3.48.82 \(\int (2 e^2 x+e^{1+3 x} (-64 x^3-48 x^4+96 x^5+48 x^6)+e^{8 x} (96 x^5+128 x^6)+e^{7 x} (384 x^5+448 x^6-512 x^7-448 x^8)+e^{1+2 x} (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8)+e^{6 x} (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10})+e^{5 x} (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12})+e^{4 x} (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e (-32 x^3-32 x^4))) \, dx\)

Optimal. Leaf size=32 \[ x^2 \left (-e+4 e^{2 x} x^2 \left (1+e^x-x^2\right )^2\right )^2 \]

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Rubi [B]  time = 4.56, antiderivative size = 230, normalized size of antiderivative = 7.19, number of steps used = 363, number of rules used = 4, integrand size = 273, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {2196, 2176, 2194, 1593} \begin {gather*} 16 e^{4 x} x^{14}-64 e^{4 x} x^{12}-64 e^{5 x} x^{12}+96 e^{4 x} x^{10}+192 e^{5 x} x^{10}+96 e^{6 x} x^{10}-64 e^{4 x} x^8-192 e^{5 x} x^8-192 e^{6 x} x^8-64 e^{7 x} x^8-8 e^{2 x+1} x^8+16 e^{4 x} x^6+64 e^{5 x} x^6+96 e^{6 x} x^6+64 e^{7 x} x^6+16 e^{8 x} x^6+16 e^{2 x+1} x^6+16 e^{3 x+1} x^6-8 e^{2 x+1} x^4-16 e^{3 x+1} x^4-8 e^{4 x+1} x^4+e^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*E^2*x + E^(1 + 3*x)*(-64*x^3 - 48*x^4 + 96*x^5 + 48*x^6) + E^(8*x)*(96*x^5 + 128*x^6) + E^(7*x)*(384*x^5
+ 448*x^6 - 512*x^7 - 448*x^8) + E^(1 + 2*x)*(-32*x^3 - 16*x^4 + 96*x^5 + 32*x^6 - 64*x^7 - 16*x^8) + E^(6*x)
*(576*x^5 + 576*x^6 - 1536*x^7 - 1152*x^8 + 960*x^9 + 576*x^10) + E^(5*x)*(384*x^5 + 320*x^6 - 1536*x^7 - 960*
x^8 + 1920*x^9 + 960*x^10 - 768*x^11 - 320*x^12) + E^(4*x)*(96*x^5 + 64*x^6 - 512*x^7 - 256*x^8 + 960*x^9 + 38
4*x^10 - 768*x^11 - 256*x^12 + 224*x^13 + 64*x^14 + E*(-32*x^3 - 32*x^4)),x]

[Out]

E^2*x^2 - 8*E^(1 + 2*x)*x^4 - 16*E^(1 + 3*x)*x^4 - 8*E^(1 + 4*x)*x^4 + 16*E^(4*x)*x^6 + 64*E^(5*x)*x^6 + 96*E^
(6*x)*x^6 + 64*E^(7*x)*x^6 + 16*E^(8*x)*x^6 + 16*E^(1 + 2*x)*x^6 + 16*E^(1 + 3*x)*x^6 - 64*E^(4*x)*x^8 - 192*E
^(5*x)*x^8 - 192*E^(6*x)*x^8 - 64*E^(7*x)*x^8 - 8*E^(1 + 2*x)*x^8 + 96*E^(4*x)*x^10 + 192*E^(5*x)*x^10 + 96*E^
(6*x)*x^10 - 64*E^(4*x)*x^12 - 64*E^(5*x)*x^12 + 16*E^(4*x)*x^14

