 x^4-60 x^5)+e^2 (-50 x^4-50 x^5))}{25 x^3} \, dx\)

Optimal. Leaf size=34 \[ e^4-e^6 \left (-\frac {3}{5}-e+\frac {3}{x}\right )^2 \left (1+\left (1+e^x\right ) x\right )^2 \]

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Rubi [B]  time = 0.64, antiderivative size = 296, normalized size of antiderivative = 8.71, number of steps used = 24, number of rules used = 9, integrand size = 183, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {12, 14, 2196, 2194, 2176, 1612, 2199, 2177, 2178} \begin {gather*} -\frac {2}{25} (3+5 e)^2 e^{x+6} x^2-\frac {1}{25} (3+5 e)^2 e^{2 x+6} x^2-\frac {1}{25} e^6 (3+5 e)^2 x^2-\frac {9 e^6}{x^2}+\frac {1}{25} \left (81+120 e-25 e^2\right ) e^{2 x+6} x+\frac {6}{25} \left (21+20 e-25 e^2\right ) e^{x+6} x+\frac {2}{25} e^6 (36+5 (9-5 e) e) x+\frac {4}{25} (3+5 e)^2 e^{x+6} x+\frac {1}{25} (3+5 e)^2 e^{2 x+6} x-\frac {3}{5} (12-5 e) e^{2 x+6}-\frac {1}{50} \left (81+120 e-25 e^2\right ) e^{2 x+6}-\frac {6}{25} \left (21+20 e-25 e^2\right ) e^{x+6}-\frac {2}{25} (54-5 (54-5 e) e) e^{x+6}-\frac {4}{25} (3+5 e)^2 e^{x+6}-\frac {1}{50} (3+5 e)^2 e^{2 x+6}-\frac {18 e^{x+6}}{x}-\frac {6 (12-5 e) e^6}{5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^6*(450 + 360*x + 72*x^3 - 18*x^4 + E*(-150*x + 90*x^3 - 60*x^4) + E^2*(-50*x^3 - 50*x^4)) + E^(6 + x)*(
450*x - 450*x^2 - 108*x^3 + 126*x^4 - 18*x^5 + E*(540*x^3 + 120*x^4 - 60*x^5) + E^2*(-50*x^3 - 150*x^4 - 50*x^
5)) + E^(6 + 2*x)*(-360*x^3 + 162*x^4 - 18*x^5 + E*(150*x^3 + 240*x^4 - 60*x^5) + E^2*(-50*x^4 - 50*x^5)))/(25
*x^3),x]

[Out]

(-3*(12 - 5*E)*E^(6 + 2*x))/5 - (4*E^(6 + x)*(3 + 5*E)^2)/25 - (E^(6 + 2*x)*(3 + 5*E)^2)/50 - (2*E^(6 + x)*(54
- 5*(54 - 5*E)*E))/25 - (6*E^(6 + x)*(21 + 20*E - 25*E^2))/25 - (E^(6 + 2*x)*(81 + 120*E - 25*E^2))/50 - (9*E
^6)/x^2 - (6*(12 - 5*E)*E^6)/(5*x) - (18*E^(6 + x))/x + (4*E^(6 + x)*(3 + 5*E)^2*x)/25 + (E^(6 + 2*x)*(3 + 5*E
)^2*x)/25 + (2*E^6*(36 + 5*(9 - 5*E)*E)*x)/25 + (6*E^(6 + x)*(21 + 20*E - 25*E^2)*x)/25 + (E^(6 + 2*x)*(81 + 1
20*E - 25*E^2)*x)/25 - (E^6*(3 + 5*E)^2*x^2)/25 - (2*E^(6 + x)*(3 + 5*E)^2*x^2)/25 - (E^(6 + 2*x)*(3 + 5*E)^2*
x^2)/25

