3.5.38 \(\int \frac {1}{16} e^{x+\frac {1}{16} e^x (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} (75 x+30 x^2+3 x^3))} (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} (75+210 x+69 x^2+6 x^3)) \, dx\)

Optimal. Leaf size=27 \[ e^{\frac {1}{16} e^x \left (-3+3 e^{-4+x}-x\right ) x (5+x)^2} \]

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Rubi [F]  time = 6.89, 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 \frac {1}{16} \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x + (E^x*(-75*x - 55*x^2 - 13*x^3 - x^4 + E^(-4 + x)*(75*x + 30*x^2 + 3*x^3)))/16)*(-75 - 185*x - 94*x
^2 - 17*x^3 - x^4 + E^(-4 + x)*(75 + 210*x + 69*x^2 + 6*x^3)))/16,x]

[Out]

(-75*Defer[Int][E^((x*(16 + 3*E^(-4 + 2*x)*(5 + x)^2 - E^x*(3 + x)*(5 + x)^2))/16), x])/16 + (75*Defer[Int][E^
(-4 + 2*x + (E^x*(-75*x - 55*x^2 - 13*x^3 - x^4 + E^(-4 + x)*(75*x + 30*x^2 + 3*x^3)))/16), x])/16 - (185*Defe
r[Int][E^((x*(16 + 3*E^(-4 + 2*x)*(5 + x)^2 - E^x*(3 + x)*(5 + x)^2))/16)*x, x])/16 + (105*Defer[Int][E^(-4 +
2*x + (E^x*(-75*x - 55*x^2 - 13*x^3 - x^4 + E^(-4 + x)*(75*x + 30*x^2 + 3*x^3)))/16)*x, x])/8 - (47*Defer[Int]
[E^((x*(16 + 3*E^(-4 + 2*x)*(5 + x)^2 - E^x*(3 + x)*(5 + x)^2))/16)*x^2, x])/8 + (69*Defer[Int][E^(-4 + 2*x +
(E^x*(-75*x - 55*x^2 - 13*x^3 - x^4 + E^(-4 + x)*(75*x + 30*x^2 + 3*x^3)))/16)*x^2, x])/16 - (17*Defer[Int][E^
((x*(16 + 3*E^(-4 + 2*x)*(5 + x)^2 - E^x*(3 + x)*(5 + x)^2))/16)*x^3, x])/16 + (3*Defer[Int][E^(-4 + 2*x + (E^
x*(-75*x - 55*x^2 - 13*x^3 - x^4 + E^(-4 + x)*(75*x + 30*x^2 + 3*x^3)))/16)*x^3, x])/8 - Defer[Int][E^((x*(16
+ 3*E^(-4 + 2*x)*(5 + x)^2 - E^x*(3 + x)*(5 + x)^2))/16)*x^4, x]/16

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{16} \int \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx\\ &=\frac {1}{16} \int \left (-75 \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right )-185 \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x-94 \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^2-17 \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^3-\exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^4+3 \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) \left (25+70 x+23 x^2+2 x^3\right )\right ) \, dx\\ &=-\left (\frac {1}{16} \int \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^4 \, dx\right )+\frac {3}{16} \int \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) \left (25+70 x+23 x^2+2 x^3\right ) \, dx-\frac {17}{16} \int \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^3 \, dx-\frac {75}{16} \int \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) \, dx-\frac {47}{8} \int \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^2 \, dx-\frac {185}{16} \int \exp \left (x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x \, dx\\ &=-\left (\frac {1}{16} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) x^4 \, dx\right )+\frac {3}{16} \int \left (25 \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right )+70 \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x+23 \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^2+2 \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^3\right ) \, dx-\frac {17}{16} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) x^3 \, dx-\frac {75}{16} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) \, dx-\frac {47}{8} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) x^2 \, dx-\frac {185}{16} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) x \, dx\\ &=-\left (\frac {1}{16} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) x^4 \, dx\right )+\frac {3}{8} \int \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^3 \, dx-\frac {17}{16} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) x^3 \, dx+\frac {69}{16} \int \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x^2 \, dx-\frac {75}{16} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) \, dx+\frac {75}{16} \int \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) \, dx-\frac {47}{8} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) x^2 \, dx-\frac {185}{16} \int \exp \left (\frac {1}{16} x \left (16+3 e^{-4+2 x} (5+x)^2-e^x (3+x) (5+x)^2\right )\right ) x \, dx+\frac {105}{8} \int \exp \left (-4+2 x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )\right ) x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.22, size = 30, normalized size = 1.11 \begin {gather*} e^{-\frac {1}{16} e^{-4+x} x (5+x)^2 \left (-3 e^x+e^4 (3+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x + (E^x*(-75*x - 55*x^2 - 13*x^3 - x^4 + E^(-4 + x)*(75*x + 30*x^2 + 3*x^3)))/16)*(-75 - 185*x
- 94*x^2 - 17*x^3 - x^4 + E^(-4 + x)*(75 + 210*x + 69*x^2 + 6*x^3)))/16,x]

