3.17.88 $\int \frac{x^6(e^{4x}(8x+4x^2)+\frac{16e^{6+3x}(24x+40x^2+12x^3)}{x^2}+\frac{256e^{12+2x}(24x+72x^2+60x^3+12x^4)}{x^4}+\frac{65536e^{24}(1+4x^2+12x^3+12x^4+4x^5)}{x^8}+\frac{4096e^{18+x}(8x+40x^2+60x^3+32x^4+4x^5)}{x^6})}{65536e^{24}}dx$

Optimal. Leaf size=24 $$3-\frac{1}{x}+{(1+x+\frac{1}{16}{e}^{-6+x}{x}^2)}^4$$

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Rubi [B]  time = 1.34, antiderivative size = 154, normalized size of antiderivative = 6.42, number of steps used = 93, number of rules used = 5, integrand size = 147, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.034, Rules used = {12, 14, 2196, 2176, 2194} $$\begin{array}{c}\frac{e^{4x-24}{x}^8}{65536}+\frac{e^{3x-18}{x}^7}{1024}+\frac{3}{128}{e}^{2x-12}{x}^6+\frac{e^{3x-18}{x}^6}{1024}+\frac{1}{4}{e}^{x-6}{x}^5+\frac{3}{64}{e}^{2x-12}{x}^5+\frac{3}{4}{e}^{x-6}{x}^4+\frac{3}{128}{e}^{2x-12}{x}^4+x^4+\frac{3}{4}{e}^{x-6}{x}^3+4x^3+\frac{1}{4}{e}^{x-6}{x}^2+6x^2+4x-\frac{1}{x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(x^6*(E^(4*x)*(8*x + 4*x^2) + (16*E^(6 + 3*x)*(24*x + 40*x^2 + 12*x^3))/x^2 + (256*E^(12 + 2*x)*(24*x + 72
*x^2 + 60*x^3 + 12*x^4))/x^4 + (65536*E^24*(1 + 4*x^2 + 12*x^3 + 12*x^4 + 4*x^5))/x^8 + (4096*E^(18 + x)*(8*x
+ 40*x^2 + 60*x^3 + 32*x^4 + 4*x^5))/x^6))/(65536*E^24),x]

[Out]

-x^(-1) + 4*x + 6*x^2 + (E^(-6 + x)*x^2)/4 + 4*x^3 + (3*E^(-6 + x)*x^3)/4 + x^4 + (3*E^(-6 + x)*x^4)/4 + (3*E^
(-12 + 2*x)*x^4)/128 + (E^(-6 + x)*x^5)/4 + (3*E^(-12 + 2*x)*x^5)/64 + (3*E^(-12 + 2*x)*x^6)/128 + (E^(-18 + 3
*x)*x^6)/1024 + (E^(-18 + 3*x)*x^7)/1024 + (E^(-24 + 4*x)*x^8)/65536