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \frac {e^6 \left (450+360 x+72 x^3-18 x^4+e \left (-150 x+90 x^3-60 x^4\right )+e^2 \left (-50 x^3-50 x^4\right )\right )+e^{6+x} \left (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e \left (540 x^3+120 x^4-60 x^5\right )+e^2 \left (-50 x^3-150 x^4-50 x^5\right )\right )+e^{6+2 x} \left (-360 x^3+162 x^4-18 x^5+e \left (150 x^3+240 x^4-60 x^5\right )+e^2 \left (-50 x^4-50 x^5\right )\right )}{x^3} \, dx\\ &=\frac {1}{25} \int \left (2 e^{6+2 x} (15-(3+5 e) x) (-12+5 e+(3+5 e) x)+\frac {2 e^6 (1+x) (15-(3+5 e) x) \left (15+(3+5 e) x^2\right )}{x^3}+\frac {2 e^{6+x} (15-(3+5 e) x) \left (15-(12-5 e) x-3 (2-5 e) x^2+(3+5 e) x^3\right )}{x^2}\right ) \, dx\\ &=\frac {2}{25} \int e^{6+2 x} (15-(3+5 e) x) (-12+5 e+(3+5 e) x) \, dx+\frac {2}{25} \int \frac {e^{6+x} (15-(3+5 e) x) \left (15-(12-5 e) x-3 (2-5 e) x^2+(3+5 e) x^3\right )}{x^2} \, dx+\frac {1}{25} \left (2 e^6\right ) \int \frac {(1+x) (15-(3+5 e) x) \left (15+(3+5 e) x^2\right )}{x^3} \, dx\\ &=\frac {2}{25} \int \left (-54 e^{6+x} \left (1+\frac {5}{54} e (-54+5 e)\right )+\frac {225 e^{6+x}}{x^2}-\frac {225 e^{6+x}}{x}-3 e^{6+x} \left (-21-20 e+25 e^2\right ) x-e^{6+x} (3+5 e)^2 x^2\right ) \, dx+\frac {2}{25} \int \left (15 e^{6+2 x} (-12+5 e)-e^{6+2 x} \left (-81-120 e+25 e^2\right ) x-e^{6+2 x} (3+5 e)^2 x^2\right ) \, dx+\frac {1}{25} \left (2 e^6\right ) \int \left (36 \left (1-\frac {5}{36} e (-9+5 e)\right )+\frac {225}{x^3}-\frac {15 (-12+5 e)}{x^2}-(3+5 e)^2 x\right ) \, dx\\ &=-\frac {9 e^6}{x^2}-\frac {6 (12-5 e) e^6}{5 x}+\frac {2}{25} e^6 (36+5 (9-5 e) e) x-\frac {1}{25} e^6 (3+5 e)^2 x^2+18 \int \frac {e^{6+x}}{x^2} \, dx-18 \int \frac {e^{6+x}}{x} \, dx-\frac {1}{5} (6 (12-5 e)) \int e^{6+2 x} \, dx-\frac {1}{25} \left (2 (3+5 e)^2\right ) \int e^{6+x} x^2 \, dx-\frac {1}{25} \left (2 (3+5 e)^2\right ) \int e^{6+2 x} x^2 \, dx-\frac {1}{25} (2 (54-5 (54-5 e) e)) \int e^{6+x} \, dx+\frac {1}{25} \left (6 \left (21+20 e-25 e^2\right )\right ) \int e^{6+x} x \, dx+\frac {1}{25} \left (2 \left (81+120 e-25 e^2\right )\right ) \int e^{6+2 x} x \, dx\\ &=-\frac {3}{5} (12-5 e) e^{6+2 x}-\frac {2}{25} e^{6+x} (54-5 (54-5 e) e)-\frac {9 e^6}{x^2}-\frac {6 (12-5 e) e^6}{5 x}-\frac {18 e^{6+x}}{x}+\frac {2}{25} e^6 (36+5 (9-5 e) e) x+\frac {6}{25} e^{6+x} \left (21+20 e-25 e^2\right ) x+\frac {1}{25} e^{6+2 x} \left (81+120 e-25 e^2\right ) x-\frac {1}{25} e^6 (3+5 e)^2 