3.68.97 \(\int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} (4+e^5+2 x)^5 (14+e^5+2 x+(4+e^5+2 x) \log (3))}{4+e^5+2 x} \, dx\)

Optimal. Leaf size=24 \[ x+e^{x+x \log (3)} \left (\frac {1}{2} \left (4+e^5\right )+x\right )^5 \]

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Rubi [A]  time = 9.86, antiderivative size = 21, normalized size of antiderivative = 0.88, number of steps used = 16, number of rules used = 5, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6741, 6742, 2196, 2176, 2194} \begin {gather*} \frac {1}{32} (3 e)^x \left (2 x+e^5+4\right )^5+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + E^5 + 2*x + (E^(x + x*Log[3])*(4 + E^5 + 2*x)^5*(14 + E^5 + 2*x + (4 + E^5 + 2*x)*Log[3]))/32)/(4 + E
^5 + 2*x),x]

[Out]

x + ((3*E)^x*(4 + E^5 + 2*x)^5)/32

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (1+\frac {e^5}{4}\right )+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx\\ &=\int \left (1+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^4 \left (14+e^5 (1+\log (3))+x (2+\log (9))+\log (81)\right )\right ) \, dx\\ &=x+\frac {1}{32} \int (3 e)^x \left (4+e^5+2 x\right )^4 \left (14+e^5 (1+\log (3))+x (2+\log (9))+\log (81)\right ) \, dx\\ &=x+\frac {1}{32} \int \left (10 (3 e)^x \left (4+e^5+2 x\right )^4+(3 e)^x \left (4+e^5+2 x\right )^5 (1+\log (3))\right ) \, dx\\ &=x+\frac {5}{16} \int (3 e)^x \left (4+e^5+2 x\right )^4 \, dx+\frac {1}{32} (1+\log (3)) \int (3 e)^x \left (4+e^5+2 x\right )^5 \, dx\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {5 (3 e)^x \left (4+e^5+2 x\right )^4}{16 (1+\log (3))}-\frac {5}{16} \int (3 e)^x \left (4+e^5+2 x\right )^4 \, dx-\frac {5 \int (3 e)^x \left (4+e^5+2 x\right )^3 \, dx}{2 (1+\log (3))}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5-\frac {5 (3 e)^x \left (4+e^5+2 x\right )^3}{2 (1+\log (3))^2}+\frac {15 \int (3 e)^x \left (4+e^5+2 x\right )^2 \, dx}{(1+\log (3))^2}+\frac {5 \int (3 e)^x \left (4+e^5+2 x\right )^3 \, dx}{2 (1+\log (3))}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {5\ 3^{1+x} e^x \left (4+e^5+2 x\right )^2}{(1+\log (3))^3}-\frac {60 \int (3 e)^x \left (4+e^5+2 x\right ) \, dx}{(1+\log (3))^3}-\frac {15 \int (3 e)^x \left (4+e^5+2 x\right )^2 \, dx}{(1+\log (3))^2}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5-\frac {20\ 3^{1+x} e^x \left (4+e^5+2 x\right )}{(1+\log (3))^4}+\frac {120 \int (3 e)^x \, dx}{(1+\log (3))^4}+\frac {60 \int (3 e)^x \left (4+e^5+2 x\right ) \, dx}{(1+\log (3))^3}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {40\ 3^{1+x} e^x}{(1+\log (3))^5}-\frac {120 \int (3 e)^x \, dx}{(1+\log (3))^4}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 4.10, size = 909, normalized size = 37.88 \begin {gather*} \frac {3^x e^{25+x} (1+\log (3))^6+32\ 3^{1+x} e^x \left (5 x^4 (1+\log (3))^5+4 x^3 (1+\log (3))^4 (-5+(1+\log (3)) \log (9))-4 \left (40+20 \log ^4(3)-2 \log ^2(3) (-80+\log (9))+\log (3) (130-3 \log (81))-2 \log ^3(3) (-40+\log (81))+\log (9) (1+\log (81))\right )+4 x (1+\log (3)) \left (40+20 \log ^4(3)-2 \log ^2(3) (-80+\log (9))+\log (3) (130-3 \log (81))-2 \log ^3(3) (-40+\log (81))+\log (9) (1+\log (81))\right )+2 x^2 (1+\log (3))^2 \left (-20-\log (9)+\log ^2(3) (-40+\log (81))+2 \log ^3(3) \log (81)+\log (27) \log (81)-2 \log (3) (25+\log (81))\right )\right )+8\ 3^x e^{15+x} (1+\log (3))^3 \left (20+50 \log (3)+12 \log ^3(3)-\log ^2(9)+x^2 \left (5+13 \log (3)+3 \log ^3(3)+\log (9)+\log ^2(9)+\log ^2(3) (11+\log (9))\right )+\log (27) \log (81)+2 \log ^2(3) (26+\log (81))+x \left (20+56 \log (3)+12 \log ^3(3)+\log ^2(9)+\log (81)+\log (9) \log (81)+2 \log ^2(3) (24+\log (81))\right )+\log (59049)\right )+3^x e^{20+x} (1+\log (3))^4 \left (20+16 \log ^2(3)+\log (9)+\log (3) (38+\log (81))+x (1+\log (3)) (10+\log (59049))\right )+16 (1+\log (3))^3 \left ((3 e)^x x^5 (1+\log (3))^2 (2+\log (9))+16 (3 e)^x x^2 (1+\log (3))^2 (25+\log (81))+16 (3 e)^x \left (64-\log (3) (-28+\log (9))+\log (9)+\log ^2(3) (14+\log (81))\right )+4 (3 e)^x x^3 \left (50-8 \log (3) (-10+\log (9))-\log (9)+\log (27) \log (81)+2 \log ^2(3) (20+\log (81))\right )+2 x (1+\log (3)) \left (1+\log ^2(3)+\log (9)-8\ 3^{1+x} e^x \log (9)+8 (3 e)^x (-50+\log (3) \log (9)+\log (9) \log (81)+\log (6561))\right )+(3 e)^x x^4 (1+\log (3)) (-10+\log (3) (8 \log (9)+\log (81))+\log (59049))\right )+16\ 3^x e^{5+x} (1+\log (3)) \left (x^4 (1+\log (3))^4 (5+\log (243))+2 x^3 (1+\log (3))^3 \left (20+4 \log ^2(3)+\log (3) (34+6 \log (9)+\log (81))+\log (729)\right )+4 x (1+\log (3)) \left (40+8 \log ^4(3)+8 \log ^3(3) (14+\log (81))+12 \log ^2(3) (16+\log (81))+\log (3) (152+6 \log (9)+9 \log (81))+\log (6561)\right )+6 x^2 (1+\log (3))^2 \left (20+50 \log (3)+4 \log ^3(3)+\log ^2(9)+\log (27) \log (81)+2 \log ^2(3) (22+\log (6561))+\log (59049)\right )+4 \left (20+4 \log ^5(3)+4 \log ^4(3) (17+\log (81))+4 \log ^2(3) (46+3 \log (81))+\log ^3(3) (152-4 \log (9)+10 \log (81))+\log (4782969)+\log (3) (86+\log (43046721))\right )\right )+8\ 3^x e^{10+x} (1+\log (3))^2 \left (2 \left (40+16 \log ^4(3)+2 \log ^3(3) (62+3 \log (81))+\log (3) (154+9 \log (81))+\log (729)+6 \log ^2(3) (34+\log (729))\right )+x^3 (1+\log (3))^3 (10+\log (59049))+6 x (1+\log (3)) \left (20+50 \log (3)+8 \log ^3(3)+\log (27) \log (81)+3 \log ^2(3) (16+\log (81))+\log (59049)\right )+3 x^2 (1+\log (3))^2 (20+\log (3) \log (81)+(2+\log (9)) \log (6561)+\log (282429536481))\right )}{32 (1+\log (3))^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + E^5 + 2*x + (E^(x + x*Log[3])*(4 + E^5 + 2*x)^5*(14 + E^5 + 2*x + (4 + E^5 + 2*x)*Log[3]))/32)/
(4 + E^5 + 2*x),x]

[Out]

