3.40.87 \(\int \frac {e^{1+e^{3+x}} (-16-12 x-2 x^2+e^{3+x} (4 x+4 x^2+x^3))}{x^5} \, dx\)

Optimal. Leaf size=27 \[ \frac {e^{1+e^{3+x}} (4+x (4+x))^2}{x^4 (2+x)^2} \]

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Rubi [A]  time = 0.09, antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2288} \begin {gather*} \frac {e^{e^{x+3}+1} \left (x^3+4 x^2+4 x\right )}{x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(1 + E^(3 + x))*(-16 - 12*x - 2*x^2 + E^(3 + x)*(4*x + 4*x^2 + x^3)))/x^5,x]

[Out]

(E^(1 + E^(3 + x))*(4*x + 4*x^2 + x^3))/x^5

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{1+e^{3+x}} \left (4 x+4 x^2+x^3\right )}{x^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 18, normalized size = 0.67 \begin {gather*} \frac {e^{1+e^{3+x}} (2+x)^2}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(1 + E^(3 + x))*(-16 - 12*x - 2*x^2 + E^(3 + x)*(4*x + 4*x^2 + x^3)))/x^5,x]

[Out]

(E^(1 + E^(3 + x))*(2 + x)^2)/x^4

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fricas [A]  time = 0.71, size = 19, normalized size = 0.70 \begin {gather*} \frac {{\left (x^{2} + 4 \, x + 4\right )} e^{\left (e^{\left (x + 3\right )} + 1\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+4*x^2+4*x)*exp(3+x)-2*x^2-12*x-16)*exp(exp(3+x)+1)/x^5,x, algorithm="fricas")

[Out]

(x^2 + 4*x + 4)*e^(e^(x + 3) + 1)/x^4

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giac [A]  time = 0.14, size = 44, normalized size = 1.63 \begin {gather*} \frac {{\left (x^{2} e^{\left (x + e^{\left (x + 3\right )} + 4\right )} + 4 \, x e^{\left (x + e^{\left (x + 3\right )} + 4\right )} + 4 \, e^{\left (x + e^{\left (x + 3\right )} + 4\right )}\right )} e^{\left (-x - 3\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+4*x^2+4*x)*exp(3+x)-2*x^2-12*x-16)*exp(exp(3+x)+1)/x^5,x, algorithm="giac")

[Out]

(x^2*e^(x + e^(x + 3) + 4) + 4*x*e^(x + e^(x + 3) + 4) + 4*e^(x + e^(x + 3) + 4))*e^(-x - 3)/x^4

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maple [A]  time = 0.05, size = 20, normalized size = 0.74

method	result	size
risch	\(\frac {\left (x^{2}+4 x +4\right ) {\mathrm e}^{{\mathrm e}^{3+x}+1}}{x^{4}}\)	\(20\)
norman	\(\frac {x^{2} {\mathrm e}^{{\mathrm e}^{3+x}+1}+4 x \,{\mathrm e}^{{\mathrm e}^{3+x}+1}+4 \,{\mathrm e}^{{\mathrm e}^{3+x}+1}}{x^{4}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3+4*x^2+4*x)*exp(3+x)-2*x^2-12*x-16)*exp(exp(3+x)+1)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/x^4*(x^2+4*x+4)*exp(exp(3+x)+1)

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maxima [A]  time = 0.55, size = 25, normalized size = 0.93 \begin {gather*} \frac {{\left (x^{2} e + 4 \, x e + 4 \, e\right )} e^{\left (e^{\left (x + 3\right )}\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+4*x^2+4*x)*exp(3+x)-2*x^2-12*x-16)*exp(exp(3+x)+1)/x^5,x, algorithm="maxima")

[Out]

(x^2*e + 4*x*e + 4*e)*e^(e^(x + 3))/x^4

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mupad [B]  time = 2.60, size = 17, normalized size = 0.63 \begin {gather*} \frac {{\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x+1}\,{\left (x+2\right )}^2}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x + 3) + 1)*(12*x + 2*x^2 - exp(x + 3)*(4*x + 4*x^2 + x^3) + 16))/x^5,x)

[Out]

(exp(exp(3)*exp(x) + 1)*(x + 2)^2)/x^4

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sympy [A]  time = 0.15, size = 19, normalized size = 0.70 \begin {gather*} \frac {\left (x^{2} + 4 x + 4\right ) e^{e^{x + 3} + 1}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3+4*x**2+4*x)*exp(3+x)-2*x**2-12*x-16)*exp(exp(3+x)+1)/x**5,x)

[Out]

(x**2 + 4*x + 4)*exp(exp(x + 3) + 1)/x**4

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