x^2-\frac {2}{25} e^{6+x} (3+5 e)^2 x^2-\frac {1}{25} e^{6+2 x} (3+5 e)^2 x^2-18 e^6 \text {Ei}(x)+18 \int \frac {e^{6+x}}{x} \, dx+\frac {1}{25} \left (2 (3+5 e)^2\right ) \int e^{6+2 x} x \, dx+\frac {1}{25} \left (4 (3+5 e)^2\right ) \int e^{6+x} x \, dx-\frac {1}{25} \left (6 \left (21+20 e-25 e^2\right )\right ) \int e^{6+x} \, dx+\frac {1}{25} \left (-81-120 e+25 e^2\right ) \int e^{6+2 x} \, dx\\ &=-\frac {3}{5} (12-5 e) e^{6+2 x}-\frac {2}{25} e^{6+x} (54-5 (54-5 e) e)-\frac {6}{25} e^{6+x} \left (21+20 e-25 e^2\right )-\frac {1}{50} e^{6+2 x} \left (81+120 e-25 e^2\right )-\frac {9 e^6}{x^2}-\frac {6 (12-5 e) e^6}{5 x}-\frac {18 e^{6+x}}{x}+\frac {4}{25} e^{6+x} (3+5 e)^2 x+\frac {1}{25} e^{6+2 x} (3+5 e)^2 x+\frac {2}{25} e^6 (36+5 (9-5 e) e) x+\frac {6}{25} e^{6+x} \left (21+20 e-25 e^2\right ) x+\frac {1}{25} e^{6+2 x} \left (81+120 e-25 e^2\right ) x-\frac {1}{25} e^6 (3+5 e)^2 x^2-\frac {2}{25} e^{6+x} (3+5 e)^2 x^2-\frac {1}{25} e^{6+2 x} (3+5 e)^2 x^2-\frac {1}{25} (3+5 e)^2 \int e^{6+2 x} \, dx-\frac {1}{25} \left (4 (3+5 e)^2\right ) \int e^{6+x} \, dx\\ &=-\frac {3}{5} (12-5 e) e^{6+2 x}-\frac {4}{25} e^{6+x} (3+5 e)^2-\frac {1}{50} e^{6+2 x} (3+5 e)^2-\frac {2}{25} e^{6+x} (54-5 (54-5 e) e)-\frac {6}{25} e^{6+x} \left (21+20 e-25 e^2\right )-\frac {1}{50} e^{6+2 x} \left (81+120 e-25 e^2\right )-\frac {9 e^6}{x^2}-\frac {6 (12-5 e) e^6}{5 x}-\frac {18 e^{6+x}}{x}+\frac {4}{25} e^{6+x} (3+5 e)^2 x+\frac {1}{25} e^{6+2 x} (3+5 e)^2 x+\frac {2}{25} e^6 (36+5 (9-5 e) e) x+\frac {6}{25} e^{6+x} \left (21+20 e-25 e^2\right ) x+\frac {1}{25} e^{6+2 x} \left (81+120 e-25 e^2\right ) x-\frac {1}{25} e^6 (3+5 e)^2 x^2-\frac {2}{25} e^{6+x} (3+5 e)^2 x^2-\frac {1}{25} e^{6+2 x} (3+5 e)^2 x^2\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.39, size = 86, normalized size = 2.53 \begin {gather*} -\frac {e^6 \left (225+30 \left (12-5 e+15 e^x\right ) x+15 e^x \left (18-20 e+15 e^x\right ) x^2-2 (3+5 e) \left (1+e^x\right ) \left (12-5 e+15 e^x\right ) x^3+(3+5 e)^2 \left (1+e^x\right )^2 x^4\right )}{25 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^6*(450 + 360*x + 72*x^3 - 18*x^4 + E*(-150*x + 90*x^3 - 60*x^4) + E^2*(-50*x^3 - 50*x^4)) + E^(6
+ x)*(450*x - 450*x^2 - 108*x^3 + 126*x^4 - 18*x^5 + E*(540*x^3 + 120*x^4 - 60*x^5) + E^2*(-50*x^3 - 150*x^4 -
50*x^5)) + E^(6 + 2*x)*(-360*x^3 + 162*x^4 - 18*x^5 + E*(150*x^3 + 240*x^4 - 60*x^5) + E^2*(-50*x^4 - 50*x^5)
))/(25*x^3),x]

