3.78.47 \(\int \frac {e^{2 x} (843780+226968 x-30576 x^2-5880 x^3+588 x^4)+e^{4 x} (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8)}{2401} \, dx\)

Optimal. Leaf size=28 \[ \left (3+e^{2 x} \left (5+x+\frac {1}{7} \left (-x-x^2\right )\right )^2\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.64, antiderivative size = 155, normalized size of antiderivative = 5.54, number of steps used = 66, number of rules used = 4, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {12, 2196, 2194, 2176} \begin {gather*} \frac {e^{4 x} x^8}{2401}-\frac {24 e^{4 x} x^7}{2401}+\frac {76 e^{4 x} x^6}{2401}+\frac {1656 e^{4 x} x^5}{2401}+\frac {6}{49} e^{2 x} x^4-\frac {6474 e^{4 x} x^4}{2401}-\frac {72}{49} e^{2 x} x^3-\frac {8280}{343} e^{4 x} x^3-\frac {204}{49} e^{2 x} x^2+\frac {1900}{49} e^{4 x} x^2+\frac {360}{7} e^{2 x} x+\frac {3000}{7} e^{4 x} x+150 e^{2 x}+625 e^{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x)*(843780 + 226968*x - 30576*x^2 - 5880*x^3 + 588*x^4) + E^(4*x)*(7031500 + 4302200*x + 198520*x^2
- 257736*x^3 - 17616*x^4 + 7080*x^5 + 136*x^6 - 88*x^7 + 4*x^8))/2401,x]

[Out]

150*E^(2*x) + 625*E^(4*x) + (360*E^(2*x)*x)/7 + (3000*E^(4*x)*x)/7 - (204*E^(2*x)*x^2)/49 + (1900*E^(4*x)*x^2)
/49 - (72*E^(2*x)*x^3)/49 - (8280*E^(4*x)*x^3)/343 + (6*E^(2*x)*x^4)/49 - (6474*E^(4*x)*x^4)/2401 + (1656*E^(4
*x)*x^5)/2401 + (76*E^(4*x)*x^6)/2401 - (24*E^(4*x)*x^7)/2401 + (E^(4*x)*x^8)/2401

