3.94.95 \(\int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} (640+320 x+20 e^{\frac {2 x^2}{5}} x+e^{\frac {x^2}{5}} (160+160 x+64 x^2))}{80 x+40 e^{\frac {x^2}{5}} x+5 e^{\frac {2 x^2}{5}} x} \, dx\)

Optimal. Leaf size=23 \[ 4 e^{-\frac {8}{\left (4+e^{\frac {x^2}{5}}\right ) x}} x \]

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Rubi [F]  time = 4.37, 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 {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} \left (640+320 x+20 e^{\frac {2 x^2}{5}} x+e^{\frac {x^2}{5}} \left (160+160 x+64 x^2\right )\right )}{80 x+40 e^{\frac {x^2}{5}} x+5 e^{\frac {2 x^2}{5}} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(640 + 320*x + 20*E^((2*x^2)/5)*x + E^(x^2/5)*(160 + 160*x + 64*x^2))/(E^(8/(4*x + E^(x^2/5)*x))*(80*x + 4
0*E^(x^2/5)*x + 5*E^((2*x^2)/5)*x)),x]

[Out]

4*Defer[Int][E^(-8/(4*x + E^(x^2/5)*x)), x] + 32*Defer[Int][1/(E^(8/(4*x + E^(x^2/5)*x))*(4 + E^(x^2/5))*x), x
] - (256*Defer[Int][x/(E^(8/(4*x + E^(x^2/5)*x))*(4 + E^(x^2/5))^2), x])/5 + (64*Defer[Int][x/(E^(8/(4*x + E^(
x^2/5)*x))*(4 + E^(x^2/5))), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} \left (640+320 x+20 e^{\frac {2 x^2}{5}} x+e^{\frac {x^2}{5}} \left (160+160 x+64 x^2\right )\right )}{5 \left (4+e^{\frac {x^2}{5}}\right )^2 x} \, dx\\ &=\frac {1}{5} \int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} \left (640+320 x+20 e^{\frac {2 x^2}{5}} x+e^{\frac {x^2}{5}} \left (160+160 x+64 x^2\right )\right )}{\left (4+e^{\frac {x^2}{5}}\right )^2 x} \, dx\\ &=\frac {1}{5} \int \left (20 e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}}-\frac {256 e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} x}{\left (4+e^{\frac {x^2}{5}}\right )^2}+\frac {32 e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} \left (5+2 x^2\right )}{\left (4+e^{\frac {x^2}{5}}\right ) x}\right ) \, dx\\ &=4 \int e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} \, dx+\frac {32}{5} \int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} \left (5+2 x^2\right )}{\left (4+e^{\frac {x^2}{5}}\right ) x} \, dx-\frac {256}{5} \int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} x}{\left (4+e^{\frac {x^2}{5}}\right )^2} \, dx\\ &=4 \int e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} \, dx+\frac {32}{5} \int \left (\frac {5 e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}}}{\left (4+e^{\frac {x^2}{5}}\right ) x}+\frac {2 e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} x}{4+e^{\frac {x^2}{5}}}\right ) \, dx-\frac {256}{5} \int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} x}{\left (4+e^{\frac {x^2}{5}}\right )^2} \, dx\\ &=4 \int e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} \, dx+\frac {64}{5} \int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} x}{4+e^{\frac {x^2}{5}}} \, dx+32 \int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}}}{\left (4+e^{\frac {x^2}{5}}\right ) x} \, dx-\frac {256}{5} \int \frac {e^{-\frac {8}{4 x+e^{\frac {x^2}{5}} x}} x}{\left (4+e^{\frac {x^2}{5}}\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 23, normalized size = 1.00 \begin {gather*} 4 e^{-\frac {8}{\left (4+e^{\frac {x^2}{5}}\right ) x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(640 + 320*x + 20*E^((2*x^2)/5)*x + E^(x^2/5)*(160 + 160*x + 64*x^2))/(E^(8/(4*x + E^(x^2/5)*x))*(80
*x + 40*E^(x^2/5)*x + 5*E^((2*x^2)/5)*x)),x]

[Out]

(4*x)/E^(8/((4 + E^(x^2/5))*x))

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fricas [A]  time = 0.52, size = 20, normalized size = 0.87 \begin {gather*} 4 \, x e^{\left (-\frac {8}{x e^{\left (\frac {1}{5} \, x^{2}\right )} + 4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x*exp(1/5*x^2)^2+(64*x^2+160*x+160)*exp(1/5*x^2)+320*x+640)*exp(-4/(x*exp(1/5*x^2)+4*x))^2/(5*x*
exp(1/5*x^2)^2+40*x*exp(1/5*x^2)+80*x),x, algorithm="fricas")