[Out]

-1/25*(E^6*(225 + 30*(12 - 5*E + 15*E^x)*x + 15*E^x*(18 - 20*E + 15*E^x)*x^2 - 2*(3 + 5*E)*(1 + E^x)*(12 - 5*E
+ 15*E^x)*x^3 + (3 + 5*E)^2*(1 + E^x)^2*x^4))/x^2

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fricas [B]  time = 0.69, size = 154, normalized size = 4.53 \begin {gather*} -\frac {{\left (25 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{14} + 30 \, {\left (x^{4} - 3 \, x^{3} - 5 \, x\right )} e^{13} + 9 \, {\left (x^{4} - 8 \, x^{3} + 40 \, x + 25\right )} e^{12} + {\left (25 \, x^{4} e^{2} + 9 \, x^{4} - 90 \, x^{3} + 225 \, x^{2} + 30 \, {\left (x^{4} - 5 \, x^{3}\right )} e\right )} e^{\left (2 \, x + 12\right )} + 2 \, {\left (25 \, {\left (x^{4} + x^{3}\right )} e^{8} + 30 \, {\left (x^{4} - 4 \, x^{3} - 5 \, x^{2}\right )} e^{7} + 9 \, {\left (x^{4} - 9 \, x^{3} + 15 \, x^{2} + 25 \, x\right )} e^{6}\right )} e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}}{25 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(((-50*x^5-50*x^4)*exp(1)^2+(-60*x^5+240*x^4+150*x^3)*exp(1)-18*x^5+162*x^4-360*x^3)*exp(3)^2*e
xp(x)^2+((-50*x^5-150*x^4-50*x^3)*exp(1)^2+(-60*x^5+120*x^4+540*x^3)*exp(1)-18*x^5+126*x^4-108*x^3-450*x^2+450
*x)*exp(3)^2*exp(x)+((-50*x^4-50*x^3)*exp(1)^2+(-60*x^4+90*x^3-150*x)*exp(1)-18*x^4+72*x^3+360*x+450)*exp(3)^2
)/x^3,x, algorithm="fricas")

[Out]

-1/25*(25*(x^4 + 2*x^3)*e^14 + 30*(x^4 - 3*x^3 - 5*x)*e^13 + 9*(x^4 - 8*x^3 + 40*x + 25)*e^12 + (25*x^4*e^2 +
9*x^4 - 90*x^3 + 225*x^2 + 30*(x^4 - 5*x^3)*e)*e^(2*x + 12) + 2*(25*(x^4 + x^3)*e^8 + 30*(x^4 - 4*x^3 - 5*x^2)
*e^7 + 9*(x^4 - 9*x^3 + 15*x^2 + 25*x)*e^6)*e^(x + 6))*e^(-6)/x^2

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giac [B]  time = 0.33, size = 207, normalized size = 6.09 \begin {gather*} -\frac {25 \, x^{4} e^{8} + 30 \, x^{4} e^{7} + 9 \, x^{4} e^{6} + 25 \, x^{4} e^{\left (2 \, x + 8\right )} + 30 \, x^{4} e^{\left (2 \, x + 7\right )} + 9 \, x^{4} e^{\left (2 \, x + 6\right )} + 50 \, x^{4} e^{\left (x + 8\right )} + 60 \, x^{4} e^{\left (x + 7\right )} + 18 \, x^{4} e^{\left (x + 6\right )} + 50 \, x^{3} e^{8} - 90 \, x^{3} e^{7} - 72 \, x^{3} e^{6} - 150 \, x^{3} e^{\left (2 \, x + 7\right )} - 90 \, x^{3} e^{\left (2 \, x + 6\right )} + 50 \, x^{3} e^{\left (x + 8\right )} - 240 \, x^{3} e^{\left (x + 7\right )} - 162 \, x^{3} e^{\left (x + 6\right )} + 225 \, x^{2} e^{\left (2 \, x + 6\right )} - 300 \, x^{2} e^{\left (x + 7\right )} + 270 \, x^{2} e^{\left (x + 6\right )} - 150 \, x e^{7} + 360 \, x e^{6} + 450 \, x e^{\left (x + 6\right )} + 225 \, e^{6}}{25 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(((-50*x^5-50*x^4)*exp(1)^2+(-60*x^5+240*x^4+150*x^3)*exp(1)-18*x^5+162*x^4-360*x^3)*exp(3)^2*e
xp(x)^2+((-50*x^5-150*x^4-50*x^3)*exp(1)^2+(-60*x^5+120*x^4+540*x^3)*exp(1)-18*x^5+126*x^4-108*x^3-450*x^2+450
*x)*exp(3)^2*exp(x)+((-50*x^4-50*x^3)*exp(1)^2+(-60*x^4+90*x^3-150*x)*exp(1)-18*x^4+72*x^3+360*x+450)*exp(3)^2
)/x^3,x, algorithm="giac")

[Out]

-1/25*(25*x^4*e^8 + 30*x^4*e^7 + 9*x^4*e^6 + 25*x^4*e^(2*x + 8) + 30*x^4*e^(2*x + 7) + 9*x^4*e^(2*x + 6) + 50*
x^4*e^(x + 8) + 60*x^4*e^(x + 7) + 18*x^4*e^(x + 6) + 50*x^3*e^8 - 90*x^3*e^7 - 72*x^3*e^6 - 150*x^3*e^(2*x +
7) - 90*x^3*e^(2*x + 6) + 50*x^3*e^(x + 8) - 240*x^3*e^(x + 7) - 162*x^3*e^(x + 6) + 225*x^2*e^(2*x + 6) - 300
*x^2*e^(x + 7) + 270*x^2*e^(x + 6) - 150*x*e^7 + 360*x*e^6 + 450*x*e^(x + 6) + 225*e^6)/x^2