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 {\int \left (e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right )+e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right )\right ) \, dx}{2401}\\ &=\frac {\int e^{2 x} \left (843780+226968 x-30576 x^2-5880 x^3+588 x^4\right ) \, dx}{2401}+\frac {\int e^{4 x} \left (7031500+4302200 x+198520 x^2-257736 x^3-17616 x^4+7080 x^5+136 x^6-88 x^7+4 x^8\right ) \, dx}{2401}\\ &=\frac {\int \left (843780 e^{2 x}+226968 e^{2 x} x-30576 e^{2 x} x^2-5880 e^{2 x} x^3+588 e^{2 x} x^4\right ) \, dx}{2401}+\frac {\int \left (7031500 e^{4 x}+4302200 e^{4 x} x+198520 e^{4 x} x^2-257736 e^{4 x} x^3-17616 e^{4 x} x^4+7080 e^{4 x} x^5+136 e^{4 x} x^6-88 e^{4 x} x^7+4 e^{4 x} x^8\right ) \, dx}{2401}\\ &=\frac {4 \int e^{4 x} x^8 \, dx}{2401}-\frac {88 \int e^{4 x} x^7 \, dx}{2401}+\frac {136 \int e^{4 x} x^6 \, dx}{2401}+\frac {12}{49} \int e^{2 x} x^4 \, dx-\frac {120}{49} \int e^{2 x} x^3 \, dx+\frac {7080 \int e^{4 x} x^5 \, dx}{2401}-\frac {17616 \int e^{4 x} x^4 \, dx}{2401}-\frac {624}{49} \int e^{2 x} x^2 \, dx+\frac {28360}{343} \int e^{4 x} x^2 \, dx+\frac {4632}{49} \int e^{2 x} x \, dx-\frac {257736 \int e^{4 x} x^3 \, dx}{2401}+\frac {2460}{7} \int e^{2 x} \, dx+\frac {87800}{49} \int e^{4 x} x \, dx+\frac {20500}{7} \int e^{4 x} \, dx\\ &=\frac {1230 e^{2 x}}{7}+\frac {5125 e^{4 x}}{7}+\frac {2316}{49} e^{2 x} x+\frac {21950}{49} e^{4 x} x-\frac {312}{49} e^{2 x} x^2+\frac {7090}{343} e^{4 x} x^2-\frac {60}{49} e^{2 x} x^3-\frac {64434 e^{4 x} x^3}{2401}+\frac {6}{49} e^{2 x} x^4-\frac {4404 e^{4 x} x^4}{2401}+\frac {1770 e^{4 x} x^5}{2401}+\frac {34 e^{4 x} x^6}{2401}-\frac {22 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}-\frac {8 \int e^{4 x} x^7 \, dx}{2401}+\frac {22}{343} \int e^{4 x} x^6 \, dx-\frac {204 \int e^{4 x} x^5 \, dx}{2401}-\frac {24}{49} \int e^{2 x} x^3 \, dx+\frac {180}{49} \int e^{2 x} x^2 \, dx-\frac {8850 \int e^{4 x} x^4 \, dx}{2401}+\frac {17616 \int e^{4 x} x^3 \, dx}{2401}+\frac {624}{49} \int e^{2 x} x \, dx-\frac {14180}{343} \int e^{4 x} x \, dx-\frac {2316}{49} \int e^{2 x} \, dx+\frac {193302 \int e^{4 x} x^2 \, dx}{2401}-\frac {21950}{49} \int e^{4 x} \, dx\\ &=\frac {7452 e^{2 x}}{49}+\frac {60775 e^{4 x}}{98}+\frac {2628}{49} e^{2 x} x+\frac {150105}{343} e^{4 x} x-\frac {222}{49} e^{2 x} x^2+\frac {195911 e^{4 x} x^2}{4802}-\frac {72}{49} e^{2 x} x^3-\frac {60030 e^{4 x} x^3}{2401}+\frac {6}{49} e^{2 x} x^4-\frac {13233 e^{4 x} x^4}{4802}+\frac {1719 e^{4 x} x^5}{2401}+\frac {145 e^{4 x} x^6}{4802}-\frac {24 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}+\frac {2}{343} \int e^{4 x} x^6 \, dx-\frac {33}{343} \int e^{4 x} x^5 \, dx+\frac {255 \int e^{4 x} x^4 \, dx}{2401}+\frac {36}{49} \int e^{2 x} x^2 \, dx-\frac {180}{49} \int e^{2 x} x \, dx+\frac {8850 \int e^{4 x} x^3 \, dx}{2401}-\frac {13212 \int e^{4 x} x^2 \, dx}{2401}-\frac {312}{49} \int e^{2 x} \, dx+\frac {3545}{343} \int e^{4 x} \, dx-\frac {96651 \int e^{4 x} x \, dx}{2401}\\ &=\frac {7296 e^{2 x}}{49}+\frac {854395 e^{4 x}}{1372}+\frac {2538}{49} e^{2 x} x+\frac {4106289 e^{4 x} x}{9604}-\frac {204}{49} e^{2 x} x^2+\frac {189305 e^{4 x} x^2}{4802}-\frac {72}{49} e^{2 x} x^3-\frac {115635 e^{4 x} x^3}{4802}+\frac {6}{49} e^{2 x} x^4-\frac {26211 e^{4 x} x^4}{9604}+\frac {6645 e^{4 x} x^5}{9604}+\frac {76 e^{4 x} x^6}{2401}-\frac {24 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}-\frac {3}{343} \int e^{4 x} x^5 \, dx-\frac {255 \int e^{4 x} x^3 \, dx}{2401}+\frac {165 \int e^{4 x} x^4 \, dx}{1372}-\frac {36}{49} \int e^{2 x} x \, dx+\frac {90}{49} \int e^{2 