3.64.91 \(\int e^{8 x} (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} (-1024-8320 x-512 x^2)+e^8 (24584+202800 x+24800 x^2+768 x^3)+e^4 (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4)) \, dx\)

Optimal. Leaf size=26 \[ e^{8 x} x \left (1-x+4 \left (16-e^4+x\right )^2\right )^2 \]

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Rubi [B]  time = 0.62, antiderivative size = 174, normalized size of antiderivative = 6.69, number of steps used = 62, number of rules used = 3, integrand size = 97, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2196, 2194, 2176} \begin {gather*} 16 e^{8 x} x^5+1016 e^{8 x} x^4-64 e^{8 x+4} x^4+24329 e^{8 x} x^3-3056 e^{8 x+4} x^3+96 e^{8 x+8} x^3+260350 e^{8 x} x^2-48912 e^{8 x+4} x^2+3064 e^{8 x+8} x^2-64 e^{8 x+12} x^2+1050625 e^{8 x} x-262400 e^{8 x+4} x+24584 e^{8 x+8} x-1024 e^{8 x+12} x-2 e^{8 x+16}+2 e^{8 x+16} (8 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(8*x)*(1050625 + 8925700*x + 2155787*x^2 + 198696*x^3 + 8208*x^4 + 128*x^5 + E^16*(16 + 128*x) + E^12*(-
1024 - 8320*x - 512*x^2) + E^8*(24584 + 202800*x + 24800*x^2 + 768*x^3) + E^4*(-262400 - 2197024*x - 400464*x^
2 - 24704*x^3 - 512*x^4)),x]

[Out]