[Out]

E^(-1/16*(E^(-4 + x)*x*(5 + x)^2*(-3*E^x + E^4*(3 + x))))

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fricas [B]  time = 0.85, size = 56, normalized size = 2.07 \begin {gather*} e^{\left (\frac {1}{16} \, {\left (16 \, x e^{4} + 3 \, {\left (x^{3} + 10 \, x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{4} + 13 \, x^{3} + 55 \, x^{2} + 75 \, x\right )} e^{\left (x + 4\right )}\right )} e^{\left (-4\right )} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*((6*x^3+69*x^2+210*x+75)*exp(x-4)-x^4-17*x^3-94*x^2-185*x-75)*exp(x)*exp(1/16*((3*x^3+30*x^2+75
*x)*exp(x-4)-x^4-13*x^3-55*x^2-75*x)*exp(x)),x, algorithm="fricas")

[Out]

e^(1/16*(16*x*e^4 + 3*(x^3 + 10*x^2 + 25*x)*e^(2*x) - (x^4 + 13*x^3 + 55*x^2 + 75*x)*e^(x + 4))*e^(-4) - x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {1}{16} \, {\left (x^{4} + 17 \, x^{3} + 94 \, x^{2} - 3 \, {\left (2 \, x^{3} + 23 \, x^{2} + 70 \, x + 25\right )} e^{\left (x - 4\right )} + 185 \, x + 75\right )} e^{\left (-\frac {1}{16} \, {\left (x^{4} + 13 \, x^{3} + 55 \, x^{2} - 3 \, {\left (x^{3} + 10 \, x^{2} + 25 \, x\right )} e^{\left (x - 4\right )} + 75 \, x\right )} e^{x} + x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*((6*x^3+69*x^2+210*x+75)*exp(x-4)-x^4-17*x^3-94*x^2-185*x-75)*exp(x)*exp(1/16*((3*x^3+30*x^2+75
*x)*exp(x-4)-x^4-13*x^3-55*x^2-75*x)*exp(x)),x, algorithm="giac")

[Out]

integrate(-1/16*(x^4 + 17*x^3 + 94*x^2 - 3*(2*x^3 + 23*x^2 + 70*x + 25)*e^(x - 4) + 185*x + 75)*e^(-1/16*(x^4
+ 13*x^3 + 55*x^2 - 3*(x^3 + 10*x^2 + 25*x)*e^(x - 4) + 75*x)*e^x + x), x)

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maple [A]  time = 0.24, size = 21, normalized size = 0.78

method	result	size
risch	\({\mathrm e}^{-\frac {x \left (5+x \right )^{2} \left (-3 \,{\mathrm e}^{x -4}+3+x \right ) {\mathrm e}^{x}}{16}}\)	\(21\)
norman	\({\mathrm e}^{\frac {\left (\left (3 x^{3}+30 x^{2}+75 x \right ) {\mathrm e}^{x} {\mathrm e}^{-4}-x^{4}-13 x^{3}-55 x^{2}-75 x \right ) {\mathrm e}^{x}}{16}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*((6*x^3+69*x^2+210*x+75)*exp(x-4)-x^4-17*x^3-94*x^2-185*x-75)*exp(x)*exp(1/16*((3*x^3+30*x^2+75*x)*ex
p(x-4)-x^4-13*x^3-55*x^2-75*x)*exp(x)),x,method=_RETURNVERBOSE)