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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} &=e^2 x^2+\int e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right ) \, dx+\int e^{8 x} \left (96 x^5+128 x^6\right ) \, dx+\int e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right ) \, dx+\int e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right ) \, dx+\int e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right ) \, dx+\int e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right ) \, dx+\int e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right ) \, dx\\ &=e^2 x^2+\int e^{8 x} x^5 (96+128 x) \, dx+\int \left (-64 e^{1+3 x} x^3-48 e^{1+3 x} x^4+96 e^{1+3 x} x^5+48 e^{1+3 x} x^6\right ) \, dx+\int \left (384 e^{7 x} x^5+448 e^{7 x} x^6-512 e^{7 x} x^7-448 e^{7 x} x^8\right ) \, dx+\int \left (-32 e^{1+2 x} x^3-16 e^{1+2 x} x^4+96 e^{1+2 x} x^5+32 e^{1+2 x} x^6-64 e^{1+2 x} x^7-16 e^{1+2 x} x^8\right ) \, dx+\int \left (576 e^{6 x} x^5+576 e^{6 x} x^6-1536 e^{6 x} x^7-1152 e^{6 x} x^8+960 e^{6 x} x^9+576 e^{6 x} x^{10}\right ) \, dx+\int \left (384 e^{5 x} x^5+320 e^{5 x} x^6-1536 e^{5 x} x^7-960 e^{5 x} x^8+1920 e^{5 x} x^9+960 e^{5 x} x^{10}-768 e^{5 x} x^{11}-320 e^{5 x} x^{12}\right ) \, dx+\int \left (96 e^{4 x} x^5+64 e^{4 x} x^6-512 e^{4 x} x^7-256 e^{4 x} x^8+960 e^{4 x} x^9+384 e^{4 x} x^{10}-768 e^{4 x} x^{11}-256 e^{4 x} x^{12}+224 e^{4 x} x^{13}+64 e^{4 x} x^{14}-32 e^{1+4 x} x^3 (1+x)\right ) \, dx\\ &=e^2 x^2-16 \int e^{1+2 x} x^4 \, dx-16 \int e^{1+2 x} x^8 \, dx-32 \int e^{1+2 x} x^3 \, dx+32 \int e^{1+2 x} x^6 \, dx-32 \int e^{1+4 x} x^3 (1+x) \, dx-48 \int e^{1+3 x} x^4 \, dx+48 \int e^{1+3 x} x^6 \, dx-64 \int e^{1+3 x} x^3 \, dx+64 \int e^{4 x} x^6 \, dx-64 \int e^{1+2 x} x^7 \, dx+64 \int e^{4 x} x^{14} \, dx+96 \int e^{4 x} x^5 \, dx+96 \int e^{1+2 x} x^5 \, dx+96 \int e^{1+3 x} x^5 \, dx+224 \int e^{4 x} x^{13} \, dx-256 \int e^{4 x} x^8 \, dx-256 \int e^{4 x} x^{12} \, dx+320 \int e^{5 x} x^6 \, dx-320 \int e^{5 x} x^{12} \, dx+384 \int e^{5 x} x^5 \, dx+384 \int e^{7 x} x^5 \, dx+384 \int e^{4 x} x^{10} \, dx+448 \int e^{7 x} x^6 \, dx-448 \int e^{7 x} x^8 \, dx-512 \int e^{4 x} x^7 \, dx-512 \int e^{7 x} x^7 \, dx+576 \int e^{6 x} x^5 \, dx+576 \int e^{6 x} x^6 \, dx+576 \int e^{6 x} x^{10} \, dx-768 \int e^{4 x} x^{11} \, dx-768 \int e^{5 x} x^{11} \, dx-960 \int e^{5 x} x^8 \, dx+960 \int e^{4 x} x^9 \, dx+960 \int e^{6 x} x^9 \, dx+960 \int e^{5 x} x^{10} \, dx-1152 \int e^{6 x} x^8 \, dx-1536 \int e^{5 x} x^7 \, dx-1536 \int e^{6 x} x^7 \, dx+1920 \int e^{5 x} x^9 \, dx+\int \left (96 e^{8 x} x^5+128 e^{8 x} x^6\right ) \, dx\\ &=e^2 x^2-16 e^{1+2 x} x^3-\frac {64}{3} e^{1+3 x} x^3-8 e^{1+2 x} x^4-16 e^{1+3 x} x^4+24 e^{4 x} x^5+\frac {384}{5} e^{5 x} x^5+96 e^{6 x} x^5+\frac {384}{7} e^{7 x} x^5+48 e^{1+2 x} x^5+32 e^{1+3 x} x^5+16 e^{4 x} x^6+64 e^{5 x} x^6+96 e^{6 x} x^6+64 e^{7 x} x^6+16 e^{1+2 x} x^6+16 e^{1+3 x} x^6-128 e^{4 x} x^7-\frac {1536}{5} e^{5 x} x^7-256 e^{6 x} x^7-\frac {512}{7} e^{7 x} x^7-32 e^{1+2 x} x^7-64 e^{4 x} x^8-192 e^{5 x} x^8-192 e^{6 x} x^8-64 e^{7 x} x^8-8 e^{1+2 x} x^8+240 e^{4 x} x^9+384 e^{5 x} x^9+160 e^{6 x} x^9+96 e^{4 x} x^{10}+192 e^{5 x} x^{10}+96 e^{6 x} x^{10}-192 e^{4 x} x^{11}-\frac {768}{5} e^{5 x} x^{11}-64 e^{4 x} x^{12}-64 e^{5 x} x^{12}+56 e^{4 x} x^{13}+16 e^{4 x} x^{14}+32 \int e^{1+2 x} x^3 \, dx-32 \int \left (e^{1+4 x} x^3+e^{1+4 x} x^4\right ) \, dx+48 \int e^{1+2 x} x^2 \, dx+64 \int e^{1+3 x} x^2 \, dx+64 \int e^{1+3 x} x^3 \, dx+64 \int e^{1+2 x} x^7 \, dx-96 \int e^{4 x} x^5 \, dx+96 \int e^{8 x} x^5 \, dx-96 \int e^{1+2 x} x^5 \, dx-96 \int e^{1+3 x} x^5 \, dx-120 \int e^{4 x} x^4 \, dx+128 \int e^{8 x} x^6 \, dx-160 \int e^{1+3 x} x^4 \, dx+224 \int e^{1+2 x} x^6 \, dx-224 \int e^{4 x} x^{13} \, dx-240 \int e^{1+2 x} x^4 \, dx-\frac {1920}{7} \int e^{7 x} x^4 \, dx-384 \int e^{5 x} x^4 \, dx-384 \int e^{5 x} x^5 \, dx-384 \int e^{7 x} x^5 \, dx-480 \int e^{6 x} x^4 \, dx+512 \int e^{7 x} x^6 \, dx+512 \int e^{4 x} x^7 \, dx+512 \int e^{7 x} x^7 \, dx-576 \int e^{6 x} x^5 \, dx-728 \int e^{4 x} x^{12} \, dx+768 \int e^{4 x} x^{11} \, dx+768 \int e^{5 x} x^{11} \, dx+896 \int e^{4 x} x^6 \, dx-960 \int e^{4 x} x^9 \, dx-960 \int e^{6 x} x^9 \, dx-1440 \int e^{6 x} x^8 \, dx+1536 \int e^{5 x} x^7 \, dx+1536 \int e^{6 x} x^7 \, dx+\frac {8448}{5} \int e^{5 x} x^{10} \, dx+1792 \int e^{6 x} x^6 \, dx-1920 \int e^{5 x} x^9 \, dx+2112 \int e^{4 x} x^{10} \, dx+\frac {10752}{5} \int e^{5 x} x^6 \, dx-2160 \int e^{4 x} x^8 \, dx-3456 \int e^{5 x} x^8 \, dx\\ &=e^2 x^2+24 e^{1+2 x} x^2+\frac {64}{3} e^{1+3 x} x^2-30 e^{4 x} x^4-\frac {384}{5} e^{5 x} x^4-80 e^{6 x} x^4-\frac {1920}{49} e^{7 x} x^4-128 e^{1+2 x} x^4-\frac {208}{3} e^{1+3 x} x^4+12 e^{8 x} x^5+240 e^{4 x} x^6+\frac {12352}{25} e^{5 x} x^6+\frac {1184}{3} e^{6 x} x^6+\frac {960}{7} e^{7 x} x^6+16 e^{8 x} x^6+128 e^{1+2 x} x^6+16 e^{1+3 x} x^6-604 e^{4 x} x^8-\frac {4416}{5} e^{5 x} x^8-432 e^{6 x} x^8-64 e^{7 x} x^8-8 e^{1+2 x} x^8+624 e^{4 x} x^{10}+\frac {13248}{25} e^{5 x} x^{10}+96 e^{6 x} x^{10}-246 e^{4 x} x^{12}-64 e^{5 x} x^{12}+16 e^{4 x} x^{14}-32 \int e^{1+4 x} x^3 \, dx-32 \int e^{1+4 x} x^4 \, dx-\frac {128}{3} \int e^{1+3 x} x \, dx-48 \int e^{1+2 x} x \, dx-48 \int e^{1+2 x} x^2 \, dx-60 \int e^{8 x} x^4 \, dx-64 \int e^{1+3 x} x^2 \, dx-96 \int e^{8 x} x^5 \, dx+120 \int e^{4 x} x^3 \, dx+120 \int e^{4 x} x^4 \, dx+\frac {7680}{49} \int e^{7 x} x^3 \, dx+160 \int e^{1+3 x} x^4 \, dx+\frac {640}{3} \int e^{1+3 x} x^3 \, dx-224 \int e^{1+2 x} x^6 \, dx+240 \int e^{1+2 x} x^4 \, dx+\frac {1920}{7} \int e^{7 x} x^4 \, dx+\frac {1536}{5} \int e^{5 x} x^3 \, dx+320 \int e^{6 x} x^3 \, dx+384 \int e^{5 x} x^4 \, dx-\frac {3072}{7} \int e^{7 x} x^5 \, dx+480 \int e^{1+2 x} x^3 \, dx+480 \int e^{6 x} x^4 \, dx-512 \int e^{7 x} x^6 \, dx-672 \int e^{1+2 x} x^5 \, dx+728 \int e^{4 x} x^{12} \, dx-896 \int e^{4 x} x^6 \, dx-1344 \int e^{4 x} x^5 \, dx+1440 \int e^{6 x} x^8 \, dx-\frac {8448}{5} \int e^{5 x} x^{10} \, dx-1792 \int e^{6 x} x^5 \, dx-1792 \int e^{6 x} x^6 \, dx+1920 \int e^{6 x} x^7 \, dx-2112 \int e^{4 x} x^{10} \, dx-\frac {10752}{5} \int e^{5 x} x^6 \, dx+2160 \int e^{4 x} x^8 \, dx+2184 \int e^{4 x} x^{11} \, dx-\frac {64512}{25} \int e^{5 x} x^5 \, dx-\frac {16896}{5} \int e^{5 x} x^9 \, dx+3456 \int e^{5 x} x^8 \, dx+4320 \int e^{4 x} x^7 \, dx-5280 \int e^{4 x} x^9 \, dx+\frac {27648}{5} \int e^{5 x} x^7 \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 48, normalized size = 1.50 \begin {gather*} \left (e x-4 e^{4 x} x^3+8 e^{3 x} x^3 \left (-1+x^2\right )-4 e^{2 x} x^3 \left (-1+x^2\right )^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*E^2*x + E^(1 + 3*x)*(-64*x^3 - 48*x^4 + 96*x^5 + 48*x^6) + E^(8*x)*(96*x^5 + 128*x^6) + E^(7*x)*(3
84*x^5 + 448*x^6 - 512*x^7 - 448*x^8) + E^(1 + 2*x)*(-32*x^3 - 16*x^4 + 96*x^5 + 32*x^6 - 64*x^7 - 16*x^8) + E
^(6*x)*(576*x^5 + 576*x^6 - 1536*x^7 - 1152*x^8 + 960*x^9 + 576*x^10) + E^(5*x)*(384*x^5 + 320*x^6 - 1536*x^7
- 960*x^8 + 1920*x^9 + 960*x^10 - 768*x^11 - 320*x^12) + E^(4*x)*(96*x^5 + 64*x^6 - 512*x^7 - 256*x^8 + 960*x^
9 + 384*x^10 - 768*x^11 - 256*x^12 + 224*x^13 + 64*x^14 + E*(-32*x^3 - 32*x^4)),x]