-2*E^(16 + 8*x) + 1050625*E^(8*x)*x - 262400*E^(4 + 8*x)*x + 24584*E^(8 + 8*x)*x - 1024*E^(12 + 8*x)*x + 26035
0*E^(8*x)*x^2 - 48912*E^(4 + 8*x)*x^2 + 3064*E^(8 + 8*x)*x^2 - 64*E^(12 + 8*x)*x^2 + 24329*E^(8*x)*x^3 - 3056*
E^(4 + 8*x)*x^3 + 96*E^(8 + 8*x)*x^3 + 1016*E^(8*x)*x^4 - 64*E^(4 + 8*x)*x^4 + 16*E^(8*x)*x^5 + 2*E^(16 + 8*x)
*(1 + 8*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} &=\int \left (1050625 e^{8 x}+8925700 e^{8 x} x+2155787 e^{8 x} x^2+198696 e^{8 x} x^3+8208 e^{8 x} x^4+128 e^{8 x} x^5+16 e^{16+8 x} (1+8 x)-128 e^{12+8 x} \left (8+65 x+4 x^2\right )+8 e^{8+8 x} \left (3073+25350 x+3100 x^2+96 x^3\right )-16 e^{4+8 x} \left (16400+137314 x+25029 x^2+1544 x^3+32 x^4\right )\right ) \, dx\\ &=8 \int e^{8+8 x} \left (3073+25350 x+3100 x^2+96 x^3\right ) \, dx+16 \int e^{16+8 x} (1+8 x) \, dx-16 \int e^{4+8 x} \left (16400+137314 x+25029 x^2+1544 x^3+32 x^4\right ) \, dx+128 \int e^{8 x} x^5 \, dx-128 \int e^{12+8 x} \left (8+65 x+4 x^2\right ) \, dx+8208 \int e^{8 x} x^4 \, dx+198696 \int e^{8 x} x^3 \, dx+1050625 \int e^{8 x} \, dx+2155787 \int e^{8 x} x^2 \, dx+8925700 \int e^{8 x} x \, dx\\ &=\frac {1050625 e^{8 x}}{8}+\frac {2231425}{2} e^{8 x} x+\frac {2155787}{8} e^{8 x} x^2+24837 e^{8 x} x^3+1026 e^{8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)+8 \int \left (3073 e^{8+8 x}+25350 e^{8+8 x} x+3100 e^{8+8 x} x^2+96 e^{8+8 x} x^3\right ) \, dx-16 \int e^{16+8 x} \, dx-16 \int \left (16400 e^{4+8 x}+137314 e^{4+8 x} x+25029 e^{4+8 x} x^2+1544 e^{4+8 x} x^3+32 e^{4+8 x} x^4\right ) \, dx-80 \int e^{8 x} x^4 \, dx-128 \int \left (8 e^{12+8 x}+65 e^{12+8 x} x+4 e^{12+8 x} x^2\right ) \, dx-4104 \int e^{8 x} x^3 \, dx-74511 \int e^{8 x} x^2 \, dx-\frac {2155787}{4} \int e^{8 x} x \, dx-\frac {2231425}{2} \int e^{8 x} \, dx\\ &=-\frac {130175 e^{8 x}}{16}-2 e^{16+8 x}+\frac {33547013}{32} e^{8 x} x+\frac {520319}{2} e^{8 x} x^2+24324 e^{8 x} x^3+1016 e^{8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)+40 \int e^{8 x} x^3 \, dx-512 \int e^{12+8 x} x^2 \, dx-512 \int e^{4+8 x} x^4 \, dx+768 \int e^{8+8 x} x^3 \, dx-1024 \int e^{12+8 x} \, dx+1539 \int e^{8 x} x^2 \, dx-8320 \int e^{12+8 x} x \, dx+\frac {74511}{4} \int e^{8 x} x \, dx+24584 \int e^{8+8 x} \, dx-24704 \int e^{4+8 x} x^3 \, dx+24800 \int e^{8+8 x} x^2 \, dx+\frac {2155787}{32} \int e^{8 x} \, dx+202800 \int e^{8+8 x} x \, dx-262400 \int e^{4+8 x} \, dx-400464 \int e^{4+8 x} x^2 \, dx-2197024 \int e^{4+8 x} x \, dx\\ &=\frac {72987 e^{8 x}}{256}-32800 e^{4+8 x}+3073 e^{8+8 x}-128 e^{12+8 x}-2 e^{16+8 x}+\frac {8405381}{8} e^{8 x} x-274628 e^{4+8 x} x+25350 e^{8+8 x} x-1040 e^{12+8 x} x+\frac {2082815}{8} e^{8 x} x^2-50058 e^{4+8 x} x^2+3100 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3088 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)-15 \int e^{8 x} x^2 \, dx+128 \int e^{12+8 x} x \, dx+256 \int e^{4+8 x} x^3 \, dx-288 \int e^{8+8 x} x^2 \, dx-\frac {1539}{4} \int e^{8 x} x \, dx+1040 \int e^{12+8 x} \, dx-\frac {74511}{32} \int e^{8 x} \, dx-6200 \int e^{8+8 x} x \, dx+9264 \int e^{4+8 x} x^2 \, dx-25350 \int e^{8+8 x} \, dx+100116 \int e^{4+8 x} x \, dx+274628 \int e^{4+8 x} \, dx\\ &=-\frac {381 e^{8 x}}{64}+\frac {3057}{2} e^{4+8 x}-\frac {383}{4} e^{8+8 x}+2 e^{12+8 x}-2 e^{16+8 x}+\frac {33619985}{32} e^{8 x} x-\frac {524227}{2} e^{4+8 x} x+24575 e^{8+8 x} x-1024 e^{12+8 x} x+260350 e^{8 x} x^2-48900 e^{4+8 x} x^2+3064 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3056 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)+\frac {15}{4} \int e^{8 x} x \, dx-16 \int e^{12+8 x} \, dx+\frac {1539}{32} \int e^{8 x} \, dx+72 \int e^{8+8 x} x \, dx-96 \int e^{4+8 x} x^2 \, dx+775 \int e^{8+8 x} \, dx-2316 \int e^{4+8 x} x \, dx-\frac {25029}{2} \int e^{4+8 x} \, dx\\ &=\frac {15 e^{8 x}}{256}-\frac {573}{16} e^{4+8 x}+\frac {9}{8} e^{8+8 x}-2 e^{16+8 x}+1050625 e^{8 x} x-262403 e^{4+8 x} x+24584 e^{8+8 x} x-1024 e^{12+8 x} x+260350 e^{8 x} x^2-48912 e^{4+8 x} x^2+3064 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3056 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)-\frac {15}{32} \int e^{8 x} \, dx-9 \int e^{8+8 x} \, dx+24 \int e^{4+8 x} x \, dx+\frac {579}{2} \int e^{4+8 x} \, dx\\ &=\frac {3}{8} e^{4+8 x}-2 e^{16+8 x}+1050625 e^{8 x} x-262400 e^{4+8 x} x+24584 e^{8+8 x} x-1024 e^{12+8 x} x+260350 e^{8 x} x^2-48912 e^{4+8 x} x^2+3064 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3056 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)-3 \int e^{4+8 x} \, dx\\ &=-2 e^{16+8 x}+1050625 e^{8 x} x-262400 e^{4+8 x} x+24584 e^{8+8 x} x-1024 e^{12+8 x} x+260350 e^{8 x} x^2-48912 e^{4+8 x} x^2+3064 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3056 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 32, normalized size = 1.23 \begin {gather*} e^{8 x} x \left (1025+4 e^8+127 x+4 x^2-8 e^4 (16+x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(8*x)*(1050625 + 8925700*x + 2155787*x^2 + 198696*x^3 + 8208*x^4 + 128*x^5 + E^16*(16 + 128*x) + E
^12*(-1024 - 8320*x - 512*x^2) + E^8*(24584 + 202800*x + 24800*x^2 + 768*x^3) + E^4*(-262400 - 2197024*x - 400
464*x^2 - 24704*x^3 - 512*x^4)),x]

