3.38.47 \(\int \frac {1}{625} (e^{2+4 x} (800 x+2200 x^2+900 x^3+100 x^4)+e^{2+2 x} (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e (-320 x-560 x^2-200 x^3-20 x^4))+e^2 (1600 x^3+1000 x^4+150 x^5+e^2 (32 x+24 x^2+4 x^3)+e (480 x^2+320 x^3+50 x^4))) \, dx\)

Optimal. Leaf size=33 \[ \frac {1}{25} e^2 (-4-x)^2 \left (-\frac {e}{5}+e^{2 x}-x\right )^2 x^2 \]

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Rubi [B]  time = 0.64, antiderivative size = 217, normalized size of antiderivative = 6.58, number of steps used = 57, number of rules used = 4, integrand size = 138, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {12, 2196, 2176, 2194} \begin {gather*} \frac {e^2 x^6}{25}-\frac {2}{25} e^{2 x+2} x^5+\frac {2 e^3 x^5}{125}+\frac {8 e^2 x^5}{25}-\frac {16}{25} e^{2 x+2} x^4-\frac {2}{125} e^{2 x+3} x^4+\frac {1}{25} e^{4 x+2} x^4+\frac {e^4 x^4}{625}+\frac {16 e^3 x^4}{125}+\frac {16 e^2 x^4}{25}-\frac {32}{25} e^{2 x+2} x^3-\frac {16}{125} e^{2 x+3} x^3+\frac {8}{25} e^{4 x+2} x^3+\frac {8 e^4 x^3}{625}+\frac {32 e^3 x^3}{125}-\frac {32}{125} e^{2 x+3} x^2+\frac {16}{25} e^{4 x+2} x^2+\frac {16 e^4 x^2}{625} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2 + 4*x)*(800*x + 2200*x^2 + 900*x^3 + 100*x^4) + E^(2 + 2*x)*(-2400*x^2 - 3200*x^3 - 1050*x^4 - 100*x
^5 + E*(-320*x - 560*x^2 - 200*x^3 - 20*x^4)) + E^2*(1600*x^3 + 1000*x^4 + 150*x^5 + E^2*(32*x + 24*x^2 + 4*x^
3) + E*(480*x^2 + 320*x^3 + 50*x^4)))/625,x]

[Out]

