3.1.98 \(\int \frac {e^{\frac {-e^5-720 x^3+160 x^4+160 x^3 \log (x)}{16 x^3}} (3 e^5+160 x^3+160 x^4)}{16 x^4} \, dx\)

Optimal. Leaf size=24 \[ e^{-\frac {e^5}{16 x^3}+5 (1+2 (-5+x+\log (x)))} \]

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Rubi [F]  time = 0.69, 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 {-e^5-720 x^3+160 x^4+160 x^3 \log (x)}{16 x^3}} \left (3 e^5+160 x^3+160 x^4\right )}{16 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-E^5 - 720*x^3 + 160*x^4 + 160*x^3*Log[x])/(16*x^3))*(3*E^5 + 160*x^3 + 160*x^4))/(16*x^4),x]

[Out]

(3*Defer[Int][E^(-40 - E^5/(16*x^3) + 10*x)*x^6, x])/16 + 10*Defer[Int][E^(-45 - E^5/(16*x^3) + 10*x)*x^9, x]
+ 10*Defer[Int][E^(-45 - E^5/(16*x^3) + 10*x)*x^10, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{16} \int \frac {e^{\frac {-e^5-720 x^3+160 x^4+160 x^3 \log (x)}{16 x^3}} \left (3 e^5+160 x^3+160 x^4\right )}{x^4} \, dx\\ &=\frac {1}{16} \int e^{-45-\frac {e^5}{16 x^3}+10 x} \left (3 e^5 x^6+160 x^9 (1+x)\right ) \, dx\\ &=\frac {1}{16} \int \left (3 e^{-40-\frac {e^5}{16 x^3}+10 x} x^6+160 e^{-45-\frac {e^5}{16 x^3}+10 x} x^9 (1+x)\right ) \, dx\\ &=\frac {3}{16} \int e^{-40-\frac {e^5}{16 x^3}+10 x} x^6 \, dx+10 \int e^{-45-\frac {e^5}{16 x^3}+10 x} x^9 (1+x) \, dx\\ &=\frac {3}{16} \int e^{-40-\frac {e^5}{16 x^3}+10 x} x^6 \, dx+10 \int \left (e^{-45-\frac {e^5}{16 x^3}+10 x} x^9+e^{-45-\frac {e^5}{16 x^3}+10 x} x^{10}\right ) \, dx\\ &=\frac {3}{16} \int e^{-40-\frac {e^5}{16 x^3}+10 x} x^6 \, dx+10 \int e^{-45-\frac {e^5}{16 x^3}+10 x} x^9 \, dx+10 \int e^{-45-\frac {e^5}{16 x^3}+10 x} x^{10} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 21, normalized size = 0.88 \begin {gather*} e^{-45-\frac {e^5}{16 x^3}+10 x} x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-E^5 - 720*x^3 + 160*x^4 + 160*x^3*Log[x])/(16*x^3))*(3*E^5 + 160*x^3 + 160*x^4))/(16*x^4),x]

[Out]

E^(-45 - E^5/(16*x^3) + 10*x)*x^10

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fricas [A]  time = 1.24, size = 28, normalized size = 1.17 \begin {gather*} e^{\left (\frac {160 \, x^{4} + 160 \, x^{3} \log \relax (x) - 720 \, x^{3} - e^{5}}{16 \, x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(3*exp(5)+160*x^4+160*x^3)*exp(1/16*(160*x^3*log(x)-exp(5)+160*x^4-720*x^3)/x^3)/x^4,x, algorit
hm="fricas")

[Out]

e^(1/16*(160*x^4 + 160*x^3*log(x) - 720*x^3 - e^5)/x^3)

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giac [A]  time = 0.17, size = 17, normalized size = 0.71 \begin {gather*} e^{\left (10 \, x - \frac {e^{5}}{16 \, x^{3}} + 10 \, \log \relax (x) - 45\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(3*exp(5)+160*x^4+160*x^3)*exp(1/16*(160*x^3*log(x)-exp(5)+160*x^4-720*x^3)/x^3)/x^4,x, algorit
hm="giac")

[Out]

e^(10*x - 1/16*e^5/x^3 + 10*log(x) - 45)

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maple [A]  time = 0.08, size = 24, normalized size = 1.00

method	result	size
risch	\(x^{10} {\mathrm e}^{-\frac {-160 x^{4}+720 x^{3}+{\mathrm e}^{5}}{16 x^{3}}}\)	\(24\)
gosper	\({\mathrm e}^{-\frac {-160 x^{3} \ln \relax (x )-160 x^{4}+720 x^{3}+{\mathrm e}^{5}}{16 x^{3}}}\)	\(27\)
norman	\({\mathrm e}^{\frac {160 x^{3} \ln \relax (x )-{\mathrm e}^{5}+160 x^{4}-720 x^{3}}{16 x^{3}}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*(3*exp(5)+160*x^4+160*x^3)*exp(1/16*(160*x^3*ln(x)-exp(5)+160*x^4-720*x^3)/x^3)/x^4,x,method=_RETURNV
ERBOSE)

[Out]

x^10*exp(-1/16*(-160*x^4+720*x^3+exp(5))/x^3)

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maxima [A]  time = 0.77, size = 17, normalized size = 0.71 \begin {gather*} x^{10} e^{\left (10 \, x - \frac {e^{5}}{16 \, x^{3}} - 45\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(3*exp(5)+160*x^4+160*x^3)*exp(1/16*(160*x^3*log(x)-exp(5)+160*x^4-720*x^3)/x^3)/x^4,x, algorit
hm="maxima")

[Out]

x^10*e^(10*x - 1/16*e^5/x^3 - 45)

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mupad [B]  time = 0.45, size = 17, normalized size = 0.71 \begin {gather*} x^{10}\,{\mathrm {e}}^{10\,x-\frac {{\mathrm {e}}^5}{16\,x^3}-45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(exp(5)/16 - 10*x^3*log(x) + 45*x^3 - 10*x^4)/x^3)*(3*exp(5) + 160*x^3 + 160*x^4))/(16*x^4),x)

[Out]

x^10*exp(10*x - exp(5)/(16*x^3) - 45)

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sympy [A]  time = 0.37, size = 27, normalized size = 1.12 \begin {gather*} e^{\frac {10 x^{4} + 10 x^{3} \log {\relax (x )} - 45 x^{3} - \frac {e^{5}}{16}}{x^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(3*exp(5)+160*x**4+160*x**3)*exp(1/16*(160*x**3*ln(x)-exp(5)+160*x**4-720*x**3)/x**3)/x**4,x)

[Out]

exp((10*x**4 + 10*x**3*log(x) - 45*x**3 - exp(5)/16)/x**3)

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