\(\int \frac {e^{-\frac {5}{x^4 \log (-3 x+2 x^5)}} (-45+150 x^4+(-180+120 x^4) \log (-3 x+2 x^5))}{(-3 x^5+2 x^9) \log ^2(-3 x+2 x^5)} \, dx\) [8620]

Optimal result

Integrand size = 70, antiderivative size = 22 \[ \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{\left (-3 x^5+2 x^9\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx=3 e^{-\frac {5}{x^4 \log \left (x+2 x \left (-2+x^4\right )\right )}} \]

[Out]

3/exp(5/x^4/ln(2*x*(x^4-2)+x))

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1607, 6838} \[ \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{\left (-3 x^5+2 x^9\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx=3 e^{-\frac {5}{x^4 \log \left (2 x^5-3 x\right )}} \]

[In]

Int[(-45 + 150*x^4 + (-180 + 120*x^4)*Log[-3*x + 2*x^5])/(E^(5/(x^4*Log[-3*x + 2*x^5]))*(-3*x^5 + 2*x^9)*Log[-
3*x + 2*x^5]^2),x]

[Out]

3/E^(5/(x^4*Log[-3*x + 2*x^5]))

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{x^5 \left (-3+2 x^4\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx \\ & = 3 e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{\left (-3 x^5+2 x^9\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx=3 e^{-\frac {5}{x^4 \log \left (x \left (-3+2 x^4\right )\right )}} \]

[In]

Integrate[(-45 + 150*x^4 + (-180 + 120*x^4)*Log[-3*x + 2*x^5])/(E^(5/(x^4*Log[-3*x + 2*x^5]))*(-3*x^5 + 2*x^9)
*Log[-3*x + 2*x^5]^2),x]

[Out]

3/E^(5/(x^4*Log[x*(-3 + 2*x^4)]))

Maple [A] (verified)

Time = 7.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
risch	\(3 \,{\mathrm e}^{-\frac {5}{x^{4} \ln \left (2 x^{5}-3 x \right )}}\)	\(21\)
parallelrisch	\(3 \,{\mathrm e}^{-\frac {5}{x^{4} \ln \left (2 x^{5}-3 x \right )}}\)	\(23\)

[In]

int(((120*x^4-180)*ln(2*x^5-3*x)+150*x^4-45)/(2*x^9-3*x^5)/ln(2*x^5-3*x)^2/exp(5/x^4/ln(2*x^5-3*x)),x,method=_
RETURNVERBOSE)

[Out]

3*exp(-5/x^4/ln(2*x^5-3*x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{\left (-3 x^5+2 x^9\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx=3 \, e^{\left (-\frac {5}{x^{4} \log \left (2 \, x^{5} - 3 \, x\right )}\right )} \]

[In]

integrate(((120*x^4-180)*log(2*x^5-3*x)+150*x^4-45)/(2*x^9-3*x^5)/log(2*x^5-3*x)^2/exp(5/x^4/log(2*x^5-3*x)),x
, algorithm="fricas")

[Out]

3*e^(-5/(x^4*log(2*x^5 - 3*x)))

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{\left (-3 x^5+2 x^9\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((120*x**4-180)*ln(2*x**5-3*x)+150*x**4-45)/(2*x**9-3*x**5)/ln(2*x**5-3*x)**2/exp(5/x**4/ln(2*x**5-3
*x)),x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{\left (-3 x^5+2 x^9\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(((120*x^4-180)*log(2*x^5-3*x)+150*x^4-45)/(2*x^9-3*x^5)/log(2*x^5-3*x)^2/exp(5/x^4/log(2*x^5-3*x)),x
, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{\left (-3 x^5+2 x^9\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx=3 \, e^{\left (-\frac {5}{x^{4} \log \left (2 \, x^{5} - 3 \, x\right )}\right )} \]

[In]

integrate(((120*x^4-180)*log(2*x^5-3*x)+150*x^4-45)/(2*x^9-3*x^5)/log(2*x^5-3*x)^2/exp(5/x^4/log(2*x^5-3*x)),x
, algorithm="giac")

[Out]

3*e^(-5/(x^4*log(2*x^5 - 3*x)))

Mupad [B] (verification not implemented)

Time = 14.61 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-\frac {5}{x^4 \log \left (-3 x+2 x^5\right )}} \left (-45+150 x^4+\left (-180+120 x^4\right ) \log \left (-3 x+2 x^5\right )\right )}{\left (-3 x^5+2 x^9\right ) \log ^2\left (-3 x+2 x^5\right )} \, dx=3\,{\mathrm {e}}^{-\frac {5}{x^4\,\ln \left (2\,x^5-3\,x\right )}} \]

[In]

int(-(exp(-5/(x^4*log(2*x^5 - 3*x)))*(log(2*x^5 - 3*x)*(120*x^4 - 180) + 150*x^4 - 45))/(log(2*x^5 - 3*x)^2*(3
*x^5 - 2*x^9)),x)

[Out]

3*exp(-5/(x^4*log(2*x^5 - 3*x)))