[Out]

(E*x - 4*E^(4*x)*x^3 + 8*E^(3*x)*x^3*(-1 + x^2) - 4*E^(2*x)*x^3*(-1 + x^2)^2)^2

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fricas [B]  time = 1.22, size = 163, normalized size = 5.09 \begin {gather*} {\left (16 \, x^{6} e^{\left (8 \, x + 4\right )} + x^{2} e^{6} - 64 \, {\left (x^{8} - x^{6}\right )} e^{\left (7 \, x + 4\right )} + 96 \, {\left (x^{10} - 2 \, x^{8} + x^{6}\right )} e^{\left (6 \, x + 4\right )} - 64 \, {\left (x^{12} - 3 \, x^{10} + 3 \, x^{8} - x^{6}\right )} e^{\left (5 \, x + 4\right )} - 8 \, {\left (x^{4} e^{3} - 2 \, {\left (x^{14} - 4 \, x^{12} + 6 \, x^{10} - 4 \, x^{8} + x^{6}\right )} e^{2}\right )} e^{\left (4 \, x + 2\right )} + 16 \, {\left (x^{6} - x^{4}\right )} e^{\left (3 \, x + 5\right )} - 8 \, {\left (x^{8} - 2 \, x^{6} + x^{4}\right )} e^{\left (2 \, x + 5\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp(x)^7+(576*x^10+960*x^9-1152*x^8-153
6*x^7+576*x^6+576*x^5)*exp(x)^6+(-320*x^12-768*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)
^5+((-32*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960*x^9-256*x^8-512*x^7+64*x^6+96*x^5)
*exp(x)^4+(48*x^6+96*x^5-48*x^4-64*x^3)*exp(1)*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*ex
p(x)^2+2*x*exp(1)^2,x, algorithm="fricas")

[Out]

(16*x^6*e^(8*x + 4) + x^2*e^6 - 64*(x^8 - x^6)*e^(7*x + 4) + 96*(x^10 - 2*x^8 + x^6)*e^(6*x + 4) - 64*(x^12 -
3*x^10 + 3*x^8 - x^6)*e^(5*x + 4) - 8*(x^4*e^3 - 2*(x^14 - 4*x^12 + 6*x^10 - 4*x^8 + x^6)*e^2)*e^(4*x + 2) + 1
6*(x^6 - x^4)*e^(3*x + 5) - 8*(x^8 - 2*x^6 + x^4)*e^(2*x + 5))*e^(-4)

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giac [B]  time = 0.15, size = 150, normalized size = 4.69 \begin {gather*} 16 \, x^{6} e^{\left (8 \, x\right )} - 8 \, x^{4} e^{\left (4 \, x + 1\right )} + x^{2} e^{2} - 64 \, {\left (x^{8} - x^{6}\right )} e^{\left (7 \, x\right )} + 96 \, {\left (x^{10} - 2 \, x^{8} + x^{6}\right )} e^{\left (6 \, x\right )} - 64 \, {\left (x^{12} - 3 \, x^{10} + 3 \, x^{8} - x^{6}\right )} e^{\left (5 \, x\right )} + 16 \, {\left (x^{14} - 4 \, x^{12} + 6 \, x^{10} - 4 \, x^{8} + x^{6}\right )} e^{\left (4 \, x\right )} + 16 \, {\left (x^{6} - x^{4}\right )} e^{\left (3 \, x + 1\right )} - 8 \, {\left (x^{8} - 2 \, x^{6} + x^{4}\right )} e^{\left (2 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp(x)^7+(576*x^10+960*x^9-1152*x^8-153
6*x^7+576*x^6+576*x^5)*exp(x)^6+(-320*x^12-768*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)
^5+((-32*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960*x^9-256*x^8-512*x^7+64*x^6+96*x^5)
*exp(x)^4+(48*x^6+96*x^5-48*x^4-64*x^3)*exp(1)*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*ex
p(x)^2+2*x*exp(1)^2,x, algorithm="giac")

