\(\int \frac {e^{\frac {4+x}{5+3 x}} (-138-111 x-27 x^2)+e^{\frac {4+x}{5+3 x}} (-46-37 x-9 x^2) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx\) [2205]

Optimal result

Integrand size = 77, antiderivative size = 28 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=3+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x} \]

[Out]

3+(ln(2*ln(2))^2+3)/(3+x)*exp((4+x)/(5+3*x))

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 25, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6820, 12, 6874, 2264, 2263, 2262, 2241, 2265, 2209, 2240} \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {e^{\frac {x+4}{3 x+5}} \left (3+\log ^2(\log (4))\right )}{x+3} \]

[In]

Int[(E^((4 + x)/(5 + 3*x))*(-138 - 111*x - 27*x^2) + E^((4 + x)/(5 + 3*x))*(-46 - 37*x - 9*x^2)*Log[Log[4]]^2)
/(225 + 420*x + 286*x^2 + 84*x^3 + 9*x^4),x]

[Out]

(E^((4 + x)/(5 + 3*x))*(3 + Log[Log[4]]^2))/(3 + x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2262

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :>
Int[(g + h*x)^m*F^((d*e + b*f)/d - f*((b*c - a*d)/(d*(c + d*x)))), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m},
x] && NeQ[b*c - a*d, 0] && EqQ[d*g - c*h, 0]

Rule 2263

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/((g_.) + (h_.)*(x_)), x_Symbol] :> Dist[d
/h, Int[F^(e + f*((a + b*x)/(c + d*x)))/(c + d*x), x], x] - Dist[(d*g - c*h)/h, Int[F^(e + f*((a + b*x)/(c + d
*x)))/((c + d*x)*(g + h*x)), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g -
c*h, 0]

Rule 2264

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_), x_Symbol] :> S
imp[(g + h*x)^(m + 1)*(F^(e + f*((a + b*x)/(c + d*x)))/(h*(m + 1))), x] - Dist[f*(b*c - a*d)*(Log[F]/(h*(m + 1
))), Int[(g + h*x)^(m + 1)*(F^(e + f*((a + b*x)/(c + d*x)))/(c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f
, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0] && ILtQ[m, -1]

Rule 2265

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/(((g_.) + (h_.)*(x_))*((i_.) + (j_.)*(x_)
)), x_Symbol] :> Dist[-d/(h*(d*i - c*j)), Subst[Int[F^(e + f*((b*i - a*j)/(d*i - c*j)) - (b*c - a*d)*f*(x/(d*i
- c*j)))/x, x], x, (i + j*x)/(c + d*x)], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && EqQ[d*g - c*h, 0]

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {4+x}{5+3 x}} \left (46+37 x+9 x^2\right ) \left (-3-\log ^2(\log (4))\right )}{\left (15+14 x+3 x^2\right )^2} \, dx \\ & = \left (-3-\log ^2(\log (4))\right ) \int \frac {e^{\frac {4+x}{5+3 x}} \left (46+37 x+9 x^2\right )}{\left (15+14 x+3 x^2\right )^2} \, dx \\ & = \left (-3-\log ^2(\log (4))\right ) \int \left (\frac {e^{\frac {4+x}{5+3 x}}}{(3+x)^2}+\frac {7 e^{\frac {4+x}{5+3 x}}}{16 (3+x)}+\frac {21 e^{\frac {4+x}{5+3 x}}}{4 (5+3 x)^2}-\frac {21 e^{\frac {4+x}{5+3 x}}}{16 (5+3 x)}\right ) \, dx \\ & = \left (-3-\log ^2(\log (4))\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(3+x)^2} \, dx-\frac {1}{16} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{3+x} \, dx+\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{5+3 x} \, dx-\frac {1}{4} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(5+3 x)^2} \, dx \\ & = \frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}-\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{5+3 x} \, dx+\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{5+3 x} \, dx+\frac {1}{4} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(3+x) (5+3 x)} \, dx-\frac {1}{4} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{(5+3 x)^2} \, dx+\left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(3+x) (5+3 x)^2} \, dx \\ & = \frac {3}{4} e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}} \left (3+\log ^2(\log (4))\right )+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}-\frac {7}{16} \sqrt [3]{e} \operatorname {ExpIntegralEi}\left (\frac {7}{3 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )-\frac {1}{16} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \text {Subst}\left (\int \frac {e^{-\frac {1}{4}+\frac {7 x}{4}}}{x} \, dx,x,\frac {3+x}{5+3 x}\right )-\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{5+3 x} \, dx+\left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \left (\frac {e^{\frac {4+x}{5+3 x}}}{16 (3+x)}+\frac {3 e^{\frac {4+x}{5+3 x}}}{4 (5+3 x)^2}-\frac {3 e^{\frac {4+x}{5+3 x}}}{16 (5+3 x)}\right ) \, dx \\ & = \frac {3}{4} e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}} \left (3+\log ^2(\log (4))\right )+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}-\frac {7 \operatorname {ExpIntegralEi}\left (\frac {7 (3+x)}{4 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )}{16 \sqrt [4]{e}}+\frac {1}{16} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{3+x} \, dx-\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{5+3 x} \, dx+\frac {1}{4} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(5+3 x)^2} \, dx \\ & = \frac {3}{4} e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}} \left (3+\log ^2(\log (4))\right )+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}-\frac {7 \operatorname {ExpIntegralEi}\left (\frac {7 (3+x)}{4 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )}{16 \sqrt [4]{e}}+\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{5+3 x} \, dx-\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{5+3 x} \, dx-\frac {1}{4} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(3+x) (5+3 x)} \, dx+\frac {1}{4} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{(5+3 x)^2} \, dx \\ & = \frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}+\frac {7}{16} \sqrt [3]{e} \operatorname {ExpIntegralEi}\left (\frac {7}{3 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )-\frac {7 \operatorname {ExpIntegralEi}\left (\frac {7 (3+x)}{4 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )}{16 \sqrt [4]{e}}+\frac {1}{16} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \text {Subst}\left (\int \frac {e^{-\frac {1}{4}+\frac {7 x}{4}}}{x} \, dx,x,\frac {3+x}{5+3 x}\right )+\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{5+3 x} \, dx \\ & = \frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x} \]