(16*E^4*x^2)/625 - (32*E^(3 + 2*x)*x^2)/125 + (16*E^(2 + 4*x)*x^2)/25 + (32*E^3*x^3)/125 + (8*E^4*x^3)/625 - (
32*E^(2 + 2*x)*x^3)/25 - (16*E^(3 + 2*x)*x^3)/125 + (8*E^(2 + 4*x)*x^3)/25 + (16*E^2*x^4)/25 + (16*E^3*x^4)/12
5 + (E^4*x^4)/625 - (16*E^(2 + 2*x)*x^4)/25 - (2*E^(3 + 2*x)*x^4)/125 + (E^(2 + 4*x)*x^4)/25 + (8*E^2*x^5)/25
+ (2*E^3*x^5)/125 - (2*E^(2 + 2*x)*x^5)/25 + (E^2*x^6)/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 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 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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx\\ &=\frac {1}{625} \int e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right ) \, dx+\frac {1}{625} \int e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right ) \, dx+\frac {1}{625} e^2 \int \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right ) \, dx\\ &=\frac {16 e^2 x^4}{25}+\frac {8 e^2 x^5}{25}+\frac {e^2 x^6}{25}+\frac {1}{625} \int \left (800 e^{2+4 x} x+2200 e^{2+4 x} x^2+900 e^{2+4 x} x^3+100 e^{2+4 x} x^4\right ) \, dx+\frac {1}{625} \int \left (-2400 e^{2+2 x} x^2-3200 e^{2+2 x} x^3-1050 e^{2+2 x} x^4-100 e^{2+2 x} x^5-20 e^{3+2 x} x \left (16+28 x+10 x^2+x^3\right )\right ) \, dx+\frac {1}{625} e^3 \int \left (480 x^2+320 x^3+50 x^4\right ) \, dx+\frac {1}{625} e^4 \int \left (32 x+24 x^2+4 x^3\right ) \, dx\\ &=\frac {16 e^4 x^2}{625}+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}+\frac {e^2 x^6}{25}-\frac {4}{125} \int e^{3+2 x} x \left (16+28 x+10 x^2+x^3\right ) \, dx+\frac {4}{25} \int e^{2+4 x} x^4 \, dx-\frac {4}{25} \int e^{2+2 x} x^5 \, dx+\frac {32}{25} \int e^{2+4 x} x \, dx+\frac {36}{25} \int e^{2+4 x} x^3 \, dx-\frac {42}{25} \int e^{2+2 x} x^4 \, dx+\frac {88}{25} \int e^{2+4 x} x^2 \, dx-\frac {96}{25} \int e^{2+2 x} x^2 \, dx-\frac {128}{25} \int e^{2+2 x} x^3 \, dx\\ &=\frac {8}{25} e^{2+4 x} x+\frac {16 e^4 x^2}{625}-\frac {48}{25} e^{2+2 x} x^2+\frac {22}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {64}{25} e^{2+2 x} x^3+\frac {9}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {21}{25} e^{2+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}-\frac {4}{125} \int \left (16 e^{3+2 x} x+28 e^{3+2 x} x^2+10 e^{3+2 x} x^3+e^{3+2 x} x^4\right ) \, dx-\frac {4}{25} \int e^{2+4 x} x^3 \, dx-\frac {8}{25} \int e^{2+4 x} \, dx+\frac {2}{5} \int e^{2+2 x} x^4 \, dx-\frac {27}{25} \int e^{2+4 x} x^2 \, dx-\frac {44}{25} \int e^{2+4 x} x \, dx+\frac {84}{25} \int e^{2+2 x} x^3 \, dx+\frac {96}{25} \int e^{2+2 x} x \, dx+\frac {192}{25} \int e^{2+2 x} x^2 \, dx\\ &=-\frac {2}{25} e^{2+4 x}+\frac {48}{25} e^{2+2 x} x-\frac {3}{25} e^{2+4 x} x+\frac {16 e^4 x^2}{625}+\frac {48}{25} e^{2+2 x} x^2+\frac {61}{100} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {22}{25} e^{2+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}-\frac {4}{125} \int e^{3+2 x} x^4 \, dx+\frac {3}{25} \int e^{2+4 x} x^2 \, dx-\frac {8}{25} \int