\(\int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx\) [8274]

Optimal result

Integrand size = 19, antiderivative size = 15 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {\left (-5+e^x+\frac {x}{12}\right ) x}{e^4} \]

[Out]

x/exp(4)*(1/12*x+exp(x)-5)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(15)=30\).

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 2207, 2225} \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {x^2}{12 e^4}-\frac {5 x}{e^4}-e^{x-4}+e^{x-4} (x+1) \]

[In]

Int[(-30 + x + E^x*(6 + 6*x))/(6*E^4),x]

[Out]

-E^(-4 + x) - (5*x)/E^4 + x^2/(12*E^4) + E^(-4 + x)*(1 + 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 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-30+x+e^x (6+6 x)\right ) \, dx}{6 e^4} \\ & = -\frac {5 x}{e^4}+\frac {x^2}{12 e^4}+\frac {\int e^x (6+6 x) \, dx}{6 e^4} \\ & = -\frac {5 x}{e^4}+\frac {x^2}{12 e^4}+e^{-4+x} (1+x)-\frac {\int e^x \, dx}{e^4} \\ & = -e^{-4+x}-\frac {5 x}{e^4}+\frac {x^2}{12 e^4}+e^{-4+x} (1+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {-30 x+6 e^x x+\frac {x^2}{2}}{6 e^4} \]

[In]

Integrate[(-30 + x + E^x*(6 + 6*x))/(6*E^4),x]

[Out]

(-30*x + 6*E^x*x + x^2/2)/(6*E^4)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33

method	result	size
risch	\(x \,{\mathrm e}^{x -4}-5 x \,{\mathrm e}^{-4}+\frac {x^{2} {\mathrm e}^{-4}}{12}\)	\(20\)
default	\(\frac {{\mathrm e}^{-4} \left (-30 x +\frac {x^{2}}{2}+6 \,{\mathrm e}^{x} x \right )}{6}\)	\(21\)
parallelrisch	\(\frac {{\mathrm e}^{-4} \left (-30 x +\frac {x^{2}}{2}+6 \,{\mathrm e}^{x} x \right )}{6}\)	\(21\)
parts	\(x \,{\mathrm e}^{-4} {\mathrm e}^{x}+\frac {{\mathrm e}^{-4} \left (-30 x +\frac {1}{2} x^{2}\right )}{6}\)	\(25\)
norman	\(x \,{\mathrm e}^{-4} {\mathrm e}^{x}-5 x \,{\mathrm e}^{-4}+\frac {x^{2} {\mathrm e}^{-4}}{12}\)	\(26\)

[In]

int(1/6*((6+6*x)*exp(x)+x-30)/exp(4),x,method=_RETURNVERBOSE)

[Out]

x*exp(x-4)-5*x*exp(-4)+1/12*x^2*exp(-4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {1}{12} \, {\left (x^{2} + 12 \, x e^{x} - 60 \, x\right )} e^{\left (-4\right )} \]

[In]

integrate(1/6*((6+6*x)*exp(x)+x-30)/exp(4),x, algorithm="fricas")

[Out]

1/12*(x^2 + 12*x*e^x - 60*x)*e^(-4)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {x^{2}}{12 e^{4}} + \frac {x e^{x}}{e^{4}} - \frac {5 x}{e^{4}} \]

[In]

integrate(1/6*((6+6*x)*exp(x)+x-30)/exp(4),x)

[Out]

x**2*exp(-4)/12 + x*exp(-4)*exp(x) - 5*x*exp(-4)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {1}{12} \, {\left (x^{2} + 12 \, x e^{x} - 60 \, x\right )} e^{\left (-4\right )} \]

[In]

integrate(1/6*((6+6*x)*exp(x)+x-30)/exp(4),x, algorithm="maxima")

[Out]

1/12*(x^2 + 12*x*e^x - 60*x)*e^(-4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {1}{12} \, {\left (x^{2} + 12 \, x e^{x} - 60 \, x\right )} e^{\left (-4\right )} \]

[In]

integrate(1/6*((6+6*x)*exp(x)+x-30)/exp(4),x, algorithm="giac")

[Out]

1/12*(x^2 + 12*x*e^x - 60*x)*e^(-4)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {x\,{\mathrm {e}}^{-4}\,\left (x+12\,{\mathrm {e}}^x-60\right )}{12} \]

[In]

int(exp(-4)*(x/6 + (exp(x)*(6*x + 6))/6 - 5),x)

[Out]

(x*exp(-4)*(x + 12*exp(x) - 60))/12