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 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{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int x^6(e^{4x}(8x+4x^2)+\frac{16e^{6+3x}(24x+40x^2+12x^3)}{x^2}+\frac{256e^{12+2x}(24x+72x^2+60x^3+12x^4)}{x^4}+\frac{65536e^{24}(1+4x^2+12x^3+12x^4+4x^5)}{x^8}+\frac{4096e^{18+x}(8x+40x^2+60x^3+32x^4+4x^5)}{x^6})dx}{65536e^{24}}\\ & =\frac{\int (4e^{4x}{x}^7(2+x)+3072e^{12+2x}{x}^3(1+x)(2+4x+x^2)+16384e^{18+x}x(1+x{)}^2(2+6x+x^2)+64e^{6+3x}{x}^5(6+10x+3x^2)+\frac{65536e^{24}(1+4x^2+12x^3+12x^4+4x^5)}{x^2})dx}{65536e^{24}}\\ & =\frac{\int e^{4x}{x}^7(2+x)dx}{16384e^{24}}+\frac{\int e^{6+3x}{x}^5(6+10x+3x^2)dx}{1024e^{24}}+\frac{3\int e^{12+2x}{x}^3(1+x)(2+4x+x^2)dx}{64e^{24}}+\frac{\int e^{18+x}x(1+x{)}^2(2+6x+x^2)dx}{4e^{24}}+\int \frac{1+4x^2+12x^3+12x^4+4x^5}{x^2}dx\\ & =\frac{\int (2e^{4x}{x}^7+e^{4x}{x}^8)dx}{16384e^{24}}+\frac{\int (6e^{6+3x}{x}^5+10e^{6+3x}{x}^6+3e^{6+3x}{x}^7)dx}{1024e^{24}}+\frac{3\int (2e^{12+2x}{x}^3+6e^{12+2x}{x}^4+5e^{12+2x}{x}^5+e^{12+2x}{x}^6)dx}{64e^{24}}+\frac{\int (2e^{18+x}x+10e^{18+x}{x}^2+15e^{18+x}{x}^3+8e^{18+x}{x}^4+e^{18+x}{x}^5)dx}{4e^{24}}+\int (4+\frac{1}{x^2}+12x+12x^2+4x^3)dx\\ & =-\frac{1}{x}+4x+6x^2+4x^3+x^4+\frac{\int e^{4x}{x}^{8}dx}{16384e^{24}}+\frac{\int e^{4x}{x}^{7}dx}{8192e^{24}}+\frac{3\int e^{6+3x}{x}^{7}dx}{1024e^{24}}+\frac{3\int e^{6+3x}{x}^{5}dx}{512e^{24}}+\frac{5\int e^{6+3x}{x}^{6}dx}{512e^{24}}+\frac{3\int e^{12+2x}{x}^{6}dx}{64e^{24}}+\frac{3\int e^{12+2x}{x}^{3}dx}{32e^{24}}+\frac{15\int e^{12+2x}{x}^{5}dx}{64e^{24}}+\frac{\int e^{18+x}{x}^{5}dx}{4e^{24}}+\frac{9\int e^{12+2x}{x}^{4}dx}{32e^{24}}+\frac{\int e^{18+x}xdx}{2e^{24}}+\frac{2\int e^{18+x}{x}^{4}dx}{e^{24}}+\frac{5\int e^{18+x}{x}^{2}dx}{2e^{24}}+\frac{15\int e^{18+x}{x}^{3}dx}{4e^{24}}\\ & =-\frac{1}{x}+4x+\frac{1}{2}{e}^{-6+x}x+6x^2+\frac{5}{2}{e}^{-6+x}{x}^2+4x^3+\frac{15}{4}{e}^{-6+x}{x}^3+\frac{3}{64}{e}^{-12+2x}{x}^3+x^4+2e^{-6+x}{x}^4+\frac{9}{64}{e}^{-12+2x}{x}^4+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{15}{128}{e}^{-12+2x}{x}^5+\frac{1}{512}{e}^{-18+3x}{x}^5+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{5e^{-18+3x}{x}^6}{1536}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^7}{32768}+\frac{e^{-24+4x}{x}^8}{65536}-\frac{\int e^{4x}{x}^{7}dx}{8192e^{24}}-\frac{7\int e^{4x}{x}^{6}dx}{32768e^{24}}-\frac{7\int e^{6+3x}{x}^{6}dx}{1024e^{24}}-\frac{5\int e^{6+3x}{x}^{4}dx}{512e^{24}}-\frac{5\int e^{6+3x}{x}^{5}dx}{256e^{24}}-\frac{9\int e^{12+2x}{x}^{2}dx}{64e^{24}}-\frac{9\int e^{12+2x}{x}^{5}dx}{64e^{24}}-\frac{\int e^{18+x}dx}{2e^{24}}-\frac{9\int e^{12+2x}{x}^{3}dx}{16e^{24}}-\frac{75\int e^{12+2x}{x}^{4}dx}{128e^{24}}-\frac{5\int e^{18+x}{x}^{4}dx}{4e^{24}}-\frac{5\int e^{18+x}xdx}{e^{24}}-\frac{8\int e^{18+x}{x}^{3}dx}{e^{24}}-\frac{45\int e^{18+x}{x}^{2}dx}{4e^{24}}\\ & =-\frac{1}{2}{e}^{-6+x}-\frac{1}{x}+4x-\frac{9}{2}{e}^{-6+x}x+6x^2-\frac{35}{4}{e}^{-6+x}{x}^2-\frac{9}{128}{e}^{-12+2x}{x}^2+4x^3-\frac{17}{4}{e}^{-6+x}{x}^3-\frac{15}{64}{e}^{-12+2x}{x}^3+x^4+\frac{3}{4}{e}^{-6+x}{x}^4-\frac{39}{256}{e}^{-12+2x}{x}^4-\frac{5e^{-18+3x}{x}^4}{1536}+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{3}{64}{e}^{-12+2x}{x}^5-\frac{7e^{-18+3x}{x}^5}{1536}+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{e^{-18+3x}{x}^6}{1024}-\frac{7e^{-24+4x}{x}^6}{131072}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^8}{65536}+\frac{7\int e^{4x}{x}^{6}dx}{32768e^{24}}+\frac{21\int e^{4x}{x}^{5}dx}{65536e^{24}}+\frac{5\int e^{6+3x}{x}^{3}dx}{384e^{24}}+\frac{7\int e^{6+3x}{x}^{5}dx}{512e^{24}}+\frac{25\int e^{6+3x}{x}^{4}dx}{768e^{24}}+\frac{9\int e^{12+2x}xdx}{64e^{24}}+\frac{45\int e^{12+2x}{x}^{4}dx}{128e^{24}}+\frac{27\int e^{12+2x}{x}^{2}dx}{32e^{24}}+\frac{75\int e^{12+2x}{x}^{3}dx}{64e^{24}}+\frac{5\int e^{18+x}dx}{e^{24}}+\frac{5\int e^{18+x}{x}^{3}dx}{e^{24}}+\frac{45\int e^{18+x}xdx}{2e^{24}}+\frac{24\int e^{18+x}{x}^{2}dx}{e^{24}}\\ & =\frac{9e^{-6+x}}{2}-\frac{1}{x}+4x+18e^{-6+x}x+\frac{9}{128}{e}^{-12+2x}x+6x^2+\frac{61}{4}{e}^{-6+x}{x}^2+\frac{45}{128}{e}^{-12+2x}{x}^2+4x^3+\frac{3}{4}{e}^{-6+x}{x}^3+\frac{45}{128}{e}^{-12+2x}{x}^3+\frac{5e^{-18+3x}{x}^3}{1152}+x^4+\frac{3}{4}{e}^{-6+x}{x}^4+\frac{3}{128}{e}^{-12+2x}{x}^4+\frac{35e^{-18+3x}{x}^4}{4608}+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{3}{64}{e}^{-12+2x}{x}^5+\frac{21e^{-24+4x}{x}^5}{262144}+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{e^{-18+3x}{x}^6}{1024}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^8}{65536}-\frac{21\int e^{4x}{x}^{5}dx}{65536e^{24}}-\frac{105\int e^{4x}{x}^{4}dx}{262144e^{24}}-\frac{5\int e^{6+3x}{x}^{2}dx}{384e^{24}}-\frac{35\int e^{6+3x}{x}^{4}dx}{1536e^{24}}-\frac{25\int e^{6+3x}{x}^{3}dx}{576e^{24}}-\frac{9\int e^{12+2x}dx}{128e^{24}}-\frac{45\int e^{12+2x}{x}^{3}dx}{64e^{24}}-\frac{27\int e^{12+2x}xdx}{32e^{24}}-\frac{225\int e^{12+2x}{x}^{2}dx}{128e^{24}}-\frac{15\int e^{18+x}{x}^{2}dx}{e^{24}}-\frac{45\int e^{18+x}dx}{2e^{24}}-\frac{48\int e^{18+x}xdx}{e^{24}}\\ & =-18e^{-6+x}-\frac{9}{256}{e}^{-12+2x}-\frac{1}{x}+4x-30e^{-6+x}x-\frac{45}{128}{e}^{-12+2x}x+6x^2+\frac{1}{4}{e}^{-6+x}{x}^2-\frac{135}{256}{e}^{-12+2x}{x}^2-\frac{5e^{-18+3x}{x}^2}{1152}+4x^3+\frac{3}{4}{e}^{-6+x}{x}^3-\frac{35e^{-18+3x}{x}^3}{3456}+x^4+\frac{3}{4}{e}^{-6+x}{x}^4+\frac{3}{128}{e}^{-12+2x}{x}^4-\frac{105e^{-24+4x}{x}^4}{1048576}+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{3}{64}{e}^{-12+2x}{x}^5+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{e^{-18+3x}{x}^6}{1024}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^8}{65536}+\frac{105\int e^{4x}{x}^{3}dx}{262144e^{24}}+\frac{105\int e^{4x}{x}^{4}dx}{262144e^{24}}+\frac{5\int e^{6+3x}xdx}{576e^{24}}+\frac{35\int e^{6+3x}{x}^{3}dx}{1152e^{24}}+\frac{25\int e^{6+3x}{x}^{2}dx}{576e^{24}}+\frac{27\int e^{12+2x}dx}{64e^{24}}+\frac{135\int e^{12+2x}{x}^{2}dx}{128e^{24}}+\frac{225\int