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maple [B]  time = 0.06, size = 158, normalized size = 4.65

method	result	size
risch	\(-x^{2} {\mathrm e}^{8}-2 x \,{\mathrm e}^{8}-\frac {6 x^{2} {\mathrm e}^{7}}{5}+\frac {18 x \,{\mathrm e}^{7}}{5}-\frac {9 x^{2} {\mathrm e}^{6}}{25}+\frac {72 x \,{\mathrm e}^{6}}{25}+\frac {\left (150 \,{\mathrm e}^{7}-360 \,{\mathrm e}^{6}\right ) x -225 \,{\mathrm e}^{6}}{25 x^{2}}+\frac {\left (-25 x^{2} {\mathrm e}^{8}-30 x^{2} {\mathrm e}^{7}+150 x \,{\mathrm e}^{7}-9 x^{2} {\mathrm e}^{6}+90 x \,{\mathrm e}^{6}-225 \,{\mathrm e}^{6}\right ) {\mathrm e}^{2 x}}{25}-\frac {2 \left (25 x^{3} {\mathrm e}^{2}+25 x^{2} {\mathrm e}^{2}+30 x^{3} {\mathrm e}-120 x^{2} {\mathrm e}+9 x^{3}-150 x \,{\mathrm e}-81 x^{2}+135 x +225\right ) {\mathrm e}^{x +6}}{25 x}\)	\(158\)
norman	\(\frac {\left (6 \,{\mathrm e} \,{\mathrm e}^{6}-\frac {72 \,{\mathrm e}^{6}}{5}\right ) x +\left (-2 \,{\mathrm e}^{6} {\mathrm e}^{2}+\frac {18 \,{\mathrm e} \,{\mathrm e}^{6}}{5}+\frac {72 \,{\mathrm e}^{6}}{25}\right ) x^{3}+\left (-{\mathrm e}^{6} {\mathrm e}^{2}-\frac {6 \,{\mathrm e} \,{\mathrm e}^{6}}{5}-\frac {9 \,{\mathrm e}^{6}}{25}\right ) x^{4}+\left (6 \,{\mathrm e} \,{\mathrm e}^{6}+\frac {18 \,{\mathrm e}^{6}}{5}\right ) x^{3} {\mathrm e}^{2 x}+\left (12 \,{\mathrm e} \,{\mathrm e}^{6}-\frac {54 \,{\mathrm e}^{6}}{5}\right ) x^{2} {\mathrm e}^{x}+\left (-2 \,{\mathrm e}^{6} {\mathrm e}^{2}-\frac {12 \,{\mathrm e} \,{\mathrm e}^{6}}{5}-\frac {18 \,{\mathrm e}^{6}}{25}\right ) x^{4} {\mathrm e}^{x}+\left (-2 \,{\mathrm e}^{6} {\mathrm e}^{2}+\frac {48 \,{\mathrm e} \,{\mathrm e}^{6}}{5}+\frac {162 \,{\mathrm e}^{6}}{25}\right ) x^{3} {\mathrm e}^{x}+\left (-{\mathrm e}^{6} {\mathrm e}^{2}-\frac {6 \,{\mathrm e} \,{\mathrm e}^{6}}{5}-\frac {9 \,{\mathrm e}^{6}}{25}\right ) x^{4} {\mathrm e}^{2 x}-9 \,{\mathrm e}^{6}-9 x^{2} {\mathrm e}^{6} {\mathrm e}^{2 x}-18 x \,{\mathrm e}^{6} {\mathrm e}^{x}}{x^{2}}\)	\(248\)
default	\(-\frac {9 x^{2} {\mathrm e}^{6}}{25}+\frac {72 x \,{\mathrm e}^{6}}{25}-\frac {36 \,{\mathrm e}^{6} {\mathrm e}^{2 x}}{5}+\frac {126 \,{\mathrm e}^{6} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )}{25}-\frac {18 \,{\mathrm e}^{6} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )}{25}-\frac {18 \,{\mathrm e}^{6} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )}{25}+\frac {162 \,{\mathrm e}^{6} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )}{25}-\frac {72 \,{\mathrm e}^{6}}{5 x}-\frac {9 \,{\mathrm e}^{6}}{x^{2}}+18 \,{\mathrm e}^{6} \expIntegralEi \left (1, -x \right )+18 \,{\mathrm e}^{6} \left (-\frac {{\mathrm e}^{x}}{x}-\expIntegralEi \left (1, -x \right )\right )-\frac {108 \,{\mathrm e}^{6} {\mathrm e}^{x}}{25}-2 \,{\mathrm e}^{6} {\mathrm e}^{x} {\mathrm e}^{2}+\frac {108 \,{\mathrm e}^{6} {\mathrm e}^{x} {\mathrm e}}{5}+3 \,{\mathrm e}^{6} {\mathrm e}^{2 x} {\mathrm e}-\frac {12 \,{\mathrm e} \,{\mathrm e}^{6} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )}{5}-2 \,{\mathrm e}^{6} {\mathrm e}^{2} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )-\frac {12 \,{\mathrm e} \,{\mathrm e}^{6} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )}{5}-2 \,{\mathrm e}^{6} {\mathrm e}^{2} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )+\frac {24 \,{\mathrm e} \,{\mathrm e}^{6} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )}{5}-6 \,{\mathrm e}^{6} {\mathrm e}^{2} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+\frac {48 \,{\mathrm e} \,{\mathrm e}^{6} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )}{5}-2 \,{\mathrm e}^{6} {\mathrm e}^{2} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-\frac {6 \,{\mathrm e} \,{\mathrm e}^{6} x^{2}}{5}-{\mathrm e}^{6} {\mathrm e}^{2} x^{2}+\frac {6 \,{\mathrm e} \,{\mathrm e}^{6}}{x}+\frac {18 \,{\mathrm e}^{6} x \,{\mathrm e}}{5}-2 x \,{\mathrm e}^{6} {\mathrm e}^{2}\)	\(457\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*(((-50*x^5-50*x^4)*exp(1)^2+(-60*x^5+240*x^4+150*x^3)*exp(1)-18*x^5+162*x^4-360*x^3)*exp(3)^2*exp(x)^
2+((-50*x^5-150*x^4-50*x^3)*exp(1)^2+(-60*x^5+120*x^4+540*x^3)*exp(1)-18*x^5+126*x^4-108*x^3-450*x^2+450*x)*ex
p(3)^2*exp(x)+((-50*x^4-50*x^3)*exp(1)^2+(-60*x^4+90*x^3-150*x)*exp(1)-18*x^4+72*x^3+360*x+450)*exp(3)^2)/x^3,
x,method=_RETURNVERBOSE)