(3^x*E^(25 + x)*(1 + Log[3])^6 + 32*3^(1 + x)*E^x*(5*x^4*(1 + Log[3])^5 + 4*x^3*(1 + Log[3])^4*(-5 + (1 + Log[
3])*Log[9]) - 4*(40 + 20*Log[3]^4 - 2*Log[3]^2*(-80 + Log[9]) + Log[3]*(130 - 3*Log[81]) - 2*Log[3]^3*(-40 + L
og[81]) + Log[9]*(1 + Log[81])) + 4*x*(1 + Log[3])*(40 + 20*Log[3]^4 - 2*Log[3]^2*(-80 + Log[9]) + Log[3]*(130
- 3*Log[81]) - 2*Log[3]^3*(-40 + Log[81]) + Log[9]*(1 + Log[81])) + 2*x^2*(1 + Log[3])^2*(-20 - Log[9] + Log[
3]^2*(-40 + Log[81]) + 2*Log[3]^3*Log[81] + Log[27]*Log[81] - 2*Log[3]*(25 + Log[81]))) + 8*3^x*E^(15 + x)*(1
+ Log[3])^3*(20 + 50*Log[3] + 12*Log[3]^3 - Log[9]^2 + x^2*(5 + 13*Log[3] + 3*Log[3]^3 + Log[9] + Log[9]^2 + L
og[3]^2*(11 + Log[9])) + Log[27]*Log[81] + 2*Log[3]^2*(26 + Log[81]) + x*(20 + 56*Log[3] + 12*Log[3]^3 + Log[9
]^2 + Log[81] + Log[9]*Log[81] + 2*Log[3]^2*(24 + Log[81])) + Log[59049]) + 3^x*E^(20 + x)*(1 + Log[3])^4*(20
+ 16*Log[3]^2 + Log[9] + Log[3]*(38 + Log[81]) + x*(1 + Log[3])*(10 + Log[59049])) + 16*(1 + Log[3])^3*((3*E)^
x*x^5*(1 + Log[3])^2*(2 + Log[9]) + 16*(3*E)^x*x^2*(1 + Log[3])^2*(25 + Log[81]) + 16*(3*E)^x*(64 - Log[3]*(-2
8 + Log[9]) + Log[9] + Log[3]^2*(14 + Log[81])) + 4*(3*E)^x*x^3*(50 - 8*Log[3]*(-10 + Log[9]) - Log[9] + Log[2
7]*Log[81] + 2*Log[3]^2*(20 + Log[81])) + 2*x*(1 + Log[3])*(1 + Log[3]^2 + Log[9] - 8*3^(1 + x)*E^x*Log[9] + 8
*(3*E)^x*(-50 + Log[3]*Log[9] + Log[9]*Log[81] + Log[6561])) + (3*E)^x*x^4*(1 + Log[3])*(-10 + Log[3]*(8*Log[9
] + Log[81]) + Log[59049])) + 16*3^x*E^(5 + x)*(1 + Log[3])*(x^4*(1 + Log[3])^4*(5 + Log[243]) + 2*x^3*(1 + Lo
g[3])^3*(20 + 4*Log[3]^2 + Log[3]*(34 + 6*Log[9] + Log[81]) + Log[729]) + 4*x*(1 + Log[3])*(40 + 8*Log[3]^4 +
8*Log[3]^3*(14 + Log[81]) + 12*Log[3]^2*(16 + Log[81]) + Log[3]*(152 + 6*Log[9] + 9*Log[81]) + Log[6561]) + 6*
x^2*(1 + Log[3])^2*(20 + 50*Log[3] + 4*Log[3]^3 + Log[9]^2 + Log[27]*Log[81] + 2*Log[3]^2*(22 + Log[6561]) + L
og[59049]) + 4*(20 + 4*Log[3]^5 + 4*Log[3]^4*(17 + Log[81]) + 4*Log[3]^2*(46 + 3*Log[81]) + Log[3]^3*(152 - 4*
Log[9] + 10*Log[81]) + Log[4782969] + Log[3]*(86 + Log[43046721]))) + 8*3^x*E^(10 + x)*(1 + Log[3])^2*(2*(40 +
16*Log[3]^4 + 2*Log[3]^3*(62 + 3*Log[81]) + Log[3]*(154 + 9*Log[81]) + Log[729] + 6*Log[3]^2*(34 + Log[729]))
+ x^3*(1 + Log[3])^3*(10 + Log[59049]) + 6*x*(1 + Log[3])*(20 + 50*Log[3] + 8*Log[3]^3 + Log[27]*Log[81] + 3*
Log[3]^2*(16 + Log[81]) + Log[59049]) + 3*x^2*(1 + Log[3])^2*(20 + Log[3]*Log[81] + (2 + Log[9])*Log[6561] + L
og[282429536481])))/(32*(1 + Log[3])^6)