e^{12+2x}xdx}{128e^{24}}+\frac{30\int e^{18+x}xdx}{e^{24}}+\frac{48\int e^{18+x}dx}{e^{24}}\\ & =30e^{-6+x}+\frac{45}{256}{e}^{-12+2x}-\frac{1}{x}+4x+\frac{135}{256}{e}^{-12+2x}x+\frac{5e^{-18+3x}x}{1728}+6x^2+\frac{1}{4}{e}^{-6+x}{x}^2+\frac{35e^{-18+3x}{x}^2}{3456}+4x^3+\frac{3}{4}{e}^{-6+x}{x}^3+\frac{105e^{-24+4x}{x}^3}{1048576}+x^4+\frac{3}{4}{e}^{-6+x}{x}^4+\frac{3}{128}{e}^{-12+2x}{x}^4+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{3}{64}{e}^{-12+2x}{x}^5+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{e^{-18+3x}{x}^6}{1024}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^8}{65536}-\frac{315\int e^{4x}{x}^{2}dx}{1048576e^{24}}-\frac{105\int e^{4x}{x}^{3}dx}{262144e^{24}}-\frac{5\int e^{6+3x}dx}{1728e^{24}}-\frac{25\int e^{6+3x}xdx}{864e^{24}}-\frac{35\int e^{6+3x}{x}^{2}dx}{1152e^{24}}-\frac{225\int e^{12+2x}dx}{256e^{24}}-\frac{135\int e^{12+2x}xdx}{128e^{24}}-\frac{30\int e^{18+x}dx}{e^{24}}\\ & =-\frac{135}{512}{e}^{-12+2x}-\frac{5e^{-18+3x}}{5184}-\frac{1}{x}+4x-\frac{35e^{-18+3x}x}{5184}+6x^2+\frac{1}{4}{e}^{-6+x}{x}^2-\frac{315e^{-24+4x}{x}^2}{4194304}+4x^3+\frac{3}{4}{e}^{-6+x}{x}^3+x^4+\frac{3}{4}{e}^{-6+x}{x}^4+\frac{3}{128}{e}^{-12+2x}{x}^4+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{3}{64}{e}^{-12+2x}{x}^5+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{e^{-18+3x}{x}^6}{1024}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^8}{65536}+\frac{315\int e^{4x}xdx}{2097152e^{24}}+\frac{315\int e^{4x}{x}^{2}dx}{1048576e^{24}}+\frac{25\int e^{6+3x}dx}{2592e^{24}}+\frac{35\int e^{6+3x}xdx}{1728e^{24}}+\frac{135\int e^{12+2x}dx}{256e^{24}}\\ & =\frac{35e^{-18+3x}}{15552}-\frac{1}{x}+4x+\frac{315e^{-24+4x}x}{8388608}+6x^2+\frac{1}{4}{e}^{-6+x}{x}^2+4x^3+\frac{3}{4}{e}^{-6+x}{x}^3+x^4+\frac{3}{4}{e}^{-6+x}{x}^4+\frac{3}{128}{e}^{-12+2x}{x}^4+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{3}{64}{e}^{-12+2x}{x}^5+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{e^{-18+3x}{x}^6}{1024}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^8}{65536}-\frac{315\int e^{4x}dx}{8388608e^{24}}-\frac{315\int e^{4x}xdx}{2097152e^{24}}-\frac{35\int e^{6+3x}dx}{5184e^{24}}\\ & =-\frac{315e^{-24+4x}}{33554432}-\frac{1}{x}+4x+6x^2+\frac{1}{4}{e}^{-6+x}{x}^2+4x^3+\frac{3}{4}{e}^{-6+x}{x}^3+x^4+\frac{3}{4}{e}^{-6+x}{x}^4+\frac{3}{128}{e}^{-12+2x}{x}^4+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{3}{64}{e}^{-12+2x}{x}^5+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{e^{-18+3x}{x}^6}{1024}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^8}{65536}+\frac{315\int e^{4x}dx}{8388608e^{24}}\\ & =-\frac{1}{x}+4x+6x^2+\frac{1}{4}{e}^{-6+x}{x}^2+4x^3+\frac{3}{4}{e}^{-6+x}{x}^3+x^4+\frac{3}{4}{e}^{-6+x}{x}^4+\frac{3}{128}{e}^{-12+2x}{x}^4+\frac{1}{4}{e}^{-6+x}{x}^5+\frac{3}{64}{e}^{-12+2x}{x}^5+\frac{3}{128}{e}^{-12+2x}{x}^6+\frac{e^{-18+3x}{x}^6}{1024}+\frac{e^{-18+3x}{x}^7}{1024}+\frac{e^{-24+4x}{x}^8}{65536}\end{array}\end{array}$$