[Out]

E^(8*x)*x*(1025 + 4*E^8 + 127*x + 4*x^2 - 8*E^4*(16 + x))^2

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fricas [B]  time = 0.82, size = 86, normalized size = 3.31 \begin {gather*} {\left (16 \, x^{5} + 1016 \, x^{4} + 24329 \, x^{3} + 260350 \, x^{2} + 16 \, x e^{16} - 64 \, {\left (x^{2} + 16 \, x\right )} e^{12} + 8 \, {\left (12 \, x^{3} + 383 \, x^{2} + 3073 \, x\right )} e^{8} - 16 \, {\left (4 \, x^{4} + 191 \, x^{3} + 3057 \, x^{2} + 16400 \, x\right )} e^{4} + 1050625 \, x\right )} e^{\left (8 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24800*x^2+202800*x+24584)*exp(4)^2+(-5
12*x^4-24704*x^3-400464*x^2-2197024*x-262400)*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625
)*exp(4*x)^2,x, algorithm="fricas")

[Out]

(16*x^5 + 1016*x^4 + 24329*x^3 + 260350*x^2 + 16*x*e^16 - 64*(x^2 + 16*x)*e^12 + 8*(12*x^3 + 383*x^2 + 3073*x)
*e^8 - 16*(4*x^4 + 191*x^3 + 3057*x^2 + 16400*x)*e^4 + 1050625*x)*e^(8*x)

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giac [B]  time = 0.19, size = 103, normalized size = 3.96 \begin {gather*} {\left (16 \, x^{5} + 1016 \, x^{4} + 24329 \, x^{3} + 260350 \, x^{2} + 1050625 \, x\right )} e^{\left (8 \, x\right )} + 16 \, x e^{\left (8 \, x + 16\right )} - 64 \, {\left (x^{2} + 16 \, x\right )} e^{\left (8 \, x + 12\right )} + 8 \, {\left (12 \, x^{3} + 383 \, x^{2} + 3073 \, x\right )} e^{\left (8 \, x + 8\right )} - 16 \, {\left (4 \, x^{4} + 191 \, x^{3} + 3057 \, x^{2} + 16400 \, x\right )} e^{\left (8 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24800*x^2+202800*x+24584)*exp(4)^2+(-5
12*x^4-24704*x^3-400464*x^2-2197024*x-262400)*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625
)*exp(4*x)^2,x, algorithm="giac")

