3.45.62 \(\int \frac {e^{\frac {-100-e^{225 x/8}+20 x^2+x^3-x^4+e^{225 x/16} (-20+2 x^2)}{x^2}} (1600+e^{225 x/8} (16-225 x)+8 x^3-16 x^4+e^{225 x/16} (320-2250 x+225 x^3))}{8 x^3} \, dx\)

Optimal. Leaf size=24 \[ e^{x-\left (-\frac {10+e^{225 x/16}}{x}+x\right )^2} \]

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Rubi [F]  time = 11.57, 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 {\exp \left (\frac {-100-e^{225 x/8}+20 x^2+x^3-x^4+e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \left (1600+e^{225 x/8} (16-225 x)+8 x^3-16 x^4+e^{225 x/16} \left (320-2250 x+225 x^3\right )\right )}{8 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-100 - E^((225*x)/8) + 20*x^2 + x^3 - x^4 + E^((225*x)/16)*(-20 + 2*x^2))/x^2)*(1600 + E^((225*x)/8)*
(16 - 225*x) + 8*x^3 - 16*x^4 + E^((225*x)/16)*(320 - 2250*x + 225*x^3)))/(8*x^3),x]

[Out]

Defer[Int][E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + x - x^2 + (E^((225*x)/16)*(-20 + 2*x^2))/x^2), x] + (225*Defe
r[Int][E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + (241*x)/16 - x^2 + (E^((225*x)/16)*(-20 + 2*x^2))/x^2), x])/8 + 2
00*Defer[Int][E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + x - x^2 + (E^((225*x)/16)*(-20 + 2*x^2))/x^2)/x^3, x] + 40
*Defer[Int][E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + (241*x)/16 - x^2 + (E^((225*x)/16)*(-20 + 2*x^2))/x^2)/x^3,
x] + 2*Defer[Int][E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + (233*x)/8 - x^2 + (E^((225*x)/16)*(-20 + 2*x^2))/x^2)/
x^3, x] - (1125*Defer[Int][E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + (241*x)/16 - x^2 + (E^((225*x)/16)*(-20 + 2*x
^2))/x^2)/x^2, x])/4 - (225*Defer[Int][E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + (233*x)/8 - x^2 + (E^((225*x)/16)
*(-20 + 2*x^2))/x^2)/x^2, x])/8 - 2*Defer[Int][E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + x - x^2 + (E^((225*x)/16)
*(-20 + 2*x^2))/x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {\exp \left (\frac {-100-e^{225 x/8}+20 x^2+x^3-x^4+e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \left (1600+e^{225 x/8} (16-225 x)+8 x^3-16 x^4+e^{225 x/16} \left (320-2250 x+225 x^3\right )\right )}{x^3} \, dx\\ &=\frac {1}{8} \int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \left (1600+e^{225 x/8} (16-225 x)+8 x^3-16 x^4+e^{225 x/16} \left (320-2250 x+225 x^3\right )\right )}{x^3} \, dx\\ &=\frac {1}{8} \int \left (-\frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {233 x}{8}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) (-16+225 x)}{x^3}+\frac {5 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {241 x}{16}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \left (64-450 x+45 x^3\right )}{x^3}-\frac {8 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \left (-200-x^3+2 x^4\right )}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{8} \int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {233 x}{8}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) (-16+225 x)}{x^3} \, dx\right )+\frac {5}{8} \int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {241 x}{16}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \left (64-450 x+45 x^3\right )}{x^3} \, dx-\int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \left (-200-x^3+2 x^4\right )}{x^3} \, dx\\ &=-\left (\frac {1}{8} \int \left (-\frac {16 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {233 x}{8}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^3}+\frac {225 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {233 x}{8}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^2}\right ) \, dx\right )+\frac {5}{8} \int \left (45 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {241 x}{16}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )+\frac {64 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {241 x}{16}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^3}-\frac {450 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {241 x}{16}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^2}\right ) \, dx-\int \left (-\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )-\frac {200 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^3}+2 \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) x\right ) \, dx\\ &=2 \int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {233 x}{8}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^3} \, dx-2 \int \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) x \, dx+\frac {225}{8} \int \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {241 x}{16}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \, dx-\frac {225}{8} \int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {233 x}{8}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^2} \, dx+40 \int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {241 x}{16}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^3} \, dx+200 \int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^3} \, dx-\frac {1125}{4} \int \frac {\exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+\frac {241 x}{16}-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right )}{x^2} \, dx+\int \exp \left (20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {e^{225 x/16} \left (-20+2 x^2\right )}{x^2}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 44, normalized size = 1.83 \begin {gather*} e^{20-\frac {100}{x^2}-\frac {e^{225 x/8}}{x^2}+x-x^2+\frac {2 e^{225 x/16} \left (-10+x^2\right )}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-100 - E^((225*x)/8) + 20*x^2 + x^3 - x^4 + E^((225*x)/16)*(-20 + 2*x^2))/x^2)*(1600 + E^((225*
x)/8)*(16 - 225*x) + 8*x^3 - 16*x^4 + E^((225*x)/16)*(320 - 2250*x + 225*x^3)))/(8*x^3),x]

[Out]

E^(20 - 100/x^2 - E^((225*x)/8)/x^2 + x - x^2 + (2*E^((225*x)/16)*(-10 + x^2))/x^2)

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fricas [A]  time = 0.59, size = 36, normalized size = 1.50 \begin {gather*} e^{\left (-\frac {x^{4} - x^{3} - 20 \, x^{2} - 2 \, {\left (x^{2} - 10\right )} e^{\left (\frac {225}{16} \, x\right )} + e^{\left (\frac {225}{8} \, x\right )} + 100}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-225*x+16)*exp(225/16*x)^2+(225*x^3-2250*x+320)*exp(225/16*x)-16*x^4+8*x^3+1600)*exp((-exp(225
/16*x)^2+(2*x^2-20)*exp(225/16*x)-x^4+x^3+20*x^2-100)/x^2)/x^3,x, algorithm="fricas")

