3.20.38 \(\int \frac {2 x^4+e^{\frac {5+e^{e^5}}{x}} (165+33 e^{e^5}+33 x)}{11 x^3} \, dx\)

Optimal. Leaf size=31 \[ \frac {x^2}{11}+\frac {x-3 \left (e^{\frac {5+e^{e^5}}{x}}+x\right )}{x} \]

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Rubi [A]  time = 0.11, antiderivative size = 26, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 14, 2288} \begin {gather*} \frac {x^2}{11}-\frac {3 e^{\frac {5+e^{e^5}}{x}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^4 + E^((5 + E^E^5)/x)*(165 + 33*E^E^5 + 33*x))/(11*x^3),x]

[Out]

(-3*E^((5 + E^E^5)/x))/x + x^2/11

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 {1}{11} \int \frac {2 x^4+e^{\frac {5+e^{e^5}}{x}} \left (165+33 e^{e^5}+33 x\right )}{x^3} \, dx\\ &=\frac {1}{11} \int \left (2 x+\frac {33 e^{\frac {5+e^{e^5}}{x}} \left (5+e^{e^5}+x\right )}{x^3}\right ) \, dx\\ &=\frac {x^2}{11}+3 \int \frac {e^{\frac {5+e^{e^5}}{x}} \left (5+e^{e^5}+x\right )}{x^3} \, dx\\ &=-\frac {3 e^{\frac {5+e^{e^5}}{x}}}{x}+\frac {x^2}{11}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 26, normalized size = 0.84 \begin {gather*} -\frac {3 e^{\frac {5+e^{e^5}}{x}}}{x}+\frac {x^2}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^4 + E^((5 + E^E^5)/x)*(165 + 33*E^E^5 + 33*x))/(11*x^3),x]

[Out]

(-3*E^((5 + E^E^5)/x))/x + x^2/11

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fricas [A]  time = 0.77, size = 21, normalized size = 0.68 \begin {gather*} \frac {x^{3} - 33 \, e^{\left (\frac {e^{\left (e^{5}\right )} + 5}{x}\right )}}{11 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/11*((33*exp(exp(5))+33*x+165)*exp((exp(exp(5))+5)/x)+2*x^4)/x^3,x, algorithm="fricas")

[Out]

1/11*(x^3 - 33*e^((e^(e^5) + 5)/x))/x

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giac [B]  time = 0.41, size = 55, normalized size = 1.77 \begin {gather*} -\frac {x^{2} {\left (\frac {33 \, {\left (e^{\left (e^{5}\right )} + 5\right )}^{3} e^{\left (\frac {e^{\left (e^{5}\right )} + 5}{x}\right )}}{x^{3}} - e^{\left (3 \, e^{5}\right )} - 15 \, e^{\left (2 \, e^{5}\right )} - 75 \, e^{\left (e^{5}\right )} - 125\right )}}{11 \, {\left (e^{\left (e^{5}\right )} + 5\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/11*((33*exp(exp(5))+33*x+165)*exp((exp(exp(5))+5)/x)+2*x^4)/x^3,x, algorithm="giac")

[Out]

-1/11*x^2*(33*(e^(e^5) + 5)^3*e^((e^(e^5) + 5)/x)/x^3 - e^(3*e^5) - 15*e^(2*e^5) - 75*e^(e^5) - 125)/(e^(e^5)
+ 5)^3

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maple [A]  time = 0.12, size = 22, normalized size = 0.71

method	result	size
risch	\(\frac {x^{2}}{11}-\frac {3 \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}}}{x}\)	\(22\)
norman	\(\frac {\frac {x^{4}}{11}-3 \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}} x}{x^{2}}\)	\(24\)
derivativedivides	\(-\frac {-\frac {625 x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}-\frac {500 \,{\mathrm e}^{{\mathrm e}^{5}} x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}-\frac {150 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}-\frac {20 \,{\mathrm e}^{3 \,{\mathrm e}^{5}} x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}-\frac {{\mathrm e}^{4 \,{\mathrm e}^{5}} x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}+\frac {165 \left ({\mathrm e}^{{\mathrm e}^{5}}+5\right ) {\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}}}{x}+33 \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}} {\mathrm e}^{{\mathrm e}^{5}}+33 \,{\mathrm e}^{{\mathrm e}^{5}} \left (\frac {\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right ) {\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}}}{x}-{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}}\right )}{11 \left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}\)	\(161\)
default	\(-\frac {-\frac {625 x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}-\frac {500 \,{\mathrm e}^{{\mathrm e}^{5}} x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}-\frac {150 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}-\frac {20 \,{\mathrm e}^{3 \,{\mathrm e}^{5}} x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}-\frac {{\mathrm e}^{4 \,{\mathrm e}^{5}} x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}+\frac {165 \left ({\mathrm e}^{{\mathrm e}^{5}}+5\right ) {\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}}}{x}+33 \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}} {\mathrm e}^{{\mathrm e}^{5}}+33 \,{\mathrm e}^{{\mathrm e}^{5}} \left (\frac {\left ({\mathrm e}^{{\mathrm e}^{5}}+5\right ) {\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}}}{x}-{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{5}}+5}{x}}\right )}{11 \left ({\mathrm e}^{{\mathrm e}^{5}}+5\right )^{2}}\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/11*((33*exp(exp(5))+33*x+165)*exp((exp(exp(5))+5)/x)+2*x^4)/x^3,x,method=_RETURNVERBOSE)

[Out]

1/11*x^2-3/x*exp((exp(exp(5))+5)/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{11} \, x^{2} + \frac {1}{11} \, \int \frac {33 \, {\left (x + e^{\left (e^{5}\right )} + 5\right )} e^{\left (\frac {e^{\left (e^{5}\right )}}{x} + \frac {5}{x}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/11*((33*exp(exp(5))+33*x+165)*exp((exp(exp(5))+5)/x)+2*x^4)/x^3,x, algorithm="maxima")

[Out]

1/11*x^2 + 1/11*integrate(33*(x + e^(e^5) + 5)*e^(e^(e^5)/x + 5/x)/x^3, x)

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mupad [B]  time = 1.17, size = 25, normalized size = 0.81 \begin {gather*} \frac {x^2}{11}-\frac {3\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^5}}{x}}\,{\mathrm {e}}^{5/x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp((exp(exp(5)) + 5)/x)*(33*x + 33*exp(exp(5)) + 165))/11 + (2*x^4)/11)/x^3,x)

[Out]

x^2/11 - (3*exp(exp(exp(5))/x)*exp(5/x))/x

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sympy [A]  time = 0.14, size = 17, normalized size = 0.55 \begin {gather*} \frac {x^{2}}{11} - \frac {3 e^{\frac {5 + e^{e^{5}}}{x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/11*((33*exp(exp(5))+33*x+165)*exp((exp(exp(5))+5)/x)+2*x**4)/x**3,x)

[Out]

x**2/11 - 3*exp((5 + exp(exp(5)))/x)/x

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