x} \, dx+\frac {6606 \int e^{4 x} x \, dx}{2401}-\frac {13275 \int e^{4 x} x^2 \, dx}{4802}+\frac {96651 \int e^{4 x} \, dx}{9604}\\ &=\frac {7341 e^{2 x}}{49}+\frac {24019711 e^{4 x}}{38416}+\frac {360}{7} e^{2 x} x+\frac {4112895 e^{4 x} x}{9604}-\frac {204}{49} e^{2 x} x^2+\frac {743945 e^{4 x} x^2}{19208}-\frac {72}{49} e^{2 x} x^3-\frac {675}{28} e^{4 x} x^3+\frac {6}{49} e^{2 x} x^4-\frac {103689 e^{4 x} x^4}{38416}+\frac {1656 e^{4 x} x^5}{2401}+\frac {76 e^{4 x} x^6}{2401}-\frac {24 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}+\frac {15 \int e^{4 x} x^4 \, dx}{1372}+\frac {765 \int e^{4 x} x^2 \, dx}{9604}-\frac {165 \int e^{4 x} x^3 \, dx}{1372}+\frac {18}{49} \int e^{2 x} \, dx-\frac {3303 \int e^{4 x} \, dx}{4802}+\frac {13275 \int e^{4 x} x \, dx}{9604}\\ &=150 e^{2 x}+\frac {24013105 e^{4 x}}{38416}+\frac {360}{7} e^{2 x} x+\frac {16464855 e^{4 x} x}{38416}-\frac {204}{49} e^{2 x} x^2+\frac {212665 e^{4 x} x^2}{5488}-\frac {72}{49} e^{2 x} x^3-\frac {132465 e^{4 x} x^3}{5488}+\frac {6}{49} e^{2 x} x^4-\frac {6474 e^{4 x} x^4}{2401}+\frac {1656 e^{4 x} x^5}{2401}+\frac {76 e^{4 x} x^6}{2401}-\frac {24 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}-\frac {15 \int e^{4 x} x^3 \, dx}{1372}-\frac {765 \int e^{4 x} x \, dx}{19208}+\frac {495 \int e^{4 x} x^2 \, dx}{5488}-\frac {13275 \int e^{4 x} \, dx}{38416}\\ &=150 e^{2 x}+\frac {96039145 e^{4 x}}{153664}+\frac {360}{7} e^{2 x} x+\frac {4704135 e^{4 x} x}{10976}-\frac {204}{49} e^{2 x} x^2+\frac {851155 e^{4 x} x^2}{21952}-\frac {72}{49} e^{2 x} x^3-\frac {8280}{343} e^{4 x} x^3+\frac {6}{49} e^{2 x} x^4-\frac {6474 e^{4 x} x^4}{2401}+\frac {1656 e^{4 x} x^5}{2401}+\frac {76 e^{4 x} x^6}{2401}-\frac {24 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}+\frac {45 \int e^{4 x} x^2 \, dx}{5488}+\frac {765 \int e^{4 x} \, dx}{76832}-\frac {495 \int e^{4 x} x \, dx}{10976}\\ &=150 e^{2 x}+\frac {27439865 e^{4 x}}{43904}+\frac {360}{7} e^{2 x} x+\frac {18816045 e^{4 x} x}{43904}-\frac {204}{49} e^{2 x} x^2+\frac {1900}{49} e^{4 x} x^2-\frac {72}{49} e^{2 x} x^3-\frac {8280}{343} e^{4 x} x^3+\frac {6}{49} e^{2 x} x^4-\frac {6474 e^{4 x} x^4}{2401}+\frac {1656 e^{4 x} x^5}{2401}+\frac {76 e^{4 x} x^6}{2401}-\frac {24 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}-\frac {45 \int e^{4 x} x \, dx}{10976}+\frac {495 \int e^{4 x} \, dx}{43904}\\ &=150 e^{2 x}+\frac {109759955 e^{4 x}}{175616}+\frac {360}{7} e^{2 x} x+\frac {3000}{7} e^{4 x} x-\frac {204}{49} e^{2 x} x^2+\frac {1900}{49} e^{4 x} x^2-\frac {72}{49} e^{2 x} x^3-\frac {8280}{343} e^{4 x} x^3+\frac {6}{49} e^{2 x} x^4-\frac {6474 e^{4 x} x^4}{2401}+\frac {1656 e^{4 x} x^5}{2401}+\frac {76 e^{4 x} x^6}{2401}-\frac {24 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}+\frac {45 \int e^{4 x} \, dx}{43904}\\ &=150 e^{2 x}+625 e^{4 x}+\frac {360}{7} e^{2 x} x+\frac {3000}{7} e^{4 x} x-\frac {204}{49} e^{2 x} x^2+\frac {1900}{49} e^{4 x} x^2-\frac {72}{49} e^{2 x} x^3-\frac {8280}{343} e^{4 x} x^3+\frac {6}{49} e^{2 x} x^4-\frac {6474 e^{4 x} x^4}{2401}+\frac {1656 e^{4 x} x^5}{2401}+\frac {76 e^{4 x} x^6}{2401}-\frac {24 e^{4 x} x^7}{2401}+\frac {e^{4 x} x^8}{2401}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.29, size = 37, normalized size = 1.32 \begin {gather*} \frac {e^{2 x} \left (-35-6 x+x^2\right )^2 \left (294+e^{2 x} \left (-35-6 x+x^2\right )^2\right )}{2401} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(843780 + 226968*x - 30576*x^2 - 5880*x^3 + 588*x^4) + E^(4*x)*(7031500 + 4302200*x + 19852
0*x^2 - 257736*x^3 - 17616*x^4 + 7080*x^5 + 136*x^6 - 88*x^7 + 4*x^8))/2401,x]