[Out]

(16*x^5 + 1016*x^4 + 24329*x^3 + 260350*x^2 + 1050625*x)*e^(8*x) + 16*x*e^(8*x + 16) - 64*(x^2 + 16*x)*e^(8*x
+ 12) + 8*(12*x^3 + 383*x^2 + 3073*x)*e^(8*x + 8) - 16*(4*x^4 + 191*x^3 + 3057*x^2 + 16400*x)*e^(8*x + 4)

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maple [A]  time = 0.16, size = 36, normalized size = 1.38

method	result	size
gosper	\(x \left (4 \,{\mathrm e}^{8}-8 x \,{\mathrm e}^{4}+4 x^{2}-128 \,{\mathrm e}^{4}+127 x +1025\right )^{2} {\mathrm e}^{8 x}\)	\(36\)
risch	\(\left (16 x \,{\mathrm e}^{16}-64 x^{2} {\mathrm e}^{12}+96 x^{3} {\mathrm e}^{8}-64 x^{4} {\mathrm e}^{4}+16 x^{5}-1024 x \,{\mathrm e}^{12}+3064 x^{2} {\mathrm e}^{8}-3056 x^{3} {\mathrm e}^{4}+1016 x^{4}+24584 x \,{\mathrm e}^{8}-48912 x^{2} {\mathrm e}^{4}+24329 x^{3}-262400 x \,{\mathrm e}^{4}+260350 x^{2}+1050625 x \right ) {\mathrm e}^{8 x}\)	\(92\)
norman	\(\left (-64 \,{\mathrm e}^{4}+1016\right ) x^{4} {\mathrm e}^{8 x}+\left (96 \,{\mathrm e}^{8}-3056 \,{\mathrm e}^{4}+24329\right ) x^{3} {\mathrm e}^{8 x}+\left (-64 \,{\mathrm e}^{12}+3064 \,{\mathrm e}^{8}-48912 \,{\mathrm e}^{4}+260350\right ) x^{2} {\mathrm e}^{8 x}+\left (16 \,{\mathrm e}^{16}-1024 \,{\mathrm e}^{12}+24584 \,{\mathrm e}^{8}-262400 \,{\mathrm e}^{4}+1050625\right ) x \,{\mathrm e}^{8 x}+16 \,{\mathrm e}^{8 x} x^{5}\)	\(111\)
meijerg	\(-\frac {33619985}{256}-\frac {\left (-196608 x^{5}+122880 x^{4}-61440 x^{3}+23040 x^{2}-5760 x +720\right ) {\mathrm e}^{8 x}}{12288}+\frac {\left (128 \,{\mathrm e}^{16}-8320 \,{\mathrm e}^{12}+202800 \,{\mathrm e}^{8}-2197024 \,{\mathrm e}^{4}+8925700\right ) \left (1-\frac {\left (-16 x +2\right ) {\mathrm e}^{8 x}}{2}\right )}{64}-\frac {\left (-512 \,{\mathrm e}^{12}+24800 \,{\mathrm e}^{8}-400464 \,{\mathrm e}^{4}+2155787\right ) \left (2-\frac {\left (192 x^{2}-48 x +6\right ) {\mathrm e}^{8 x}}{3}\right )}{512}-\frac {\left (-512 \,{\mathrm e}^{4}+8208\right ) \left (24-\frac {\left (20480 x^{4}-10240 x^{3}+3840 x^{2}-960 x +120\right ) {\mathrm e}^{8 x}}{5}\right )}{32768}+\frac {\left (768 \,{\mathrm e}^{8}-24704 \,{\mathrm e}^{4}+198696\right ) \left (6-\frac {\left (-2048 x^{3}+768 x^{2}-192 x +24\right ) {\mathrm e}^{8 x}}{4}\right )}{4096}-2 \,{\mathrm e}^{16} \left (1-{\mathrm e}^{8 x}\right )+128 \,{\mathrm e}^{12} \left (1-{\mathrm e}^{8 x}\right )-3073 \,{\mathrm e}^{8} \left (1-{\mathrm e}^{8 x}\right )+32800 \,{\mathrm e}^{4} \left (1-{\mathrm e}^{8 x}\right )+\frac {1050625 \,{\mathrm e}^{8 x}}{8}\)	\(226\)
