3.32.79 \(\int e^{-15+2 e^{-3+x}} (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4)) \, dx\)

Optimal. Leaf size=22 \[ 4-20736 e^{2 e^{-3+x}} \left (x+\frac {x}{e^3}\right )^4 \]

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Rubi [B]  time = 0.04, antiderivative size = 48, normalized size of antiderivative = 2.18, number of steps used = 5, number of rules used = 2, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6, 2288} \begin {gather*} -20736 e^{2 e^{x-3}-12} \left (e^{12} x^4+4 e^9 x^4+6 e^6 x^4+4 e^3 x^4+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-15 + 2*E^(-3 + x))*(-82944*E^3*x^3 - 331776*E^6*x^3 - 497664*E^9*x^3 - 331776*E^12*x^3 - 82944*E^15*x^
3 + E^x*(-41472*x^4 - 165888*E^3*x^4 - 248832*E^6*x^4 - 165888*E^9*x^4 - 41472*E^12*x^4)),x]

[Out]

-20736*E^(-12 + 2*E^(-3 + x))*(x^4 + 4*E^3*x^4 + 6*E^6*x^4 + 4*E^9*x^4 + E^12*x^4)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-15+2 e^{-3+x}} \left (-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+\left (-82944 e^3-331776 e^6\right ) x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx\\ &=\int e^{-15+2 e^{-3+x}} \left (-82944 e^{15} x^3+\left (-82944 e^3-331776 e^6\right ) x^3+\left (-497664 e^9-331776 e^{12}\right ) x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx\\ &=\int e^{-15+2 e^{-3+x}} \left (\left (-497664 e^9-331776 e^{12}\right ) x^3+\left (-82944 e^3-331776 e^6-82944 e^{15}\right ) x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx\\ &=\int e^{-15+2 e^{-3+x}} \left (\left (-82944 e^3-331776 e^6-497664 e^9-331776 e^{12}-82944 e^{15}\right ) x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx\\ &=-20736 e^{-12+2 e^{-3+x}} \left (x^4+4 e^3 x^4+6 e^6 x^4+4 e^9 x^4+e^{12} x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.05 \begin {gather*} -20736 e^{-12+2 e^{-3+x}} \left (1+e^3\right )^4 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-15 + 2*E^(-3 + x))*(-82944*E^3*x^3 - 331776*E^6*x^3 - 497664*E^9*x^3 - 331776*E^12*x^3 - 82944*E
^15*x^3 + E^x*(-41472*x^4 - 165888*E^3*x^4 - 248832*E^6*x^4 - 165888*E^9*x^4 - 41472*E^12*x^4)),x]

[Out]

-20736*E^(-12 + 2*E^(-3 + x))*(1 + E^3)^4*x^4

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fricas [B]  time = 0.47, size = 50, normalized size = 2.27 \begin {gather*} -20736 \, {\left (x^{4} e^{15} + 4 \, x^{4} e^{12} + 6 \, x^{4} e^{9} + 4 \, x^{4} e^{6} + x^{4} e^{3}\right )} e^{\left (-{\left (15 \, e^{3} - 2 \, e^{x}\right )} e^{\left (-3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-165888*x^4*exp(3)-41472*x^4)*exp(x)-82
944*x^3*exp(3)^5-331776*x^3*exp(3)^4-497664*x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(
3))^2/exp(3)^5,x, algorithm="fricas")

[Out]

-20736*(x^4*e^15 + 4*x^4*e^12 + 6*x^4*e^9 + 4*x^4*e^6 + x^4*e^3)*e^(-(15*e^3 - 2*e^x)*e^(-3))

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giac [B]  time = 0.21, size = 71, normalized size = 3.23 \begin {gather*} -20736 \, {\left (x^{4} e^{\left (2 \, e^{\left (x - 3\right )}\right )} + x^{4} e^{\left (2 \, e^{\left (x - 3\right )} + 12\right )} + 4 \, x^{4} e^{\left (2 \, e^{\left (x - 3\right )} + 9\right )} + 6 \, x^{4} e^{\left (2 \, e^{\left (x - 3\right )} + 6\right )} + 4 \, x^{4} e^{\left (2 \, e^{\left (x - 3\right )} + 3\right )}\right )} e^{\left (-12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-165888*x^4*exp(3)-41472*x^4)*exp(x)-82
944*x^3*exp(3)^5-331776*x^3*exp(3)^4-497664*x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(
3))^2/exp(3)^5,x, algorithm="giac")