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fricas [B]  time = 0.72, size = 96, normalized size = 4.00 \begin {gather*} \frac {1}{32} \, {\left (32 \, x^{5} + 320 \, x^{4} + 1280 \, x^{3} + 2560 \, x^{2} + 10 \, {\left (x + 2\right )} e^{20} + 40 \, {\left (x^{2} + 4 \, x + 4\right )} e^{15} + 80 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} e^{10} + 80 \, {\left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 32 \, x + 16\right )} e^{5} + 2560 \, x + e^{25} + 1024\right )} e^{\left (x \log \relax (3) + x\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(5)+2*x+4)*log(3)+exp(5)+2*x+14)*exp(x*log(3)+x)*(1/2*exp(5)+x+2)^5+exp(5)+2*x+4)/(exp(5)+2*x+
4),x, algorithm="fricas")

[Out]

1/32*(32*x^5 + 320*x^4 + 1280*x^3 + 2560*x^2 + 10*(x + 2)*e^20 + 40*(x^2 + 4*x + 4)*e^15 + 80*(x^3 + 6*x^2 + 1
2*x + 8)*e^10 + 80*(x^4 + 8*x^3 + 24*x^2 + 32*x + 16)*e^5 + 2560*x + e^25 + 1024)*e^(x*log(3) + x) + x

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giac [B]  time = 0.32, size = 240, normalized size = 10.00 \begin {gather*} x^{5} e^{\left (x \log \relax (3) + x\right )} + \frac {5}{2} \, x^{4} e^{\left (x \log \relax (3) + x + 5\right )} + 10 \, x^{4} e^{\left (x \log \relax (3) + x\right )} + \frac {5}{2} \, x^{3} e^{\left (x \log \relax (3) + x + 10\right )} + 20 \, x^{3} e^{\left (x \log \relax (3) + x + 5\right )} + 40 \, x^{3} e^{\left (x \log \relax (3) + x\right )} + \frac {5}{4} \, x^{2} e^{\left (x \log \relax (3) + x + 15\right )} + 15 \, x^{2} e^{\left (x \log \relax (3) + x + 10\right )} + 60 \, x^{2} e^{\left (x \log \relax (3) + x + 5\right )} + 80 \, x^{2} e^{\left (x \log \relax (3) + x\right )} + \frac {5}{16} \, x e^{\left (x \log \relax (3) + x + 20\right )} + 5 \, x e^{\left (x \log \relax (3) + x + 15\right )} + 30 \, x e^{\left (x \log \relax (3) + x + 10\right )} + 80 \, x e^{\left (x \log \relax (3) + x + 5\right )} + 80 \, x e^{\left (x \log \relax (3) + x\right )} + x + \frac {1}{32} \, e^{\left (x \log \relax (3) + x + 25\right )} + \frac {5}{8} \, e^{\left (x \log \relax (3) + x + 20\right )} + 5 \, e^{\left (x \log \relax (3) + x + 15\right )} + 20 \, e^{\left (x \log \relax (3) + x + 10\right )} + 40 \, e^{\left (x \log \relax (3) + x + 5\right )} + 32 \, e^{\left (x \log \relax (3) + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(5)+2*x+4)*log(3)+exp(5)+2*x+14)*exp(x*log(3)+x)*(1/2*exp(5)+x+2)^5+exp(5)+2*x+4)/(exp(5)+2*x+
4),x, algorithm="giac")