derivativedivides	\(1050625 x \,{\mathrm e}^{8 x}+260350 x^{2} {\mathrm e}^{8 x}+24329 \,{\mathrm e}^{8 x} x^{3}+1016 \,{\mathrm e}^{8 x} x^{4}+16 \,{\mathrm e}^{8 x} x^{5}-32800 \,{\mathrm e}^{8 x} {\mathrm e}^{4}+3073 \,{\mathrm e}^{8 x} {\mathrm e}^{8}-128 \,{\mathrm e}^{8 x} {\mathrm e}^{12}+2 \,{\mathrm e}^{8 x} {\mathrm e}^{16}-137314 \,{\mathrm e}^{4} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )-\frac {25029 \,{\mathrm e}^{4} \left (8 x^{2} {\mathrm e}^{8 x}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )}{4}-\frac {193 \,{\mathrm e}^{4} \left (32 \,{\mathrm e}^{8 x} x^{3}-12 x^{2} {\mathrm e}^{8 x}+3 x \,{\mathrm e}^{8 x}-\frac {3 \,{\mathrm e}^{8 x}}{8}\right )}{2}-\frac {{\mathrm e}^{4} \left (128 \,{\mathrm e}^{8 x} x^{4}-64 \,{\mathrm e}^{8 x} x^{3}+24 x^{2} {\mathrm e}^{8 x}-6 x \,{\mathrm e}^{8 x}+\frac {3 \,{\mathrm e}^{8 x}}{4}\right )}{2}+12675 \,{\mathrm e}^{8} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )+\frac {775 \,{\mathrm e}^{8} \left (8 x^{2} {\mathrm e}^{8 x}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )}{2}+3 \,{\mathrm e}^{8} \left (32 \,{\mathrm e}^{8 x} x^{3}-12 x^{2} {\mathrm e}^{8 x}+3 x \,{\mathrm e}^{8 x}-\frac {3 \,{\mathrm e}^{8 x}}{8}\right )-520 \,{\mathrm e}^{12} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )-8 \,{\mathrm e}^{12} \left (8 x^{2} {\mathrm e}^{8 x}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )+8 \,{\mathrm e}^{16} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )\)	\(443\)
default	\(1050625 x \,{\mathrm e}^{8 x}+260350 x^{2} {\mathrm e}^{8 x}+24329 \,{\mathrm e}^{8 x} x^{3}+1016 \,{\mathrm e}^{8 x} x^{4}+16 \,{\mathrm e}^{8 x} x^{5}-32800 \,{\mathrm e}^{8 x} {\mathrm e}^{4}+3073 \,{\mathrm e}^{8 x} {\mathrm e}^{8}-128 \,{\mathrm e}^{8 x} {\mathrm e}^{12}+2 \,{\mathrm e}^{8 x} {\mathrm e}^{16}-137314 \,{\mathrm e}^{4} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )-\frac {25029 \,{\mathrm e}^{4} \left (8 x^{2} {\mathrm e}^{8 x}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )}{4}-\frac {193 \,{\mathrm e}^{4} \left (32 \,{\mathrm e}^{8 x} x^{3}-12 x^{2} {\mathrm e}^{8 x}+3 x \,{\mathrm e}^{8 x}-\frac {3 \,{\mathrm e}^{8 x}}{8}\right )}{2}-\frac {{\mathrm e}^{4} \left (128 \,{\mathrm e}^{8 x} x^{4}-64 \,{\mathrm e}^{8 x} x^{3}+24 x^{2} {\mathrm e}^{8 x}-6 x \,{\mathrm e}^{8 x}+\frac {3 \,{\mathrm e}^{8 x}}{4}\right )}{2}+12675 \,{\mathrm e}^{8} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )+\frac {775 \,{\mathrm e}^{8} \left (8 x^{2} {\mathrm e}^{8 x}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )}{2}+3 \,{\mathrm e}^{8} \left (32 \,{\mathrm e}^{8 x} x^{3}-12 x^{2} {\mathrm e}^{8 x}+3 x \,{\mathrm e}^{8 x}-\frac {3 \,{\mathrm e}^{8 x}}{8}\right )-520 \,{\mathrm e}^{12} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )-8 \,{\mathrm e}^{12} \left (8 x^{2} {\mathrm e}^{8 x}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )+8 \,{\mathrm e}^{16} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )\)	\(443\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24800*x^2+202800*x+24584)*exp(4)^2+(-512*x^4
-24704*x^3-400464*x^2-2197024*x-262400)*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625)*exp(
4*x)^2,x,method=_RETURNVERBOSE)