[Out]

16*x^6*e^(8*x) - 8*x^4*e^(4*x + 1) + x^2*e^2 - 64*(x^8 - x^6)*e^(7*x) + 96*(x^10 - 2*x^8 + x^6)*e^(6*x) - 64*(
x^12 - 3*x^10 + 3*x^8 - x^6)*e^(5*x) + 16*(x^14 - 4*x^12 + 6*x^10 - 4*x^8 + x^6)*e^(4*x) + 16*(x^6 - x^4)*e^(3
*x + 1) - 8*(x^8 - 2*x^6 + x^4)*e^(2*x + 1)

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maple [B]  time = 0.17, size = 159, normalized size = 4.97

method	result	size
risch	\(16 x^{6} {\mathrm e}^{8 x}+\left (-64 x^{8}+64 x^{6}\right ) {\mathrm e}^{7 x}+\left (96 x^{10}-192 x^{8}+96 x^{6}\right ) {\mathrm e}^{6 x}+\left (-64 x^{12}+192 x^{10}-192 x^{8}+64 x^{6}\right ) {\mathrm e}^{5 x}+\left (16 x^{14}-64 x^{12}+96 x^{10}-64 x^{8}+16 x^{6}-8 x^{4} {\mathrm e}\right ) {\mathrm e}^{4 x}+\left (16 x^{6}-16 x^{4}\right ) {\mathrm e}^{3 x +1}+\left (-8 x^{8}+16 x^{6}-8 x^{4}\right ) {\mathrm e}^{2 x +1}+x^{2} {\mathrm e}^{2}\)	\(159\)
default	\(16 \,{\mathrm e}^{4 x} x^{14}-64 \,{\mathrm e}^{4 x} x^{12}+96 \,{\mathrm e}^{4 x} x^{10}-64 x^{8} {\mathrm e}^{4 x}+16 x^{6} {\mathrm e}^{4 x}-32 \,{\mathrm e} \left (\frac {x^{3} {\mathrm e}^{4 x}}{4}-\frac {3 x^{2} {\mathrm e}^{4 x}}{16}+\frac {3 x \,{\mathrm e}^{4 x}}{32}-\frac {3 \,{\mathrm e}^{4 x}}{128}\right )-32 \,{\mathrm e} \left (\frac {x^{4} {\mathrm e}^{4 x}}{4}-\frac {x^{3} {\mathrm e}^{4 x}}{4}+\frac {3 x^{2} {\mathrm e}^{4 x}}{16}-\frac {3 x \,{\mathrm e}^{4 x}}{32}+\frac {3 \,{\mathrm e}^{4 x}}{128}\right )+16 x^{6} {\mathrm e}^{8 x}+64 \,{\mathrm e}^{7 x} x^{6}-64 \,{\mathrm e}^{7 x} x^{8}+96 \,{\mathrm e}^{6 x} x^{6}-192 \,{\mathrm e}^{6 x} x^{8}+96 \,{\mathrm e}^{6 x} x^{10}-64 \,{\mathrm e}^{5 x} x^{12}+64 x^{6} {\mathrm e}^{5 x}-192 \,{\mathrm e}^{5 x} x^{8}+192 \,{\mathrm e}^{5 x} x^{10}+16 \,{\mathrm e} \left (-x^{4} {\mathrm e}^{3 x}+x^{6} {\mathrm e}^{3 x}\right )+16 \,{\mathrm e} \left (-\frac {{\mathrm e}^{2 x} x^{4}}{2}+x^{6} {\mathrm e}^{2 x}-\frac {x^{8} {\mathrm e}^{2 x}}{2}\right )+x^{2} {\mathrm e}^{2}\)	\(279\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp(x)^7+(576*x^10+960*x^9-1152*x^8-1536*x^7+
576*x^6+576*x^5)*exp(x)^6+(-320*x^12-768*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)^5+((-
32*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960*x^9-256*x^8-512*x^7+64*x^6+96*x^5)*exp(x
)^4+(48*x^6+96*x^5-48*x^4-64*x^3)*exp(1)*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*exp(x)^2
+2*x*exp(1)^2,x,method=_RETURNVERBOSE)

[Out]

16*x^6*exp(8*x)+(-64*x^8+64*x^6)*exp(7*x)+(96*x^10-192*x^8+96*x^6)*exp(6*x)+(-64*x^12+192*x^10-192*x^8+64*x^6)
*exp(5*x)+(16*x^14-64*x^12+96*x^10-64*x^8+16*x^6-8*x^4*exp(1))*exp(4*x)+(16*x^6-16*x^4)*exp(3*x+1)+(-8*x^8+16*
x^6-8*x^4)*exp(2*x+1)+x^2*exp(2)

