3.89.47 \(\int \frac {e^{\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}} (16+48 x+56 x^2+13 x^3)}{16 x+16 x^2+4 x^3} \, dx\)

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

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Rubi [F]  time = 2.15, 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 {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right ) \left (16+48 x+56 x^2+13 x^3\right )}{16 x+16 x^2+4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((44 + E*(-16 - 8*x) + 38*x + 13*x^2 + (8 + 4*x)*Log[x])/(8 + 4*x))*(16 + 48*x + 56*x^2 + 13*x^3))/(16*
x + 16*x^2 + 4*x^3),x]

[Out]

(13*Defer[Int][E^((44 + E*(-16 - 8*x) + 38*x + 13*x^2 + (8 + 4*x)*Log[x])/(8 + 4*x)), x])/4 + Defer[Int][E^((4
4 + E*(-16 - 8*x) + 38*x + 13*x^2 + (8 + 4*x)*Log[x])/(8 + 4*x))/x, x] - 5*Defer[Int][E^((44 + E*(-16 - 8*x) +
38*x + 13*x^2 + (8 + 4*x)*Log[x])/(8 + 4*x))/(2 + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right ) \left (16+48 x+56 x^2+13 x^3\right )}{x \left (16+16 x+4 x^2\right )} \, dx\\ &=\int \frac {\exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right ) \left (16+48 x+56 x^2+13 x^3\right )}{4 x (2+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {\exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right ) \left (16+48 x+56 x^2+13 x^3\right )}{x (2+x)^2} \, dx\\ &=\frac {1}{4} \int \left (13 \exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right )+\frac {4 \exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right )}{x}-\frac {20 \exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right )}{(2+x)^2}\right ) \, dx\\ &=\frac {13}{4} \int \exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right ) \, dx-5 \int \frac {\exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right )}{(2+x)^2} \, dx+\int \frac {\exp \left (\frac {44+e (-16-8 x)+38 x+13 x^2+(8+4 x) \log (x)}{8+4 x}\right )}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 21, normalized size = 0.78 \begin {gather*} e^{3-2 e+\frac {13 x}{4}+\frac {5}{2+x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((44 + E*(-16 - 8*x) + 38*x + 13*x^2 + (8 + 4*x)*Log[x])/(8 + 4*x))*(16 + 48*x + 56*x^2 + 13*x^3)
)/(16*x + 16*x^2 + 4*x^3),x]

[Out]

E^(3 - 2*E + (13*x)/4 + 5/(2 + x))*x

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fricas [A]  time = 0.47, size = 32, normalized size = 1.19 \begin {gather*} e^{\left (\frac {13 \, x^{2} - 8 \, {\left (x + 2\right )} e + 4 \, {\left (x + 2\right )} \log \relax (x) + 38 \, x + 44}{4 \, {\left (x + 2\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*x^3+56*x^2+48*x+16)*exp(((4*x+8)*log(x)+(-8*x-16)*exp(1)+13*x^2+38*x+44)/(4*x+8))/(4*x^3+16*x^2+
16*x),x, algorithm="fricas")

[Out]

e^(1/4*(13*x^2 - 8*(x + 2)*e + 4*(x + 2)*log(x) + 38*x + 44)/(x + 2))

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giac [B]  time = 0.20, size = 64, normalized size = 2.37 \begin {gather*} e^{\left (\frac {13 \, x^{2}}{4 \, {\left (x + 2\right )}} - \frac {2 \, x e}{x + 2} + \frac {x \log \relax (x)}{x + 2} + \frac {19 \, x}{2 \, {\left (x + 2\right )}} - \frac {4 \, e}{x + 2} + \frac {2 \, \log \relax (x)}{x + 2} + \frac {11}{x + 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*x^3+56*x^2+48*x+16)*exp(((4*x+8)*log(x)+(-8*x-16)*exp(1)+13*x^2+38*x+44)/(4*x+8))/(4*x^3+16*x^2+
16*x),x, algorithm="giac")

