3.67.81 \(\int \frac {480-180 x+5 x^2+e^{5+x} x^4}{e^5 x^4} \, dx\)

Optimal. Leaf size=18 \[ e^x-\frac {5 (-16+x) (-2+x)}{e^5 x^3} \]

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.56, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2194} \begin {gather*} -\frac {160}{e^5 x^3}+\frac {90}{e^5 x^2}+e^x-\frac {5}{e^5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(480 - 180*x + 5*x^2 + E^(5 + x)*x^4)/(E^5*x^4),x]

[Out]

E^x - 160/(E^5*x^3) + 90/(E^5*x^2) - 5/(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]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {480-180 x+5 x^2+e^{5+x} x^4}{x^4} \, dx}{e^5}\\ &=\frac {\int \left (e^{5+x}+\frac {5 \left (96-36 x+x^2\right )}{x^4}\right ) \, dx}{e^5}\\ &=\frac {\int e^{5+x} \, dx}{e^5}+\frac {5 \int \frac {96-36 x+x^2}{x^4} \, dx}{e^5}\\ &=e^x+\frac {5 \int \left (\frac {96}{x^4}-\frac {36}{x^3}+\frac {1}{x^2}\right ) \, dx}{e^5}\\ &=e^x-\frac {160}{e^5 x^3}+\frac {90}{e^5 x^2}-\frac {5}{e^5 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.39 \begin {gather*} \frac {e^{5+x}-\frac {160}{x^3}+\frac {90}{x^2}-\frac {5}{x}}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(480 - 180*x + 5*x^2 + E^(5 + x)*x^4)/(E^5*x^4),x]

[Out]

(E^(5 + x) - 160/x^3 + 90/x^2 - 5/x)/E^5

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fricas [A]  time = 0.48, size = 24, normalized size = 1.33 \begin {gather*} \frac {{\left (x^{3} e^{\left (x + 5\right )} - 5 \, x^{2} + 90 \, x - 160\right )} e^{\left (-5\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*exp(5)*exp(x)+5*x^2-180*x+480)/x^4/exp(5),x, algorithm="fricas")

[Out]

(x^3*e^(x + 5) - 5*x^2 + 90*x - 160)*e^(-5)/x^3

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giac [A]  time = 0.18, size = 24, normalized size = 1.33 \begin {gather*} \frac {{\left (x^{3} e^{\left (x + 5\right )} - 5 \, x^{2} + 90 \, x - 160\right )} e^{\left (-5\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*exp(5)*exp(x)+5*x^2-180*x+480)/x^4/exp(5),x, algorithm="giac")

[Out]

(x^3*e^(x + 5) - 5*x^2 + 90*x - 160)*e^(-5)/x^3

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maple [A]  time = 0.04, size = 20, normalized size = 1.11

method	result	size
risch	\(\frac {{\mathrm e}^{-5} \left (-5 x^{2}+90 x -160\right )}{x^{3}}+{\mathrm e}^{x}\)	\(20\)
default	\({\mathrm e}^{-5} \left ({\mathrm e}^{5} {\mathrm e}^{x}-\frac {160}{x^{3}}+\frac {90}{x^{2}}-\frac {5}{x}\right )\)	\(27\)
norman	\(\frac {{\mathrm e}^{x} x^{3}-160 \,{\mathrm e}^{-5}+90 x \,{\mathrm e}^{-5}-5 x^{2} {\mathrm e}^{-5}}{x^{3}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*exp(5)*exp(x)+5*x^2-180*x+480)/x^4/exp(5),x,method=_RETURNVERBOSE)

[Out]

exp(-5)*(-5*x^2+90*x-160)/x^3+exp(x)

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maxima [A]  time = 0.44, size = 26, normalized size = 1.44 \begin {gather*} -{\left (\frac {5}{x} - \frac {90}{x^{2}} + \frac {160}{x^{3}} - e^{\left (x + 5\right )}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*exp(5)*exp(x)+5*x^2-180*x+480)/x^4/exp(5),x, algorithm="maxima")

[Out]

-(5/x - 90/x^2 + 160/x^3 - e^(x + 5))*e^(-5)

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mupad [B]  time = 4.17, size = 25, normalized size = 1.39 \begin {gather*} {\mathrm {e}}^x-\frac {5\,{\mathrm {e}}^{-5}\,x^2-90\,{\mathrm {e}}^{-5}\,x+160\,{\mathrm {e}}^{-5}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-5)*(5*x^2 - 180*x + x^4*exp(5)*exp(x) + 480))/x^4,x)

[Out]

exp(x) - (160*exp(-5) - 90*x*exp(-5) + 5*x^2*exp(-5))/x^3

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sympy [A]  time = 0.11, size = 19, normalized size = 1.06 \begin {gather*} e^{x} + \frac {- 5 x^{2} + 90 x - 160}{x^{3} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4*exp(5)*exp(x)+5*x**2-180*x+480)/x**4/exp(5),x)

[Out]

exp(x) + (-5*x**2 + 90*x - 160)*exp(-5)/x**3

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