[Out]

x^5*e^(x*log(3) + x) + 5/2*x^4*e^(x*log(3) + x + 5) + 10*x^4*e^(x*log(3) + x) + 5/2*x^3*e^(x*log(3) + x + 10)
+ 20*x^3*e^(x*log(3) + x + 5) + 40*x^3*e^(x*log(3) + x) + 5/4*x^2*e^(x*log(3) + x + 15) + 15*x^2*e^(x*log(3) +
x + 10) + 60*x^2*e^(x*log(3) + x + 5) + 80*x^2*e^(x*log(3) + x) + 5/16*x*e^(x*log(3) + x + 20) + 5*x*e^(x*log
(3) + x + 15) + 30*x*e^(x*log(3) + x + 10) + 80*x*e^(x*log(3) + x + 5) + 80*x*e^(x*log(3) + x) + x + 1/32*e^(x
*log(3) + x + 25) + 5/8*e^(x*log(3) + x + 20) + 5*e^(x*log(3) + x + 15) + 20*e^(x*log(3) + x + 10) + 40*e^(x*l
og(3) + x + 5) + 32*e^(x*log(3) + x)

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maple [B]  time = 0.24, size = 114, normalized size = 4.75

method	result	size
risch	\(x +\left (32+80 x +40 \,{\mathrm e}^{5}+x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+\frac {5 x^{4} {\mathrm e}^{5}}{2}+\frac {5 x \,{\mathrm e}^{20}}{16}+\frac {5 x^{2} {\mathrm e}^{15}}{4}+\frac {5 x^{3} {\mathrm e}^{10}}{2}+20 \,{\mathrm e}^{10}+30 x \,{\mathrm e}^{10}+5 x \,{\mathrm e}^{15}+15 x^{2} {\mathrm e}^{10}+20 x^{3} {\mathrm e}^{5}+60 x^{2} {\mathrm e}^{5}+80 x \,{\mathrm e}^{5}+5 \,{\mathrm e}^{15}+\frac {5 \,{\mathrm e}^{20}}{8}+\frac {{\mathrm e}^{25}}{32}\right ) 3^{x} {\mathrm e}^{x}\)	\(114\)
norman	\(x +\left (\frac {{\mathrm e}^{25}}{32}+\frac {5 \,{\mathrm e}^{20}}{8}+5 \,{\mathrm e}^{15}+20 \,{\mathrm e}^{10}+40 \,{\mathrm e}^{5}+32\right ) {\mathrm e}^{x \ln \relax (3)+x}+{\mathrm e}^{x \ln \relax (3)+x} x^{5}+\left (\frac {5 \,{\mathrm e}^{5}}{2}+10\right ) x^{4} {\mathrm e}^{x \ln \relax (3)+x}+\left (\frac {5 \,{\mathrm e}^{10}}{2}+20 \,{\mathrm e}^{5}+40\right ) x^{3} {\mathrm e}^{x \ln \relax (3)+x}+\left (\frac {5 \,{\mathrm e}^{15}}{4}+15 \,{\mathrm e}^{10}+60 \,{\mathrm e}^{5}+80\right ) x^{2} {\mathrm e}^{x \ln \relax (3)+x}+\left (\frac {5 \,{\mathrm e}^{20}}{16}+5 \,{\mathrm e}^{15}+30 \,{\mathrm e}^{10}+80 \,{\mathrm e}^{5}+80\right ) x \,{\mathrm e}^{x \ln \relax (3)+x}\)	\(154\)
derivativedivides	\(\text {Expression too large to display}\)	\(26214\)
default	\(\text {Expression too large to display}\)	\(26214\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((exp(5)+2*x+4)*ln(3)+exp(5)+2*x+14)*exp(x*ln(3)+x)*(1/2*exp(5)+x+2)^5+exp(5)+2*x+4)/(exp(5)+2*x+4),x,met
hod=_RETURNVERBOSE)

[Out]

