3.97.77 \(\int \frac {8 x^3+4 e x^3+4 x^4+e^2 x^4+e^2 (8 x^3+4 e x^3+4 x^4) \log (e^3 (2+e+x))}{2+e+x} \, dx\)

Optimal. Leaf size=19 \[ x^4 \left (1+e^2 \log \left (e^3 (2+e+x)\right )\right ) \]

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Rubi [A]  time = 0.25, antiderivative size = 37, normalized size of antiderivative = 1.95, number of steps used = 9, number of rules used = 5, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {6, 6742, 77, 2395, 43} \begin {gather*} \frac {1}{4} \left (4+e^2\right ) x^4-\frac {e^2 x^4}{4}+e^2 x^4 (\log (x+e+2)+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*x^3 + 4*E*x^3 + 4*x^4 + E^2*x^4 + E^2*(8*x^3 + 4*E*x^3 + 4*x^4)*Log[E^3*(2 + E + x)])/(2 + E + x),x]

[Out]

-1/4*(E^2*x^4) + ((4 + E^2)*x^4)/4 + E^2*x^4*(3 + Log[2 + E + x])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

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 {(8+4 e) x^3+4 x^4+e^2 x^4+e^2 \left (8 x^3+4 e x^3+4 x^4\right ) \log \left (e^3 (2+e+x)\right )}{2+e+x} \, dx\\ &=\int \frac {(8+4 e) x^3+\left (4+e^2\right ) x^4+e^2 \left (8 x^3+4 e x^3+4 x^4\right ) \log \left (e^3 (2+e+x)\right )}{2+e+x} \, dx\\ &=\int \left (\frac {x^3 \left (4 (2+e)+\left (4+e^2\right ) x\right )}{2+e+x}+4 e^2 x^3 (3+\log (2+e+x))\right ) \, dx\\ &=\left (4 e^2\right ) \int x^3 (3+\log (2+e+x)) \, dx+\int \frac {x^3 \left (4 (2+e)+\left (4+e^2\right ) x\right )}{2+e+x} \, dx\\ &=e^2 x^4 (3+\log (2+e+x))-e^2 \int \frac {x^4}{2+e+x} \, dx+\int \left (-e^2 (2+e)^3+e^2 (2+e)^2 x-e^2 (2+e) x^2+\left (4+e^2\right ) x^3+\frac {e^2 (2+e)^4}{2+e+x}\right ) \, dx\\ &=-e^2 (2+e)^3 x+\frac {1}{2} e^2 (2+e)^2 x^2-\frac {1}{3} e^2 (2+e) x^3+\frac {1}{4} \left (4+e^2\right ) x^4+e^2 (2+e)^4 \log (2+e+x)+e^2 x^4 (3+\log (2+e+x))-e^2 \int \left (-(2+e)^3+(2+e)^2 x-(2+e) x^2+x^3+\frac {(2+e)^4}{2+e+x}\right ) \, dx\\ &=-\frac {1}{4} e^2 x^4+\frac {1}{4} \left (4+e^2\right ) x^4+e^2 x^4 (3+\log (2+e+x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 24, normalized size = 1.26 \begin {gather*} x^4+3 e^2 x^4+e^2 x^4 \log (2+e+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x^3 + 4*E*x^3 + 4*x^4 + E^2*x^4 + E^2*(8*x^3 + 4*E*x^3 + 4*x^4)*Log[E^3*(2 + E + x)])/(2 + E + x)
,x]

[Out]

x^4 + 3*E^2*x^4 + E^2*x^4*Log[2 + E + x]

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fricas [A]  time = 0.57, size = 20, normalized size = 1.05 \begin {gather*} x^{4} e^{2} \log \left ({\left (x + 2\right )} e^{3} + e^{4}\right ) + x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(1)+4*x^4+8*x^3)*exp(2)*log((2+x+exp(1))*exp(3))+x^4*exp(2)+4*x^3*exp(1)+4*x^4+8*x^3)/(2+
x+exp(1)),x, algorithm="fricas")