[Out]

x*(4*exp(4)^2-8*x*exp(4)+4*x^2-128*exp(4)+127*x+1025)^2*exp(4*x)^2

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maxima [B]  time = 0.37, size = 368, normalized size = 14.15 \begin {gather*} \frac {1}{256} \, {\left (4096 \, x^{5} - 2560 \, x^{4} + 1280 \, x^{3} - 480 \, x^{2} + 120 \, x - 15\right )} e^{\left (8 \, x\right )} - \frac {1}{8} \, {\left (512 \, x^{4} e^{4} - 256 \, x^{3} e^{4} + 96 \, x^{2} e^{4} - 24 \, x e^{4} + 3 \, e^{4}\right )} e^{\left (8 \, x\right )} + \frac {513}{256} \, {\left (512 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 24 \, x + 3\right )} e^{\left (8 \, x\right )} + \frac {3}{8} \, {\left (256 \, x^{3} e^{8} - 96 \, x^{2} e^{8} + 24 \, x e^{8} - 3 \, e^{8}\right )} e^{\left (8 \, x\right )} - \frac {193}{16} \, {\left (256 \, x^{3} e^{4} - 96 \, x^{2} e^{4} + 24 \, x e^{4} - 3 \, e^{4}\right )} e^{\left (8 \, x\right )} + \frac {24837}{256} \, {\left (256 \, x^{3} - 96 \, x^{2} + 24 \, x - 3\right )} e^{\left (8 \, x\right )} - 2 \, {\left (32 \, x^{2} e^{12} - 8 \, x e^{12} + e^{12}\right )} e^{\left (8 \, x\right )} + \frac {775}{8} \, {\left (32 \, x^{2} e^{8} - 8 \, x e^{8} + e^{8}\right )} e^{\left (8 \, x\right )} - \frac {25029}{16} \, {\left (32 \, x^{2} e^{4} - 8 \, x e^{4} + e^{4}\right )} e^{\left (8 \, x\right )} + \frac {2155787}{256} \, {\left (32 \, x^{2} - 8 \, x + 1\right )} e^{\left (8 \, x\right )} + 2 \, {\left (8 \, x e^{16} - e^{16}\right )} e^{\left (8 \, x\right )} - 130 \, {\left (8 \, x e^{12} - e^{12}\right )} e^{\left (8 \, x\right )} + \frac {12675}{4} \, {\left (8 \, x e^{8} - e^{8}\right )} e^{\left (8 \, x\right )} - \frac {68657}{2} \, {\left (8 \, x e^{4} - e^{4}\right )} e^{\left (8 \, x\right )} + \frac {2231425}{16} \, {\left (8 \, x - 1\right )} e^{\left (8 \, x\right )} + \frac {1050625}{8} \, e^{\left (8 \, x\right )} + 2 \, e^{\left (8 \, x + 16\right )} - 128 \, e^{\left (8 \, x + 12\right )} + 3073 \, e^{\left (8 \, x + 8\right )} - 32800 \, e^{\left (8 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24800*x^2+202800*x+24584)*exp(4)^2+(-5
12*x^4-24704*x^3-400464*x^2-2197024*x-262400)*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625
)*exp(4*x)^2,x, algorithm="maxima")

[Out]