e^{3+2 x} x^3 \, dx+\frac {11}{25} \int e^{2+4 x} \, dx-\frac {64}{125} \int e^{3+2 x} x \, dx+\frac {27}{50} \int e^{2+4 x} x \, dx-\frac {4}{5} \int e^{2+2 x} x^3 \, dx-\frac {112}{125} \int e^{3+2 x} x^2 \, dx-\frac {48}{25} \int e^{2+2 x} \, dx-\frac {126}{25} \int e^{2+2 x} x^2 \, dx-\frac {192}{25} \int e^{2+2 x} x \, dx\\ &=-\frac {24}{25} e^{2+2 x}+\frac {3}{100} e^{2+4 x}-\frac {48}{25} e^{2+2 x} x-\frac {32}{125} e^{3+2 x} x+\frac {3}{200} e^{2+4 x} x+\frac {16 e^4 x^2}{625}-\frac {3}{5} e^{2+2 x} x^2-\frac {56}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {4}{25} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}-\frac {3}{50} \int e^{2+4 x} x \, dx+\frac {8}{125} \int e^{3+2 x} x^3 \, dx-\frac {27}{200} \int e^{2+4 x} \, dx+\frac {32}{125} \int e^{3+2 x} \, dx+\frac {12}{25} \int e^{3+2 x} x^2 \, dx+\frac {112}{125} \int e^{3+2 x} x \, dx+\frac {6}{5} \int e^{2+2 x} x^2 \, dx+\frac {96}{25} \int e^{2+2 x} \, dx+\frac {126}{25} \int e^{2+2 x} x \, dx\\ &=\frac {24}{25} e^{2+2 x}+\frac {16}{125} e^{3+2 x}-\frac {3}{800} e^{2+4 x}+\frac {3}{5} e^{2+2 x} x+\frac {24}{125} e^{3+2 x} x+\frac {16 e^4 x^2}{625}-\frac {26}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {16}{125} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}+\frac {3}{200} \int e^{2+4 x} \, dx-\frac {12}{125} \int e^{3+2 x} x^2 \, dx-\frac {56}{125} \int e^{3+2 x} \, dx-\frac {12}{25} \int e^{3+2 x} x \, dx-\frac {6}{5} \int e^{2+2 x} x \, dx-\frac {63}{25} \int e^{2+2 x} \, dx\\ &=-\frac {3}{10} e^{2+2 x}-\frac {12}{125} e^{3+2 x}-\frac {6}{125} e^{3+2 x} x+\frac {16 e^4 x^2}{625}-\frac {32}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {16}{125} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}+\frac {12}{125} \int e^{3+2 x} x \, dx+\frac {6}{25} \int e^{3+2 x} \, dx+\frac {3}{5} \int e^{2+2 x} \, dx\\ &=\frac {3}{125} e^{3+2 x}+\frac {16 e^4 x^2}{625}-\frac {32}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {16}{125} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}-\frac {6}{125} \int e^{3+2 x} \, dx\\ &=\frac {16 e^4 x^2}{625}-\frac {32}{125} e^{3+2 x} x^2+\frac {16}{25} e^{2+4 x} x^2+\frac {32 e^3 x^3}{125}+\frac {8 e^4 x^3}{625}-\frac {32}{25} e^{2+2 x} x^3-\frac {16}{125} e^{3+2 x} x^3+\frac {8}{25} e^{2+4 x} x^3+\frac {16 e^2 x^4}{25}+\frac {16 e^3 x^4}{125}+\frac {e^4 x^4}{625}-\frac {16}{25} e^{2+2 x} x^4-\frac {2}{125} e^{3+2 x} x^4+\frac {1}{25} e^{2+4 x} x^4+\frac {8 e^2 x^5}{25}+\frac {2 e^3 x^5}{125}-\frac {2}{25} e^{2+2 x} x^5+\frac {e^2 x^6}{25}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{625} e^2 x^2 (4+x)^2 \left (e-5 e^{2 x}+5 x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2 + 4*x)*(800*x + 2200*x^2 + 900*x^3 + 100*x^4) + E^(2 + 2*x)*(-2400*x^2 - 3200*x^3 - 1050*x^4 -
100*x^5 + E*(-320*x - 560*x^2 - 200*x^3 - 20*x^4)) + E^2*(1600*x^3 + 1000*x^4 + 150*x^5 + E^2*(32*x + 24*x^2
+ 4*x^3) + E*(480*x^2 + 320*x^3 + 50*x^4)))/625,x]