[Out]

x^4*e^2*log((x + 2)*e^3 + e^4) + x^4

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giac [A]  time = 0.16, size = 22, normalized size = 1.16 \begin {gather*} x^{4} e^{2} \log \left (x e^{3} + e^{4} + 2 \, e^{3}\right ) + x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(1)+4*x^4+8*x^3)*exp(2)*log((2+x+exp(1))*exp(3))+x^4*exp(2)+4*x^3*exp(1)+4*x^4+8*x^3)/(2+
x+exp(1)),x, algorithm="giac")

[Out]

x^4*e^2*log(x*e^3 + e^4 + 2*e^3) + x^4

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maple [A]  time = 0.08, size = 20, normalized size = 1.05

method	result	size
norman	\(x^{4}+x^{4} {\mathrm e}^{2} \ln \left (\left (2+x +{\mathrm e}\right ) {\mathrm e}^{3}\right )\)	\(20\)
risch	\(x^{4}+x^{4} {\mathrm e}^{2} \ln \left (\left (2+x +{\mathrm e}\right ) {\mathrm e}^{3}\right )\)	\(20\)
derivativedivides	\({\mathrm e}^{-3} \left (-32 \left ({\mathrm e}+2\right ) {\mathrm e}^{3}-32 x \,{\mathrm e}^{3}+12 \,{\mathrm e}^{-3} {\mathrm e} \,{\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}+{\mathrm e}^{3} {\mathrm e}^{4} {\mathrm e}^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-\frac {4 \,{\mathrm e}^{-6} {\mathrm e} \,{\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}}{3}+12 \,{\mathrm e}^{-3} \left ({\mathrm e}^{2}\right )^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{2}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}}{4}\right )+48 \,{\mathrm e}^{-3} {\mathrm e} \,{\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{2}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}}{4}\right )-12 \,{\mathrm e}^{-6} {\mathrm e} \,{\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{3}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}}{9}\right )+8 \left ({\mathrm e}^{3}\right )^{2} {\mathrm e}^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+24 \,{\mathrm e}^{3} \left ({\mathrm e}^{2}\right )^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+3 \,{\mathrm e}^{-3} \left ({\mathrm e}^{2}\right )^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}+32 \,{\mathrm e}^{2} {\mathrm e} \,{\mathrm e}^{3} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-48 \,{\mathrm e} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-56 \,{\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-32 \,{\mathrm e}^{2} \left (\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right ) \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-x \,{\mathrm e}^{3}-\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-4 \,{\mathrm e}^{3} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+24 \,{\mathrm e}^{-3} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}-8 \,{\mathrm e}^{-6} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}+{\mathrm e}^{-9} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{4}+18 \,{\mathrm e}^{-3} {\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}+24 \,{\mathrm e}^{-3} {\mathrm e} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}-4 \,{\mathrm e}^{-6} {\mathrm e} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}-\frac {8 \,{\mathrm e}^{-6} {\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}}{3}-24 \left ({\mathrm e}^{2}\right )^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+48 \,{\mathrm e}^{-3} {\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{2}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}}{4}\right )-24 \,{\mathrm e}^{-6} {\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{3}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}}{9}\right )-4 \,{\mathrm e}^{3} {\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+4 \,{\mathrm e}^{-9} {\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{4} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{4}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{4}}{16}\right )-4 \,{\mathrm e}^{3} {\mathrm e}^{2} \left (\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right ) \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-x \,{\mathrm e}^{3}-\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-24 \left ({\mathrm e}^{2}\right )^{2} \left (\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right ) \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-x \,{\mathrm e}^{3}-\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+\frac {{\mathrm e}^{-9} {\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{4}}{4}-48 \,{\mathrm e} \,{\mathrm e}^{2} \left (\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right ) \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-x \,{\mathrm e}^{3}-\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+16 \,{\mathrm e}^{2} {\mathrm e}^{3} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-48 \,{\mathrm e} \,{\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )\right )\)	\(1039\)