1/256*(4096*x^5 - 2560*x^4 + 1280*x^3 - 480*x^2 + 120*x - 15)*e^(8*x) - 1/8*(512*x^4*e^4 - 256*x^3*e^4 + 96*x^
2*e^4 - 24*x*e^4 + 3*e^4)*e^(8*x) + 513/256*(512*x^4 - 256*x^3 + 96*x^2 - 24*x + 3)*e^(8*x) + 3/8*(256*x^3*e^8
- 96*x^2*e^8 + 24*x*e^8 - 3*e^8)*e^(8*x) - 193/16*(256*x^3*e^4 - 96*x^2*e^4 + 24*x*e^4 - 3*e^4)*e^(8*x) + 248
37/256*(256*x^3 - 96*x^2 + 24*x - 3)*e^(8*x) - 2*(32*x^2*e^12 - 8*x*e^12 + e^12)*e^(8*x) + 775/8*(32*x^2*e^8 -
8*x*e^8 + e^8)*e^(8*x) - 25029/16*(32*x^2*e^4 - 8*x*e^4 + e^4)*e^(8*x) + 2155787/256*(32*x^2 - 8*x + 1)*e^(8*
x) + 2*(8*x*e^16 - e^16)*e^(8*x) - 130*(8*x*e^12 - e^12)*e^(8*x) + 12675/4*(8*x*e^8 - e^8)*e^(8*x) - 68657/2*(
8*x*e^4 - e^4)*e^(8*x) + 2231425/16*(8*x - 1)*e^(8*x) + 1050625/8*e^(8*x) + 2*e^(8*x + 16) - 128*e^(8*x + 12)
+ 3073*e^(8*x + 8) - 32800*e^(8*x + 4)

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mupad [B]  time = 4.36, size = 31, normalized size = 1.19 \begin {gather*} x\,{\mathrm {e}}^{8\,x}\,{\left (127\,x-128\,{\mathrm {e}}^4+4\,{\mathrm {e}}^8-8\,x\,{\mathrm {e}}^4+4\,x^2+1025\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(8*x)*(8925700*x - exp(12)*(8320*x + 512*x^2 + 1024) + exp(8)*(202800*x + 24800*x^2 + 768*x^3 + 24584)
- exp(4)*(2197024*x + 400464*x^2 + 24704*x^3 + 512*x^4 + 262400) + 2155787*x^2 + 198696*x^3 + 8208*x^4 + 128*x
^5 + exp(16)*(128*x + 16) + 1050625),x)

[Out]

x*exp(8*x)*(127*x - 128*exp(4) + 4*exp(8) - 8*x*exp(4) + 4*x^2 + 1025)^2

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sympy [B]  time = 0.22, size = 105, normalized size = 4.04 \begin {gather*} \left (16 x^{5} - 64 x^{4} e^{4} + 1016 x^{4} - 3056 x^{3} e^{4} + 24329 x^{3} + 96 x^{3} e^{8} - 64 x^{2} e^{12} - 48912 x^{2} e^{4} + 260350 x^{2} + 3064 x^{2} e^{8} - 1024 x e^{12} - 262400 x e^{4} + 1050625 x + 24584 x e^{8} + 16 x e^{16}\right ) e^{8 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x+16)*exp(4)**4+(-512*x**2-8320*x-1024)*exp(4)**3+(768*x**3+24800*x**2+202800*x+24584)*exp(4)*
*2+(-512*x**4-24704*x**3-400464*x**2-2197024*x-262400)*exp(4)+128*x**5+8208*x**4+198696*x**3+2155787*x**2+8925
700*x+1050625)*exp(4*x)**2,x)

[Out]

(16*x**5 - 64*x**4*exp(4) + 1016*x**4 - 3056*x**3*exp(4) + 24329*x**3 + 96*x**3*exp(8) - 64*x**2*exp(12) - 489
12*x**2*exp(4) + 260350*x**2 + 3064*x**2*exp(8) - 1024*x*exp(12) - 262400*x*exp(4) + 1050625*x + 24584*x*exp(8
) + 16*x*exp(16))*exp(8*x)

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