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

Optimal. Leaf size=22 \[ 3 e^{-\frac {5}{x^4 \log \left (x+2 x \left (-2+x^4\right )\right )}} \]

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Rubi [A]  time = 0.94, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1593, 6706} \begin {gather*} 3 e^{-\frac {5}{x^4 \log \left (2 x^5-3 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[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 1593

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 6706

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.61, size = 21, normalized size = 0.95 \begin {gather*} 3 e^{-\frac {5}{x^4 \log \left (x \left (-3+2 x^4\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[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)]))

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fricas [A]  time = 1.24, size = 20, normalized size = 0.91 \begin {gather*} 3 \, e^{\left (-\frac {5}{x^{4} \log \left (2 \, x^{5} - 3 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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giac [A]  time = 0.49, size = 20, normalized size = 0.91 \begin {gather*} 3 \, e^{\left (-\frac {5}{x^{4} \log \left (2 \, x^{5} - 3 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [A]  time = 0.41, size = 21, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 5.46, size = 20, normalized size = 0.91 \begin {gather*} 3\,{\mathrm {e}}^{-\frac {5}{x^4\,\ln \left (2\,x^5-3\,x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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sympy [A]  time = 0.56, size = 17, normalized size = 0.77 \begin {gather*} 3 e^{- \frac {5}{x^{4} \log {\left (2 x^{5} - 3 x \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

3*exp(-5/(x**4*log(2*x**5 - 3*x)))

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