x+(32+80*x+40*exp(5)+x^5+10*x^4+40*x^3+80*x^2+5/2*x^4*exp(5)+5/16*x*exp(20)+5/4*x^2*exp(15)+5/2*x^3*exp(10)+20
*exp(10)+30*x*exp(10)+5*x*exp(15)+15*x^2*exp(10)+20*x^3*exp(5)+60*x^2*exp(5)+80*x*exp(5)+5*exp(15)+5/8*exp(20)
+1/32*exp(25))*3^x*exp(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -64 \, e^{\left (-\frac {1}{2} \, {\left (e^{5} + 4\right )} {\left (\log \relax (3) + 1\right )}\right )} E_{1}\left (-\frac {1}{2} \, {\left (2 \, x + e^{5} + 4\right )} {\left (\log \relax (3) + 1\right )}\right ) \log \relax (3) - 224 \, e^{\left (-\frac {1}{2} \, {\left (e^{5} + 4\right )} {\left (\log \relax (3) + 1\right )}\right )} E_{1}\left (-\frac {1}{2} \, {\left (2 \, x + e^{5} + 4\right )} {\left (\log \relax (3) + 1\right )}\right ) - \frac {1}{2} \, {\left (e^{5} + 4\right )} \log \left (2 \, x + e^{5} + 4\right ) + \frac {1}{2} \, e^{5} \log \left (2 \, x + e^{5} + 4\right ) + x + \frac {1}{32} \, \int \frac {{\left (64 \, x^{6} {\left (\log \relax (3) + 1\right )} + 64 \, {\left (3 \, {\left (\log \relax (3) + 1\right )} e^{5} + 12 \, \log \relax (3) + 17\right )} x^{5} + 80 \, {\left (3 \, {\left (\log \relax (3) + 1\right )} e^{10} + 2 \, {\left (12 \, \log \relax (3) + 17\right )} e^{5} + 48 \, \log \relax (3) + 88\right )} x^{4} + 160 \, {\left ({\left (\log \relax (3) + 1\right )} e^{15} + {\left (12 \, \log \relax (3) + 17\right )} e^{10} + 8 \, {\left (6 \, \log \relax (3) + 11\right )} e^{5} + 64 \, \log \relax (3) + 144\right )} x^{3} + 20 \, {\left (3 \, {\left (\log \relax (3) + 1\right )} e^{20} + 4 \, {\left (12 \, \log \relax (3) + 17\right )} e^{15} + 48 \, {\left (6 \, \log \relax (3) + 11\right )} e^{10} + 192 \, {\left (4 \, \log \relax (3) + 9\right )} e^{5} + 768 \, \log \relax (3) + 2048\right )} x^{2} + 4 \, {\left (3 \, {\left (\log \relax (3) + 1\right )} e^{25} + 5 \, {\left (12 \, \log \relax (3) + 17\right )} e^{20} + 80 \, {\left (6 \, \log \relax (3) + 11\right )} e^{15} + 480 \, {\left (4 \, \log \relax (3) + 9\right )} e^{10} + 1280 \, {\left (3 \, \log \relax (3) + 8\right )} e^{5} + 3072 \, \log \relax (3) + 9472\right )} x + {\left (\log \relax (3) + 1\right )} e^{30} + 2 \, {\left (12 \, \log \relax (3) + 17\right )} e^{25} + 40 \, {\left (6 \, \log \relax (3) + 11\right )} e^{20} + 320 \, {\left (4 \, \log \relax (3) + 9\right )} e^{15} + 1280 \, {\left (3 \, \log \relax (3) + 8\right )} e^{10} + 512 \, {\left (12 \, \log \relax (3) + 37\right )} e^{5}\right )} e^{\left (x \log \relax (3) + x\right )}}{2 \, x + e^{5} + 4}\,{d x} + 2 \, \log \left (2 \, x + e^{5} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(5)+2*x+4)*log(3)+exp(5)+2*x+14)*exp(x*log(3)+x)*(1/2*exp(5)+x+2)^5+exp(5)+2*x+4)/(exp(5)+2*x+
4),x, algorithm="maxima")

[Out]

