3.60.68 \(\int e^{-x+\frac {1}{5} e^{-x} (1125 x+150 x^2+5 x^3+e^x (225 x^3+30 x^4+x^5))} (225-165 x-27 x^2-x^3+e^x (135 x^2+24 x^3+x^4)) \, dx\)

Optimal. Leaf size=26 \[ e^{\left (\frac {e^{-x}}{x}+\frac {x}{5}\right ) x^2 (15+x)^2} \]

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Rubi [F]  time = 4.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) \left (225-165 x-27 x^2-x^3+e^x \left (135 x^2+24 x^3+x^4\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-x + (1125*x + 150*x^2 + 5*x^3 + E^x*(225*x^3 + 30*x^4 + x^5))/(5*E^x))*(225 - 165*x - 27*x^2 - x^3 + E
^x*(135*x^2 + 24*x^3 + x^4)),x]

[Out]

225*Defer[Int][E^(-x + 45*x^3 + 6*x^4 + x^5/5 + (x*(15 + x)^2)/E^x), x] - 165*Defer[Int][E^(-x + 45*x^3 + 6*x^
4 + x^5/5 + (x*(15 + x)^2)/E^x)*x, x] + 135*Defer[Int][E^((x*(15 + x)^2*(5 + E^x*x^2))/(5*E^x))*x^2, x] - 27*D
efer[Int][E^(-x + 45*x^3 + 6*x^4 + x^5/5 + (x*(15 + x)^2)/E^x)*x^2, x] + 24*Defer[Int][E^((x*(15 + x)^2*(5 + E
^x*x^2))/(5*E^x))*x^3, x] - Defer[Int][E^(-x + 45*x^3 + 6*x^4 + x^5/5 + (x*(15 + x)^2)/E^x)*x^3, x] + Defer[In
t][E^((x*(15 + x)^2*(5 + E^x*x^2))/(5*E^x))*x^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) (15+x) \left (15-12 x-x^2+9 e^x x^2+e^x x^3\right ) \, dx\\ &=\int \left (225 \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right )-165 \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) x-27 \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) x^2-\exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) x^3+\exp \left (\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) x^2 \left (135+24 x+x^2\right )\right ) \, dx\\ &=-\left (27 \int \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) x^2 \, dx\right )-165 \int \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) x \, dx+225 \int \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) \, dx-\int \exp \left (-x+\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) x^3 \, dx+\int \exp \left (\frac {1}{5} e^{-x} \left (1125 x+150 x^2+5 x^3+e^x \left (225 x^3+30 x^4+x^5\right )\right )\right ) x^2 \left (135+24 x+x^2\right ) \, dx\\ &=-\left (27 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x^2 \, dx\right )-165 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x \, dx+225 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) \, dx-\int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x^3 \, dx+\int e^{\frac {1}{5} e^{-x} x (15+x)^2 \left (5+e^x x^2\right )} x^2 \left (135+24 x+x^2\right ) \, dx\\ &=-\left (27 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x^2 \, dx\right )-165 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x \, dx+225 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) \, dx-\int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x^3 \, dx+\int \left (135 e^{\frac {1}{5} e^{-x} x (15+x)^2 \left (5+e^x x^2\right )} x^2+24 e^{\frac {1}{5} e^{-x} x (15+x)^2 \left (5+e^x x^2\right )} x^3+e^{\frac {1}{5} e^{-x} x (15+x)^2 \left (5+e^x x^2\right )} x^4\right ) \, dx\\ &=24 \int e^{\frac {1}{5} e^{-x} x (15+x)^2 \left (5+e^x x^2\right )} x^3 \, dx-27 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x^2 \, dx+135 \int e^{\frac {1}{5} e^{-x} x (15+x)^2 \left (5+e^x x^2\right )} x^2 \, dx-165 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x \, dx+225 \int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) \, dx-\int \exp \left (-x+45 x^3+6 x^4+\frac {x^5}{5}+e^{-x} x (15+x)^2\right ) x^3 \, dx+\int e^{\frac {1}{5} e^{-x} x (15+x)^2 \left (5+e^x x^2\right )} x^4 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.94, size = 26, normalized size = 1.00 \begin {gather*} e^{\frac {1}{5} e^{-x} x (15+x)^2 \left (5+e^x x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-x + (1125*x + 150*x^2 + 5*x^3 + E^x*(225*x^3 + 30*x^4 + x^5))/(5*E^x))*(225 - 165*x - 27*x^2 - x
^3 + E^x*(135*x^2 + 24*x^3 + x^4)),x]

[Out]

E^((x*(15 + x)^2*(5 + E^x*x^2))/(5*E^x))

