3.22.92 \(\int \frac {e^{4 x-5 x^2-x^3+x \log (5)-x \log (3-x)} (-12+33 x-x^2-3 x^3+(-3+x) \log (5)+(3-x) \log (3-x))}{-3+x} \, dx\)

Optimal. Leaf size=24 \[ e^{x \left (4+\log (5)-x \left (5+x+\frac {\log (3-x)}{x}\right )\right )} \]

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Rubi [A]  time = 0.55, antiderivative size = 29, normalized size of antiderivative = 1.21, number of steps used = 1, number of rules used = 1, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6706} \begin {gather*} 5^x e^{-x^3-5 x^2+4 x} (3-x)^{-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4*x - 5*x^2 - x^3 + x*Log[5] - x*Log[3 - x])*(-12 + 33*x - x^2 - 3*x^3 + (-3 + x)*Log[5] + (3 - x)*Log
[3 - x]))/(-3 + x),x]

[Out]

(5^x*E^(4*x - 5*x^2 - x^3))/(3 - x)^x

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} &=5^x e^{4 x-5 x^2-x^3} (3-x)^{-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 4.34, size = 26, normalized size = 1.08 \begin {gather*} 5^x e^{-x \left (-4+5 x+x^2\right )} (3-x)^{-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*x - 5*x^2 - x^3 + x*Log[5] - x*Log[3 - x])*(-12 + 33*x - x^2 - 3*x^3 + (-3 + x)*Log[5] + (3 -
x)*Log[3 - x]))/(-3 + x),x]

[Out]

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

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fricas [A]  time = 0.72, size = 28, normalized size = 1.17 \begin {gather*} e^{\left (-x^{3} - 5 \, x^{2} + x \log \relax (5) - x \log \left (-x + 3\right ) + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3-x)*log(3-x)+(x-3)*log(5)-3*x^3-x^2+33*x-12)*exp(-x*log(3-x)+x*log(5)-x^3-5*x^2+4*x)/(x-3),x, alg
orithm="fricas")

[Out]

e^(-x^3 - 5*x^2 + x*log(5) - x*log(-x + 3) + 4*x)

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giac [A]  time = 0.25, size = 28, normalized size = 1.17 \begin {gather*} e^{\left (-x^{3} - 5 \, x^{2} + x \log \relax (5) - x \log \left (-x + 3\right ) + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3-x)*log(3-x)+(x-3)*log(5)-3*x^3-x^2+33*x-12)*exp(-x*log(3-x)+x*log(5)-x^3-5*x^2+4*x)/(x-3),x, alg
orithm="giac")

[Out]

e^(-x^3 - 5*x^2 + x*log(5) - x*log(-x + 3) + 4*x)

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maple [A]  time = 0.37, size = 26, normalized size = 1.08

method	result	size
risch	\(\left (3-x \right )^{-x} 5^{x} {\mathrm e}^{-x \left (x^{2}+5 x -4\right )}\)	\(26\)
norman	\({\mathrm e}^{-x \ln \left (3-x \right )+x \ln \relax (5)-x^{3}-5 x^{2}+4 x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3-x)*ln(3-x)+(x-3)*ln(5)-3*x^3-x^2+33*x-12)*exp(-x*ln(3-x)+x*ln(5)-x^3-5*x^2+4*x)/(x-3),x,method=_RETURN
VERBOSE)

[Out]

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

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maxima [A]  time = 0.60, size = 28, normalized size = 1.17 \begin {gather*} e^{\left (-x^{3} - 5 \, x^{2} + x \log \relax (5) - x \log \left (-x + 3\right ) + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3-x)*log(3-x)+(x-3)*log(5)-3*x^3-x^2+33*x-12)*exp(-x*log(3-x)+x*log(5)-x^3-5*x^2+4*x)/(x-3),x, alg
orithm="maxima")

[Out]

e^(-x^3 - 5*x^2 + x*log(5) - x*log(-x + 3) + 4*x)

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mupad [B]  time = 1.48, size = 29, normalized size = 1.21 \begin {gather*} \frac {5^x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{-x^3}\,{\mathrm {e}}^{-5\,x^2}}{{\left (3-x\right )}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4*x + x*log(5) - x*log(3 - x) - 5*x^2 - x^3)*(x^2 - log(5)*(x - 3) - 33*x + 3*x^3 + log(3 - x)*(x -
3) + 12))/(x - 3),x)

[Out]

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

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sympy [A]  time = 0.39, size = 24, normalized size = 1.00 \begin {gather*} e^{- x^{3} - 5 x^{2} - x \log {\left (3 - x \right )} + x \log {\relax (5 )} + 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3-x)*ln(3-x)+(x-3)*ln(5)-3*x**3-x**2+33*x-12)*exp(-x*ln(3-x)+x*ln(5)-x**3-5*x**2+4*x)/(x-3),x)

[Out]

exp(-x**3 - 5*x**2 - x*log(3 - x) + x*log(5) + 4*x)

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