3.2.22 \(\int \frac {e^{\frac {4+x^2-2 x^3-6 x^5}{x}} (-20+5 x^2-20 x^3-120 x^5)}{x^2} \, dx\)

Optimal. Leaf size=23 \[ 5 e^{\frac {4}{x}+x-2 x^3 \left (\frac {1}{x}+3 x\right )} \]

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Rubi [A]  time = 0.22, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6706} \begin {gather*} 5 e^{\frac {-6 x^5-2 x^3+x^2+4}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((4 + x^2 - 2*x^3 - 6*x^5)/x)*(-20 + 5*x^2 - 20*x^3 - 120*x^5))/x^2,x]

[Out]

5*E^((4 + x^2 - 2*x^3 - 6*x^5)/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 e^{\frac {4+x^2-2 x^3-6 x^5}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 21, normalized size = 0.91 \begin {gather*} 5 e^{\frac {4}{x}+x-2 x^2-6 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 + x^2 - 2*x^3 - 6*x^5)/x)*(-20 + 5*x^2 - 20*x^3 - 120*x^5))/x^2,x]

[Out]

5*E^(4/x + x - 2*x^2 - 6*x^4)

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fricas [A]  time = 0.56, size = 25, normalized size = 1.09 \begin {gather*} 5 \, e^{\left (-\frac {6 \, x^{5} + 2 \, x^{3} - x^{2} - 4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-120*x^5-20*x^3+5*x^2-20)*exp((-6*x^5-2*x^3+x^2+4)/x)/x^2,x, algorithm="fricas")

[Out]

5*e^(-(6*x^5 + 2*x^3 - x^2 - 4)/x)

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giac [A]  time = 0.41, size = 20, normalized size = 0.87 \begin {gather*} 5 \, e^{\left (-6 \, x^{4} - 2 \, x^{2} + x + \frac {4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-120*x^5-20*x^3+5*x^2-20)*exp((-6*x^5-2*x^3+x^2+4)/x)/x^2,x, algorithm="giac")

[Out]

5*e^(-6*x^4 - 2*x^2 + x + 4/x)

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maple [A]  time = 0.10, size = 23, normalized size = 1.00

method	result	size
norman	\(5 \,{\mathrm e}^{\frac {-6 x^{5}-2 x^{3}+x^{2}+4}{x}}\)	\(23\)
gosper	\(5 \,{\mathrm e}^{-\frac {6 x^{5}+2 x^{3}-x^{2}-4}{x}}\)	\(26\)
risch	\(5 \,{\mathrm e}^{-\frac {6 x^{5}+2 x^{3}-x^{2}-4}{x}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-120*x^5-20*x^3+5*x^2-20)*exp((-6*x^5-2*x^3+x^2+4)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

5*exp((-6*x^5-2*x^3+x^2+4)/x)

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maxima [A]  time = 0.83, size = 20, normalized size = 0.87 \begin {gather*} 5 \, e^{\left (-6 \, x^{4} - 2 \, x^{2} + x + \frac {4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-120*x^5-20*x^3+5*x^2-20)*exp((-6*x^5-2*x^3+x^2+4)/x)/x^2,x, algorithm="maxima")

[Out]

5*e^(-6*x^4 - 2*x^2 + x + 4/x)

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mupad [B]  time = 0.32, size = 22, normalized size = 0.96 \begin {gather*} 5\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^{4/x}\,{\mathrm {e}}^{-6\,x^4}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x^2 - 2*x^3 - 6*x^5 + 4)/x)*(20*x^3 - 5*x^2 + 120*x^5 + 20))/x^2,x)

[Out]

5*exp(-2*x^2)*exp(4/x)*exp(-6*x^4)*exp(x)

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sympy [A]  time = 0.15, size = 19, normalized size = 0.83 \begin {gather*} 5 e^{\frac {- 6 x^{5} - 2 x^{3} + x^{2} + 4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-120*x**5-20*x**3+5*x**2-20)*exp((-6*x**5-2*x**3+x**2+4)/x)/x**2,x)

[Out]

5*exp((-6*x**5 - 2*x**3 + x**2 + 4)/x)

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