default	\({\mathrm e}^{-3} \left (-32 \left ({\mathrm e}+2\right ) {\mathrm e}^{3}-32 x \,{\mathrm e}^{3}+12 \,{\mathrm e}^{-3} {\mathrm e} \,{\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}+{\mathrm e}^{3} {\mathrm e}^{4} {\mathrm e}^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-\frac {4 \,{\mathrm e}^{-6} {\mathrm e} \,{\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}}{3}+12 \,{\mathrm e}^{-3} \left ({\mathrm e}^{2}\right )^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{2}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}}{4}\right )+48 \,{\mathrm e}^{-3} {\mathrm e} \,{\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{2}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}}{4}\right )-12 \,{\mathrm e}^{-6} {\mathrm e} \,{\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{3}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}}{9}\right )+8 \left ({\mathrm e}^{3}\right )^{2} {\mathrm e}^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+24 \,{\mathrm e}^{3} \left ({\mathrm e}^{2}\right )^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+3 \,{\mathrm e}^{-3} \left ({\mathrm e}^{2}\right )^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}+32 \,{\mathrm e}^{2} {\mathrm e} \,{\mathrm e}^{3} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-48 \,{\mathrm e} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-56 \,{\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-32 \,{\mathrm e}^{2} \left (\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right ) \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-x \,{\mathrm e}^{3}-\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-4 \,{\mathrm e}^{3} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+24 \,{\mathrm e}^{-3} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}-8 \,{\mathrm e}^{-6} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}+{\mathrm e}^{-9} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{4}+18 \,{\mathrm e}^{-3} {\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}+24 \,{\mathrm e}^{-3} {\mathrm e} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}-4 \,{\mathrm e}^{-6} {\mathrm e} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}-\frac {8 \,{\mathrm e}^{-6} {\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}}{3}-24 \left ({\mathrm e}^{2}\right )^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+48 \,{\mathrm e}^{-3} {\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{2}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{2}}{4}\right )-24 \,{\mathrm e}^{-6} {\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{3}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{3}}{9}\right )-4 \,{\mathrm e}^{3} {\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+4 \,{\mathrm e}^{-9} {\mathrm e}^{2} \left (\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{4} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )}{4}-\frac {\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{4}}{16}\right )-4 \,{\mathrm e}^{3} {\mathrm e}^{2} \left (\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right ) \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-x \,{\mathrm e}^{3}-\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-24 \left ({\mathrm e}^{2}\right )^{2} \left (\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right ) \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-x \,{\mathrm e}^{3}-\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+\frac {{\mathrm e}^{-9} {\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )^{4}}{4}-48 \,{\mathrm e} \,{\mathrm e}^{2} \left (\left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right ) \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-x \,{\mathrm e}^{3}-\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )+16 \,{\mathrm e}^{2} {\mathrm e}^{3} \ln \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )-48 \,{\mathrm e} \,{\mathrm e}^{2} \left (x \,{\mathrm e}^{3}+\left ({\mathrm e}+2\right ) {\mathrm e}^{3}\right )\right )\)	\(1039\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3*exp(1)+4*x^4+8*x^3)*exp(2)*ln((2+x+exp(1))*exp(3))+x^4*exp(2)+4*x^3*exp(1)+4*x^4+8*x^3)/(2+x+exp(1
)),x,method=_RETURNVERBOSE)

[Out]

x^4+x^4*exp(2)*ln((2+x+exp(1))*exp(3))

