\(\int e^{-320-340 x-20 x^2+e^{1+x} (80 x+85 x^2+5 x^3)} (-340-40 x+e^{1+x} (80+250 x+100 x^2+5 x^3)) \, dx\) [4409]

Optimal result

Integrand size = 59, antiderivative size = 19 \[ \int e^{-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx=e^{5 (1+x) (16+x) \left (-4+e^{1+x} x\right )} \]

[Out]

exp(5*(x*exp(1+x)-4)*(1+x)*(x+16))

Rubi [F]

\[ \int e^{-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx=\int \exp \left (-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )\right ) \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx \]

[In]

Int[E^(-320 - 340*x - 20*x^2 + E^(1 + x)*(80*x + 85*x^2 + 5*x^3))*(-340 - 40*x + E^(1 + x)*(80 + 250*x + 100*x
^2 + 5*x^3)),x]

[Out]

-340*Defer[Int][E^(5*(-4 + E^(1 + x)*x)*(16 + 17*x + x^2)), x] + 80*Defer[Int][E^(1 + x + 5*(-4 + E^(1 + x)*x)
*(16 + 17*x + x^2)), x] - 40*Defer[Int][E^(5*(-4 + E^(1 + x)*x)*(16 + 17*x + x^2))*x, x] + 250*Defer[Int][E^(1
+ x + 5*(-4 + E^(1 + x)*x)*(16 + 17*x + x^2))*x, x] + 100*Defer[Int][E^(1 + x + 5*(-4 + E^(1 + x)*x)*(16 + 17
*x + x^2))*x^2, x] + 5*Defer[Int][E^(1 + x + 5*(-4 + E^(1 + x)*x)*(16 + 17*x + x^2))*x^3, x]

Rubi steps \begin{align*} \text {integral}& = \int e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx \\ & = \int \left (-340 e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )}-40 e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x+5 e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} \left (16+50 x+20 x^2+x^3\right )\right ) \, dx \\ & = 5 \int e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} \left (16+50 x+20 x^2+x^3\right ) \, dx-40 \int e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x \, dx-340 \int e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} \, dx \\ & = 5 \int \left (16 e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )}+50 e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x+20 e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x^2+e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x^3\right ) \, dx-40 \int e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x \, dx-340 \int e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} \, dx \\ & = 5 \int e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x^3 \, dx-40 \int e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x \, dx+80 \int e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} \, dx+100 \int e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x^2 \, dx+250 \int e^{1+x+5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} x \, dx-340 \int e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx=e^{5 \left (-4+e^{1+x} x\right ) \left (16+17 x+x^2\right )} \]

[In]

Integrate[E^(-320 - 340*x - 20*x^2 + E^(1 + x)*(80*x + 85*x^2 + 5*x^3))*(-340 - 40*x + E^(1 + x)*(80 + 250*x +
100*x^2 + 5*x^3)),x]

[Out]

E^(5*(-4 + E^(1 + x)*x)*(16 + 17*x + x^2))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
risch	\({\mathrm e}^{5 \left (x \,{\mathrm e}^{1+x}-4\right ) \left (1+x \right ) \left (x +16\right )}\)	\(18\)
norman	\({\mathrm e}^{\left (5 x^{3}+85 x^{2}+80 x \right ) {\mathrm e}^{1+x}-20 x^{2}-340 x -320}\)	\(31\)
parallelrisch	\({\mathrm e}^{\left (5 x^{3}+85 x^{2}+80 x \right ) {\mathrm e}^{1+x}-20 x^{2}-340 x -320}\)	\(31\)

[In]

int(((5*x^3+100*x^2+250*x+80)*exp(1+x)-40*x-340)*exp((5*x^3+85*x^2+80*x)*exp(1+x)-20*x^2-340*x-320),x,method=_
RETURNVERBOSE)

[Out]

exp(5*(x*exp(1+x)-4)*(1+x)*(x+16))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int e^{-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx=e^{\left (-20 \, x^{2} + 5 \, {\left (x^{3} + 17 \, x^{2} + 16 \, x\right )} e^{\left (x + 1\right )} - 340 \, x - 320\right )} \]

[In]

integrate(((5*x^3+100*x^2+250*x+80)*exp(1+x)-40*x-340)*exp((5*x^3+85*x^2+80*x)*exp(1+x)-20*x^2-340*x-320),x, a
lgorithm="fricas")

[Out]

e^(-20*x^2 + 5*(x^3 + 17*x^2 + 16*x)*e^(x + 1) - 340*x - 320)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int e^{-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx=e^{- 20 x^{2} - 340 x + \left (5 x^{3} + 85 x^{2} + 80 x\right ) e^{x + 1} - 320} \]

[In]

integrate(((5*x**3+100*x**2+250*x+80)*exp(1+x)-40*x-340)*exp((5*x**3+85*x**2+80*x)*exp(1+x)-20*x**2-340*x-320)
,x)

[Out]

exp(-20*x**2 - 340*x + (5*x**3 + 85*x**2 + 80*x)*exp(x + 1) - 320)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int e^{-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx=e^{\left (5 \, x^{3} e^{\left (x + 1\right )} + 85 \, x^{2} e^{\left (x + 1\right )} - 20 \, x^{2} + 80 \, x e^{\left (x + 1\right )} - 340 \, x - 320\right )} \]

[In]

integrate(((5*x^3+100*x^2+250*x+80)*exp(1+x)-40*x-340)*exp((5*x^3+85*x^2+80*x)*exp(1+x)-20*x^2-340*x-320),x, a
lgorithm="maxima")

[Out]

e^(5*x^3*e^(x + 1) + 85*x^2*e^(x + 1) - 20*x^2 + 80*x*e^(x + 1) - 340*x - 320)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int e^{-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx=e^{\left (5 \, x^{3} e^{\left (x + 1\right )} + 85 \, x^{2} e^{\left (x + 1\right )} - 20 \, x^{2} + 80 \, x e^{\left (x + 1\right )} - 340 \, x - 320\right )} \]

[In]

integrate(((5*x^3+100*x^2+250*x+80)*exp(1+x)-40*x-340)*exp((5*x^3+85*x^2+80*x)*exp(1+x)-20*x^2-340*x-320),x, a
lgorithm="giac")

[Out]

e^(5*x^3*e^(x + 1) + 85*x^2*e^(x + 1) - 20*x^2 + 80*x*e^(x + 1) - 340*x - 320)

Mupad [B] (verification not implemented)

Time = 10.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.16 \[ \int e^{-320-340 x-20 x^2+e^{1+x} \left (80 x+85 x^2+5 x^3\right )} \left (-340-40 x+e^{1+x} \left (80+250 x+100 x^2+5 x^3\right )\right ) \, dx={\mathrm {e}}^{-340\,x}\,{\mathrm {e}}^{-320}\,{\mathrm {e}}^{80\,x\,\mathrm {e}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-20\,x^2}\,{\mathrm {e}}^{5\,x^3\,\mathrm {e}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{85\,x^2\,\mathrm {e}\,{\mathrm {e}}^x} \]

[In]

int(-exp(exp(x + 1)*(80*x + 85*x^2 + 5*x^3) - 340*x - 20*x^2 - 320)*(40*x - exp(x + 1)*(250*x + 100*x^2 + 5*x^
3 + 80) + 340),x)

[Out]

exp(-340*x)*exp(-320)*exp(80*x*exp(1)*exp(x))*exp(-20*x^2)*exp(5*x^3*exp(1)*exp(x))*exp(85*x^2*exp(1)*exp(x))