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maxima [B]  time = 0.39, size = 722, normalized size = 22.56 \begin {gather*} 16 \, x^{6} e^{\left (8 \, x\right )} + x^{2} e^{2} - 64 \, {\left (x^{8} - x^{6}\right )} e^{\left (7 \, x\right )} + 96 \, {\left (x^{10} - 2 \, x^{8} + x^{6}\right )} e^{\left (6 \, x\right )} - 64 \, {\left (x^{12} - 3 \, x^{10} + 3 \, x^{8} - x^{6}\right )} e^{\left (5 \, x\right )} + \frac {1}{8192} \, {\left (131072 \, x^{14} - 458752 \, x^{13} + 1490944 \, x^{12} - 4472832 \, x^{11} + 12300288 \, x^{10} - 30750720 \, x^{9} + 69189120 \, x^{8} - 138378240 \, x^{7} + 242161920 \, x^{6} - 363242880 \, x^{5} + 454053600 \, x^{4} - 454053600 \, x^{3} + 340540200 \, x^{2} - 170270100 \, x + 42567525\right )} e^{\left (4 \, x\right )} + \frac {7}{8192} \, {\left (65536 \, x^{13} - 212992 \, x^{12} + 638976 \, x^{11} - 1757184 \, x^{10} + 4392960 \, x^{9} - 9884160 \, x^{8} + 19768320 \, x^{7} - 34594560 \, x^{6} + 51891840 \, x^{5} - 64864800 \, x^{4} + 64864800 \, x^{3} - 48648600 \, x^{2} + 24324300 \, x - 6081075\right )} e^{\left (4 \, x\right )} - \frac {1}{256} \, {\left (16384 \, x^{12} - 49152 \, x^{11} + 135168 \, x^{10} - 337920 \, x^{9} + 760320 \, x^{8} - 1520640 \, x^{7} + 2661120 \, x^{6} - 3991680 \, x^{5} + 4989600 \, x^{4} - 4989600 \, x^{3} + 3742200 \, x^{2} - 1871100 \, x + 467775\right )} e^{\left (4 \, x\right )} - \frac {3}{256} \, {\left (16384 \, x^{11} - 45056 \, x^{10} + 112640 \, x^{9} - 253440 \, x^{8} + 506880 \, x^{7} - 887040 \, x^{6} + 1330560 \, x^{5} - 1663200 \, x^{4} + 1663200 \, x^{3} - 1247400 \, x^{2} + 623700 \, x - 155925\right )} e^{\left (4 \, x\right )} + \frac {3}{128} \, {\left (4096 \, x^{10} - 10240 \, x^{9} + 23040 \, x^{8} - 46080 \, x^{7} + 80640 \, x^{6} - 120960 \, x^{5} + 151200 \, x^{4} - 151200 \, x^{3} + 113400 \, x^{2} - 56700 \, x + 14175\right )} e^{\left (4 \, x\right )} + \frac {15}{128} \, {\left (2048 \, x^{9} - 4608 \, x^{8} + 9216 \, x^{7} - 16128 \, x^{6} + 24192 \, x^{5} - 30240 \, x^{4} + 30240 \, x^{3} - 22680 \, x^{2} + 11340 \, x - 2835\right )} e^{\left (4 \, x\right )} - \frac {1}{8} \, {\left (512 \, x^{8} - 1024 \, x^{7} + 1792 \, x^{6} - 2688 \, x^{5} + 3360 \, x^{4} - 3360 \, x^{3} + 2520 \, x^{2} - 1260 \, x + 315\right )} e^{\left (4 \, x\right )} - \frac {1}{8} \, {\left (1024 \, x^{7} - 1792 \, x^{6} + 2688 \, x^{5} - 3360 \, x^{4} + 3360 \, x^{3} - 2520 \, x^{2} + 1260 \, x - 315\right )} e^{\left (4 \, x\right )} + \frac {1}{16} \, {\left (256 \, x^{6} - 384 \, x^{5} + 480 \, x^{4} - 480 \, x^{3} + 360 \, x^{2} - 180 \, x + 45\right )} e^{\left (4 \, x\right )} + \frac {3}{16} \, {\left (128 \, x^{5} - 160 \, x^{4} + 160 \, x^{3} - 120 \, x^{2} + 60 \, x - 15\right )} e^{\left (4 \, x\right )} - \frac {1}{4} \, {\left (32 \, x^{4} e - 32 \, x^{3} e + 24 \, x^{2} e - 12 \, x e + 3 \, e\right )} e^{\left (4 \, x\right )} - \frac {1}{4} \, {\left (32 \, x^{3} e - 24 \, x^{2} e + 12 \, x e - 3 \, e\right )} e^{\left (4 \, x\right )} + 16 \, {\left (x^{6} e - x^{4} e\right )} e^{\left (3 \, x\right )} - 8 \, {\left (x^{8} e - 2 \, x^{6} e + x^{4} e\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp(x)^7+(576*x^10+960*x^9-1152*x^8-153
6*x^7+576*x^6+576*x^5)*exp(x)^6+(-320*x^12-768*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)
^5+((-32*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960*x^9-256*x^8-512*x^7+64*x^6+96*x^5)
*exp(x)^4+(48*x^6+96*x^5-48*x^4-64*x^3)*exp(1)*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*ex
p(x)^2+2*x*exp(1)^2,x, algorithm="maxima")