[Out]

4*x*e^(-8/(x*e^(1/5*x^2) + 4*x))

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giac [A]  time = 0.16, size = 20, normalized size = 0.87 \begin {gather*} 4 \, x e^{\left (-\frac {8}{x e^{\left (\frac {1}{5} \, x^{2}\right )} + 4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x*exp(1/5*x^2)^2+(64*x^2+160*x+160)*exp(1/5*x^2)+320*x+640)*exp(-4/(x*exp(1/5*x^2)+4*x))^2/(5*x*
exp(1/5*x^2)^2+40*x*exp(1/5*x^2)+80*x),x, algorithm="giac")

[Out]

4*x*e^(-8/(x*e^(1/5*x^2) + 4*x))

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maple [A]  time = 0.08, size = 20, normalized size = 0.87

method	result	size
risch	\(4 x \,{\mathrm e}^{-\frac {8}{\left (4+{\mathrm e}^{\frac {x^{2}}{5}}\right ) x}}\)	\(20\)
norman	\(\frac {16 x \,{\mathrm e}^{-\frac {8}{x \,{\mathrm e}^{\frac {x^{2}}{5}}+4 x}}+4 x \,{\mathrm e}^{\frac {x^{2}}{5}} {\mathrm e}^{-\frac {8}{x \,{\mathrm e}^{\frac {x^{2}}{5}}+4 x}}}{4+{\mathrm e}^{\frac {x^{2}}{5}}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x*exp(1/5*x^2)^2+(64*x^2+160*x+160)*exp(1/5*x^2)+320*x+640)*exp(-4/(x*exp(1/5*x^2)+4*x))^2/(5*x*exp(1/
5*x^2)^2+40*x*exp(1/5*x^2)+80*x),x,method=_RETURNVERBOSE)

[Out]

4*x*exp(-8/(4+exp(1/5*x^2))/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {4}{5} \, \int \frac {{\left (5 \, x e^{\left (\frac {2}{5} \, x^{2}\right )} + 8 \, {\left (2 \, x^{2} + 5 \, x + 5\right )} e^{\left (\frac {1}{5} \, x^{2}\right )} + 80 \, x + 160\right )} e^{\left (-\frac {8}{x e^{\left (\frac {1}{5} \, x^{2}\right )} + 4 \, x}\right )}}{x e^{\left (\frac {2}{5} \, x^{2}\right )} + 8 \, x e^{\left (\frac {1}{5} \, x^{2}\right )} + 16 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x*exp(1/5*x^2)^2+(64*x^2+160*x+160)*exp(1/5*x^2)+320*x+640)*exp(-4/(x*exp(1/5*x^2)+4*x))^2/(5*x*
exp(1/5*x^2)^2+40*x*exp(1/5*x^2)+80*x),x, algorithm="maxima")

[Out]

4/5*integrate((5*x*e^(2/5*x^2) + 8*(2*x^2 + 5*x + 5)*e^(1/5*x^2) + 80*x + 160)*e^(-8/(x*e^(1/5*x^2) + 4*x))/(x
*e^(2/5*x^2) + 8*x*e^(1/5*x^2) + 16*x), x)

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mupad [B]  time = 8.16, size = 20, normalized size = 0.87 \begin {gather*} 4\,x\,{\mathrm {e}}^{-\frac {8}{4\,x+x\,{\mathrm {e}}^{\frac {x^2}{5}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-8/(4*x + x*exp(x^2/5)))*(320*x + exp(x^2/5)*(160*x + 64*x^2 + 160) + 20*x*exp((2*x^2)/5) + 640))/(80
*x + 40*x*exp(x^2/5) + 5*x*exp((2*x^2)/5)),x)

[Out]

4*x*exp(-8/(4*x + x*exp(x^2/5)))

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sympy [A]  time = 39.96, size = 17, normalized size = 0.74 \begin {gather*} 4 x e^{- \frac {8}{x e^{\frac {x^{2}}{5}} + 4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x*exp(1/5*x**2)**2+(64*x**2+160*x+160)*exp(1/5*x**2)+320*x+640)*exp(-4/(x*exp(1/5*x**2)+4*x))**2
/(5*x*exp(1/5*x**2)**2+40*x*exp(1/5*x**2)+80*x),x)

[Out]

4*x*exp(-8/(x*exp(x**2/5) + 4*x))

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