3.79.38 \(\int \frac {3^{\frac {1}{2 x^3}} e^{\frac {2}{x^3}} (-\frac {1}{8+3 x})^{\frac {1}{2 x^3}} (-3 x+(-24-9 x) \log (-\frac {3 e^4}{8+3 x}))}{16 x^4+6 x^5} \, dx\)

Optimal. Leaf size=25 \[ e^{\frac {2}{x^3}} \left (-\frac {1}{\frac {8}{3}+x}\right )^{\frac {1}{2 x^3}} \]

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Rubi [F]  time = 2.58, 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 {3^{\frac {1}{2 x^3}} e^{\frac {2}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}} \left (-3 x+(-24-9 x) \log \left (-\frac {3 e^4}{8+3 x}\right )\right )}{16 x^4+6 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3^(1/(2*x^3))*E^(2/x^3)*(-(8 + 3*x)^(-1))^(1/(2*x^3))*(-3*x + (-24 - 9*x)*Log[(-3*E^4)/(8 + 3*x)]))/(16*x
^4 + 6*x^5),x]

[Out]

(-3*Log[(-3*E^4)/(8 + 3*x)]*Defer[Int][(E^((2 + Log[3]/2)/x^3)*(-(8 + 3*x)^(-1))^(1/(2*x^3)))/x^4, x])/2 - (3*
Defer[Int][(E^((2 + Log[3]/2)/x^3)*(-(8 + 3*x)^(-1))^(1/(2*x^3)))/x^3, x])/16 + (9*Defer[Int][(E^((2 + Log[3]/
2)/x^3)*(-(8 + 3*x)^(-1))^(1/(2*x^3)))/x^2, x])/128 - (27*Defer[Int][(E^((2 + Log[3]/2)/x^3)*(-(8 + 3*x)^(-1))
^(1/(2*x^3)))/x, x])/1024 + (81*Defer[Int][(E^((2 + Log[3]/2)/x^3)*(-(8 + 3*x)^(-1))^(1/(2*x^3)))/(8 + 3*x), x
])/1024 + (9*Defer[Int][Defer[Int][(E^((4 + Log[3])/(2*x^3))*((-8 - 3*x)^(-1))^(1/(2*x^3)))/x^4, x]/(-8 - 3*x)
, x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3^{\frac {1}{2 x^3}} e^{\frac {2}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}} \left (-3 x+(-24-9 x) \log \left (-\frac {3 e^4}{8+3 x}\right )\right )}{x^4 (16+6 x)} \, dx\\ &=\int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}} \left (-3 x+(-24-9 x) \log \left (-\frac {3 e^4}{8+3 x}\right )\right )}{x^4 (16+6 x)} \, dx\\ &=\int \left (-\frac {3 e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{2 x^3 (8+3 x)}-\frac {3 e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}} \log \left (-\frac {3 e^4}{8+3 x}\right )}{2 x^4}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{x^3 (8+3 x)} \, dx\right )-\frac {3}{2} \int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}} \log \left (-\frac {3 e^4}{8+3 x}\right )}{x^4} \, dx\\ &=-\left (\frac {3}{2} \int \left (\frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{8 x^3}-\frac {3 e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{64 x^2}+\frac {9 e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{512 x}-\frac {27 e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{512 (8+3 x)}\right ) \, dx\right )+\frac {3}{2} \int \frac {3 \int \frac {e^{\frac {4+\log (3)}{2 x^3}} \left (\frac {1}{-8-3 x}\right )^{\frac {1}{2 x^3}}}{x^4} \, dx}{-8-3 x} \, dx-\frac {1}{2} \left (3 \log \left (-\frac {3 e^4}{8+3 x}\right )\right ) \int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{x^4} \, dx\\ &=-\frac {27 \int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{x} \, dx}{1024}+\frac {9}{128} \int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{x^2} \, dx+\frac {81 \int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{8+3 x} \, dx}{1024}-\frac {3}{16} \int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{x^3} \, dx+\frac {9}{2} \int \frac {\int \frac {e^{\frac {4+\log (3)}{2 x^3}} \left (\frac {1}{-8-3 x}\right )^{\frac {1}{2 x^3}}}{x^4} \, dx}{-8-3 x} \, dx-\frac {1}{2} \left (3 \log \left (-\frac {3 e^4}{8+3 x}\right )\right ) \int \frac {e^{\frac {2+\frac {\log (3)}{2}}{x^3}} \left (-\frac {1}{8+3 x}\right )^{\frac {1}{2 x^3}}}{x^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.90, size = 32, normalized size = 1.28 \begin {gather*} 3^{\frac {1}{2 x^3}} e^{\frac {2}{x^3}} \left (\frac {1}{-8-3 x}\right )^{\frac {1}{2 x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3^(1/(2*x^3))*E^(2/x^3)*(-(8 + 3*x)^(-1))^(1/(2*x^3))*(-3*x + (-24 - 9*x)*Log[(-3*E^4)/(8 + 3*x)]))
/(16*x^4 + 6*x^5),x]