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maxima [B]  time = 0.36, size = 679, normalized size = 35.74 \begin {gather*} x^{4} - \frac {4}{3} \, x^{3} {\left (e + 2\right )} + \frac {8}{3} \, x^{3} + 2 \, x^{2} {\left (e^{2} + 4 \, e + 4\right )} - 4 \, x^{2} {\left (e + 2\right )} + \frac {2}{3} \, {\left (2 \, x^{3} - 3 \, x^{2} {\left (e + 2\right )} + 6 \, x {\left (e^{2} + 4 \, e + 4\right )} - 6 \, {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} \log \left (x + e + 2\right )\right )} e^{3} \log \left (x e^{3} + e^{4} + 2 \, e^{3}\right ) + \frac {1}{3} \, {\left (3 \, x^{4} - 4 \, x^{3} {\left (e + 2\right )} + 6 \, x^{2} {\left (e^{2} + 4 \, e + 4\right )} - 12 \, x {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} + 12 \, {\left (e^{4} + 8 \, e^{3} + 24 \, e^{2} + 32 \, e + 16\right )} \log \left (x + e + 2\right )\right )} e^{2} \log \left (x e^{3} + e^{4} + 2 \, e^{3}\right ) + \frac {4}{3} \, {\left (2 \, x^{3} - 3 \, x^{2} {\left (e + 2\right )} + 6 \, x {\left (e^{2} + 4 \, e + 4\right )} - 6 \, {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} \log \left (x + e + 2\right )\right )} e^{2} \log \left (x e^{3} + e^{4} + 2 \, e^{3}\right ) - 4 \, x {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} + 8 \, x {\left (e^{2} + 4 \, e + 4\right )} - \frac {1}{9} \, {\left (4 \, x^{3} - 15 \, x^{2} {\left (e + 2\right )} - 18 \, {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} \log \left (x + e + 2\right )^{2} + 66 \, x {\left (e^{2} + 4 \, e + 4\right )} - 66 \, {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} \log \left (x + e + 2\right )\right )} e^{3} - \frac {1}{36} \, {\left (9 \, x^{4} - 28 \, x^{3} {\left (e + 2\right )} + 78 \, x^{2} {\left (e^{2} + 4 \, e + 4\right )} + 72 \, {\left (e^{4} + 8 \, e^{3} + 24 \, e^{2} + 32 \, e + 16\right )} \log \left (x + e + 2\right )^{2} - 300 \, x {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} + 300 \, {\left (e^{4} + 8 \, e^{3} + 24 \, e^{2} + 32 \, e + 16\right )} \log \left (x + e + 2\right )\right )} e^{2} + \frac {1}{12} \, {\left (3 \, x^{4} - 4 \, x^{3} {\left (e + 2\right )} + 6 \, x^{2} {\left (e^{2} + 4 \, e + 4\right )} - 12 \, x {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} + 12 \, {\left (e^{4} + 8 \, e^{3} + 24 \, e^{2} + 32 \, e + 16\right )} \log \left (x + e + 2\right )\right )} e^{2} - \frac {2}{9} \, {\left (4 \, x^{3} - 15 \, x^{2} {\left (e + 2\right )} - 18 \, {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} \log \left (x + e + 2\right )^{2} + 66 \, x {\left (e^{2} + 4 \, e + 4\right )} - 66 \, {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} \log \left (x + e + 2\right )\right )} e^{2} + \frac {2}{3} \, {\left (2 \, x^{3} - 3 \, x^{2} {\left (e + 2\right )} + 6 \, x {\left (e^{2} + 4 \, e + 4\right )} - 6 \, {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} \log \left (x + e + 2\right )\right )} e + 4 \, {\left (e^{4} + 8 \, e^{3} + 24 \, e^{2} + 32 \, e + 16\right )} \log \left (x + e + 2\right ) - 8 \, {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )} \log \left (x + e + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(1)+4*x^4+8*x^3)*exp(2)*log((2+x+exp(1))*exp(3))+x^4*exp(2)+4*x^3*exp(1)+4*x^4+8*x^3)/(2+
x+exp(1)),x, algorithm="maxima")

