3.44.12 \(\int \frac {e^{\frac {2 (3+x+x^2+x^3)}{x}} (-6+2 x^2+4 x^3)}{x^2} \, dx\)

Optimal. Leaf size=19 \[ e^{2 x+2 x^2+\frac {2 (3+x)}{x}} \]

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Rubi [A]  time = 0.16, antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6706} \begin {gather*} e^{\frac {2 \left (x^3+x^2+x+3\right )}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((2*(3 + x + x^2 + x^3))/x)*(-6 + 2*x^2 + 4*x^3))/x^2,x]

[Out]

E^((2*(3 + x + x^2 + x^3))/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} &=e^{\frac {2 \left (3+x+x^2+x^3\right )}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 17, normalized size = 0.89 \begin {gather*} e^{2+\frac {6}{x}+2 x+2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(3 + x + x^2 + x^3))/x)*(-6 + 2*x^2 + 4*x^3))/x^2,x]

[Out]

E^(2 + 6/x + 2*x + 2*x^2)

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fricas [A]  time = 0.76, size = 15, normalized size = 0.79 \begin {gather*} e^{\left (\frac {2 \, {\left (x^{3} + x^{2} + x + 3\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+2*x^2-6)*exp((x^3+x^2+x+3)/x)^2/x^2,x, algorithm="fricas")

[Out]

e^(2*(x^3 + x^2 + x + 3)/x)

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giac [A]  time = 0.12, size = 16, normalized size = 0.84 \begin {gather*} e^{\left (2 \, x^{2} + 2 \, x + \frac {6}{x} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+2*x^2-6)*exp((x^3+x^2+x+3)/x)^2/x^2,x, algorithm="giac")

[Out]

e^(2*x^2 + 2*x + 6/x + 2)

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maple [A]  time = 0.09, size = 16, normalized size = 0.84

method	result	size
risch	\({\mathrm e}^{\frac {2 x^{3}+2 x^{2}+2 x +6}{x}}\)	\(16\)
gosper	\({\mathrm e}^{\frac {2 x^{3}+2 x^{2}+2 x +6}{x}}\)	\(17\)
norman	\({\mathrm e}^{\frac {2 x^{3}+2 x^{2}+2 x +6}{x}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3+2*x^2-6)*exp((x^3+x^2+x+3)/x)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(2*(x^3+x^2+x+3)/x)

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maxima [A]  time = 0.41, size = 16, normalized size = 0.84 \begin {gather*} e^{\left (2 \, x^{2} + 2 \, x + \frac {6}{x} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+2*x^2-6)*exp((x^3+x^2+x+3)/x)^2/x^2,x, algorithm="maxima")

[Out]

e^(2*x^2 + 2*x + 6/x + 2)

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mupad [B]  time = 3.02, size = 19, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^{6/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*(x + x^2 + x^3 + 3))/x)*(2*x^2 + 4*x^3 - 6))/x^2,x)

[Out]

exp(2*x)*exp(2)*exp(2*x^2)*exp(6/x)

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sympy [A]  time = 0.13, size = 17, normalized size = 0.89 \begin {gather*} e^{\frac {2 \left (x^{3} + x^{2} + x + 3\right )}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3+2*x**2-6)*exp((x**3+x**2+x+3)/x)**2/x**2,x)

[Out]

exp(2*(x**3 + x**2 + x + 3)/x)

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