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Mathematica [B]  time = 0.30, size = 89, normalized size = 3.71 $$\begin{array}{c}-\frac{1}{x}+4x+6x^2+4x^3+x^4+\frac{e^{4(-6+x)}{x}^8}{65536}+\frac{e^{3(-6+x)}{x}^6(1+x)}{1024}+\frac{3}{128}{e}^{2(-6+x)}{x}^4(1+x{)}^2+\frac{1}{4}{e}^{-6+x}{x}^2(1+x{)}^3\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(E^(4*x)*(8*x + 4*x^2) + (16*E^(6 + 3*x)*(24*x + 40*x^2 + 12*x^3))/x^2 + (256*E^(12 + 2*x)*(24*
x + 72*x^2 + 60*x^3 + 12*x^4))/x^4 + (65536*E^24*(1 + 4*x^2 + 12*x^3 + 12*x^4 + 4*x^5))/x^8 + (4096*E^(18 + x)
*(8*x + 40*x^2 + 60*x^3 + 32*x^4 + 4*x^5))/x^6))/(65536*E^24),x]

[Out]

-x^(-1) + 4*x + 6*x^2 + 4*x^3 + x^4 + (E^(4*(-6 + x))*x^8)/65536 + (E^(3*(-6 + x))*x^6*(1 + x))/1024 + (3*E^(2
*(-6 + x))*x^4*(1 + x)^2)/128 + (E^(-6 + x)*x^2*(1 + x)^3)/4

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fricas [B]  time = 0.85, size = 132, normalized size = 5.50 $$\begin{array}{c}\frac{(x^{33}{e}^{(4x+24\log \left(\frac{4}{x}\right)+72)}+18446744073709551616(x^5+4x^4+6x^3+4x^2-1)e^{96}+262144(x^{26}+x^{25})e^{(3x+18\log \left(\frac{4}{x}\right)+78)}[phone]\mathrm{}(x^{19}+2x^{18}+x^{17})e^{(2x+12\log \left(\frac{4}{x}\right)+84)}+1125899906842624(x^{12}+3x^{11}+3x^{10}+x^9)e^{(x+6\log \left(\frac{4}{x}\right)+90)})e^{(-96)}}{18446744073709551616x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+12*x^4+12*x^3+4*x^2+1)*exp(log(4/x)+3)^8+(4*x^5+32*x^4+60*x^3+40*x^2+8*x)*exp(x)*exp(log(4/x
)+3)^6+(12*x^4+60*x^3+72*x^2+24*x)*exp(x)^2*exp(log(4/x)+3)^4+(12*x^3+40*x^2+24*x)*exp(x)^3*exp(log(4/x)+3)^2+
(4*x^2+8*x)*exp(x)^4)/x^2/exp(log(4/x)+3)^8,x, algorithm="fricas")

[Out]

1/18446744073709551616*(x^33*e^(4*x + 24*log(4/x) + 72) + 18446744073709551616*(x^5 + 4*x^4 + 6*x^3 + 4*x^2 -
1)*e^96 + 262144*(x^26 + x^25)*e^(3*x + 18*log(4/x) + 78) + 25769803776*(x^19 + 2*x^18 + x^17)*e^(2*x + 12*log
(4/x) + 84) + 1125899906842624*(x^12 + 3*x^11 + 3*x^10 + x^9)*e^(x + 6*log(4/x) + 90))*e^(-96)/x