[Out]

16*x^6*e^(8*x) + x^2*e^2 - 64*(x^8 - x^6)*e^(7*x) + 96*(x^10 - 2*x^8 + x^6)*e^(6*x) - 64*(x^12 - 3*x^10 + 3*x^
8 - x^6)*e^(5*x) + 1/8192*(131072*x^14 - 458752*x^13 + 1490944*x^12 - 4472832*x^11 + 12300288*x^10 - 30750720*
x^9 + 69189120*x^8 - 138378240*x^7 + 242161920*x^6 - 363242880*x^5 + 454053600*x^4 - 454053600*x^3 + 340540200
*x^2 - 170270100*x + 42567525)*e^(4*x) + 7/8192*(65536*x^13 - 212992*x^12 + 638976*x^11 - 1757184*x^10 + 43929
60*x^9 - 9884160*x^8 + 19768320*x^7 - 34594560*x^6 + 51891840*x^5 - 64864800*x^4 + 64864800*x^3 - 48648600*x^2
+ 24324300*x - 6081075)*e^(4*x) - 1/256*(16384*x^12 - 49152*x^11 + 135168*x^10 - 337920*x^9 + 760320*x^8 - 15
20640*x^7 + 2661120*x^6 - 3991680*x^5 + 4989600*x^4 - 4989600*x^3 + 3742200*x^2 - 1871100*x + 467775)*e^(4*x)
- 3/256*(16384*x^11 - 45056*x^10 + 112640*x^9 - 253440*x^8 + 506880*x^7 - 887040*x^6 + 1330560*x^5 - 1663200*x
^4 + 1663200*x^3 - 1247400*x^2 + 623700*x - 155925)*e^(4*x) + 3/128*(4096*x^10 - 10240*x^9 + 23040*x^8 - 46080
*x^7 + 80640*x^6 - 120960*x^5 + 151200*x^4 - 151200*x^3 + 113400*x^2 - 56700*x + 14175)*e^(4*x) + 15/128*(2048
*x^9 - 4608*x^8 + 9216*x^7 - 16128*x^6 + 24192*x^5 - 30240*x^4 + 30240*x^3 - 22680*x^2 + 11340*x - 2835)*e^(4*
x) - 1/8*(512*x^8 - 1024*x^7 + 1792*x^6 - 2688*x^5 + 3360*x^4 - 3360*x^3 + 2520*x^2 - 1260*x + 315)*e^(4*x) -
1/8*(1024*x^7 - 1792*x^6 + 2688*x^5 - 3360*x^4 + 3360*x^3 - 2520*x^2 + 1260*x - 315)*e^(4*x) + 1/16*(256*x^6 -
384*x^5 + 480*x^4 - 480*x^3 + 360*x^2 - 180*x + 45)*e^(4*x) + 3/16*(128*x^5 - 160*x^4 + 160*x^3 - 120*x^2 + 6
0*x - 15)*e^(4*x) - 1/4*(32*x^4*e - 32*x^3*e + 24*x^2*e - 12*x*e + 3*e)*e^(4*x) - 1/4*(32*x^3*e - 24*x^2*e + 1
2*x*e - 3*e)*e^(4*x) + 16*(x^6*e - x^4*e)*e^(3*x) - 8*(x^8*e - 2*x^6*e + x^4*e)*e^(2*x)

