3.86.55 \(\int \frac {e^{4-\frac {20+9 x-16 x^4-4 x^5+(4+x) \log (3 x)}{4+x}} (-16-24 x-x^2+256 x^4+128 x^5+16 x^6)}{16 x+8 x^2+x^3} \, dx\)

Optimal. Leaf size=26 \[ \frac {e^{-1-4 \left (-x^4+\frac {x}{4+x}\right )}}{3 x} \]

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Rubi [F]  time = 3.47, 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 (4-\frac {20+9 x-16 x^4-4 x^5+(4+x) \log (3 x)}{4+x}\right ) \left (-16-24 x-x^2+256 x^4+128 x^5+16 x^6\right )}{16 x+8 x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4 - (20 + 9*x - 16*x^4 - 4*x^5 + (4 + x)*Log[3*x])/(4 + x))*(-16 - 24*x - x^2 + 256*x^4 + 128*x^5 + 16
*x^6))/(16*x + 8*x^2 + x^3),x]

[Out]

-Defer[Int][E^((-4 - 5*x + 16*x^4 + 4*x^5 - 4*Log[3*x] - x*Log[3*x])/(4 + x))/x, x] + 16*Defer[Int][E^((-4 - 5
*x + 16*x^4 + 4*x^5 - 4*Log[3*x] - x*Log[3*x])/(4 + x))*x^3, x] - 16*Defer[Int][E^((-4 - 5*x + 16*x^4 + 4*x^5
- 4*Log[3*x] - x*Log[3*x])/(4 + x))/(4 + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (4-\frac {20+9 x-16 x^4-4 x^5+(4+x) \log (3 x)}{4+x}\right ) \left (-16-24 x-x^2+256 x^4+128 x^5+16 x^6\right )}{x \left (16+8 x+x^2\right )} \, dx\\ &=\int \frac {\exp \left (4-\frac {20+9 x-16 x^4-4 x^5+(4+x) \log (3 x)}{4+x}\right ) \left (-16-24 x-x^2+256 x^4+128 x^5+16 x^6\right )}{x (4+x)^2} \, dx\\ &=\int \frac {\exp \left (\frac {-4-5 x+16 x^4+4 x^5-4 \log (3 x)-x \log (3 x)}{4+x}\right ) \left (-16-24 x-x^2+256 x^4+128 x^5+16 x^6\right )}{x (4+x)^2} \, dx\\ &=\int \left (-\frac {\exp \left (\frac {-4-5 x+16 x^4+4 x^5-4 \log (3 x)-x \log (3 x)}{4+x}\right )}{x}+16 \exp \left (\frac {-4-5 x+16 x^4+4 x^5-4 \log (3 x)-x \log (3 x)}{4+x}\right ) x^3-\frac {16 \exp \left (\frac {-4-5 x+16 x^4+4 x^5-4 \log (3 x)-x \log (3 x)}{4+x}\right )}{(4+x)^2}\right ) \, dx\\ &=16 \int \exp \left (\frac {-4-5 x+16 x^4+4 x^5-4 \log (3 x)-x \log (3 x)}{4+x}\right ) x^3 \, dx-16 \int \frac {\exp \left (\frac {-4-5 x+16 x^4+4 x^5-4 \log (3 x)-x \log (3 x)}{4+x}\right )}{(4+x)^2} \, dx-\int \frac {\exp \left (\frac {-4-5 x+16 x^4+4 x^5-4 \log (3 x)-x \log (3 x)}{4+x}\right )}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 23, normalized size = 0.88 \begin {gather*} \frac {e^{-5+4 x^4+\frac {16}{4+x}}}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4 - (20 + 9*x - 16*x^4 - 4*x^5 + (4 + x)*Log[3*x])/(4 + x))*(-16 - 24*x - x^2 + 256*x^4 + 128*x^
5 + 16*x^6))/(16*x + 8*x^2 + x^3),x]

[Out]

E^(-5 + 4*x^4 + 16/(4 + x))/(3*x)

