3.18.47 \(\int \frac {2 e^2+e^{16} (-x^2+e^5 x^2)}{e^{16} x^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac {2}{e^{14} x}-x+e^5 x \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {12, 14} \begin {gather*} -\left (\left (1-e^5\right ) x\right )-\frac {2}{e^{14} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(E^14*x) - (1 - E^5)*x

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {2 e^2+e^{16} \left (-x^2+e^5 x^2\right )}{x^2} \, dx}{e^{16}}\\ &=\frac {\int \left (e^{16} \left (-1+e^5\right )+\frac {2 e^2}{x^2}\right ) \, dx}{e^{16}}\\ &=-\frac {2}{e^{14} x}-\left (1-e^5\right ) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {2}{e^{14} x}-x+e^5 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(E^14*x) - x + E^5*x

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fricas [A]  time = 0.59, size = 21, normalized size = 1.24 \begin {gather*} \frac {{\left (x^{2} e^{19} - x^{2} e^{14} - 2\right )} e^{\left (-14\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*e^19 - x^2*e^14 - 2)*e^(-14)/x

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giac [A]  time = 0.26, size = 20, normalized size = 1.18 \begin {gather*} {\left (x e^{21} - x e^{16} - \frac {2 \, e^{2}}{x}\right )} e^{\left (-16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e^21 - x*e^16 - 2*e^2/x)*e^(-16)

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maple [A]  time = 0.05, size = 22, normalized size = 1.29

method	result	size
norman	\(\frac {\left ({\mathrm e}^{5}-1\right ) x^{2}-2 \,{\mathrm e}^{-16} {\mathrm e}^{2}}{x}\)	\(22\)
risch	\({\mathrm e}^{-16} x \,{\mathrm e}^{21}-{\mathrm e}^{-16} x \,{\mathrm e}^{16}-\frac {2 \,{\mathrm e}^{-14}}{x}\)	\(22\)
default	\({\mathrm e}^{-16} \left (x \,{\mathrm e}^{21}-x \,{\mathrm e}^{16}-\frac {2 \,{\mathrm e}^{2}}{x}\right )\)	\(23\)
gosper	\(\frac {\left ({\mathrm e}^{5} {\mathrm e}^{16} x^{2}-x^{2} {\mathrm e}^{16}-2 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-16}}{x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((exp(5)-1)*x^2-2/exp(16)*exp(2))/x

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maxima [A]  time = 0.35, size = 20, normalized size = 1.18 \begin {gather*} {\left (x {\left (e^{21} - e^{16}\right )} - \frac {2 \, e^{2}}{x}\right )} e^{\left (-16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(e^21 - e^16) - 2*e^2/x)*e^(-16)

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mupad [B]  time = 0.06, size = 14, normalized size = 0.82 \begin {gather*} x\,\left ({\mathrm {e}}^5-1\right )-\frac {2\,{\mathrm {e}}^{-14}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-16)*(2*exp(2) + exp(16)*(x^2*exp(5) - x^2)))/x^2,x)

[Out]

x*(exp(5) - 1) - (2*exp(-14))/x

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sympy [A]  time = 0.09, size = 15, normalized size = 0.88 \begin {gather*} \frac {- x \left (- e^{19} + e^{14}\right ) - \frac {2}{x}}{e^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2*exp(5)-x**2)*exp(16)+2*exp(2))/x**2/exp(16),x)

[Out]

(-x*(-exp(19) + exp(14)) - 2/x)*exp(-14)

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