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giac [B]  time = 0.25, size = 141, normalized size = 5.88 $$\begin{array}{c}\frac{(x^9{e}^{(4x+36)}+64x^8{e}^{(3x+42)}+64x^7{e}^{(3x+42)}+1536x^7{e}^{(2x+48)}+3072x^6{e}^{(2x+48)}+16384x^6{e}^{(x+54)}+65536x^5{e}^{60}+1536x^5{e}^{(2x+48)}+49152x^5{e}^{(x+54)}+262144x^4{e}^{60}+49152x^4{e}^{(x+54)}+393216x^3{e}^{60}+16384x^3{e}^{(x+54)}+262144x^2{e}^{60}-65536e^{60})e^{(-60)}}{65536x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+12*x^4+12*x^3+4*x^2+1)*exp(log(4/x)+3)^8+(4*x^5+32*x^4+60*x^3+40*x^2+8*x)*exp(x)*exp(log(4/x
)+3)^6+(12*x^4+60*x^3+72*x^2+24*x)*exp(x)^2*exp(log(4/x)+3)^4+(12*x^3+40*x^2+24*x)*exp(x)^3*exp(log(4/x)+3)^2+
(4*x^2+8*x)*exp(x)^4)/x^2/exp(log(4/x)+3)^8,x, algorithm="giac")

[Out]

1/65536*(x^9*e^(4*x + 36) + 64*x^8*e^(3*x + 42) + 64*x^7*e^(3*x + 42) + 1536*x^7*e^(2*x + 48) + 3072*x^6*e^(2*
x + 48) + 16384*x^6*e^(x + 54) + 65536*x^5*e^60 + 1536*x^5*e^(2*x + 48) + 49152*x^5*e^(x + 54) + 262144*x^4*e^
60 + 49152*x^4*e^(x + 54) + 393216*x^3*e^60 + 16384*x^3*e^(x + 54) + 262144*x^2*e^60 - 65536*e^60)*e^(-60)/x

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maple [B]  time = 0.13, size = 89, normalized size = 3.71

method	result	size
risch	$x^4+4x^3+6x^2+4x-\frac{1}{x}+\frac{x^2(x^3+3x^2+3x+1){\mathrm{e}}^{x-6}}{4}+\frac{3x^4(x^2+2x+1){\mathrm{e}}^{2x-12}}{128}+\frac{x^6(x+1){\mathrm{e}}^{3x-18}}{1024}+\frac{x^8{\mathrm{e}}^{4x-24}}{65536}$	$89$
default	$\text{Expression too large to display}$	$11630$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^5+12*x^4+12*x^3+4*x^2+1)*exp(ln(4/x)+3)^8+(4*x^5+32*x^4+60*x^3+40*x^2+8*x)*exp(x)*exp(ln(4/x)+3)^6+(
12*x^4+60*x^3+72*x^2+24*x)*exp(x)^2*exp(ln(4/x)+3)^4+(12*x^3+40*x^2+24*x)*exp(x)^3*exp(ln(4/x)+3)^2+(4*x^2+8*x
)*exp(x)^4)/x^2/exp(ln(4/x)+3)^8,x,method=_RETURNVERBOSE)

[Out]

x^4+4*x^3+6*x^2+4*x-1/x+1/4*x^2*(x^3+3*x^2+3*x+1)*exp(x-6)+3/128*x^4*(x^2+2*x+1)*exp(2*x-12)+1/1024*x^6*(x+1)*
exp(3*x-18)+1/65536*x^8*exp(4*x-24)