[Out]

(E^2*x^2*(4 + x)^2*(E - 5*E^(2*x) + 5*x)^2)/625

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fricas [B]  time = 0.76, size = 124, normalized size = 3.76 \begin {gather*} \frac {1}{625} \, {\left ({\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{6} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{5} + 25 \, {\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} e^{4} + 25 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (4 \, x + 4\right )} - 10 \, {\left ({\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{3} + 5 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{2}\right )} e^{\left (2 \, x + 2\right )}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((-20*x^4-200*x^3-560*x^2-320*x)*exp(1)
-100*x^5-1050*x^4-3200*x^3-2400*x^2)*exp(2)*exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x
^2)*exp(1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x, algorithm="fricas")

[Out]

1/625*((x^4 + 8*x^3 + 16*x^2)*e^6 + 10*(x^5 + 8*x^4 + 16*x^3)*e^5 + 25*(x^6 + 8*x^5 + 16*x^4)*e^4 + 25*(x^4 +
8*x^3 + 16*x^2)*e^(4*x + 4) - 10*((x^4 + 8*x^3 + 16*x^2)*e^3 + 5*(x^5 + 8*x^4 + 16*x^3)*e^2)*e^(2*x + 2))*e^(-
2)

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giac [B]  time = 0.15, size = 122, normalized size = 3.70 \begin {gather*} \frac {1}{625} \, {\left (25 \, x^{6} + 200 \, x^{5} + 400 \, x^{4} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{2} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e\right )} e^{2} + \frac {1}{25} \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (4 \, x + 2\right )} - \frac {2}{125} \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (2 \, x + 3\right )} - \frac {2}{25} \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{\left (2 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((-20*x^4-200*x^3-560*x^2-320*x)*exp(1)
-100*x^5-1050*x^4-3200*x^3-2400*x^2)*exp(2)*exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x
^2)*exp(1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x, algorithm="giac")

[Out]

1/625*(25*x^6 + 200*x^5 + 400*x^4 + (x^4 + 8*x^3 + 16*x^2)*e^2 + 10*(x^5 + 8*x^4 + 16*x^3)*e)*e^2 + 1/25*(x^4
+ 8*x^3 + 16*x^2)*e^(4*x + 2) - 2/125*(x^4 + 8*x^3 + 16*x^2)*e^(2*x + 3) - 2/25*(x^5 + 8*x^4 + 16*x^3)*e^(2*x
+ 2)

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maple [B]  time = 0.09, size = 144, normalized size = 4.36

method	result	size
risch	\(\frac {\left (25 x^{4}+200 x^{3}+400 x^{2}\right ) {\mathrm e}^{4 x +2}}{625}+\frac {\left (-10 x^{4} {\mathrm e}-50 x^{5}-80 x^{3} {\mathrm e}-400 x^{4}-160 x^{2} {\mathrm e}-800 x^{3}\right ) {\mathrm e}^{2 x +2}}{625}+\frac {x^{6} {\mathrm e}^{2}}{25}+\frac {2 \,{\mathrm e}^{2} x^{5} {\mathrm e}}{125}+\frac {8 \,{\mathrm e}^{2} x^{5}}{25}+\frac {x^{4} {\mathrm e}^{4}}{625}+\frac {16 \,{\mathrm e}^{2} x^{4} {\mathrm e}}{125}+\frac {16 x^{4} {\mathrm e}^{2}}{25}+\frac {8 x^{3} {\mathrm e}^{4}}{625}+\frac {32 \,{\mathrm e}^{2} x^{3} {\mathrm e}}{125}+\frac {16 x^{2} {\mathrm e}^{4}}{625}\)	\(144\)
norman	\(\left (\frac {2 \,{\mathrm e} \,{\mathrm e}^{2}}{125}+\frac {8 \,{\mathrm e}^{2}}{25}\right ) x^{5}+\left (\frac {8 \left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {32 \,{\mathrm e} \,{\mathrm e}^{2}}{125}\right ) x^{3}+\left (\frac {\left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {16 \,{\mathrm e} \,{\mathrm e}^{2}}{125}+\frac {16 \,{\mathrm e}^{2}}{25}\right ) x^{4}+\left (-\frac {32 \,{\mathrm e}^{2}}{25}-\frac {16 \,{\mathrm e} \,{\mathrm e}^{2}}{125}\right ) x^{3} {\mathrm e}^{2 x}+\left (-\frac {16 \,{\mathrm e}^{2}}{25}-\frac {2 \,{\mathrm e} \,{\mathrm e}^{2}}{125}\right ) x^{4} {\mathrm e}^{2 x}+\frac {x^{6} {\mathrm e}^{2}}{25}+\frac {16 x^{2} \left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {16 x^{2} {\mathrm e}^{2} {\mathrm e}^{4 x}}{25}+\frac {8 x^{3} {\mathrm e}^{2} {\mathrm e}^{4 x}}{25}+\frac {x^{4} {\mathrm e}^{2} {\mathrm e}^{4 x}}{25}-\frac {2 \,{\mathrm e}^{2} x^{5} {\mathrm e}^{2 x}}{25}-\frac {32 x^{2} {\mathrm e} \,{\mathrm e}^{2} {\mathrm e}^{2 x}}{125}\)	\(172\)
default	\(\frac {{\mathrm e}^{2} \left ({\mathrm e}^{2} \left (x^{4}+8 x^{3}+16 x^{2}\right )+{\mathrm e} \left (10 x^{5}+80 x^{4}+160 x^{3}\right )+25 x^{6}+200 x^{5}+400 x^{4}\right )}{625}+\frac {{\mathrm e}^{2} \left (-800 \,{\mathrm e}^{2 x} x^{3}-50 x^{5} {\mathrm e}^{2 x}-400 \,{\mathrm e}^{2 x} x^{4}-200 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )-320 \,{\mathrm e} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-560 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )-20 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{4}}{2}-{\mathrm e}^{2 x} x^{3}+\frac {3 \,{\mathrm e}^{2 x} x^{2}}{2}-\frac {3 x \,{\mathrm e}^{2 x}}{2}+\frac {3 \,{\mathrm e}^{2 x}}{4}\right )\right )}{625}+\frac {4 \,{\mathrm e}^{2} \left (\frac {x^{4} {\mathrm e}^{4 x}}{4}+2 x^{3} {\mathrm e}^{4 x}+4 x^{2} {\mathrm e}^{4 x}\right )}{25}\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((-20*x^4-200*x^3-560*x^2-320*x)*exp(1)-100*x
^5-1050*x^4-3200*x^3-2400*x^2)*exp(2)*exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x^2)*ex
p(1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x,method=_RETURNVERBOSE)