[Out]

exp(-1/16*x*(5+x)^2*(-3*exp(x-4)+3+x)*exp(x))

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maxima [B]  time = 1.08, size = 59, normalized size = 2.19 \begin {gather*} e^{\left (-\frac {1}{16} \, x^{4} e^{x} + \frac {3}{16} \, x^{3} e^{\left (2 \, x - 4\right )} - \frac {13}{16} \, x^{3} e^{x} + \frac {15}{8} \, x^{2} e^{\left (2 \, x - 4\right )} - \frac {55}{16} \, x^{2} e^{x} + \frac {75}{16} \, x e^{\left (2 \, x - 4\right )} - \frac {75}{16} \, x e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*((6*x^3+69*x^2+210*x+75)*exp(x-4)-x^4-17*x^3-94*x^2-185*x-75)*exp(x)*exp(1/16*((3*x^3+30*x^2+75
*x)*exp(x-4)-x^4-13*x^3-55*x^2-75*x)*exp(x)),x, algorithm="maxima")

[Out]

e^(-1/16*x^4*e^x + 3/16*x^3*e^(2*x - 4) - 13/16*x^3*e^x + 15/8*x^2*e^(2*x - 4) - 55/16*x^2*e^x + 75/16*x*e^(2*
x - 4) - 75/16*x*e^x)

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mupad [B]  time = 0.89, size = 65, normalized size = 2.41 \begin {gather*} {\mathrm {e}}^{-\frac {75\,x\,{\mathrm {e}}^x}{16}}\,{\mathrm {e}}^{\frac {3\,x^3\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-4}}{16}}\,{\mathrm {e}}^{\frac {15\,x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-4}}{8}}\,{\mathrm {e}}^{-\frac {x^4\,{\mathrm {e}}^x}{16}}\,{\mathrm {e}}^{-\frac {13\,x^3\,{\mathrm {e}}^x}{16}}\,{\mathrm {e}}^{-\frac {55\,x^2\,{\mathrm {e}}^x}{16}}\,{\mathrm {e}}^{\frac {75\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-4}}{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(exp(x)*(75*x - exp(x - 4)*(75*x + 30*x^2 + 3*x^3) + 55*x^2 + 13*x^3 + x^4))/16)*exp(x)*(185*x - ex
p(x - 4)*(210*x + 69*x^2 + 6*x^3 + 75) + 94*x^2 + 17*x^3 + x^4 + 75))/16,x)

[Out]

exp(-(75*x*exp(x))/16)*exp((3*x^3*exp(2*x)*exp(-4))/16)*exp((15*x^2*exp(2*x)*exp(-4))/8)*exp(-(x^4*exp(x))/16)
*exp(-(13*x^3*exp(x))/16)*exp(-(55*x^2*exp(x))/16)*exp((75*x*exp(2*x)*exp(-4))/16)

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sympy [A]  time = 0.52, size = 49, normalized size = 1.81 \begin {gather*} e^{\left (- \frac {x^{4}}{16} - \frac {13 x^{3}}{16} - \frac {55 x^{2}}{16} - \frac {75 x}{16} + \frac {\left (3 x^{3} + 30 x^{2} + 75 x\right ) e^{x}}{16 e^{4}}\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*((6*x**3+69*x**2+210*x+75)*exp(x-4)-x**4-17*x**3-94*x**2-185*x-75)*exp(x)*exp(1/16*((3*x**3+30*
x**2+75*x)*exp(x-4)-x**4-13*x**3-55*x**2-75*x)*exp(x)),x)

[Out]

exp((-x**4/16 - 13*x**3/16 - 55*x**2/16 - 75*x/16 + (3*x**3 + 30*x**2 + 75*x)*exp(-4)*exp(x)/16)*exp(x))

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