∫ −32+e2x(−2+4x)+ex(20−20x−2x2)_________________________

Optimal. Leaf size=19 \[ \frac {\left (4-2 e^x\right ) \left (8-e^x+x\right )}{x} \]

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Rubi [A] time = 0.11, antiderivative size = 29, normalized size of antiderivative = 1.53, number of steps used = 9, number of rules used = 6, integrand size = 31, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.194, Rules used = {14, 2197, 2199, 2194, 2177, 2178} \begin {gather*} -2 e^x-\frac {20 e^x}{x}+\frac {2 e^{2 x}}{x}+\frac {32}{x} \end {gather*}

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[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m

+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), 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] && LtQ[m, -1] && IntegerQ[2*m] && !$UseGamma ===

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral

Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !$UseGamma === True

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],

e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c

*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo

werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {32}{x^2}+\frac {2 e^{2 x} (-1+2 x)}{x^2}-\frac {2 e^x \left (-10+10 x+x^2\right )}{x^2}\right ) \, dx\\ &=\frac {32}{x}+2 \int \frac {e^{2 x} (-1+2 x)}{x^2} \, dx-2 \int \frac {e^x \left (-10+10 x+x^2\right )}{x^2} \, dx\\ &=\frac {32}{x}+\frac {2 e^{2 x}}{x}-2 \int \left (e^x-\frac {10 e^x}{x^2}+\frac {10 e^x}{x}\right ) \, dx\\ &=\frac {32}{x}+\frac {2 e^{2 x}}{x}-2 \int e^x \, dx+20 \int \frac {e^x}{x^2} \, dx-20 \int \frac {e^x}{x} \, dx\\ &=-2 e^x+\frac {32}{x}-\frac {20 e^x}{x}+\frac {2 e^{2 x}}{x}-20 \text {Ei}(x)+20 \int \frac {e^x}{x} \, dx\\ &=-2 e^x+\frac {32}{x}-\frac {20 e^x}{x}+\frac {2 e^{2 x}}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.03, size = 21, normalized size = 1.11 \begin {gather*} -\frac {2 \left (-16-e^{2 x}+e^x (10+x)\right )}{x} \end {gather*}

Integrate[(-32 + E^(2*x)*(-2 + 4*x) + E^x*(20 - 20*x - 2*x^2))/x^2,x]

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fricas [A] time = 0.73, size = 19, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left ({\left (x + 10\right )} e^{x} - e^{\left (2 \, x\right )} - 16\right )}}{x} \end {gather*}

integrate(((4*x-2)*exp(x)^2+(-2*x^2-20*x+20)*exp(x)-32)/x^2,x, algorithm="fricas")

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giac [A] time = 0.13, size = 21, normalized size = 1.11 \begin {gather*} -\frac {2 \, {\left (x e^{x} - e^{\left (2 \, x\right )} + 10 \, e^{x} - 16\right )}}{x} \end {gather*}

integrate(((4*x-2)*exp(x)^2+(-2*x^2-20*x+20)*exp(x)-32)/x^2,x, algorithm="giac")

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method	result	size

norman	$\frac {32+2 \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x -20 \,{\mathrm e}^{x}}{x}$	$22$
risch	$\frac {32}{x}+\frac {2 \,{\mathrm e}^{2 x}}{x}-\frac {2 \left (x +10\right ) {\mathrm e}^{x}}{x}$	$26$
default	$\frac {32}{x}-\frac {20 \,{\mathrm e}^{x}}{x}+\frac {2 \,{\mathrm e}^{2 x}}{x}-2 \,{\mathrm e}^{x}$	$27$

int(((4*x-2)*exp(x)^2+(-2*x^2-20*x+20)*exp(x)-32)/x^2,x,method=_RETURNVERBOSE)

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maxima [C] time = 0.40, size = 34, normalized size = 1.79 \begin {gather*} \frac {32}{x} + 4 \, {\rm Ei}\left (2 \, x\right ) - 20 \, {\rm Ei}\relax (x) - 2 \, e^{x} + 20 \, \Gamma \left (-1, -x\right ) - 4 \, \Gamma \left (-1, -2 \, x\right ) \end {gather*}

integrate(((4*x-2)*exp(x)^2+(-2*x^2-20*x+20)*exp(x)-32)/x^2,x, algorithm="maxima")

32/x + 4*Ei(2*x) - 20*Ei(x) - 2*e^x + 20*gamma(-1, -x) - 4*gamma(-1, -2*x)

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mupad [B] time = 0.05, size = 21, normalized size = 1.11 \begin {gather*} \frac {2\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^x+32}{x}-2\,{\mathrm {e}}^x \end {gather*}

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sympy [A] time = 0.11, size = 27, normalized size = 1.42 \begin {gather*} \frac {32}{x} + \frac {2 x e^{2 x} + \left (- 2 x^{2} - 20 x\right ) e^{x}}{x^{2}} \end {gather*}

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3.65.40 \(\int \frac {-32+e^{2 x} (-2+4 x)+e^x (20-20 x-2 x^2)}{x^2} \, dx\)