[Out]

1/625*(25*x^4+200*x^3+400*x^2)*exp(4*x+2)+1/625*(-10*x^4*exp(1)-50*x^5-80*x^3*exp(1)-400*x^4-160*x^2*exp(1)-80
0*x^3)*exp(2*x+2)+1/25*x^6*exp(2)+2/125*exp(2)*x^5*exp(1)+8/25*exp(2)*x^5+1/625*x^4*exp(2)^2+16/125*exp(2)*x^4
*exp(1)+16/25*x^4*exp(2)+8/625*x^3*exp(2)^2+32/125*exp(2)*x^3*exp(1)+16/625*x^2*exp(4)

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maxima [B]  time = 0.41, size = 127, normalized size = 3.85 \begin {gather*} \frac {1}{625} \, {\left (25 \, x^{6} + 200 \, x^{5} + 400 \, x^{4} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{2} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e\right )} e^{2} + \frac {1}{25} \, {\left (x^{4} e^{2} + 8 \, x^{3} e^{2} + 16 \, x^{2} e^{2}\right )} e^{\left (4 \, x\right )} - \frac {2}{125} \, {\left (5 \, x^{5} e^{2} + x^{4} {\left (e^{3} + 40 \, e^{2}\right )} + 8 \, x^{3} {\left (e^{3} + 10 \, e^{2}\right )} + 16 \, x^{2} e^{3}\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((-20*x^4-200*x^3-560*x^2-320*x)*exp(1)
-100*x^5-1050*x^4-3200*x^3-2400*x^2)*exp(2)*exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x
^2)*exp(1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x, algorithm="maxima")

[Out]

1/625*(25*x^6 + 200*x^5 + 400*x^4 + (x^4 + 8*x^3 + 16*x^2)*e^2 + 10*(x^5 + 8*x^4 + 16*x^3)*e)*e^2 + 1/25*(x^4*
e^2 + 8*x^3*e^2 + 16*x^2*e^2)*e^(4*x) - 2/125*(5*x^5*e^2 + x^4*(e^3 + 40*e^2) + 8*x^3*(e^3 + 10*e^2) + 16*x^2*
e^3)*e^(2*x)

