∫ −45−80x4+e2x(−9−16x4+32x5)_______________________

Optimal. Leaf size=24 \[ 1+\frac {9}{64 x^4}-\log \left (\frac {4 x}{5+e^{2 x}}\right ) \]

________________________________________________________________________________________

Rubi [A] time = 0.39, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 10, number of rules used = 8, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6741, 12, 6742, 2282, 36, 29, 31, 14} \begin {gather*} \frac {9}{64 x^4}+\log \left (e^{2 x}+5\right )-\log (x) \end {gather*}

Int[(-45 - 80*x^4 + E^(2*x)*(-9 - 16*x^4 + 32*x^5))/(80*x^5 + 16*E^(2*x)*x^5),x]

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

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]]

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -

Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu

nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F

reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x

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

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-45-80 x^4+e^{2 x} \left (-9-16 x^4+32 x^5\right )}{16 \left (5+e^{2 x}\right ) x^5} \, dx\\ &=\frac {1}{16} \int \frac {-45-80 x^4+e^{2 x} \left (-9-16 x^4+32 x^5\right )}{\left (5+e^{2 x}\right ) x^5} \, dx\\ &=\frac {1}{16} \int \left (-\frac {160}{5+e^{2 x}}+\frac {-9-16 x^4+32 x^5}{x^5}\right ) \, dx\\ &=\frac {1}{16} \int \frac {-9-16 x^4+32 x^5}{x^5} \, dx-10 \int \frac {1}{5+e^{2 x}} \, dx\\ &=\frac {1}{16} \int \left (32-\frac {9}{x^5}-\frac {16}{x}\right ) \, dx-5 \operatorname {Subst}\left (\int \frac {1}{x (5+x)} \, dx,x,e^{2 x}\right )\\ &=\frac {9}{64 x^4}+2 x-\log (x)-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x}\right )+\operatorname {Subst}\left (\int \frac {1}{5+x} \, dx,x,e^{2 x}\right )\\ &=\frac {9}{64 x^4}+\log \left (5+e^{2 x}\right )-\log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 0.05, size = 26, normalized size = 1.08 \begin {gather*} \frac {1}{16} \left (\frac {9}{4 x^4}+16 \log \left (5+e^{2 x}\right )-16 \log (x)\right ) \end {gather*}

Integrate[(-45 - 80*x^4 + E^(2*x)*(-9 - 16*x^4 + 32*x^5))/(80*x^5 + 16*E^(2*x)*x^5),x]

(9/(4*x^4) + 16*Log[5 + E^(2*x)] - 16*Log[x])/16

________________________________________________________________________________________

fricas [A] time = 0.80, size = 26, normalized size = 1.08 \begin {gather*} -\frac {64 \, x^{4} \log \relax (x) - 64 \, x^{4} \log \left (e^{\left (2 \, x\right )} + 5\right ) - 9}{64 \, x^{4}} \end {gather*}

integrate(((32*x^5-16*x^4-9)*exp(x)^2-80*x^4-45)/(16*x^5*exp(x)^2+80*x^5),x, algorithm="fricas")

-1/64*(64*x^4*log(x) - 64*x^4*log(e^(2*x) + 5) - 9)/x^4

________________________________________________________________________________________

giac [A] time = 0.30, size = 26, normalized size = 1.08 \begin {gather*} -\frac {64 \, x^{4} \log \relax (x) - 64 \, x^{4} \log \left (e^{\left (2 \, x\right )} + 5\right ) - 9}{64 \, x^{4}} \end {gather*}

integrate(((32*x^5-16*x^4-9)*exp(x)^2-80*x^4-45)/(16*x^5*exp(x)^2+80*x^5),x, algorithm="giac")

-1/64*(64*x^4*log(x) - 64*x^4*log(e^(2*x) + 5) - 9)/x^4

________________________________________________________________________________________


method	result	size

norman	\(\frac {9}{64 x^{4}}-\ln \relax (x )+\ln \left (5+{\mathrm e}^{2 x}\right )\)	\(18\)
risch	\(\frac {9}{64 x^{4}}-\ln \relax (x )+\ln \left (5+{\mathrm e}^{2 x}\right )\)	\(18\)

int(((32*x^5-16*x^4-9)*exp(x)^2-80*x^4-45)/(16*x^5*exp(x)^2+80*x^5),x,method=_RETURNVERBOSE)

________________________________________________________________________________________

maxima [A] time = 0.46, size = 17, normalized size = 0.71 \begin {gather*} \frac {9}{64 \, x^{4}} - \log \relax (x) + \log \left (e^{\left (2 \, x\right )} + 5\right ) \end {gather*}

integrate(((32*x^5-16*x^4-9)*exp(x)^2-80*x^4-45)/(16*x^5*exp(x)^2+80*x^5),x, algorithm="maxima")

________________________________________________________________________________________

mupad [B] time = 0.33, size = 17, normalized size = 0.71 \begin {gather*} \ln \left ({\mathrm {e}}^{2\,x}+5\right )-\ln \relax (x)+\frac {9}{64\,x^4} \end {gather*}

int(-(exp(2*x)*(16*x^4 - 32*x^5 + 9) + 80*x^4 + 45)/(16*x^5*exp(2*x) + 80*x^5),x)

________________________________________________________________________________________

sympy [A] time = 0.14, size = 17, normalized size = 0.71 \begin {gather*} - \log {\relax (x )} + \log {\left (e^{2 x} + 5 \right )} + \frac {9}{64 x^{4}} \end {gather*}

integrate(((32*x**5-16*x**4-9)*exp(x)**2-80*x**4-45)/(16*x**5*exp(x)**2+80*x**5),x)

-log(x) + log(exp(2*x) + 5) + 9/(64*x**4)

________________________________________________________________________________________

3.3.48 \(\int \frac {-45-80 x^4+e^{2 x} (-9-16 x^4+32 x^5)}{80 x^5+16 e^{2 x} x^5} \, dx\)