[Out]

-x^2*exp(8)-2*x*exp(8)-6/5*x^2*exp(7)+18/5*x*exp(7)-9/25*x^2*exp(6)+72/25*x*exp(6)+1/25*((150*exp(7)-360*exp(6
))*x-225*exp(6))/x^2+1/25*(-25*x^2*exp(8)-30*x^2*exp(7)+150*x*exp(7)-9*x^2*exp(6)+90*x*exp(6)-225*exp(6))*exp(
2*x)-2/25/x*(25*x^3*exp(2)+25*x^2*exp(2)+30*x^3*exp(1)-120*x^2*exp(1)+9*x^3-150*x*exp(1)-81*x^2+135*x+225)*exp
(x+6)

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maxima [C]  time = 0.60, size = 317, normalized size = 9.32 \begin {gather*} -x^{2} e^{8} - \frac {6}{5} \, x^{2} e^{7} - \frac {9}{25} \, x^{2} e^{6} - 2 \, x e^{8} + \frac {18}{5} \, x e^{7} + \frac {72}{25} \, x e^{6} - 18 \, {\rm Ei}\relax (x) e^{6} - \frac {1}{2} \, {\left (2 \, x^{2} e^{8} - 2 \, x e^{8} + e^{8}\right )} e^{\left (2 \, x\right )} - \frac {3}{5} \, {\left (2 \, x^{2} e^{7} - 2 \, x e^{7} + e^{7}\right )} e^{\left (2 \, x\right )} - \frac {9}{50} \, {\left (2 \, x^{2} e^{6} - 2 \, x e^{6} + e^{6}\right )} e^{\left (2 \, x\right )} - \frac {1}{2} \, {\left (2 \, x e^{8} - e^{8}\right )} e^{\left (2 \, x\right )} + \frac {12}{5} \, {\left (2 \, x e^{7} - e^{7}\right )} e^{\left (2 \, x\right )} + \frac {81}{50} \, {\left (2 \, x e^{6} - e^{6}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{2} e^{8} - 2 \, x e^{8} + 2 \, e^{8}\right )} e^{x} - \frac {12}{5} \, {\left (x^{2} e^{7} - 2 \, x e^{7} + 2 \, e^{7}\right )} e^{x} - \frac {18}{25} \, {\left (x^{2} e^{6} - 2 \, x e^{6} + 2 \, e^{6}\right )} e^{x} - 6 \, {\left (x e^{8} - e^{8}\right )} e^{x} + \frac {24}{5} \, {\left (x e^{7} - e^{7}\right )} e^{x} + \frac {126}{25} \, {\left (x e^{6} - e^{6}\right )} e^{x} + 18 \, e^{6} \Gamma \left (-1, -x\right ) + \frac {6 \, e^{7}}{x} - \frac {72 \, e^{6}}{5 \, x} - \frac {9 \, e^{6}}{x^{2}} + 3 \, e^{\left (2 \, x + 7\right )} - \frac {36}{5} \, e^{\left (2 \, x + 6\right )} - 2 \, e^{\left (x + 8\right )} + \frac {108}{5} \, e^{\left (x + 7\right )} - \frac {108}{25} \, e^{\left (x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(((-50*x^5-50*x^4)*exp(1)^2+(-60*x^5+240*x^4+150*x^3)*exp(1)-18*x^5+162*x^4-360*x^3)*exp(3)^2*e
xp(x)^2+((-50*x^5-150*x^4-50*x^3)*exp(1)^2+(-60*x^5+120*x^4+540*x^3)*exp(1)-18*x^5+126*x^4-108*x^3-450*x^2+450
