3.93.82 \(\int \frac {e^{\frac {4-x \log (5+3 x)}{\log (5+3 x)}} (-48+(-20-12 x) \log ^2(5+3 x))}{(5+3 x) \log ^2(5+3 x)} \, dx\)

Optimal. Leaf size=18 \[ 4 e^{-x+\frac {4}{\log (5+3 x)}} \]

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Rubi [F]  time = 0.65, 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 {4-x \log (5+3 x)}{\log (5+3 x)}} \left (-48+(-20-12 x) \log ^2(5+3 x)\right )}{(5+3 x) \log ^2(5+3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4 - x*Log[5 + 3*x])/Log[5 + 3*x])*(-48 + (-20 - 12*x)*Log[5 + 3*x]^2))/((5 + 3*x)*Log[5 + 3*x]^2),x]

[Out]

-4*Defer[Int][E^(-x + 4/Log[5 + 3*x]), x] - 48*Defer[Int][E^(-x + 4/Log[5 + 3*x])/((5 + 3*x)*Log[5 + 3*x]^2),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-4 e^{\frac {4-x \log (5+3 x)}{\log (5+3 x)}}-\frac {48 e^{\frac {4-x \log (5+3 x)}{\log (5+3 x)}}}{(5+3 x) \log ^2(5+3 x)}\right ) \, dx\\ &=-\left (4 \int e^{\frac {4-x \log (5+3 x)}{\log (5+3 x)}} \, dx\right )-48 \int \frac {e^{\frac {4-x \log (5+3 x)}{\log (5+3 x)}}}{(5+3 x) \log ^2(5+3 x)} \, dx\\ &=-\left (4 \int e^{-x+\frac {4}{\log (5+3 x)}} \, dx\right )-48 \int \frac {e^{-x+\frac {4}{\log (5+3 x)}}}{(5+3 x) \log ^2(5+3 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 18, normalized size = 1.00 \begin {gather*} 4 e^{-x+\frac {4}{\log (5+3 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 - x*Log[5 + 3*x])/Log[5 + 3*x])*(-48 + (-20 - 12*x)*Log[5 + 3*x]^2))/((5 + 3*x)*Log[5 + 3*x]^
2),x]

[Out]

4*E^(-x + 4/Log[5 + 3*x])

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fricas [A]  time = 0.74, size = 23, normalized size = 1.28 \begin {gather*} 4 \, e^{\left (-\frac {x \log \left (3 \, x + 5\right ) - 4}{\log \left (3 \, x + 5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x-20)*log(3*x+5)^2-48)*exp((-x*log(3*x+5)+4)/log(3*x+5))/(3*x+5)/log(3*x+5)^2,x, algorithm="fr
icas")

[Out]

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

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giac [A]  time = 0.17, size = 17, normalized size = 0.94 \begin {gather*} 4 \, e^{\left (-x + \frac {4}{\log \left (3 \, x + 5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x-20)*log(3*x+5)^2-48)*exp((-x*log(3*x+5)+4)/log(3*x+5))/(3*x+5)/log(3*x+5)^2,x, algorithm="gi
ac")

[Out]

4*e^(-x + 4/log(3*x + 5))

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maple [A]  time = 0.51, size = 24, normalized size = 1.33

method	result	size
norman	\(4 \,{\mathrm e}^{\frac {-x \ln \left (3 x +5\right )+4}{\ln \left (3 x +5\right )}}\)	\(24\)
risch	\(4 \,{\mathrm e}^{-\frac {x \ln \left (3 x +5\right )-4}{\ln \left (3 x +5\right )}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x-20)*ln(3*x+5)^2-48)*exp((-x*ln(3*x+5)+4)/ln(3*x+5))/(3*x+5)/ln(3*x+5)^2,x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x-20)*log(3*x+5)^2-48)*exp((-x*log(3*x+5)+4)/log(3*x+5))/(3*x+5)/log(3*x+5)^2,x, algorithm="ma
xima")

[Out]

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

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mupad [B]  time = 6.14, size = 17, normalized size = 0.94 \begin {gather*} 4\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {4}{\ln \left (3\,x+5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(x*log(3*x + 5) - 4)/log(3*x + 5))*(log(3*x + 5)^2*(12*x + 20) + 48))/(log(3*x + 5)^2*(3*x + 5)),x)

[Out]

4*exp(-x)*exp(4/log(3*x + 5))

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sympy [A]  time = 0.38, size = 19, normalized size = 1.06 \begin {gather*} 4 e^{\frac {- x \log {\left (3 x + 5 \right )} + 4}{\log {\left (3 x + 5 \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x-20)*ln(3*x+5)**2-48)*exp((-x*ln(3*x+5)+4)/ln(3*x+5))/(3*x+5)/ln(3*x+5)**2,x)

[Out]

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

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