\(\int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx\) [6438]

Optimal result

Integrand size = 24, antiderivative size = 20 \[ \int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx=-x+\frac {20 e^7 x (1+x)}{4+e^5} \]

[Out]

4*exp(7)*x*(1+x)/(4/5+1/5*exp(5))-x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12} \[ \int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx=\frac {5 e^7 (2 x+1)^2}{4+e^5}-x \]

[In]

Int[(-4 - E^5 + E^7*(20 + 40*x))/(4 + E^5),x]

[Out]

-x + (5*E^7*(1 + 2*x)^2)/(4 + E^5)

Rule 12

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-4-e^5+e^7 (20+40 x)\right ) \, dx}{4+e^5} \\ & = -x+\frac {5 e^7 (1+2 x)^2}{4+e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx=-\frac {4 x+e^5 x-20 e^7 x-20 e^7 x^2}{4+e^5} \]

[In]

Integrate[(-4 - E^5 + E^7*(20 + 40*x))/(4 + E^5),x]

[Out]

-((4*x + E^5*x - 20*E^7*x - 20*E^7*x^2)/(4 + E^5))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20

method	result	size
gosper	\(\frac {x \left (20 x \,{\mathrm e}^{7}+20 \,{\mathrm e}^{7}-{\mathrm e}^{5}-4\right )}{4+{\mathrm e}^{5}}\)	\(24\)
default	\(\frac {20 x^{2} {\mathrm e}^{7}+20 x \,{\mathrm e}^{7}-x \,{\mathrm e}^{5}-4 x}{4+{\mathrm e}^{5}}\)	\(29\)
parallelrisch	\(\frac {{\mathrm e}^{7} \left (20 x^{2}+20 x \right )+\left (-4-{\mathrm e}^{5}\right ) x}{4+{\mathrm e}^{5}}\)	\(29\)
norman	\(\frac {\left (20 \,{\mathrm e}^{7}-{\mathrm e}^{5}-4\right ) x}{4+{\mathrm e}^{5}}+\frac {20 \,{\mathrm e}^{7} x^{2}}{4+{\mathrm e}^{5}}\)	\(33\)
risch	\(\frac {20 \,{\mathrm e}^{7} x^{2}}{4+{\mathrm e}^{5}}+\frac {20 \,{\mathrm e}^{7} x}{4+{\mathrm e}^{5}}-\frac {x \,{\mathrm e}^{5}}{4+{\mathrm e}^{5}}-\frac {4 x}{4+{\mathrm e}^{5}}\)	\(46\)

[In]

int(((40*x+20)*exp(7)-exp(5)-4)/(4+exp(5)),x,method=_RETURNVERBOSE)

[Out]

x*(20*x*exp(7)+20*exp(7)-exp(5)-4)/(4+exp(5))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx=\frac {20 \, {\left (x^{2} + x\right )} e^{7} - x e^{5} - 4 \, x}{e^{5} + 4} \]

[In]

integrate(((40*x+20)*exp(7)-exp(5)-4)/(4+exp(5)),x, algorithm="fricas")

[Out]

(20*(x^2 + x)*e^7 - x*e^5 - 4*x)/(e^5 + 4)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx=\frac {20 x^{2} e^{7}}{4 + e^{5}} + \frac {x \left (- e^{5} - 4 + 20 e^{7}\right )}{4 + e^{5}} \]

[In]

integrate(((40*x+20)*exp(7)-exp(5)-4)/(4+exp(5)),x)

[Out]

20*x**2*exp(7)/(4 + exp(5)) + x*(-exp(5) - 4 + 20*exp(7))/(4 + exp(5))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx=\frac {20 \, {\left (x^{2} + x\right )} e^{7} - x e^{5} - 4 \, x}{e^{5} + 4} \]

[In]

integrate(((40*x+20)*exp(7)-exp(5)-4)/(4+exp(5)),x, algorithm="maxima")

[Out]

(20*(x^2 + x)*e^7 - x*e^5 - 4*x)/(e^5 + 4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx=\frac {20 \, {\left (x^{2} + x\right )} e^{7} - x e^{5} - 4 \, x}{e^{5} + 4} \]

[In]

integrate(((40*x+20)*exp(7)-exp(5)-4)/(4+exp(5)),x, algorithm="giac")

[Out]

(20*(x^2 + x)*e^7 - x*e^5 - 4*x)/(e^5 + 4)

Mupad [B] (verification not implemented)

Time = 12.91 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {-4-e^5+e^7 (20+40 x)}{4+e^5} \, dx=\frac {{\mathrm {e}}^{-7}\,{\left ({\mathrm {e}}^5-{\mathrm {e}}^7\,\left (40\,x+20\right )+4\right )}^2}{80\,\left ({\mathrm {e}}^5+4\right )} \]

[In]

int(-(exp(5) - exp(7)*(40*x + 20) + 4)/(exp(5) + 4),x)

[Out]

(exp(-7)*(exp(5) - exp(7)*(40*x + 20) + 4)^2)/(80*(exp(5) + 4))