*x)*exp(3)^2*exp(x)+((-50*x^4-50*x^3)*exp(1)^2+(-60*x^4+90*x^3-150*x)*exp(1)-18*x^4+72*x^3+360*x+450)*exp(3)^2
)/x^3,x, algorithm="maxima")

[Out]

-x^2*e^8 - 6/5*x^2*e^7 - 9/25*x^2*e^6 - 2*x*e^8 + 18/5*x*e^7 + 72/25*x*e^6 - 18*Ei(x)*e^6 - 1/2*(2*x^2*e^8 - 2
*x*e^8 + e^8)*e^(2*x) - 3/5*(2*x^2*e^7 - 2*x*e^7 + e^7)*e^(2*x) - 9/50*(2*x^2*e^6 - 2*x*e^6 + e^6)*e^(2*x) - 1
/2*(2*x*e^8 - e^8)*e^(2*x) + 12/5*(2*x*e^7 - e^7)*e^(2*x) + 81/50*(2*x*e^6 - e^6)*e^(2*x) - 2*(x^2*e^8 - 2*x*e
^8 + 2*e^8)*e^x - 12/5*(x^2*e^7 - 2*x*e^7 + 2*e^7)*e^x - 18/25*(x^2*e^6 - 2*x*e^6 + 2*e^6)*e^x - 6*(x*e^8 - e^
8)*e^x + 24/5*(x*e^7 - e^7)*e^x + 126/25*(x*e^6 - e^6)*e^x + 18*e^6*gamma(-1, -x) + 6*e^7/x - 72/5*e^6/x - 9*e
^6/x^2 + 3*e^(2*x + 7) - 36/5*e^(2*x + 6) - 2*e^(x + 8) + 108/5*e^(x + 7) - 108/25*e^(x + 6)

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mupad [B]  time = 0.87, size = 139, normalized size = 4.09 \begin {gather*} {\mathrm {e}}^{x+6}\,\left (12\,\mathrm {e}-\frac {54}{5}\right )-x^2\,\left (\frac {9\,{\mathrm {e}}^6}{25}+\frac {6\,{\mathrm {e}}^7}{5}+{\mathrm {e}}^8+{\mathrm {e}}^{2\,x+6}\,\left (\frac {6\,\mathrm {e}}{5}+{\mathrm {e}}^2+\frac {9}{25}\right )+{\mathrm {e}}^{x+6}\,\left (\frac {12\,\mathrm {e}}{5}+2\,{\mathrm {e}}^2+\frac {18}{25}\right )\right )-\frac {9\,{\mathrm {e}}^6+x\,\left (18\,{\mathrm {e}}^{x+6}+\frac {72\,{\mathrm {e}}^6}{5}-6\,{\mathrm {e}}^7\right )}{x^2}-9\,{\mathrm {e}}^{2\,x+6}+x\,\left (\frac {72\,{\mathrm {e}}^6}{25}+\frac {18\,{\mathrm {e}}^7}{5}-2\,{\mathrm {e}}^8+{\mathrm {e}}^{x+6}\,\left (\frac {48\,\mathrm {e}}{5}-2\,{\mathrm {e}}^2+\frac {162}{25}\right )+{\mathrm {e}}^{2\,x+12}\,\left (6\,{\mathrm {e}}^{-5}+\frac {18\,{\mathrm {e}}^{-6}}{5}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(6)*exp(x)*(exp(2)*(50*x^3 + 150*x^4 + 50*x^5) - 450*x - exp(1)*(540*x^3 + 120*x^4 - 60*x^5) + 450*x
^2 + 108*x^3 - 126*x^4 + 18*x^5))/25 - (exp(6)*(360*x - exp(1)*(150*x - 90*x^3 + 60*x^4) - exp(2)*(50*x^3 + 50
*x^4) + 72*x^3 - 18*x^4 + 450))/25 + (exp(2*x)*exp(6)*(exp(2)*(50*x^4 + 50*x^5) - exp(1)*(150*x^3 + 240*x^4 -
60*x^5) + 360*x^3 - 162*x^4 + 18*x^5))/25)/x^3,x)

