3.39.11 \(\int \frac {x^5+e^{4+\frac {e^4 (4+4 x+5 x^2+2 x^3+x^4)}{x^4}} (16+12 x+10 x^2+2 x^3)}{x^5} \, dx\)

Optimal. Leaf size=25 \[ 3-e^{\frac {e^4 \left (x+\frac {2+x}{x}\right )^2}{x^2}}+x \]

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Rubi [A]  time = 0.29, antiderivative size = 21, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {14, 6706} \begin {gather*} x-e^{\frac {e^4 \left (x^2+x+2\right )^2}{x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5 + E^(4 + (E^4*(4 + 4*x + 5*x^2 + 2*x^3 + x^4))/x^4)*(16 + 12*x + 10*x^2 + 2*x^3))/x^5,x]

[Out]

-E^((E^4*(2 + x + x^2)^2)/x^4) + x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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} &=\int \left (1+\frac {2 e^{4+\frac {e^4 \left (2+x+x^2\right )^2}{x^4}} (4+x) \left (2+x+x^2\right )}{x^5}\right ) \, dx\\ &=x+2 \int \frac {e^{4+\frac {e^4 \left (2+x+x^2\right )^2}{x^4}} (4+x) \left (2+x+x^2\right )}{x^5} \, dx\\ &=-e^{\frac {e^4 \left (2+x+x^2\right )^2}{x^4}}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 21, normalized size = 0.84 \begin {gather*} -e^{\frac {e^4 \left (2+x+x^2\right )^2}{x^4}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5 + E^(4 + (E^4*(4 + 4*x + 5*x^2 + 2*x^3 + x^4))/x^4)*(16 + 12*x + 10*x^2 + 2*x^3))/x^5,x]

[Out]

-E^((E^4*(2 + x + x^2)^2)/x^4) + x

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fricas [A]  time = 0.53, size = 42, normalized size = 1.68 \begin {gather*} {\left (x e^{4} - e^{\left (\frac {4 \, x^{4} + {\left (x^{4} + 2 \, x^{3} + 5 \, x^{2} + 4 \, x + 4\right )} e^{4}}{x^{4}}\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+10*x^2+12*x+16)*exp(4)*exp((x^4+2*x^3+5*x^2+4*x+4)*exp(4)/x^4)+x^5)/x^5,x, algorithm="fricas
")

[Out]

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

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giac [A]  time = 0.20, size = 36, normalized size = 1.44 \begin {gather*} x - e^{\left (\frac {2 \, e^{4}}{x} + \frac {5 \, e^{4}}{x^{2}} + \frac {4 \, e^{4}}{x^{3}} + \frac {4 \, e^{4}}{x^{4}} + e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+10*x^2+12*x+16)*exp(4)*exp((x^4+2*x^3+5*x^2+4*x+4)*exp(4)/x^4)+x^5)/x^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.18, size = 20, normalized size = 0.80

method	result	size
risch	\(x -{\mathrm e}^{\frac {\left (x^{2}+x +2\right )^{2} {\mathrm e}^{4}}{x^{4}}}\)	\(20\)
norman	\(\frac {x^{5}-x^{4} {\mathrm e}^{\frac {\left (x^{4}+2 x^{3}+5 x^{2}+4 x +4\right ) {\mathrm e}^{4}}{x^{4}}}}{x^{4}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+10*x^2+12*x+16)*exp(4)*exp((x^4+2*x^3+5*x^2+4*x+4)*exp(4)/x^4)+x^5)/x^5,x,method=_RETURNVERBOSE)

[Out]

x-exp((x^2+x+2)^2*exp(4)/x^4)

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maxima [A]  time = 0.60, size = 36, normalized size = 1.44 \begin {gather*} x - e^{\left (\frac {2 \, e^{4}}{x} + \frac {5 \, e^{4}}{x^{2}} + \frac {4 \, e^{4}}{x^{3}} + \frac {4 \, e^{4}}{x^{4}} + e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+10*x^2+12*x+16)*exp(4)*exp((x^4+2*x^3+5*x^2+4*x+4)*exp(4)/x^4)+x^5)/x^5,x, algorithm="maxima
")

[Out]

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

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mupad [B]  time = 3.04, size = 39, normalized size = 1.56 \begin {gather*} x-{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^4}{x}}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^4}{x^2}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^4}{x^3}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^4}{x^4}}\,{\mathrm {e}}^{{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5 + exp(4)*exp((exp(4)*(4*x + 5*x^2 + 2*x^3 + x^4 + 4))/x^4)*(12*x + 10*x^2 + 2*x^3 + 16))/x^5,x)

[Out]

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

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sympy [A]  time = 0.32, size = 27, normalized size = 1.08 \begin {gather*} x - e^{\frac {\left (x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 4\right ) e^{4}}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+10*x**2+12*x+16)*exp(4)*exp((x**4+2*x**3+5*x**2+4*x+4)*exp(4)/x**4)+x**5)/x**5,x)

[Out]

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

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