∫ 220−5x+e2+x(−44+21x)_________________

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

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Rubi [A] time = 0.36, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {6741, 12, 6742, 2282, 36, 31, 29, 43} \begin {gather*} \frac {x}{4}+5 \log \left (5-e^{x+2}\right )-11 \log (x) \end {gather*}

Int[(220 - 5*x + E^(2 + x)*(-44 + 21*x))/(-20*x + 4*E^(2 + x)*x),x]

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

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[((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])

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 {-220+5 x-e^{2+x} (-44+21 x)}{4 \left (5-e^{2+x}\right ) x} \, dx\\ &=\frac {1}{4} \int \frac {-220+5 x-e^{2+x} (-44+21 x)}{\left (5-e^{2+x}\right ) x} \, dx\\ &=\frac {1}{4} \int \left (\frac {100}{-5+e^{2+x}}+\frac {-44+21 x}{x}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-44+21 x}{x} \, dx+25 \int \frac {1}{-5+e^{2+x}} \, dx\\ &=\frac {1}{4} \int \left (21-\frac {44}{x}\right ) \, dx+25 \operatorname {Subst}\left (\int \frac {1}{(-5+x) x} \, dx,x,e^{2+x}\right )\\ &=\frac {21 x}{4}-11 \log (x)+5 \operatorname {Subst}\left (\int \frac {1}{-5+x} \, dx,x,e^{2+x}\right )-5 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2+x}\right )\\ &=\frac {x}{4}+5 \log \left (5-e^{2+x}\right )-11 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.05, size = 22, normalized size = 0.92 \begin {gather*} \frac {x}{4}+5 \log \left (5-e^{2+x}\right )-11 \log (x) \end {gather*}

Integrate[(220 - 5*x + E^(2 + x)*(-44 + 21*x))/(-20*x + 4*E^(2 + x)*x),x]

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fricas [A] time = 0.80, size = 17, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, x - 11 \, \log \relax (x) + 5 \, \log \left (e^{\left (x + 2\right )} - 5\right ) \end {gather*}

integrate(((21*x-44)*exp(2+x)-5*x+220)/(4*x*exp(2+x)-20*x),x, algorithm="fricas")

1/4*x - 11*log(x) + 5*log(e^(x + 2) - 5)

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giac [A] time = 0.14, size = 17, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, x - 11 \, \log \relax (x) + 5 \, \log \left (e^{\left (x + 2\right )} - 5\right ) \end {gather*}

integrate(((21*x-44)*exp(2+x)-5*x+220)/(4*x*exp(2+x)-20*x),x, algorithm="giac")

1/4*x - 11*log(x) + 5*log(e^(x + 2) - 5)

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method	result	size

norman	\(\frac {x}{4}-11 \ln \relax (x )+5 \ln \left ({\mathrm e}^{2+x}-5\right )\)	\(18\)
risch	\(\frac {x}{4}-11 \ln \relax (x )-10+5 \ln \left ({\mathrm e}^{2+x}-5\right )\)	\(19\)

int(((21*x-44)*exp(2+x)-5*x+220)/(4*x*exp(2+x)-20*x),x,method=_RETURNVERBOSE)

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maxima [A] time = 0.37, size = 20, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, x + 5 \, \log \left ({\left (e^{\left (x + 2\right )} - 5\right )} e^{\left (-2\right )}\right ) - 11 \, \log \relax (x) \end {gather*}

integrate(((21*x-44)*exp(2+x)-5*x+220)/(4*x*exp(2+x)-20*x),x, algorithm="maxima")

1/4*x + 5*log((e^(x + 2) - 5)*e^(-2)) - 11*log(x)

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mupad [B] time = 0.08, size = 18, normalized size = 0.75 \begin {gather*} \frac {x}{4}+5\,\ln \left ({\mathrm {e}}^2\,{\mathrm {e}}^x-5\right )-11\,\ln \relax (x) \end {gather*}

int(-(exp(x + 2)*(21*x - 44) - 5*x + 220)/(20*x - 4*x*exp(x + 2)),x)

x/4 + 5*log(exp(2)*exp(x) - 5) - 11*log(x)

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sympy [A] time = 0.15, size = 17, normalized size = 0.71 \begin {gather*} \frac {x}{4} - 11 \log {\relax (x )} + 5 \log {\left (e^{x + 2} - 5 \right )} \end {gather*}

integrate(((21*x-44)*exp(2+x)-5*x+220)/(4*x*exp(2+x)-20*x),x)

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3.90.89 \(\int \frac {220-5 x+e^{2+x} (-44+21 x)}{-20 x+4 e^{2+x} x} \, dx\)