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mupad [B]  time = 3.81, size = 208, normalized size = 6.50 \begin {gather*} 16\,x^6\,{\mathrm {e}}^{4\,x}+64\,x^6\,{\mathrm {e}}^{5\,x}+96\,x^6\,{\mathrm {e}}^{6\,x}-64\,x^8\,{\mathrm {e}}^{4\,x}+64\,x^6\,{\mathrm {e}}^{7\,x}-192\,x^8\,{\mathrm {e}}^{5\,x}+16\,x^6\,{\mathrm {e}}^{8\,x}-192\,x^8\,{\mathrm {e}}^{6\,x}+96\,x^{10}\,{\mathrm {e}}^{4\,x}-64\,x^8\,{\mathrm {e}}^{7\,x}+192\,x^{10}\,{\mathrm {e}}^{5\,x}+96\,x^{10}\,{\mathrm {e}}^{6\,x}-64\,x^{12}\,{\mathrm {e}}^{4\,x}-64\,x^{12}\,{\mathrm {e}}^{5\,x}+16\,x^{14}\,{\mathrm {e}}^{4\,x}+x^2\,{\mathrm {e}}^2-8\,x^4\,{\mathrm {e}}^{2\,x+1}-16\,x^4\,{\mathrm {e}}^{3\,x+1}-8\,x^4\,{\mathrm {e}}^{4\,x+1}+16\,x^6\,{\mathrm {e}}^{2\,x+1}+16\,x^6\,{\mathrm {e}}^{3\,x+1}-8\,x^8\,{\mathrm {e}}^{2\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*exp(2) + exp(6*x)*(576*x^5 + 576*x^6 - 1536*x^7 - 1152*x^8 + 960*x^9 + 576*x^10) + exp(8*x)*(96*x^5 +
128*x^6) + exp(4*x)*(96*x^5 - exp(1)*(32*x^3 + 32*x^4) + 64*x^6 - 512*x^7 - 256*x^8 + 960*x^9 + 384*x^10 - 768
*x^11 - 256*x^12 + 224*x^13 + 64*x^14) + exp(5*x)*(384*x^5 + 320*x^6 - 1536*x^7 - 960*x^8 + 1920*x^9 + 960*x^1
0 - 768*x^11 - 320*x^12) + exp(7*x)*(384*x^5 + 448*x^6 - 512*x^7 - 448*x^8) - exp(3*x)*exp(1)*(64*x^3 + 48*x^4
- 96*x^5 - 48*x^6) - exp(2*x)*exp(1)*(32*x^3 + 16*x^4 - 96*x^5 - 32*x^6 + 64*x^7 + 16*x^8),x)

[Out]

16*x^6*exp(4*x) + 64*x^6*exp(5*x) + 96*x^6*exp(6*x) - 64*x^8*exp(4*x) + 64*x^6*exp(7*x) - 192*x^8*exp(5*x) + 1
6*x^6*exp(8*x) - 192*x^8*exp(6*x) + 96*x^10*exp(4*x) - 64*x^8*exp(7*x) + 192*x^10*exp(5*x) + 96*x^10*exp(6*x)
- 64*x^12*exp(4*x) - 64*x^12*exp(5*x) + 16*x^14*exp(4*x) + x^2*exp(2) - 8*x^4*exp(2*x + 1) - 16*x^4*exp(3*x +
1) - 8*x^4*exp(4*x + 1) + 16*x^6*exp(2*x + 1) + 16*x^6*exp(3*x + 1) - 8*x^8*exp(2*x + 1)

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sympy [B]  time = 0.39, size = 168, normalized size = 5.25 \begin {gather*} 16 x^{6} e^{8 x} + x^{2} e^{2} + \left (- 64 x^{8} + 64 x^{6}\right ) e^{7 x} + \left (16 e x^{6} - 16 e x^{4}\right ) e^{3 x} + \left (96 x^{10} - 192 x^{8} + 96 x^{6}\right ) e^{6 x} + \left (- 8 e x^{8} + 16 e x^{6} - 8 e x^{4}\right ) e^{2 x} + \left (- 64 x^{12} + 192 x^{10} - 192 x^{8} + 64 x^{6}\right ) e^{5 x} + \left (16 x^{14} - 64 x^{12} + 96 x^{10} - 64 x^{8} + 16 x^{6} - 8 e x^{4}\right ) e^{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x**6+96*x**5)*exp(x)**8+(-448*x**8-512*x**7+448*x**6+384*x**5)*exp(x)**7+(576*x**10+960*x**9-11
52*x**8-1536*x**7+576*x**6+576*x**5)*exp(x)**6+(-320*x**12-768*x**11+960*x**10+1920*x**9-960*x**8-1536*x**7+32
0*x**6+384*x**5)*exp(x)**5+((-32*x**4-32*x**3)*exp(1)+64*x**14+224*x**13-256*x**12-768*x**11+384*x**10+960*x**
9-256*x**8-512*x**7+64*x**6+96*x**5)*exp(x)**4+(48*x**6+96*x**5-48*x**4-64*x**3)*exp(1)*exp(x)**3+(-16*x**8-64
*x**7+32*x**6+96*x**5-16*x**4-32*x**3)*exp(1)*exp(x)**2+2*x*exp(1)**2,x)

[Out]

16*x**6*exp(8*x) + x**2*exp(2) + (-64*x**8 + 64*x**6)*exp(7*x) + (16*E*x**6 - 16*E*x**4)*exp(3*x) + (96*x**10
- 192*x**8 + 96*x**6)*exp(6*x) + (-8*E*x**8 + 16*E*x**6 - 8*E*x**4)*exp(2*x) + (-64*x**12 + 192*x**10 - 192*x*
*8 + 64*x**6)*exp(5*x) + (16*x**14 - 64*x**12 + 96*x**10 - 64*x**8 + 16*x**6 - 8*E*x**4)*exp(4*x)

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