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

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} $$\begin{array}{c}-\frac{160}{e^5{x}^3}+\frac{90}{e^5{x}^2}+e^x-\frac{5}{e^{5}x}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int \frac{480-180x+5x^2+e^{5+x}{x}^4}{x^4}dx}{e^5}\\ & =\frac{\int (e^{5+x}+\frac{5(96-36x+x^2)}{x^4})dx}{e^5}\\ & =\frac{\int e^{5+x}dx}{e^5}+\frac{5\int \frac{96-36x+x^2}{x^4}dx}{e^5}\\ & =e^x+\frac{5\int (\frac{96}{x^4}-\frac{36}{x^3}+\frac{1}{x^2})dx}{e^5}\\ & =e^x-\frac{160}{e^5{x}^3}+\frac{90}{e^5{x}^2}-\frac{5}{e^{5}x}\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{(x^3{e}^{(x+5)}-5x^2+90x-160)e^{(-5)}}{x^3}\end{array}$$

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 $$\begin{array}{c}\frac{(x^3{e}^{(x+5)}-5x^2+90x-160)e^{(-5)}}{x^3}\end{array}$$

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	$\frac{{\mathrm{e}}^{-5}(-5x^2+90x-160)}{x^3}+{\mathrm{e}}^x$	$20$
default	${\mathrm{e}}^{-5}({\mathrm{e}}^5{\mathrm{e}}^x-\frac{160}{x^3}+\frac{90}{x^2}-\frac{5}{x})$	$27$
norman	$\frac{{\mathrm{e}}^x{x}^3-160{\mathrm{e}}^{-5}+90x{\mathrm{e}}^{-5}-5x^2{\mathrm{e}}^{-5}}{x^3}$	$34$

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 $$\begin{array}{c}-(\frac{5}{x}-\frac{90}{x^2}+\frac{160}{x^3}-e^{(x+5)})e^{(-5)}\end{array}$$

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 $$\begin{array}{c}{\mathrm{e}}^x-\frac{5{\mathrm{e}}^{-5}{x}^2-90{\mathrm{e}}^{-5}x+160{\mathrm{e}}^{-5}}{x^3}\end{array}$$

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 $$\begin{array}{c}{e}^x+\frac{-5x^2+90x-160}{x^3{e}^5}\end{array}$$

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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