3.23.95 \(\int \frac {-360+8 x+e^4 (-1800-40 e^2+80 x)+e^8 (-200 e^2+200 x)}{75 e^8} \, dx\)

Optimal. Leaf size=24 \[ \frac {4}{3} \left (-e^2+\frac {-9+\frac {x}{5}}{e^4}+x\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 2.04, number of steps used = 2, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {12} \begin {gather*} \frac {4 x^2}{75 e^8}+\frac {2 \left (-2 x+e^2+45\right )^2}{15 e^4}+\frac {4}{3} \left (e^2-x\right )^2-\frac {24 x}{5 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-360 + 8*x + E^4*(-1800 - 40*E^2 + 80*x) + E^8*(-200*E^2 + 200*x))/(75*E^8),x]

[Out]

(2*(45 + E^2 - 2*x)^2)/(15*E^4) + (4*(E^2 - x)^2)/3 - (24*x)/(5*E^8) + (4*x^2)/(75*E^8)

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 {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (-360+8 x+e^4 \left (-1800-40 e^2+80 x\right )+e^8 \left (-200 e^2+200 x\right )\right ) \, dx}{75 e^8}\\ &=\frac {2 \left (45+e^2-2 x\right )^2}{15 e^4}+\frac {4}{3} \left (e^2-x\right )^2-\frac {24 x}{5 e^8}+\frac {4 x^2}{75 e^8}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 41, normalized size = 1.71 \begin {gather*} -\frac {8 \left (1+5 e^4\right ) \left (45 x+5 e^6 x-\frac {x^2}{2}-\frac {5 e^4 x^2}{2}\right )}{75 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-360 + 8*x + E^4*(-1800 - 40*E^2 + 80*x) + E^8*(-200*E^2 + 200*x))/(75*E^8),x]

[Out]

(-8*(1 + 5*E^4)*(45*x + 5*E^6*x - x^2/2 - (5*E^4*x^2)/2))/(75*E^8)

________________________________________________________________________________________

fricas [B]  time = 0.86, size = 39, normalized size = 1.62 \begin {gather*} \frac {4}{75} \, {\left (25 \, x^{2} e^{8} + x^{2} - 50 \, x e^{10} - 10 \, x e^{6} + 10 \, {\left (x^{2} - 45 \, x\right )} e^{4} - 90 \, x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/75*((-200*exp(1)^2+200*x)*exp(4)^2+(-40*exp(1)^2+80*x-1800)*exp(4)+8*x-360)/exp(4)^2,x, algorithm=
"fricas")

[Out]

4/75*(25*x^2*e^8 + x^2 - 50*x*e^10 - 10*x*e^6 + 10*(x^2 - 45*x)*e^4 - 90*x)*e^(-8)

________________________________________________________________________________________

giac [B]  time = 0.18, size = 40, normalized size = 1.67 \begin {gather*} \frac {4}{75} \, {\left (x^{2} + 25 \, {\left (x^{2} - 2 \, x e^{2}\right )} e^{8} + 10 \, {\left (x^{2} - x e^{2} - 45 \, x\right )} e^{4} - 90 \, x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/75*((-200*exp(1)^2+200*x)*exp(4)^2+(-40*exp(1)^2+80*x-1800)*exp(4)+8*x-360)/exp(4)^2,x, algorithm=
"giac")

[Out]

4/75*(x^2 + 25*(x^2 - 2*x*e^2)*e^8 + 10*(x^2 - x*e^2 - 45*x)*e^4 - 90*x)*e^(-8)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 32, normalized size = 1.33