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fricas [A]  time = 0.96, size = 31, normalized size = 1.19 \begin {gather*} e^{\left (\frac {4 \, x^{5} + 16 \, x^{4} - {\left (x + 4\right )} \log \left (3 \, x\right ) - 5 \, x - 4}{x + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^6+128*x^5+256*x^4-x^2-24*x-16)*exp(2)^2/(x^3+8*x^2+16*x)/exp(((4+x)*log(3*x)-4*x^5-16*x^4+9*x+
20)/(4+x)),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.17, size = 60, normalized size = 2.31 \begin {gather*} e^{\left (\frac {4 \, x^{5}}{x + 4} + \frac {16 \, x^{4}}{x + 4} - \frac {x \log \left (3 \, x\right )}{x + 4} - \frac {5 \, x}{x + 4} - \frac {4 \, \log \left (3 \, x\right )}{x + 4} - \frac {4}{x + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^6+128*x^5+256*x^4-x^2-24*x-16)*exp(2)^2/(x^3+8*x^2+16*x)/exp(((4+x)*log(3*x)-4*x^5-16*x^4+9*x+
20)/(4+x)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 36, normalized size = 1.38

method	result	size
risch	\({\mathrm e}^{-\frac {-4 x^{5}-16 x^{4}+x \ln \left (3 x \right )+4 \ln \left (3 x \right )+5 x +4}{4+x}}\)	\(36\)
gosper	\({\mathrm e}^{4} {\mathrm e}^{-\frac {-4 x^{5}-16 x^{4}+x \ln \left (3 x \right )+4 \ln \left (3 x \right )+9 x +20}{4+x}}\)	\(42\)
norman	\(\frac {\left (x \,{\mathrm e}^{4}+4 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-\frac {\left (4+x \right ) \ln \left (3 x \right )-4 x^{5}-16 x^{4}+9 x +20}{4+x}}}{4+x}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^6+128*x^5+256*x^4-x^2-24*x-16)*exp(2)^2/(x^3+8*x^2+16*x)/exp(((4+x)*ln(3*x)-4*x^5-16*x^4+9*x+20)/(4+
x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.75, size = 20, normalized size = 0.77 \begin {gather*} \frac {e^{\left (4 \, x^{4} + \frac {16}{x + 4} - 5\right )}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^6+128*x^5+256*x^4-x^2-24*x-16)*exp(2)^2/(x^3+8*x^2+16*x)/exp(((4+x)*log(3*x)-4*x^5-16*x^4+9*x+
20)/(4+x)),x, algorithm="maxima")

[Out]

1/3*e^(4*x^4 + 16/(x + 4) - 5)/x

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mupad [B]  time = 5.34, size = 46, normalized size = 1.77 \begin {gather*} \frac {{\mathrm {e}}^4\,{\mathrm {e}}^{-\frac {9\,x}{x+4}}\,{\mathrm {e}}^{\frac {4\,x^5}{x+4}}\,{\mathrm {e}}^{\frac {16\,x^4}{x+4}}\,{\mathrm {e}}^{-\frac {20}{x+4}}}{3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(9*x + log(3*x)*(x + 4) - 16*x^4 - 4*x^5 + 20)/(x + 4))*exp(4)*(24*x + x^2 - 256*x^4 - 128*x^5 - 16
*x^6 + 16))/(16*x + 8*x^2 + x^3),x)

[Out]

(exp(4)*exp(-(9*x)/(x + 4))*exp((4*x^5)/(x + 4))*exp((16*x^4)/(x + 4))*exp(-20/(x + 4)))/(3*x)

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sympy [A]  time = 0.65, size = 31, normalized size = 1.19 \begin {gather*} e^{4} e^{- \frac {- 4 x^{5} - 16 x^{4} + 9 x + \left (x + 4\right ) \log {\left (3 x \right )} + 20}{x + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x**6+128*x**5+256*x**4-x**2-24*x-16)*exp(2)**2/(x**3+8*x**2+16*x)/exp(((4+x)*ln(3*x)-4*x**5-16*x
**4+9*x+20)/(4+x)),x)

[Out]

exp(4)*exp(-(-4*x**5 - 16*x**4 + 9*x + (x + 4)*log(3*x) + 20)/(x + 4))

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