\(\int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx\) [6557]

Optimal result

Integrand size = 19, antiderivative size = 18 \[ \int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx=x+\frac {28 \left (2+x-x^2\right )}{e^5 x^2} \]

[Out]

x+28*(-x^2+x+2)/x^2/exp(5)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 14} \[ \int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx=\frac {56}{e^5 x^2}+x+\frac {28}{e^5 x} \]

[In]

Int[(-112 - 28*x + E^5*x^3)/(E^5*x^3),x]

[Out]

56/(E^5*x^2) + 28/(E^5*x) + 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]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-112-28 x+e^5 x^3}{x^3} \, dx}{e^5} \\ & = \frac {\int \left (e^5-\frac {112}{x^3}-\frac {28}{x^2}\right ) \, dx}{e^5} \\ & = \frac {56}{e^5 x^2}+\frac {28}{e^5 x}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx=\frac {\frac {56}{x^2}+\frac {28}{x}+e^5 x}{e^5} \]

[In]

Integrate[(-112 - 28*x + E^5*x^3)/(E^5*x^3),x]

[Out]

(56/x^2 + 28/x + E^5*x)/E^5

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78

method	result	size
risch	\(x +\frac {{\mathrm e}^{-5} \left (28 x +56\right )}{x^{2}}\)	\(14\)
gosper	\(\frac {\left (x^{3} {\mathrm e}^{5}+56+28 x \right ) {\mathrm e}^{-5}}{x^{2}}\)	\(20\)
parallelrisch	\(\frac {\left (x^{3} {\mathrm e}^{5}+56+28 x \right ) {\mathrm e}^{-5}}{x^{2}}\)	\(20\)
default	\({\mathrm e}^{-5} \left (x \,{\mathrm e}^{5}+\frac {28}{x}+\frac {56}{x^{2}}\right )\)	\(21\)
norman	\(\frac {x^{3}+28 \,{\mathrm e}^{-5} x +56 \,{\mathrm e}^{-5}}{x^{2}}\)	\(22\)

[In]

int((x^3*exp(5)-28*x-112)/x^3/exp(5),x,method=_RETURNVERBOSE)

[Out]

x+exp(-5)*(28*x+56)/x^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx=\frac {{\left (x^{3} e^{5} + 28 \, x + 56\right )} e^{\left (-5\right )}}{x^{2}} \]

[In]

integrate((x^3*exp(5)-28*x-112)/x^3/exp(5),x, algorithm="fricas")

[Out]

(x^3*e^5 + 28*x + 56)*e^(-5)/x^2

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx=\frac {x e^{5} + \frac {28 x + 56}{x^{2}}}{e^{5}} \]

[In]

integrate((x**3*exp(5)-28*x-112)/x**3/exp(5),x)

[Out]

(x*exp(5) + (28*x + 56)/x**2)*exp(-5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx={\left (x e^{5} + \frac {28 \, {\left (x + 2\right )}}{x^{2}}\right )} e^{\left (-5\right )} \]

[In]

integrate((x^3*exp(5)-28*x-112)/x^3/exp(5),x, algorithm="maxima")

[Out]

(x*e^5 + 28*(x + 2)/x^2)*e^(-5)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx={\left (x e^{5} + \frac {28 \, {\left (x + 2\right )}}{x^{2}}\right )} e^{\left (-5\right )} \]

[In]

integrate((x^3*exp(5)-28*x-112)/x^3/exp(5),x, algorithm="giac")

[Out]

(x*e^5 + 28*(x + 2)/x^2)*e^(-5)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {-112-28 x+e^5 x^3}{e^5 x^3} \, dx=x+\frac {{\mathrm {e}}^{-5}\,\left (28\,x+56\right )}{x^2} \]

[In]

int(-(exp(-5)*(28*x - x^3*exp(5) + 112))/x^3,x)

[Out]

x + (exp(-5)*(28*x + 56))/x^2