method	result	size
gosper	\(-\frac {4 \left (1+5 \,{\mathrm e}^{4}\right ) x \left (10 \,{\mathrm e}^{2} {\mathrm e}^{4}-5 x \,{\mathrm e}^{4}-x +90\right ) {\mathrm e}^{-8}}{75}\)	\(32\)
risch	\(-\frac {8 \,{\mathrm e}^{2} x}{3}+\frac {4 x^{2}}{3}-\frac {8 x \,{\mathrm e}^{-2}}{15}+\frac {8 x^{2} {\mathrm e}^{-4}}{15}-24 x \,{\mathrm e}^{-4}+\frac {4 x^{2} {\mathrm e}^{-8}}{75}-\frac {24 \,{\mathrm e}^{-8} x}{5}\)	\(41\)
default	\(\frac {{\mathrm e}^{-8} \left ({\mathrm e}^{8} \left (-200 \,{\mathrm e}^{2} x +100 x^{2}\right )+{\mathrm e}^{4} \left (-40 \,{\mathrm e}^{2} x +40 x^{2}-1800 x \right )+4 x^{2}-360 x \right )}{75}\)	\(53\)
norman	\(\left (\frac {4 \left (25 \,{\mathrm e}^{8}+10 \,{\mathrm e}^{4}+1\right ) {\mathrm e}^{-4} x^{2}}{75}-\frac {8 \left (5 \,{\mathrm e}^{2} {\mathrm e}^{8}+{\mathrm e}^{2} {\mathrm e}^{4}+45 \,{\mathrm e}^{4}+9\right ) {\mathrm e}^{-4} x}{15}\right ) {\mathrm e}^{-4}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/75*((-200*exp(1)^2+200*x)*exp(4)^2+(-40*exp(1)^2+80*x-1800)*exp(4)+8*x-360)/exp(4)^2,x,method=_RETURNVER
BOSE)

[Out]

-4/75*(1+5*exp(4))*x*(10*exp(1)^2*exp(4)-5*x*exp(4)-x+90)/exp(4)^2

________________________________________________________________________________________

maxima [B]  time = 0.43, size = 40, normalized size = 1.67 \begin {gather*} \frac {4}{75} \, {\left (x^{2} + 25 \, {\left (x^{2} - 2 \, x e^{2}\right )} e^{8} + 10 \, {\left (x^{2} - x e^{2} - 45 \, x\right )} e^{4} - 90 \, x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/75*((-200*exp(1)^2+200*x)*exp(4)^2+(-40*exp(1)^2+80*x-1800)*exp(4)+8*x-360)/exp(4)^2,x, algorithm=
"maxima")

[Out]

4/75*(x^2 + 25*(x^2 - 2*x*e^2)*e^8 + 10*(x^2 - x*e^2 - 45*x)*e^4 - 90*x)*e^(-8)

________________________________________________________________________________________

mupad [B]  time = 1.37, size = 23, normalized size = 0.96 \begin {gather*} \frac {4\,x\,{\mathrm {e}}^{-8}\,\left (5\,{\mathrm {e}}^4+1\right )\,\left (x-10\,{\mathrm {e}}^6+5\,x\,{\mathrm {e}}^4-90\right )}{75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-8)*((8*x)/75 + (exp(8)*(200*x - 200*exp(2)))/75 - (exp(4)*(40*exp(2) - 80*x + 1800))/75 - 24/5),x)

[Out]

(4*x*exp(-8)*(5*exp(4) + 1)*(x - 10*exp(6) + 5*x*exp(4) - 90))/75

________________________________________________________________________________________

sympy [B]  time = 0.08, size = 44, normalized size = 1.83 \begin {gather*} \frac {x^{2} \left (4 + 40 e^{4} + 100 e^{8}\right )}{75 e^{8}} + \frac {x \left (- 40 e^{10} - 360 e^{4} - 8 e^{6} - 72\right )}{15 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/75*((-200*exp(1)**2+200*x)*exp(4)**2+(-40*exp(1)**2+80*x-1800)*exp(4)+8*x-360)/exp(4)**2,x)

[Out]

x**2*(4 + 40*exp(4) + 100*exp(8))*exp(-8)/75 + x*(-40*exp(10) - 360*exp(4) - 8*exp(6) - 72)*exp(-8)/15

________________________________________________________________________________________