3.3.47 \(\int \frac {e^{\frac {1089+132 x+e x-194 x^2-12 x^3+9 x^4}{x}} (-39204-6984 x^2-864 x^3+972 x^4)}{x^2} \, dx\)

Optimal. Leaf size=26 \[ 36 e^{e+\frac {(4-3 (5-x))^2 (3+x)^2}{x}} \]

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Rubi [A]  time = 0.34, antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6706} \begin {gather*} 36 e^{\frac {9 x^4-12 x^3-194 x^2+e x+132 x+1089}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((1089 + 132*x + E*x - 194*x^2 - 12*x^3 + 9*x^4)/x)*(-39204 - 6984*x^2 - 864*x^3 + 972*x^4))/x^2,x]

[Out]

36*E^((1089 + 132*x + E*x - 194*x^2 - 12*x^3 + 9*x^4)/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} &=36 e^{\frac {1089+132 x+e x-194 x^2-12 x^3+9 x^4}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 22, normalized size = 0.85 \begin {gather*} 36 e^{e+\frac {\left (33+2 x-3 x^2\right )^2}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((1089 + 132*x + E*x - 194*x^2 - 12*x^3 + 9*x^4)/x)*(-39204 - 6984*x^2 - 864*x^3 + 972*x^4))/x^2,
x]

[Out]

36*E^(E + (33 + 2*x - 3*x^2)^2/x)

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fricas [A]  time = 0.90, size = 31, normalized size = 1.19 \begin {gather*} 36 \, e^{\left (\frac {9 \, x^{4} - 12 \, x^{3} - 194 \, x^{2} + x e + 132 \, x + 1089}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((972*x^4-864*x^3-6984*x^2-39204)*exp((x*exp(1)+9*x^4-12*x^3-194*x^2+132*x+1089)/x)/x^2,x, algorithm=
"fricas")

[Out]

36*e^((9*x^4 - 12*x^3 - 194*x^2 + x*e + 132*x + 1089)/x)

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giac [A]  time = 0.37, size = 25, normalized size = 0.96 \begin {gather*} 36 \, e^{\left (9 \, x^{3} - 12 \, x^{2} - 194 \, x + \frac {1089}{x} + e + 132\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((972*x^4-864*x^3-6984*x^2-39204)*exp((x*exp(1)+9*x^4-12*x^3-194*x^2+132*x+1089)/x)/x^2,x, algorithm=
"giac")

[Out]

36*e^(9*x^3 - 12*x^2 - 194*x + 1089/x + e + 132)

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maple [A]  time = 0.38, size = 32, normalized size = 1.23

method	result	size
gosper	\(36 \,{\mathrm e}^{\frac {x \,{\mathrm e}+9 x^{4}-12 x^{3}-194 x^{2}+132 x +1089}{x}}\)	\(32\)
norman	\(36 \,{\mathrm e}^{\frac {x \,{\mathrm e}+9 x^{4}-12 x^{3}-194 x^{2}+132 x +1089}{x}}\)	\(32\)
risch	\(36 \,{\mathrm e}^{\frac {x \,{\mathrm e}+9 x^{4}-12 x^{3}-194 x^{2}+132 x +1089}{x}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((972*x^4-864*x^3-6984*x^2-39204)*exp((x*exp(1)+9*x^4-12*x^3-194*x^2+132*x+1089)/x)/x^2,x,method=_RETURNVER
BOSE)

[Out]

36*exp((x*exp(1)+9*x^4-12*x^3-194*x^2+132*x+1089)/x)

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maxima [A]  time = 0.72, size = 25, normalized size = 0.96 \begin {gather*} 36 \, e^{\left (9 \, x^{3} - 12 \, x^{2} - 194 \, x + \frac {1089}{x} + e + 132\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((972*x^4-864*x^3-6984*x^2-39204)*exp((x*exp(1)+9*x^4-12*x^3-194*x^2+132*x+1089)/x)/x^2,x, algorithm=
"maxima")

[Out]

36*e^(9*x^3 - 12*x^2 - 194*x + 1089/x + e + 132)

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mupad [B]  time = 0.38, size = 29, normalized size = 1.12 \begin {gather*} 36\,{\mathrm {e}}^{-194\,x}\,{\mathrm {e}}^{132}\,{\mathrm {e}}^{9\,x^3}\,{\mathrm {e}}^{-12\,x^2}\,{\mathrm {e}}^{1089/x}\,{\mathrm {e}}^{\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((132*x + x*exp(1) - 194*x^2 - 12*x^3 + 9*x^4 + 1089)/x)*(6984*x^2 + 864*x^3 - 972*x^4 + 39204))/x^2,
x)

[Out]

36*exp(-194*x)*exp(132)*exp(9*x^3)*exp(-12*x^2)*exp(1089/x)*exp(exp(1))

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sympy [A]  time = 0.16, size = 29, normalized size = 1.12 \begin {gather*} 36 e^{\frac {9 x^{4} - 12 x^{3} - 194 x^{2} + e x + 132 x + 1089}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((972*x**4-864*x**3-6984*x**2-39204)*exp((x*exp(1)+9*x**4-12*x**3-194*x**2+132*x+1089)/x)/x**2,x)

[Out]

36*exp((9*x**4 - 12*x**3 - 194*x**2 + E*x + 132*x + 1089)/x)

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