[In]

Integrate[(E^((4 + x)/(5 + 3*x))*(-138 - 111*x - 27*x^2) + E^((4 + x)/(5 + 3*x))*(-46 - 37*x - 9*x^2)*Log[Log[
4]]^2)/(225 + 420*x + 286*x^2 + 84*x^3 + 9*x^4),x]

[Out]

(E^((4 + x)/(5 + 3*x))*(3 + Log[Log[4]]^2))/(3 + x)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {\left (\ln \left (2 \ln \left (2\right )\right )^{2}+3\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}}{3+x}\)	\(28\)
risch	\(\frac {\left (\ln \left (2\right )^{2}+2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+\ln \left (\ln \left (2\right )\right )^{2}+3\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}}{3+x}\)	\(37\)
parallelrisch	\(\frac {81 \,{\mathrm e}^{\frac {4+x}{3 x +5}} \ln \left (2 \ln \left (2\right )\right )^{2}+243 \,{\mathrm e}^{\frac {4+x}{3 x +5}}}{243+81 x}\)	\(44\)
norman	\(\frac {\left (15+10 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+5 \ln \left (2\right )^{2}+5 \ln \left (\ln \left (2\right )\right )^{2}\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}+\left (9+6 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+3 \ln \left (2\right )^{2}+3 \ln \left (\ln \left (2\right )\right )^{2}\right ) x \,{\mathrm e}^{\frac {4+x}{3 x +5}}}{3 x^{2}+14 x +15}\)	\(86\)
derivativedivides	\(\frac {9 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{4}-\frac {21 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{16 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+588 \ln \left (2\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )+588 \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )-\frac {35 \ln \left (2\right )^{2} {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{192 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {119 \ln \left (2\right )^{2} {\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{192}+3087 \ln \left (\ln \left (2\right )\right )^{2} \left (-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{7056 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{7056}\right )+1029 \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{\frac {9408}{3 x +5}+2352}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{28224}\right )+1176 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )+6174 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{7056 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{7056}\right )+2058 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{\frac {9408}{3 x +5}+2352}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{28224}\right )\)	\(500\)
default	\(\frac {9 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{4}-\frac {21 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{16 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+588 \ln \left (2\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )+588 \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )-\frac {35 \ln \left (2\right )^{2} {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{192 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {119 \ln \left (2\right )^{2} {\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{192}+3087 \ln \left (\ln \left (2\right )\right )^{2} \left (-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{7056 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{7056}\right )+1029 \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{\frac {9408}{3 x +5}+2352}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{28224}\right )+1176 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )+6174 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{7056 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{7056}\right )+2058 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{\frac {9408}{3 x +5}+2352}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{28224}\right )\)	\(500\)