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maxima [B]  time = 0.38, size = 575, normalized size = 23.96 $$\begin{array}{c}\frac{1}{8153726976}(8153726976x^4{e}^{24}+32614907904x^3{e}^{24}+48922361856x^2{e}^{24}+32614907904x{e}^{24}+243(512x^8-1024x^7+1792x^6-2688x^5+3360x^4-3360x^3+2520x^2-1260x+315)e^{\left(4x\right)}+243(1024x^7-1792x^6+2688x^5-3360x^4+3360x^3-2520x^2+1260x-315)e^{\left(4x\right)}+32768(243x^7{e}^6-567x^6{e}^6+1134x^5{e}^6-1890x^4{e}^6+2520x^3{e}^6-2520x^2{e}^6+1680x{e}^6-560e^6)e^{\left(3x\right)}+327680(81x^6{e}^6-162x^5{e}^6+270x^4{e}^6-360x^3{e}^6+360x^2{e}^6-240x{e}^6+80e^6)e^{\left(3x\right)}+196608(81x^5{e}^6-135x^4{e}^6+180x^3{e}^6-180x^2{e}^6+120x{e}^6-40e^6)e^{\left(3x\right)}[phone]\mathrm{}(4x^6{e}^{12}-12x^5{e}^{12}+30x^4{e}^{12}-60x^3{e}^{12}+90x^2{e}^{12}-90x{e}^{12}+45e^{12})e^{\left(2x\right)}[phone]\mathrm{}(4x^5{e}^{12}-10x^4{e}^{12}+20x^3{e}^{12}-30x^2{e}^{12}+30x{e}^{12}-15e^{12})e^{\left(2x\right)}[phone]\mathrm{}(2x^4{e}^{12}-4x^3{e}^{12}+6x^2{e}^{12}-6x{e}^{12}+3e^{12})e^{\left(2x\right)}[phone]\mathrm{}(4x^3{e}^{12}-6x^2{e}^{12}+6x{e}^{12}-3e^{12})e^{\left(2x\right)}[phone]\mathrm{}(x^5{e}^{18}-5x^4{e}^{18}+20x^3{e}^{18}-60x^2{e}^{18}+120x{e}^{18}-120e^{18})e^x+16307453952(x^4{e}^{18}-4x^3{e}^{18}+12x^2{e}^{18}-24x{e}^{18}+24e^{18})e^x+30576476160(x^3{e}^{18}-3x^2{e}^{18}+6x{e}^{18}-6e^{18})e^x+20384317440(x^2{e}^{18}-2x{e}^{18}+2e^{18})e^x+4076863488(x{e}^{18}-e^{18})e^x-\frac{8153726976e^{24}}{x})e^{(-24)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+12*x^4+12*x^3+4*x^2+1)*exp(log(4/x)+3)^8+(4*x^5+32*x^4+60*x^3+40*x^2+8*x)*exp(x)*exp(log(4/x
)+3)^6+(12*x^4+60*x^3+72*x^2+24*x)*exp(x)^2*exp(log(4/x)+3)^4+(12*x^3+40*x^2+24*x)*exp(x)^3*exp(log(4/x)+3)^2+
(4*x^2+8*x)*exp(x)^4)/x^2/exp(log(4/x)+3)^8,x, algorithm="maxima")

[Out]

1/8153726976*(8153726976*x^4*e^24 + 32614907904*x^3*e^24 + 48922361856*x^2*e^24 + 32614907904*x*e^24 + 243*(51
2*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) + 243*(1024*x^
7 - 1792*x^6 + 2688*x^5 - 3360*x^4 + 3360*x^3 - 2520*x^2 + 1260*x - 315)*e^(4*x) + 32768*(243*x^7*e^6 - 567*x^
6*e^6 + 1134*x^5*e^6 - 1890*x^4*e^6 + 2520*x^3*e^6 - 2520*x^2*e^6 + 1680*x*e^6 - 560*e^6)*e^(3*x) + 327680*(81
*x^6*e^6 - 162*x^5*e^6 + 270*x^4*e^6 - 360*x^3*e^6 + 360*x^2*e^6 - 240*x*e^6 + 80*e^6)*e^(3*x) + 196608*(81*x^
5*e^6 - 135*x^4*e^6 + 180*x^3*e^6 - 180*x^2*e^6 + 120*x*e^6 - 40*e^6)*e^(3*x) + 47775744*(4*x^6*e^12 - 12*x^5*
e^12 + 30*x^4*e^12 - 60*x^3*e^12 + 90*x^2*e^12 - 90*x*e^12 + 45*e^12)*e^(2*x) + 238878720*(4*x^5*e^12 - 10*x^4
*e^12 + 20*x^3*e^12 - 30*x^2*e^12 + 30*x*e^12 - 15*e^12)*e^(2*x) + 573308928*(2*x^4*e^12 - 4*x^3*e^12 + 6*x^2*
e^12 - 6*x*e^12 + 3*e^12)*e^(2*x) + 95551488*(4*x^3*e^12 - 6*x^2*e^12 + 6*x*e^12 - 3*e^12)*e^(2*x) + 203843174
4*(x^5*e^18 - 5*x^4*e^18 + 20*x^3*e^18 - 60*x^2*e^18 + 120*x*e^18 - 120*e^18)*e^x + 16307453952*(x^4*e^18 - 4*
x^3*e^18 + 12*x^2*e^18 - 24*x*e^18 + 24*e^18)*e^x + 30576476160*(x^3*e^18 - 3*x^2*e^18 + 6*x*e^18 - 6*e^18)*e^
x + 20384317440*(x^2*e^18 - 2*x*e^18 + 2*e^18)*e^x + 4076863488*(x*e^18 - e^18)*e^x - 8153726976*e^24/x)*e^(-2
4)