[Out]

x^4 - 4/3*x^3*(e + 2) + 8/3*x^3 + 2*x^2*(e^2 + 4*e + 4) - 4*x^2*(e + 2) + 2/3*(2*x^3 - 3*x^2*(e + 2) + 6*x*(e^
2 + 4*e + 4) - 6*(e^3 + 6*e^2 + 12*e + 8)*log(x + e + 2))*e^3*log(x*e^3 + e^4 + 2*e^3) + 1/3*(3*x^4 - 4*x^3*(e
+ 2) + 6*x^2*(e^2 + 4*e + 4) - 12*x*(e^3 + 6*e^2 + 12*e + 8) + 12*(e^4 + 8*e^3 + 24*e^2 + 32*e + 16)*log(x +
e + 2))*e^2*log(x*e^3 + e^4 + 2*e^3) + 4/3*(2*x^3 - 3*x^2*(e + 2) + 6*x*(e^2 + 4*e + 4) - 6*(e^3 + 6*e^2 + 12*
e + 8)*log(x + e + 2))*e^2*log(x*e^3 + e^4 + 2*e^3) - 4*x*(e^3 + 6*e^2 + 12*e + 8) + 8*x*(e^2 + 4*e + 4) - 1/9
*(4*x^3 - 15*x^2*(e + 2) - 18*(e^3 + 6*e^2 + 12*e + 8)*log(x + e + 2)^2 + 66*x*(e^2 + 4*e + 4) - 66*(e^3 + 6*e
^2 + 12*e + 8)*log(x + e + 2))*e^3 - 1/36*(9*x^4 - 28*x^3*(e + 2) + 78*x^2*(e^2 + 4*e + 4) + 72*(e^4 + 8*e^3 +
24*e^2 + 32*e + 16)*log(x + e + 2)^2 - 300*x*(e^3 + 6*e^2 + 12*e + 8) + 300*(e^4 + 8*e^3 + 24*e^2 + 32*e + 16
)*log(x + e + 2))*e^2 + 1/12*(3*x^4 - 4*x^3*(e + 2) + 6*x^2*(e^2 + 4*e + 4) - 12*x*(e^3 + 6*e^2 + 12*e + 8) +
12*(e^4 + 8*e^3 + 24*e^2 + 32*e + 16)*log(x + e + 2))*e^2 - 2/9*(4*x^3 - 15*x^2*(e + 2) - 18*(e^3 + 6*e^2 + 12
*e + 8)*log(x + e + 2)^2 + 66*x*(e^2 + 4*e + 4) - 66*(e^3 + 6*e^2 + 12*e + 8)*log(x + e + 2))*e^2 + 2/3*(2*x^3
- 3*x^2*(e + 2) + 6*x*(e^2 + 4*e + 4) - 6*(e^3 + 6*e^2 + 12*e + 8)*log(x + e + 2))*e + 4*(e^4 + 8*e^3 + 24*e^
2 + 32*e + 16)*log(x + e + 2) - 8*(e^3 + 6*e^2 + 12*e + 8)*log(x + e + 2)

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mupad [B]  time = 6.42, size = 18, normalized size = 0.95 \begin {gather*} x^4\,\left ({\mathrm {e}}^2\,\ln \left ({\mathrm {e}}^3\,\left (x+\mathrm {e}+2\right )\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*exp(1) + x^4*exp(2) + 8*x^3 + 4*x^4 + exp(2)*log(exp(3)*(x + exp(1) + 2))*(4*x^3*exp(1) + 8*x^3 + 4
*x^4))/(x + exp(1) + 2),x)

[Out]

x^4*(exp(2)*log(exp(3)*(x + exp(1) + 2)) + 1)

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sympy [A]  time = 0.26, size = 20, normalized size = 1.05 \begin {gather*} x^{4} e^{2} \log {\left (\left (x + 2 + e\right ) e^{3} \right )} + x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3*exp(1)+4*x**4+8*x**3)*exp(2)*ln((2+x+exp(1))*exp(3))+x**4*exp(2)+4*x**3*exp(1)+4*x**4+8*x**
3)/(2+x+exp(1)),x)

[Out]

x**4*exp(2)*log((x + 2 + E)*exp(3)) + x**4

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