[Out]

exp(x + 6)*(12*exp(1) - 54/5) - x^2*((9*exp(6))/25 + (6*exp(7))/5 + exp(8) + exp(2*x + 6)*((6*exp(1))/5 + exp(
2) + 9/25) + exp(x + 6)*((12*exp(1))/5 + 2*exp(2) + 18/25)) - (9*exp(6) + x*(18*exp(x + 6) + (72*exp(6))/5 - 6
*exp(7)))/x^2 - 9*exp(2*x + 6) + x*((72*exp(6))/25 + (18*exp(7))/5 - 2*exp(8) + exp(x + 6)*((48*exp(1))/5 - 2*
exp(2) + 162/25) + exp(2*x + 12)*(6*exp(-5) + (18*exp(-6))/5))

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sympy [B]  time = 0.44, size = 190, normalized size = 5.59 \begin {gather*} - \frac {x^{2} \left (9 e^{6} + 30 e^{7} + 25 e^{8}\right )}{25} - \frac {x \left (- 90 e^{7} - 72 e^{6} + 50 e^{8}\right )}{25} + \frac {\left (- 625 x^{3} e^{8} - 750 x^{3} e^{7} - 225 x^{3} e^{6} + 2250 x^{2} e^{6} + 3750 x^{2} e^{7} - 5625 x e^{6}\right ) e^{2 x} + \left (- 1250 x^{3} e^{8} - 1500 x^{3} e^{7} - 450 x^{3} e^{6} - 1250 x^{2} e^{8} + 4050 x^{2} e^{6} + 6000 x^{2} e^{7} - 6750 x e^{6} + 7500 x e^{7} - 11250 e^{6}\right ) e^{x}}{625 x} - \frac {x \left (- 150 e^{7} + 360 e^{6}\right ) + 225 e^{6}}{25 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(((-50*x**5-50*x**4)*exp(1)**2+(-60*x**5+240*x**4+150*x**3)*exp(1)-18*x**5+162*x**4-360*x**3)*e
xp(3)**2*exp(x)**2+((-50*x**5-150*x**4-50*x**3)*exp(1)**2+(-60*x**5+120*x**4+540*x**3)*exp(1)-18*x**5+126*x**4
-108*x**3-450*x**2+450*x)*exp(3)**2*exp(x)+((-50*x**4-50*x**3)*exp(1)**2+(-60*x**4+90*x**3-150*x)*exp(1)-18*x*
*4+72*x**3+360*x+450)*exp(3)**2)/x**3,x)

[Out]

-x**2*(9*exp(6) + 30*exp(7) + 25*exp(8))/25 - x*(-90*exp(7) - 72*exp(6) + 50*exp(8))/25 + ((-625*x**3*exp(8) -
750*x**3*exp(7) - 225*x**3*exp(6) + 2250*x**2*exp(6) + 3750*x**2*exp(7) - 5625*x*exp(6))*exp(2*x) + (-1250*x*
*3*exp(8) - 1500*x**3*exp(7) - 450*x**3*exp(6) - 1250*x**2*exp(8) + 4050*x**2*exp(6) + 6000*x**2*exp(7) - 6750
*x*exp(6) + 7500*x*exp(7) - 11250*exp(6))*exp(x))/(625*x) - (x*(-150*exp(7) + 360*exp(6)) + 225*exp(6))/(25*x*
*2)

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