[Out]

3^(1/(2*x^3))*E^(2/x^3)*((-8 - 3*x)^(-1))^(1/(2*x^3))

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fricas [A]  time = 0.65, size = 17, normalized size = 0.68 \begin {gather*} \left (-\frac {3 \, e^{4}}{3 \, x + 8}\right )^{\frac {1}{2 \, x^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x-24)*log(-3*exp(4)/(3*x+8))-3*x)*exp(1/2*log(-3*exp(4)/(3*x+8))/x^3)/(6*x^5+16*x^4),x, algorit
hm="fricas")

[Out]

(-3*e^4/(3*x + 8))^(1/2/x^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, {\left ({\left (3 \, x + 8\right )} \log \left (-\frac {3 \, e^{4}}{3 \, x + 8}\right ) + x\right )} \left (-\frac {3 \, e^{4}}{3 \, x + 8}\right )^{\frac {1}{2 \, x^{3}}}}{2 \, {\left (3 \, x^{5} + 8 \, x^{4}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x-24)*log(-3*exp(4)/(3*x+8))-3*x)*exp(1/2*log(-3*exp(4)/(3*x+8))/x^3)/(6*x^5+16*x^4),x, algorit
hm="giac")

[Out]

integrate(-3/2*((3*x + 8)*log(-3*e^4/(3*x + 8)) + x)*(-3*e^4/(3*x + 8))^(1/2/x^3)/(3*x^5 + 8*x^4), x)

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maple [A]  time = 0.22, size = 18, normalized size = 0.72

method	result	size
risch	\(\left (-\frac {3 \,{\mathrm e}^{4}}{3 x +8}\right )^{\frac {1}{2 x^{3}}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*x-24)*ln(-3*exp(4)/(3*x+8))-3*x)*exp(1/2*ln(-3*exp(4)/(3*x+8))/x^3)/(6*x^5+16*x^4),x,method=_RETURNVE
RBOSE)

[Out]

(-3*exp(4)/(3*x+8))^(1/2/x^3)

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maxima [A]  time = 0.50, size = 25, normalized size = 1.00 \begin {gather*} e^{\left (\frac {\log \relax (3)}{2 \, x^{3}} - \frac {\log \left (-3 \, x - 8\right )}{2 \, x^{3}} + \frac {2}{x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x-24)*log(-3*exp(4)/(3*x+8))-3*x)*exp(1/2*log(-3*exp(4)/(3*x+8))/x^3)/(6*x^5+16*x^4),x, algorit
hm="maxima")

[Out]

e^(1/2*log(3)/x^3 - 1/2*log(-3*x - 8)/x^3 + 2/x^3)

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mupad [B]  time = 6.03, size = 22, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{\frac {2}{x^3}}\,{\left (-\frac {3}{3\,x+8}\right )}^{\frac {1}{2\,x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2/x^3)*(-3/(3*x + 8))^(1/(2*x^3))

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sympy [A]  time = 0.40, size = 19, normalized size = 0.76 \begin {gather*} e^{\frac {\log {\left (- \frac {3 e^{4}}{3 x + 8} \right )}}{2 x^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x-24)*ln(-3*exp(4)/(3*x+8))-3*x)*exp(1/2*ln(-3*exp(4)/(3*x+8))/x**3)/(6*x**5+16*x**4),x)

[Out]

exp(log(-3*exp(4)/(3*x + 8))/(2*x**3))

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