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mupad [B]  time = 0.19, size = 147, normalized size = 4.45 \begin {gather*} x^5\,\left (\frac {8\,{\mathrm {e}}^2}{25}+\frac {2\,{\mathrm {e}}^3}{125}\right )+x^3\,\left (\frac {32\,{\mathrm {e}}^3}{125}+\frac {8\,{\mathrm {e}}^4}{625}\right )+\frac {16\,x^2\,{\mathrm {e}}^4}{625}+\frac {x^6\,{\mathrm {e}}^2}{25}-\frac {32\,x^2\,{\mathrm {e}}^{2\,x+3}}{125}+\frac {16\,x^2\,{\mathrm {e}}^{4\,x+2}}{25}+\frac {8\,x^3\,{\mathrm {e}}^{4\,x+2}}{25}-\frac {2\,x^5\,{\mathrm {e}}^{2\,x+2}}{25}+\frac {x^4\,{\mathrm {e}}^{4\,x+2}}{25}+x^4\,\left (\frac {16\,{\mathrm {e}}^2}{25}+\frac {16\,{\mathrm {e}}^3}{125}+\frac {{\mathrm {e}}^4}{625}\right )-\frac {x^4\,{\mathrm {e}}^{2\,x+2}\,\left (10\,\mathrm {e}+400\right )}{625}-\frac {x^3\,{\mathrm {e}}^{2\,x+2}\,\left (80\,\mathrm {e}+800\right )}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)*(exp(2)*(32*x + 24*x^2 + 4*x^3) + exp(1)*(480*x^2 + 320*x^3 + 50*x^4) + 1600*x^3 + 1000*x^4 + 150*
x^5))/625 - (exp(2*x)*exp(2)*(exp(1)*(320*x + 560*x^2 + 200*x^3 + 20*x^4) + 2400*x^2 + 3200*x^3 + 1050*x^4 + 1
00*x^5))/625 + (exp(4*x)*exp(2)*(800*x + 2200*x^2 + 900*x^3 + 100*x^4))/625,x)

[Out]

x^5*((8*exp(2))/25 + (2*exp(3))/125) + x^3*((32*exp(3))/125 + (8*exp(4))/625) + (16*x^2*exp(4))/625 + (x^6*exp
(2))/25 - (32*x^2*exp(2*x + 3))/125 + (16*x^2*exp(4*x + 2))/25 + (8*x^3*exp(4*x + 2))/25 - (2*x^5*exp(2*x + 2)
)/25 + (x^4*exp(4*x + 2))/25 + x^4*((16*exp(2))/25 + (16*exp(3))/125 + exp(4)/625) - (x^4*exp(2*x + 2)*(10*exp
(1) + 400))/625 - (x^3*exp(2*x + 2)*(80*exp(1) + 800))/625

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sympy [B]  time = 0.33, size = 165, normalized size = 5.00 \begin {gather*} \frac {x^{6} e^{2}}{25} + x^{5} \left (\frac {2 e^{3}}{125} + \frac {8 e^{2}}{25}\right ) + x^{4} \left (\frac {e^{4}}{625} + \frac {16 e^{3}}{125} + \frac {16 e^{2}}{25}\right ) + x^{3} \left (\frac {8 e^{4}}{625} + \frac {32 e^{3}}{125}\right ) + \frac {16 x^{2} e^{4}}{625} + \frac {\left (125 x^{4} e^{2} + 1000 x^{3} e^{2} + 2000 x^{2} e^{2}\right ) e^{4 x}}{3125} + \frac {\left (- 250 x^{5} e^{2} - 2000 x^{4} e^{2} - 50 x^{4} e^{3} - 4000 x^{3} e^{2} - 400 x^{3} e^{3} - 800 x^{2} e^{3}\right ) e^{2 x}}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(100*x**4+900*x**3+2200*x**2+800*x)*exp(2)*exp(x)**4+1/625*((-20*x**4-200*x**3-560*x**2-320*x)
*exp(1)-100*x**5-1050*x**4-3200*x**3-2400*x**2)*exp(2)*exp(x)**2+1/625*((4*x**3+24*x**2+32*x)*exp(1)**2+(50*x*
*4+320*x**3+480*x**2)*exp(1)+150*x**5+1000*x**4+1600*x**3)*exp(2),x)

[Out]

x**6*exp(2)/25 + x**5*(2*exp(3)/125 + 8*exp(2)/25) + x**4*(exp(4)/625 + 16*exp(3)/125 + 16*exp(2)/25) + x**3*(
8*exp(4)/625 + 32*exp(3)/125) + 16*x**2*exp(4)/625 + (125*x**4*exp(2) + 1000*x**3*exp(2) + 2000*x**2*exp(2))*e
xp(4*x)/3125 + (-250*x**5*exp(2) - 2000*x**4*exp(2) - 50*x**4*exp(3) - 4000*x**3*exp(2) - 400*x**3*exp(3) - 80
0*x**2*exp(3))*exp(2*x)/3125

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