\(\int \frac {-8+24 x^2+e^x (-4+4 x-x^2)}{3 x^2} \, dx\) [6645]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{3 x^2} \, dx=\frac {1}{3} \left (2+e^x\right ) \left (\frac {3}{x}+\frac {1-x}{x}\right )+8 x \]

[Out]

1/3*(exp(x)+2)*(3/x+(1-x)/x)+8*x

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 14, 2230, 2225, 2208, 2209} \[ \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{3 x^2} \, dx=8 x-\frac {e^x}{3}+\frac {4 e^x}{3 x}+\frac {8}{3 x} \]

[In]

Int[(-8 + 24*x^2 + E^x*(-4 + 4*x - x^2))/(3*x^2),x]

[Out]

-1/3*E^x + 8/(3*x) + (4*E^x)/(3*x) + 8*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 14

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]]

Rule 2208

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] &&  !TrueQ[$UseGamm
a]

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 2225

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

Rule 2230

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]
&& IntegerQ[m] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{x^2} \, dx \\ & = \frac {1}{3} \int \left (-\frac {e^x (-2+x)^2}{x^2}+\frac {8 \left (-1+3 x^2\right )}{x^2}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {e^x (-2+x)^2}{x^2} \, dx\right )+\frac {8}{3} \int \frac {-1+3 x^2}{x^2} \, dx \\ & = -\left (\frac {1}{3} \int \left (e^x+\frac {4 e^x}{x^2}-\frac {4 e^x}{x}\right ) \, dx\right )+\frac {8}{3} \int \left (3-\frac {1}{x^2}\right ) \, dx \\ & = \frac {8}{3 x}+8 x-\frac {\int e^x \, dx}{3}-\frac {4}{3} \int \frac {e^x}{x^2} \, dx+\frac {4}{3} \int \frac {e^x}{x} \, dx \\ & = -\frac {e^x}{3}+\frac {8}{3 x}+\frac {4 e^x}{3 x}+8 x+\frac {4 \text {Ei}(x)}{3}-\frac {4}{3} \int \frac {e^x}{x} \, dx \\ & = -\frac {e^x}{3}+\frac {8}{3 x}+\frac {4 e^x}{3 x}+8 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{3 x^2} \, dx=\frac {1}{3} \left (-e^x \left (1-\frac {4}{x}\right )+\frac {8}{x}+24 x\right ) \]

[In]

Integrate[(-8 + 24*x^2 + E^x*(-4 + 4*x - x^2))/(3*x^2),x]

[Out]

(-(E^x*(1 - 4/x)) + 8/x + 24*x)/3

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71

method	result	size
risch	\(8 x +\frac {8}{3 x}-\frac {\left (x -4\right ) {\mathrm e}^{x}}{3 x}\)	\(20\)
default	\(8 x +\frac {8}{3 x}+\frac {4 \,{\mathrm e}^{x}}{3 x}-\frac {{\mathrm e}^{x}}{3}\)	\(21\)
norman	\(\frac {\frac {8}{3}+8 x^{2}-\frac {{\mathrm e}^{x} x}{3}+\frac {4 \,{\mathrm e}^{x}}{3}}{x}\)	\(21\)
parts	\(8 x +\frac {8}{3 x}+\frac {4 \,{\mathrm e}^{x}}{3 x}-\frac {{\mathrm e}^{x}}{3}\)	\(21\)
parallelrisch	\(\frac {24 x^{2}-{\mathrm e}^{x} x +8+4 \,{\mathrm e}^{x}}{3 x}\)	\(22\)

[In]

int(1/3*((-x^2+4*x-4)*exp(x)+24*x^2-8)/x^2,x,method=_RETURNVERBOSE)

[Out]

8*x+8/3/x-1/3*(x-4)/x*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{3 x^2} \, dx=\frac {24 \, x^{2} - {\left (x - 4\right )} e^{x} + 8}{3 \, x} \]

[In]

integrate(1/3*((-x^2+4*x-4)*exp(x)+24*x^2-8)/x^2,x, algorithm="fricas")

[Out]

1/3*(24*x^2 - (x - 4)*e^x + 8)/x

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{3 x^2} \, dx=8 x + \frac {\left (4 - x\right ) e^{x}}{3 x} + \frac {8}{3 x} \]

[In]

integrate(1/3*((-x**2+4*x-4)*exp(x)+24*x**2-8)/x**2,x)

[Out]

8*x + (4 - x)*exp(x)/(3*x) + 8/(3*x)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{3 x^2} \, dx=8 \, x + \frac {8}{3 \, x} + \frac {4}{3} \, {\rm Ei}\left (x\right ) - \frac {1}{3} \, e^{x} - \frac {4}{3} \, \Gamma \left (-1, -x\right ) \]

[In]

integrate(1/3*((-x^2+4*x-4)*exp(x)+24*x^2-8)/x^2,x, algorithm="maxima")

[Out]

8*x + 8/3/x + 4/3*Ei(x) - 1/3*e^x - 4/3*gamma(-1, -x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{3 x^2} \, dx=\frac {24 \, x^{2} - x e^{x} + 4 \, e^{x} + 8}{3 \, x} \]

[In]

integrate(1/3*((-x^2+4*x-4)*exp(x)+24*x^2-8)/x^2,x, algorithm="giac")

[Out]

1/3*(24*x^2 - x*e^x + 4*e^x + 8)/x

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64 \[ \int \frac {-8+24 x^2+e^x \left (-4+4 x-x^2\right )}{3 x^2} \, dx=8\,x-\frac {{\mathrm {e}}^x}{3}+\frac {\frac {4\,{\mathrm {e}}^x}{3}+\frac {8}{3}}{x} \]

[In]

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

[Out]

8*x - exp(x)/3 + ((4*exp(x))/3 + 8/3)/x