[Out]

(E^(2*x)*(-35 - 6*x + x^2)^2*(294 + E^(2*x)*(-35 - 6*x + x^2)^2))/2401

________________________________________________________________________________________

fricas [B]  time = 0.83, size = 69, normalized size = 2.46 \begin {gather*} \frac {1}{2401} \, {\left (x^{8} - 24 \, x^{7} + 76 \, x^{6} + 1656 \, x^{5} - 6474 \, x^{4} - 57960 \, x^{3} + 93100 \, x^{2} + 1029000 \, x + 1500625\right )} e^{\left (4 \, x\right )} + \frac {6}{49} \, {\left (x^{4} - 12 \, x^{3} - 34 \, x^{2} + 420 \, x + 1225\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+198520*x^2+4302200*x+7031500)*exp(x)^4+1/
2401*(588*x^4-5880*x^3-30576*x^2+226968*x+843780)*exp(x)^2,x, algorithm="fricas")

[Out]

1/2401*(x^8 - 24*x^7 + 76*x^6 + 1656*x^5 - 6474*x^4 - 57960*x^3 + 93100*x^2 + 1029000*x + 1500625)*e^(4*x) + 6
/49*(x^4 - 12*x^3 - 34*x^2 + 420*x + 1225)*e^(2*x)

________________________________________________________________________________________

giac [B]  time = 0.23, size = 69, normalized size = 2.46 \begin {gather*} \frac {1}{2401} \, {\left (x^{8} - 24 \, x^{7} + 76 \, x^{6} + 1656 \, x^{5} - 6474 \, x^{4} - 57960 \, x^{3} + 93100 \, x^{2} + 1029000 \, x + 1500625\right )} e^{\left (4 \, x\right )} + \frac {6}{49} \, {\left (x^{4} - 12 \, x^{3} - 34 \, x^{2} + 420 \, x + 1225\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+198520*x^2+4302200*x+7031500)*exp(x)^4+1/
2401*(588*x^4-5880*x^3-30576*x^2+226968*x+843780)*exp(x)^2,x, algorithm="giac")

[Out]

