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

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

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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 {gather*} \frac {e^{4 x-24} x^8}{65536}+\frac {e^{3 x-18} x^7}{1024}+\frac {3}{128} e^{2 x-12} x^6+\frac {e^{3 x-18} x^6}{1024}+\frac {1}{4} e^{x-6} x^5+\frac {3}{64} e^{2 x-12} x^5+\frac {3}{4} e^{x-6} x^4+\frac {3}{128} e^{2 x-12} x^4+x^4+\frac {3}{4} e^{x-6} x^3+4 x^3+\frac {1}{4} e^{x-6} x^2+6 x^2+4 x-\frac {1}{x} \end {gather*}

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

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Mathematica [B]  time = 0.30, size = 89, normalized size = 3.71 \begin {gather*} -\frac {1}{x}+4 x+6 x^2+4 x^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 {gather*}

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 {gather*} \frac {{\left (x^{33} e^{\left (4 \, x + 24 \, \log \left (\frac {4}{x}\right ) + 72\right )} + 18446744073709551616 \, {\left (x^{5} + 4 \, x^{4} + 6 \, x^{3} + 4 \, x^{2} - 1\right )} e^{96} + 262144 \, {\left (x^{26} + x^{25}\right )} e^{\left (3 \, x + 18 \, \log \left (\frac {4}{x}\right ) + 78\right )} + 25769803776 \, {\left (x^{19} + 2 \, x^{18} + x^{17}\right )} e^{\left (2 \, x + 12 \, \log \left (\frac {4}{x}\right ) + 84\right )} + 1125899906842624 \, {\left (x^{12} + 3 \, x^{11} + 3 \, x^{10} + x^{9}\right )} e^{\left (x + 6 \, \log \left (\frac {4}{x}\right ) + 90\right )}\right )} e^{\left (-96\right )}}{18446744073709551616 \, x} \end {gather*}

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 {gather*} \frac {{\left (x^{9} e^{\left (4 \, x + 36\right )} + 64 \, x^{8} e^{\left (3 \, x + 42\right )} + 64 \, x^{7} e^{\left (3 \, x + 42\right )} + 1536 \, x^{7} e^{\left (2 \, x + 48\right )} + 3072 \, x^{6} e^{\left (2 \, x + 48\right )} + 16384 \, x^{6} e^{\left (x + 54\right )} + 65536 \, x^{5} e^{60} + 1536 \, x^{5} e^{\left (2 \, x + 48\right )} + 49152 \, x^{5} e^{\left (x + 54\right )} + 262144 \, x^{4} e^{60} + 49152 \, x^{4} e^{\left (x + 54\right )} + 393216 \, x^{3} e^{60} + 16384 \, x^{3} e^{\left (x + 54\right )} + 262144 \, x^{2} e^{60} - 65536 \, e^{60}\right )} e^{\left (-60\right )}}{65536 \, x} \end {gather*}

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	\(x^{4}+4 x^{3}+6 x^{2}+4 x -\frac {1}{x}+\frac {x^{2} \left (x^{3}+3 x^{2}+3 x +1\right ) {\mathrm e}^{x -6}}{4}+\frac {3 x^{4} \left (x^{2}+2 x +1\right ) {\mathrm e}^{2 x -12}}{128}+\frac {x^{6} \left (x +1\right ) {\mathrm e}^{3 x -18}}{1024}+\frac {x^{8} {\mathrm e}^{4 x -24}}{65536}\)	\(89\)
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 {gather*} \frac {1}{8153726976} \, {\left (8153726976 \, x^{4} e^{24} + 32614907904 \, x^{3} e^{24} + 48922361856 \, x^{2} e^{24} + 32614907904 \, x e^{24} + 243 \, {\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 )} + 243 \, {\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 )} + 32768 \, {\left (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}\right )} e^{\left (3 \, x\right )} + 327680 \, {\left (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}\right )} e^{\left (3 \, x\right )} + 196608 \, {\left (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}\right )} e^{\left (3 \, x\right )} + 47775744 \, {\left (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}\right )} e^{\left (2 \, x\right )} + 238878720 \, {\left (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}\right )} e^{\left (2 \, x\right )} + 573308928 \, {\left (2 \, x^{4} e^{12} - 4 \, x^{3} e^{12} + 6 \, x^{2} e^{12} - 6 \, x e^{12} + 3 \, e^{12}\right )} e^{\left (2 \, x\right )} + 95551488 \, {\left (4 \, x^{3} e^{12} - 6 \, x^{2} e^{12} + 6 \, x e^{12} - 3 \, e^{12}\right )} e^{\left (2 \, x\right )} + 2038431744 \, {\left (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}\right )} e^{x} + 16307453952 \, {\left (x^{4} e^{18} - 4 \, x^{3} e^{18} + 12 \, x^{2} e^{18} - 24 \, x e^{18} + 24 \, e^{18}\right )} e^{x} + 30576476160 \, {\left (x^{3} e^{18} - 3 \, x^{2} e^{18} + 6 \, x e^{18} - 6 \, e^{18}\right )} e^{x} + 20384317440 \, {\left (x^{2} e^{18} - 2 \, x e^{18} + 2 \, e^{18}\right )} e^{x} + 4076863488 \, {\left (x e^{18} - e^{18}\right )} e^{x} - \frac {8153726976 \, e^{24}}{x}\right )} e^{\left (-24\right )} \end {gather*}

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 {gather*} 4\,x+\frac {x^2\,{\mathrm {e}}^{x-6}}{4}+\frac {3\,x^3\,{\mathrm {e}}^{x-6}}{4}+\frac {3\,x^4\,{\mathrm {e}}^{x-6}}{4}+\frac {x^5\,{\mathrm {e}}^{x-6}}{4}+\frac {3\,x^4\,{\mathrm {e}}^{2\,x-12}}{128}+\frac {3\,x^5\,{\mathrm {e}}^{2\,x-12}}{64}+\frac {3\,x^6\,{\mathrm {e}}^{2\,x-12}}{128}+\frac {x^6\,{\mathrm {e}}^{3\,x-18}}{1024}+\frac {x^7\,{\mathrm {e}}^{3\,x-18}}{1024}+\frac {x^8\,{\mathrm {e}}^{4\,x-24}}{65536}-\frac {1}{x}+6\,x^2+4\,x^3+x^4 \end {gather*}

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 {gather*} x^{4} + 4 x^{3} + 6 x^{2} + 4 x + \frac {524288 x^{8} e^{36} e^{4 x} + \left (33554432 x^{7} e^{42} + 33554432 x^{6} e^{42}\right ) e^{3 x} + \left (805306368 x^{6} e^{48} + 1610612736 x^{5} e^{48} + 805306368 x^{4} e^{48}\right ) e^{2 x} + \left (8589934592 x^{5} e^{54} + 25769803776 x^{4} e^{54} + 25769803776 x^{3} e^{54} + 8589934592 x^{2} e^{54}\right ) e^{x}}{34359738368 e^{60}} - \frac {1}{x} \end {gather*}

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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