[In]

int(((-9*x^2-37*x-46)*exp((4+x)/(3*x+5))*ln(2*ln(2))^2+(-27*x^2-111*x-138)*exp((4+x)/(3*x+5)))/(9*x^4+84*x^3+2
86*x^2+420*x+225),x,method=_RETURNVERBOSE)

[Out]

(ln(2*ln(2))^2+3)/(3+x)*exp((4+x)/(3*x+5))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2 \, \log \left (2\right )\right )^{2} + 3 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{x + 3} \]

[In]

integrate(((-9*x^2-37*x-46)*exp((4+x)/(3*x+5))*log(2*log(2))^2+(-27*x^2-111*x-138)*exp((4+x)/(3*x+5)))/(9*x^4+
84*x^3+286*x^2+420*x+225),x, algorithm="fricas")

[Out]

(e^((x + 4)/(3*x + 5))*log(2*log(2))^2 + 3*e^((x + 4)/(3*x + 5)))/(x + 3)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {\left (2 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + \log {\left (\log {\left (2 \right )} \right )}^{2} + \log {\left (2 \right )}^{2} + 3\right ) e^{\frac {x + 4}{3 x + 5}}}{x + 3} \]

[In]

integrate(((-9*x**2-37*x-46)*exp((4+x)/(3*x+5))*ln(2*ln(2))**2+(-27*x**2-111*x-138)*exp((4+x)/(3*x+5)))/(9*x**
4+84*x**3+286*x**2+420*x+225),x)

[Out]

(2*log(2)*log(log(2)) + log(log(2))**2 + log(2)**2 + 3)*exp((x + 4)/(3*x + 5))/(x + 3)

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {{\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2} + 3\right )} e^{\left (\frac {7}{3 \, {\left (3 \, x + 5\right )}} + \frac {1}{3}\right )}}{x + 3} \]

[In]

integrate(((-9*x^2-37*x-46)*exp((4+x)/(3*x+5))*log(2*log(2))^2+(-27*x^2-111*x-138)*exp((4+x)/(3*x+5)))/(9*x^4+
84*x^3+286*x^2+420*x+225),x, algorithm="maxima")

[Out]

(log(2)^2 + 2*log(2)*log(log(2)) + log(log(2))^2 + 3)*e^(7/3/(3*x + 5) + 1/3)/(x + 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (29) = 58\).

Time = 0.56 (sec) , antiderivative size = 198, normalized size of antiderivative = 7.07 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {\frac {3 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2\right )^{2}}{3 \, x + 5} - e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2\right )^{2} + \frac {6 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2\right ) \log \left (\log \left (2\right )\right )}{3 \, x + 5} - 2 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \frac {3 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (\log \left (2\right )\right )^{2}}{3 \, x + 5} - e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (\log \left (2\right )\right )^{2} + \frac {9 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{3 \, x + 5} - 3 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{\frac {4 \, {\left (x + 4\right )}}{3 \, x + 5} + 1} \]

[In]

integrate(((-9*x^2-37*x-46)*exp((4+x)/(3*x+5))*log(2*log(2))^2+(-27*x^2-111*x-138)*exp((4+x)/(3*x+5)))/(9*x^4+
84*x^3+286*x^2+420*x+225),x, algorithm="giac")

[Out]

(3*(x + 4)*e^((x + 4)/(3*x + 5))*log(2)^2/(3*x + 5) - e^((x + 4)/(3*x + 5))*log(2)^2 + 6*(x + 4)*e^((x + 4)/(3
*x + 5))*log(2)*log(log(2))/(3*x + 5) - 2*e^((x + 4)/(3*x + 5))*log(2)*log(log(2)) + 3*(x + 4)*e^((x + 4)/(3*x
+ 5))*log(log(2))^2/(3*x + 5) - e^((x + 4)/(3*x + 5))*log(log(2))^2 + 9*(x + 4)*e^((x + 4)/(3*x + 5))/(3*x +
5) - 3*e^((x + 4)/(3*x + 5)))/(4*(x + 4)/(3*x + 5) + 1)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {{\mathrm {e}}^{\frac {x+4}{3\,x+5}}\,\left ({\ln \left (\ln \left (4\right )\right )}^2+3\right )}{x+3} \]

[In]

int(-(exp((x + 4)/(3*x + 5))*(111*x + 27*x^2 + 138) + log(2*log(2))^2*exp((x + 4)/(3*x + 5))*(37*x + 9*x^2 + 4
6))/(420*x + 286*x^2 + 84*x^3 + 9*x^4 + 225),x)

[Out]

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