1/2401*(x^8 - 24*x^7 + 76*x^6 + 1656*x^5 - 6474*x^4 - 57960*x^3 + 93100*x^2 + 1029000*x + 1500625)*e^(4*x) + 6
/49*(x^4 - 12*x^3 - 34*x^2 + 420*x + 1225)*e^(2*x)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 72, normalized size = 2.57

method	result	size
risch	\(\frac {\left (x^{8}-24 x^{7}+76 x^{6}+1656 x^{5}-6474 x^{4}-57960 x^{3}+93100 x^{2}+1029000 x +1500625\right ) {\mathrm e}^{4 x}}{2401}+\frac {\left (294 x^{4}-3528 x^{3}-9996 x^{2}+123480 x +360150\right ) {\mathrm e}^{2 x}}{2401}\)	\(72\)
default	\(\frac {6 \,{\mathrm e}^{2 x} x^{4}}{49}-\frac {72 \,{\mathrm e}^{2 x} x^{3}}{49}-\frac {204 \,{\mathrm e}^{2 x} x^{2}}{49}+\frac {360 x \,{\mathrm e}^{2 x}}{7}+150 \,{\mathrm e}^{2 x}+\frac {x^{8} {\mathrm e}^{4 x}}{2401}-\frac {24 \,{\mathrm e}^{4 x} x^{7}}{2401}+\frac {76 x^{6} {\mathrm e}^{4 x}}{2401}+\frac {1656 x^{5} {\mathrm e}^{4 x}}{2401}-\frac {6474 x^{4} {\mathrm e}^{4 x}}{2401}-\frac {8280 x^{3} {\mathrm e}^{4 x}}{343}+\frac {1900 x^{2} {\mathrm e}^{4 x}}{49}+\frac {3000 x \,{\mathrm e}^{4 x}}{7}+625 \,{\mathrm e}^{4 x}\)	\(118\)
meijerg	\(-775+\frac {5125 \,{\mathrm e}^{4 x}}{7}+\frac {\left (589824 x^{8}-1179648 x^{7}+2064384 x^{6}-3096576 x^{5}+3870720 x^{4}-3870720 x^{3}+2903040 x^{2}-1451520 x +362880\right ) {\mathrm e}^{4 x}}{1416167424}+\frac {11 \left (-131072 x^{7}+229376 x^{6}-344064 x^{5}+430080 x^{4}-430080 x^{3}+322560 x^{2}-161280 x +40320\right ) {\mathrm e}^{4 x}}{157351936}+\frac {17 \left (28672 x^{6}-43008 x^{5}+53760 x^{4}-53760 x^{3}+40320 x^{2}-20160 x +5040\right ) {\mathrm e}^{4 x}}{34420736}-\frac {295 \left (-6144 x^{5}+7680 x^{4}-7680 x^{3}+5760 x^{2}-2880 x +720\right ) {\mathrm e}^{4 x}}{2458624}-\frac {1101 \left (1280 x^{4}-1280 x^{3}+960 x^{2}-480 x +120\right ) {\mathrm e}^{4 x}}{768320}+\frac {32217 \left (-256 x^{3}+192 x^{2}-96 x +24\right ) {\mathrm e}^{4 x}}{307328}+\frac {3545 \left (48 x^{2}-24 x +6\right ) {\mathrm e}^{4 x}}{8232}-\frac {10975 \left (-8 x +2\right ) {\mathrm e}^{4 x}}{196}+\frac {1230 \,{\mathrm e}^{2 x}}{7}+\frac {3 \left (80 x^{4}-160 x^{3}+240 x^{2}-240 x +120\right ) {\mathrm e}^{2 x}}{1960}+\frac {15 \left (-32 x^{3}+48 x^{2}-48 x +24\right ) {\mathrm e}^{2 x}}{392}-\frac {26 \left (12 x^{2}-12 x +6\right ) {\mathrm e}^{2 x}}{49}-\frac {579 \left (-4 x +2\right ) {\mathrm e}^{2 x}}{49}\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+198520*x^2+4302200*x+7031500)*exp(x)^4+1/2401*(
588*x^4-5880*x^3-30576*x^2+226968*x+843780)*exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/2401*(x^8-24*x^7+76*x^6+1656*x^5-6474*x^4-57960*x^3+93100*x^2+1029000*x+1500625)*exp(4*x)+1/2401*(294*x^4-35
28*x^3-9996*x^2+123480*x+360150)*exp(2*x)