[Out]

e^(13/4*x^2/(x + 2) - 2*x*e/(x + 2) + x*log(x)/(x + 2) + 19/2*x/(x + 2) - 4*e/(x + 2) + 2*log(x)/(x + 2) + 11/
(x + 2))

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maple [A]  time = 0.05, size = 37, normalized size = 1.37

method	result	size
gosper	\({\mathrm e}^{\frac {4 x \ln \relax (x )-8 x \,{\mathrm e}+13 x^{2}+8 \ln \relax (x )-16 \,{\mathrm e}+38 x +44}{4 x +8}}\)	\(37\)
risch	\({\mathrm e}^{-\frac {-4 x \ln \relax (x )+8 x \,{\mathrm e}-13 x^{2}-8 \ln \relax (x )+16 \,{\mathrm e}-38 x -44}{4 \left (2+x \right )}}\)	\(37\)
norman	\(\frac {x \,{\mathrm e}^{\frac {\left (4 x +8\right ) \ln \relax (x )+\left (-8 x -16\right ) {\mathrm e}+13 x^{2}+38 x +44}{4 x +8}}+2 \,{\mathrm e}^{\frac {\left (4 x +8\right ) \ln \relax (x )+\left (-8 x -16\right ) {\mathrm e}+13 x^{2}+38 x +44}{4 x +8}}}{2+x}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13*x^3+56*x^2+48*x+16)*exp(((4*x+8)*ln(x)+(-8*x-16)*exp(1)+13*x^2+38*x+44)/(4*x+8))/(4*x^3+16*x^2+16*x),x
,method=_RETURNVERBOSE)

[Out]

exp(1/4*(4*x*ln(x)-8*x*exp(1)+13*x^2+8*ln(x)-16*exp(1)+38*x+44)/(2+x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{4} \, \int \frac {{\left (13 \, x^{3} + 56 \, x^{2} + 48 \, x + 16\right )} e^{\left (\frac {13 \, x^{2} - 8 \, {\left (x + 2\right )} e + 4 \, {\left (x + 2\right )} \log \relax (x) + 38 \, x + 44}{4 \, {\left (x + 2\right )}}\right )}}{x^{3} + 4 \, x^{2} + 4 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*x^3+56*x^2+48*x+16)*exp(((4*x+8)*log(x)+(-8*x-16)*exp(1)+13*x^2+38*x+44)/(4*x+8))/(4*x^3+16*x^2+
16*x),x, algorithm="maxima")

[Out]

1/4*integrate((13*x^3 + 56*x^2 + 48*x + 16)*e^(1/4*(13*x^2 - 8*(x + 2)*e + 4*(x + 2)*log(x) + 38*x + 44)/(x +
2))/(x^3 + 4*x^2 + 4*x), x)

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mupad [B]  time = 5.49, size = 55, normalized size = 2.04 \begin {gather*} x\,{\mathrm {e}}^{\frac {13\,x^2}{4\,x+8}}\,{\mathrm {e}}^{-\frac {4\,\mathrm {e}}{x+2}}\,{\mathrm {e}}^{\frac {19\,x}{2\,x+4}}\,{\mathrm {e}}^{-\frac {2\,x\,\mathrm {e}}{x+2}}\,{\mathrm {e}}^{\frac {11}{x+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((38*x + log(x)*(4*x + 8) + 13*x^2 - exp(1)*(8*x + 16) + 44)/(4*x + 8))*(48*x + 56*x^2 + 13*x^3 + 16))
/(16*x + 16*x^2 + 4*x^3),x)

[Out]

x*exp((13*x^2)/(4*x + 8))*exp(-(4*exp(1))/(x + 2))*exp((19*x)/(2*x + 4))*exp(-(2*x*exp(1))/(x + 2))*exp(11/(x
+ 2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*x**3+56*x**2+48*x+16)*exp(((4*x+8)*ln(x)+(-8*x-16)*exp(1)+13*x**2+38*x+44)/(4*x+8))/(4*x**3+16*x
**2+16*x),x)

[Out]

Timed out

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