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mupad [B]  time = 1.35, size = 124, normalized size = 5.17 $$\begin{array}{c}4x+\frac{x^2{\mathrm{e}}^{x-6}}{4}+\frac{3x^3{\mathrm{e}}^{x-6}}{4}+\frac{3x^4{\mathrm{e}}^{x-6}}{4}+\frac{x^5{\mathrm{e}}^{x-6}}{4}+\frac{3x^4{\mathrm{e}}^{2x-12}}{128}+\frac{3x^5{\mathrm{e}}^{2x-12}}{64}+\frac{3x^6{\mathrm{e}}^{2x-12}}{128}+\frac{x^6{\mathrm{e}}^{3x-18}}{1024}+\frac{x^7{\mathrm{e}}^{3x-18}}{1024}+\frac{x^8{\mathrm{e}}^{4x-24}}{65536}-\frac{1}{x}+6x^2+4x^3+x^4\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- 8*log(4/x) - 24)*(exp(4*x)*(8*x + 4*x^2) + exp(8*log(4/x) + 24)*(4*x^2 + 12*x^3 + 12*x^4 + 4*x^5 +
1) + exp(3*x)*exp(2*log(4/x) + 6)*(24*x + 40*x^2 + 12*x^3) + exp(6*log(4/x) + 18)*exp(x)*(8*x + 40*x^2 + 60*x^
3 + 32*x^4 + 4*x^5) + exp(2*x)*exp(4*log(4/x) + 12)*(24*x + 72*x^2 + 60*x^3 + 12*x^4)))/x^2,x)

[Out]

4*x + (x^2*exp(x - 6))/4 + (3*x^3*exp(x - 6))/4 + (3*x^4*exp(x - 6))/4 + (x^5*exp(x - 6))/4 + (3*x^4*exp(2*x -
12))/128 + (3*x^5*exp(2*x - 12))/64 + (3*x^6*exp(2*x - 12))/128 + (x^6*exp(3*x - 18))/1024 + (x^7*exp(3*x - 1
8))/1024 + (x^8*exp(4*x - 24))/65536 - 1/x + 6*x^2 + 4*x^3 + x^4

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sympy [B]  time = 0.37, size = 128, normalized size = 5.33 $$\begin{array}{c}{x}^4+4x^3+6x^2+4x+\frac{524288x^8{e}^{36}{e}^{4x}+(33554432x^7{e}^{42}+33554432x^6{e}^{42})e^{3x}+(805306368x^6{e}^{48}+1610612736x^5{e}^{48}+805306368x^4{e}^{48})e^{2x}+(8589934592x^5{e}^{54}+25769803776x^4{e}^{54}+25769803776x^3{e}^{54}+8589934592x^2{e}^{54})e^x}{34359738368e^{60}}-\frac{1}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**5+12*x**4+12*x**3+4*x**2+1)*exp(ln(4/x)+3)**8+(4*x**5+32*x**4+60*x**3+40*x**2+8*x)*exp(x)*exp
(ln(4/x)+3)**6+(12*x**4+60*x**3+72*x**2+24*x)*exp(x)**2*exp(ln(4/x)+3)**4+(12*x**3+40*x**2+24*x)*exp(x)**3*exp
(ln(4/x)+3)**2+(4*x**2+8*x)*exp(x)**4)/x**2/exp(ln(4/x)+3)**8,x)

[Out]

x**4 + 4*x**3 + 6*x**2 + 4*x + (524288*x**8*exp(36)*exp(4*x) + (33554432*x**7*exp(42) + 33554432*x**6*exp(42))
*exp(3*x) + (805306368*x**6*exp(48) + 1610612736*x**5*exp(48) + 805306368*x**4*exp(48))*exp(2*x) + (8589934592
*x**5*exp(54) + 25769803776*x**4*exp(54) + 25769803776*x**3*exp(54) + 8589934592*x**2*exp(54))*exp(x))*exp(-60
)/34359738368 - 1/x

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