________________________________________________________________________________________

maxima [B]  time = 0.37, size = 125, normalized size = 4.46 \begin {gather*} \frac {1}{2401} \, {\left (x^{8} - 24 \, x^{7} + 76 \, x^{6} + 1656 \, x^{5} - 6474 \, x^{4} - 57960 \, x^{3} + 93100 \, x^{2} + 1029000 \, x + 1500625\right )} e^{\left (4 \, x\right )} + \frac {3}{49} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} - \frac {15}{49} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} - \frac {156}{49} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1158}{49} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {1230}{7} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2401*(4*x^8-88*x^7+136*x^6+7080*x^5-17616*x^4-257736*x^3+198520*x^2+4302200*x+7031500)*exp(x)^4+1/
2401*(588*x^4-5880*x^3-30576*x^2+226968*x+843780)*exp(x)^2,x, algorithm="maxima")

[Out]

1/2401*(x^8 - 24*x^7 + 76*x^6 + 1656*x^5 - 6474*x^4 - 57960*x^3 + 93100*x^2 + 1029000*x + 1500625)*e^(4*x) + 3
/49*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3)*e^(2*x) - 15/49*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) - 156/49*(2*x^2 - 2*x
+ 1)*e^(2*x) + 1158/49*(2*x - 1)*e^(2*x) + 1230/7*e^(2*x)

________________________________________________________________________________________

mupad [B]  time = 7.83, size = 59, normalized size = 2.11 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,{\left (-x^2+6\,x+35\right )}^2\,\left (1225\,{\mathrm {e}}^{2\,x}+420\,x\,{\mathrm {e}}^{2\,x}-34\,x^2\,{\mathrm {e}}^{2\,x}-12\,x^3\,{\mathrm {e}}^{2\,x}+x^4\,{\mathrm {e}}^{2\,x}+294\right )}{2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*(4302200*x + 198520*x^2 - 257736*x^3 - 17616*x^4 + 7080*x^5 + 136*x^6 - 88*x^7 + 4*x^8 + 7031500
))/2401 + (exp(2*x)*(226968*x - 30576*x^2 - 5880*x^3 + 588*x^4 + 843780))/2401,x)

[Out]

(exp(2*x)*(6*x - x^2 + 35)^2*(1225*exp(2*x) + 420*x*exp(2*x) - 34*x^2*exp(2*x) - 12*x^3*exp(2*x) + x^4*exp(2*x
) + 294))/2401

________________________________________________________________________________________

sympy [B]  time = 0.21, size = 73, normalized size = 2.61 \begin {gather*} \frac {\left (14406 x^{4} - 172872 x^{3} - 489804 x^{2} + 6050520 x + 17647350\right ) e^{2 x}}{117649} + \frac {\left (49 x^{8} - 1176 x^{7} + 3724 x^{6} + 81144 x^{5} - 317226 x^{4} - 2840040 x^{3} + 4561900 x^{2} + 50421000 x + 73530625\right ) e^{4 x}}{117649} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2401*(4*x**8-88*x**7+136*x**6+7080*x**5-17616*x**4-257736*x**3+198520*x**2+4302200*x+7031500)*exp(
x)**4+1/2401*(588*x**4-5880*x**3-30576*x**2+226968*x+843780)*exp(x)**2,x)

[Out]

(14406*x**4 - 172872*x**3 - 489804*x**2 + 6050520*x + 17647350)*exp(2*x)/117649 + (49*x**8 - 1176*x**7 + 3724*
x**6 + 81144*x**5 - 317226*x**4 - 2840040*x**3 + 4561900*x**2 + 50421000*x + 73530625)*exp(4*x)/117649

________________________________________________________________________________________