[Out]

e^(-(x^4 - x^3 - 20*x^2 - 2*(x^2 - 10)*e^(225/16*x) + e^(225/8*x) + 100)/x^2)

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giac [A]  time = 0.39, size = 38, normalized size = 1.58 \begin {gather*} e^{\left (-x^{2} + x - \frac {e^{\left (\frac {225}{8} \, x\right )}}{x^{2}} - \frac {20 \, e^{\left (\frac {225}{16} \, x\right )}}{x^{2}} - \frac {100}{x^{2}} + 2 \, e^{\left (\frac {225}{16} \, x\right )} + 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-225*x+16)*exp(225/16*x)^2+(225*x^3-2250*x+320)*exp(225/16*x)-16*x^4+8*x^3+1600)*exp((-exp(225
/16*x)^2+(2*x^2-20)*exp(225/16*x)-x^4+x^3+20*x^2-100)/x^2)/x^3,x, algorithm="giac")

[Out]

e^(-x^2 + x - e^(225/8*x)/x^2 - 20*e^(225/16*x)/x^2 - 100/x^2 + 2*e^(225/16*x) + 20)

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maple [A]  time = 0.09, size = 41, normalized size = 1.71

method	result	size
norman	\({\mathrm e}^{\frac {-{\mathrm e}^{\frac {225 x}{8}}+\left (2 x^{2}-20\right ) {\mathrm e}^{\frac {225 x}{16}}-x^{4}+x^{3}+20 x^{2}-100}{x^{2}}}\)	\(41\)
risch	\({\mathrm e}^{-\frac {x^{4}-2 \,{\mathrm e}^{\frac {225 x}{16}} x^{2}-x^{3}-20 x^{2}+20 \,{\mathrm e}^{\frac {225 x}{16}}+{\mathrm e}^{\frac {225 x}{8}}+100}{x^{2}}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*((-225*x+16)*exp(225/16*x)^2+(225*x^3-2250*x+320)*exp(225/16*x)-16*x^4+8*x^3+1600)*exp((-exp(225/16*x)
^2+(2*x^2-20)*exp(225/16*x)-x^4+x^3+20*x^2-100)/x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

exp((-exp(225/16*x)^2+(2*x^2-20)*exp(225/16*x)-x^4+x^3+20*x^2-100)/x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{8} \, \int \frac {{\left (16 \, x^{4} - 8 \, x^{3} + {\left (225 \, x - 16\right )} e^{\left (\frac {225}{8} \, x\right )} - 5 \, {\left (45 \, x^{3} - 450 \, x + 64\right )} e^{\left (\frac {225}{16} \, x\right )} - 1600\right )} e^{\left (-\frac {x^{4} - x^{3} - 20 \, x^{2} - 2 \, {\left (x^{2} - 10\right )} e^{\left (\frac {225}{16} \, x\right )} + e^{\left (\frac {225}{8} \, x\right )} + 100}{x^{2}}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-225*x+16)*exp(225/16*x)^2+(225*x^3-2250*x+320)*exp(225/16*x)-16*x^4+8*x^3+1600)*exp((-exp(225
/16*x)^2+(2*x^2-20)*exp(225/16*x)-x^4+x^3+20*x^2-100)/x^2)/x^3,x, algorithm="maxima")

[Out]

-1/8*integrate((16*x^4 - 8*x^3 + (225*x - 16)*e^(225/8*x) - 5*(45*x^3 - 450*x + 64)*e^(225/16*x) - 1600)*e^(-(
x^4 - x^3 - 20*x^2 - 2*(x^2 - 10)*e^(225/16*x) + e^(225/8*x) + 100)/x^2)/x^3, x)

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mupad [B]  time = 3.56, size = 44, normalized size = 1.83 \begin {gather*} {\mathrm {e}}^{2\,{\mathrm {e}}^{\frac {225\,x}{16}}}\,{\mathrm {e}}^{20}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{-\frac {100}{x^2}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{\frac {225\,x}{8}}}{x^2}}\,{\mathrm {e}}^{-\frac {20\,{\mathrm {e}}^{\frac {225\,x}{16}}}{x^2}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(exp((225*x)/8) - exp((225*x)/16)*(2*x^2 - 20) - 20*x^2 - x^3 + x^4 + 100)/x^2)*(exp((225*x)/16)*(22
5*x^3 - 2250*x + 320) - exp((225*x)/8)*(225*x - 16) + 8*x^3 - 16*x^4 + 1600))/(8*x^3),x)

[Out]

exp(2*exp((225*x)/16))*exp(20)*exp(-x^2)*exp(-100/x^2)*exp(-exp((225*x)/8)/x^2)*exp(-(20*exp((225*x)/16))/x^2)
*exp(x)

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sympy [B]  time = 0.37, size = 37, normalized size = 1.54 \begin {gather*} e^{\frac {- x^{4} + x^{3} + 20 x^{2} + \left (2 x^{2} - 20\right ) e^{\frac {225 x}{16}} - e^{\frac {225 x}{8}} - 100}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-225*x+16)*exp(225/16*x)**2+(225*x**3-2250*x+320)*exp(225/16*x)-16*x**4+8*x**3+1600)*exp((-exp
(225/16*x)**2+(2*x**2-20)*exp(225/16*x)-x**4+x**3+20*x**2-100)/x**2)/x**3,x)

[Out]

exp((-x**4 + x**3 + 20*x**2 + (2*x**2 - 20)*exp(225*x/16) - exp(225*x/8) - 100)/x**2)

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