[Out]

-20736*(x^4*e^(2*e^(x - 3)) + x^4*e^(2*e^(x - 3) + 12) + 4*x^4*e^(2*e^(x - 3) + 9) + 6*x^4*e^(2*e^(x - 3) + 6)
+ 4*x^4*e^(2*e^(x - 3) + 3))*e^(-12)

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maple [A]  time = 0.10, size = 31, normalized size = 1.41

method	result	size
risch	\(-20736 \left ({\mathrm e}^{12}+4 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{6}+4 \,{\mathrm e}^{3}+1\right ) x^{4} {\mathrm e}^{-12+2 \,{\mathrm e}^{x -3}}\)	\(31\)
norman	\(\left (-20736 \,{\mathrm e}^{12}-82944 \,{\mathrm e}^{9}-124416 \,{\mathrm e}^{6}-82944 \,{\mathrm e}^{3}-20736\right ) x^{4} {\mathrm e}^{2 \,{\mathrm e}^{x -3}} {\mathrm e}^{-12}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-165888*x^4*exp(3)-41472*x^4)*exp(x)-82944*x^
3*exp(3)^5-331776*x^3*exp(3)^4-497664*x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(3))^2/
exp(3)^5,x,method=_RETURNVERBOSE)

[Out]

-20736*(exp(12)+4*exp(9)+6*exp(6)+4*exp(3)+1)*x^4*exp(-12+2*exp(x-3))

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maxima [A]  time = 0.76, size = 30, normalized size = 1.36 \begin {gather*} -20736 \, x^{4} {\left (e^{12} + 4 \, e^{9} + 6 \, e^{6} + 4 \, e^{3} + 1\right )} e^{\left (2 \, e^{\left (x - 3\right )} - 12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-165888*x^4*exp(3)-41472*x^4)*exp(x)-82
944*x^3*exp(3)^5-331776*x^3*exp(3)^4-497664*x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(
3))^2/exp(3)^5,x, algorithm="maxima")

[Out]

-20736*x^4*(e^12 + 4*e^9 + 6*e^6 + 4*e^3 + 1)*e^(2*e^(x - 3) - 12)

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mupad [B]  time = 1.97, size = 30, normalized size = 1.36 \begin {gather*} -20736\,x^4\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-12}\,\left (4\,{\mathrm {e}}^3+6\,{\mathrm {e}}^6+4\,{\mathrm {e}}^9+{\mathrm {e}}^{12}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(2*exp(-3)*exp(x))*exp(-15)*(exp(x)*(165888*x^4*exp(3) + 248832*x^4*exp(6) + 165888*x^4*exp(9) + 41472
*x^4*exp(12) + 41472*x^4) + 82944*x^3*exp(3) + 331776*x^3*exp(6) + 497664*x^3*exp(9) + 331776*x^3*exp(12) + 82
944*x^3*exp(15)),x)

[Out]

-20736*x^4*exp(2*exp(-3)*exp(x))*exp(-12)*(4*exp(3) + 6*exp(6) + 4*exp(9) + exp(12) + 1)

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sympy [A]  time = 0.36, size = 53, normalized size = 2.41 \begin {gather*} \frac {\left (- 20736 x^{4} e^{12} - 82944 x^{4} e^{9} - 124416 x^{4} e^{6} - 82944 x^{4} e^{3} - 20736 x^{4}\right ) e^{\frac {2 e^{x}}{e^{3}}}}{e^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-41472*x**4*exp(3)**4-165888*x**4*exp(3)**3-248832*x**4*exp(3)**2-165888*x**4*exp(3)-41472*x**4)*e
xp(x)-82944*x**3*exp(3)**5-331776*x**3*exp(3)**4-497664*x**3*exp(3)**3-331776*x**3*exp(3)**2-82944*x**3*exp(3)
)*exp(exp(x)/exp(3))**2/exp(3)**5,x)

[Out]

(-20736*x**4*exp(12) - 82944*x**4*exp(9) - 124416*x**4*exp(6) - 82944*x**4*exp(3) - 20736*x**4)*exp(-12)*exp(2
*exp(-3)*exp(x))

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