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fricas [A]  time = 0.74, size = 43, normalized size = 1.65 \begin {gather*} e^{\left (\frac {1}{5} \, {\left (5 \, x^{3} + 150 \, x^{2} + {\left (x^{5} + 30 \, x^{4} + 225 \, x^{3} - 5 \, x\right )} e^{x} + 1125 \, x\right )} e^{\left (-x\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+24*x^3+135*x^2)*exp(x)-x^3-27*x^2-165*x+225)*exp(1/5*((x^5+30*x^4+225*x^3)*exp(x)+5*x^3+150*x^
2+1125*x)/exp(x))/exp(x),x, algorithm="fricas")

[Out]

e^(1/5*(5*x^3 + 150*x^2 + (x^5 + 30*x^4 + 225*x^3 - 5*x)*e^x + 1125*x)*e^(-x) + x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -{\left (x^{3} + 27 \, x^{2} - {\left (x^{4} + 24 \, x^{3} + 135 \, x^{2}\right )} e^{x} + 165 \, x - 225\right )} e^{\left (\frac {1}{5} \, {\left (5 \, x^{3} + 150 \, x^{2} + {\left (x^{5} + 30 \, x^{4} + 225 \, x^{3}\right )} e^{x} + 1125 \, x\right )} e^{\left (-x\right )} - x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+24*x^3+135*x^2)*exp(x)-x^3-27*x^2-165*x+225)*exp(1/5*((x^5+30*x^4+225*x^3)*exp(x)+5*x^3+150*x^
2+1125*x)/exp(x))/exp(x),x, algorithm="giac")

[Out]

integrate(-(x^3 + 27*x^2 - (x^4 + 24*x^3 + 135*x^2)*e^x + 165*x - 225)*e^(1/5*(5*x^3 + 150*x^2 + (x^5 + 30*x^4
+ 225*x^3)*e^x + 1125*x)*e^(-x) - x), x)

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maple [A]  time = 0.08, size = 22, normalized size = 0.85

method	result	size
risch	\({\mathrm e}^{\frac {x \left (x +15\right )^{2} \left ({\mathrm e}^{x} x^{2}+5\right ) {\mathrm e}^{-x}}{5}}\)	\(22\)
norman	\({\mathrm e}^{\frac {\left (\left (x^{5}+30 x^{4}+225 x^{3}\right ) {\mathrm e}^{x}+5 x^{3}+150 x^{2}+1125 x \right ) {\mathrm e}^{-x}}{5}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4+24*x^3+135*x^2)*exp(x)-x^3-27*x^2-165*x+225)*exp(1/5*((x^5+30*x^4+225*x^3)*exp(x)+5*x^3+150*x^2+1125
*x)/exp(x))/exp(x),x,method=_RETURNVERBOSE)

[Out]

exp(1/5*x*(x+15)^2*(exp(x)*x^2+5)*exp(-x))

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maxima [A]  time = 0.59, size = 41, normalized size = 1.58 \begin {gather*} e^{\left (\frac {1}{5} \, x^{5} + 6 \, x^{4} + x^{3} e^{\left (-x\right )} + 45 \, x^{3} + 30 \, x^{2} e^{\left (-x\right )} + 225 \, x e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4+24*x^3+135*x^2)*exp(x)-x^3-27*x^2-165*x+225)*exp(1/5*((x^5+30*x^4+225*x^3)*exp(x)+5*x^3+150*x^
2+1125*x)/exp(x))/exp(x),x, algorithm="maxima")

[Out]

e^(1/5*x^5 + 6*x^4 + x^3*e^(-x) + 45*x^3 + 30*x^2*e^(-x) + 225*x*e^(-x))

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mupad [B]  time = 4.41, size = 46, normalized size = 1.77 \begin {gather*} {\mathrm {e}}^{225\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{6\,x^4}\,{\mathrm {e}}^{\frac {x^5}{5}}\,{\mathrm {e}}^{45\,x^3}\,{\mathrm {e}}^{x^3\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{30\,x^2\,{\mathrm {e}}^{-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(exp(-x)*(225*x + 30*x^2 + x^3 + (exp(x)*(225*x^3 + 30*x^4 + x^5))/5))*exp(-x)*(165*x + 27*x^2 + x^3 -
exp(x)*(135*x^2 + 24*x^3 + x^4) - 225),x)

[Out]

exp(225*x*exp(-x))*exp(6*x^4)*exp(x^5/5)*exp(45*x^3)*exp(x^3*exp(-x))*exp(30*x^2*exp(-x))

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sympy [A]  time = 0.26, size = 34, normalized size = 1.31 \begin {gather*} e^{\left (x^{3} + 30 x^{2} + 225 x + \frac {\left (x^{5} + 30 x^{4} + 225 x^{3}\right ) e^{x}}{5}\right ) e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4+24*x**3+135*x**2)*exp(x)-x**3-27*x**2-165*x+225)*exp(1/5*((x**5+30*x**4+225*x**3)*exp(x)+5*x*
*3+150*x**2+1125*x)/exp(x))/exp(x),x)

[Out]

exp((x**3 + 30*x**2 + 225*x + (x**5 + 30*x**4 + 225*x**3)*exp(x)/5)*exp(-x))

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