-64*e^(-1/2*(e^5 + 4)*(log(3) + 1))*exp_integral_e(1, -1/2*(2*x + e^5 + 4)*(log(3) + 1))*log(3) - 224*e^(-1/2*
(e^5 + 4)*(log(3) + 1))*exp_integral_e(1, -1/2*(2*x + e^5 + 4)*(log(3) + 1)) - 1/2*(e^5 + 4)*log(2*x + e^5 + 4
) + 1/2*e^5*log(2*x + e^5 + 4) + x + 1/32*integrate((64*x^6*(log(3) + 1) + 64*(3*(log(3) + 1)*e^5 + 12*log(3)
+ 17)*x^5 + 80*(3*(log(3) + 1)*e^10 + 2*(12*log(3) + 17)*e^5 + 48*log(3) + 88)*x^4 + 160*((log(3) + 1)*e^15 +
(12*log(3) + 17)*e^10 + 8*(6*log(3) + 11)*e^5 + 64*log(3) + 144)*x^3 + 20*(3*(log(3) + 1)*e^20 + 4*(12*log(3)
+ 17)*e^15 + 48*(6*log(3) + 11)*e^10 + 192*(4*log(3) + 9)*e^5 + 768*log(3) + 2048)*x^2 + 4*(3*(log(3) + 1)*e^2
5 + 5*(12*log(3) + 17)*e^20 + 80*(6*log(3) + 11)*e^15 + 480*(4*log(3) + 9)*e^10 + 1280*(3*log(3) + 8)*e^5 + 30
72*log(3) + 9472)*x + (log(3) + 1)*e^30 + 2*(12*log(3) + 17)*e^25 + 40*(6*log(3) + 11)*e^20 + 320*(4*log(3) +
9)*e^15 + 1280*(3*log(3) + 8)*e^10 + 512*(12*log(3) + 37)*e^5)*e^(x*log(3) + x)/(2*x + e^5 + 4), x) + 2*log(2*
x + e^5 + 4)

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mupad [B]  time = 4.22, size = 100, normalized size = 4.17 \begin {gather*} x+3^x\,x^5\,{\mathrm {e}}^x+\frac {3^x\,{\mathrm {e}}^x\,\left (1280\,{\mathrm {e}}^5+640\,{\mathrm {e}}^{10}+160\,{\mathrm {e}}^{15}+20\,{\mathrm {e}}^{20}+{\mathrm {e}}^{25}+1024\right )}{32}+\frac {3^x\,x^4\,{\mathrm {e}}^x\,\left (80\,{\mathrm {e}}^5+320\right )}{32}+\frac {5\,3^x\,x^2\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^3}{4}+\frac {5\,3^x\,x^3\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^2}{2}+\frac {5\,3^x\,x\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^4}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(5) + exp(x + x*log(3))*(x + exp(5)/2 + 2)^5*(2*x + exp(5) + log(3)*(2*x + exp(5) + 4) + 14) + 4
)/(2*x + exp(5) + 4),x)

[Out]

x + 3^x*x^5*exp(x) + (3^x*exp(x)*(1280*exp(5) + 640*exp(10) + 160*exp(15) + 20*exp(20) + exp(25) + 1024))/32 +
(3^x*x^4*exp(x)*(80*exp(5) + 320))/32 + (5*3^x*x^2*exp(x)*(exp(5) + 4)^3)/4 + (5*3^x*x^3*exp(x)*(exp(5) + 4)^
2)/2 + (5*3^x*x*exp(x)*(exp(5) + 4)^4)/16

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sympy [B]  time = 0.55, size = 138, normalized size = 5.75 \begin {gather*} x + \frac {\left (32 x^{5} + 320 x^{4} + 80 x^{4} e^{5} + 1280 x^{3} + 640 x^{3} e^{5} + 80 x^{3} e^{10} + 2560 x^{2} + 1920 x^{2} e^{5} + 480 x^{2} e^{10} + 40 x^{2} e^{15} + 2560 x + 2560 x e^{5} + 960 x e^{10} + 160 x e^{15} + 10 x e^{20} + 1024 + 1280 e^{5} + 640 e^{10} + 160 e^{15} + 20 e^{20} + e^{25}\right ) e^{x + x \log {\relax (3 )}}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(5)+2*x+4)*ln(3)+exp(5)+2*x+14)*exp(x*ln(3)+x)*(1/2*exp(5)+x+2)**5+exp(5)+2*x+4)/(exp(5)+2*x+4
),x)

[Out]

x + (32*x**5 + 320*x**4 + 80*x**4*exp(5) + 1280*x**3 + 640*x**3*exp(5) + 80*x**3*exp(10) + 2560*x**2 + 1920*x*
*2*exp(5) + 480*x**2*exp(10) + 40*x**2*exp(15) + 2560*x + 2560*x*exp(5) + 960*x*exp(10) + 160*x*exp(15) + 10*x
*exp(20) + 1024 + 1280*exp(5) + 640*exp(10) + 160*exp(15) + 20*exp(20) + exp(25))*exp(x + x*log(3))/32

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