3.89.83 $\int \frac{4x^2-2x^3-2x^5+e^{17}(-4+6x+10x^2-8x^3-6x^5+2x^6)}{e^{17}{x}^5}dx$

Optimal. Leaf size=19 $${(-3-\frac{1}{e^{17}}+\frac{1-x}{x^2}+x)}^2$$

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Rubi [B]  time = 0.05, antiderivative size = 42, normalized size of antiderivative = 2.21, number of steps used = 3, number of rules used = 2, integrand size = 52, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.038, Rules used = {12, 14} $$\begin{array}{c}\frac{1}{x^4}-\frac{2}{x^3}+x^2-\frac{5+\frac{2}{e^{17}}}{x^2}-2(3+\frac{1}{e^{17}})x+\frac{2(4+\frac{1}{e^{17}})}{x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(4*x^2 - 2*x^3 - 2*x^5 + E^17*(-4 + 6*x + 10*x^2 - 8*x^3 - 6*x^5 + 2*x^6))/(E^17*x^5),x]

[Out]

x^(-4) - 2/x^3 - (5 + 2/E^17)/x^2 + (2*(4 + E^(-17)))/x - 2*(3 + E^(-17))*x + x^2

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int \frac{4x^2-2x^3-2x^5+e^{17}(-4+6x+10x^2-8x^3-6x^5+2x^6)}{x^5}dx}{e^{17}}\\ & =\frac{\int (-2(1+3e^{17})-\frac{4e^{17}}{x^5}+\frac{6e^{17}}{x^4}+\frac{2(2+5e^{17})}{x^3}-\frac{2(1+4e^{17})}{x^2}+2e^{17}x)dx}{e^{17}}\\ & =\frac{1}{x^4}-\frac{2}{x^3}-\frac{5+\frac{2}{e^{17}}}{x^2}+\frac{2(4+\frac{1}{e^{17}})}{x}-2(3+\frac{1}{e^{17}})x+x^2\end{array}\end{array}$$

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Mathematica [B]  time = 0.03, size = 43, normalized size = 2.26 $$\begin{array}{c}\frac{1-2x-5x^2+8x^3-6x^5+x^6-\frac{2x^2(1-x+x^3)}{e^{17}}}{x^4}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(4*x^2 - 2*x^3 - 2*x^5 + E^17*(-4 + 6*x + 10*x^2 - 8*x^3 - 6*x^5 + 2*x^6))/(E^17*x^5),x]

[Out]

(1 - 2*x - 5*x^2 + 8*x^3 - 6*x^5 + x^6 - (2*x^2*(1 - x + x^3))/E^17)/x^4

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fricas [B]  time = 0.46, size = 50, normalized size = 2.63 $$\begin{array}{c}-\frac{(2x^5-2x^3+2x^2-(x^6-6x^5+8x^3-5x^2-2x+1)e^{17})e^{(-17)}}{x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-6*x^5-8*x^3+10*x^2+6*x-4)*exp(17)-2*x^5-2*x^3+4*x^2)/x^5/exp(17),x, algorithm="fricas")

[Out]

-(2*x^5 - 2*x^3 + 2*x^2 - (x^6 - 6*x^5 + 8*x^3 - 5*x^2 - 2*x + 1)*e^17)*e^(-17)/x^4

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giac [B]  time = 0.47, size = 54, normalized size = 2.84 $$\begin{array}{c}(x^2{e}^{17}-6x{e}^{17}-2x+\frac{8x^3{e}^{17}+2x^3-5x^2{e}^{17}-2x^2-2x{e}^{17}+e^{17}}{x^4})e^{(-17)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-6*x^5-8*x^3+10*x^2+6*x-4)*exp(17)-2*x^5-2*x^3+4*x^2)/x^5/exp(17),x, algorithm="giac")

[Out]

(x^2*e^17 - 6*x*e^17 - 2*x + (8*x^3*e^17 + 2*x^3 - 5*x^2*e^17 - 2*x^2 - 2*x*e^17 + e^17)/x^4)*e^(-17)

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maple [B]  time = 0.07, size = 47, normalized size = 2.47

method	result	size
risch	$x^2-6x-2{\mathrm{e}}^{-17}x+\frac{{\mathrm{e}}^{-17}((8{\mathrm{e}}^{17}+2)x^3+(-5{\mathrm{e}}^{17}-2)x^2-2{\mathrm{e}}^{17}x+{\mathrm{e}}^{17})}{x^4}$	$47$
default	${\mathrm{e}}^{-17}({\mathrm{e}}^{17}{x}^2-6{\mathrm{e}}^{17}x-2x+\frac{{\mathrm{e}}^{17}}{x^4}-\frac{5{\mathrm{e}}^{17}+2}{x^2}-\frac{2(-4{\mathrm{e}}^{17}-1)}{x}-\frac{2{\mathrm{e}}^{17}}{x^3})$	$56$
norman	$\frac{1+x^6-2x-2(3{\mathrm{e}}^{17}+1){\mathrm{e}}^{-17}{x}^5+2(4{\mathrm{e}}^{17}+1){\mathrm{e}}^{-17}{x}^3-(5{\mathrm{e}}^{17}+2){\mathrm{e}}^{-17}{x}^2}{x^4}$	$58$
gosper	$\frac{(x^6{\mathrm{e}}^{17}-6x^5{\mathrm{e}}^{17}-2x^5+8{\mathrm{e}}^{17}{x}^3-5{\mathrm{e}}^{17}{x}^2+2x^3-2{\mathrm{e}}^{17}x-2x^2+{\mathrm{e}}^{17}){\mathrm{e}}^{-17}}{x^4}$	$59$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6-6*x^5-8*x^3+10*x^2+6*x-4)*exp(17)-2*x^5-2*x^3+4*x^2)/x^5/exp(17),x,method=_RETURNVERBOSE)

[Out]

x^2-6*x-2*exp(-17)*x+exp(-17)*((8*exp(17)+2)*x^3+(-5*exp(17)-2)*x^2-2*exp(17)*x+exp(17))/x^4

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maxima [B]  time = 0.38, size = 53, normalized size = 2.79 $$\begin{array}{c}(x^2{e}^{17}-2x(3e^{17}+1)+\frac{2x^3(4e^{17}+1)-x^2(5e^{17}+2)-2x{e}^{17}+e^{17}}{x^4})e^{(-17)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-6*x^5-8*x^3+10*x^2+6*x-4)*exp(17)-2*x^5-2*x^3+4*x^2)/x^5/exp(17),x, algorithm="maxima")

[Out]

(x^2*e^17 - 2*x*(3*e^17 + 1) + (2*x^3*(4*e^17 + 1) - x^2*(5*e^17 + 2) - 2*x*e^17 + e^17)/x^4)*e^(-17)

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mupad [B]  time = 0.11, size = 50, normalized size = 2.63 $$\begin{array}{c}{x}^2+\frac{{\mathrm{e}}^{-17}((8{\mathrm{e}}^{17}+2)x^3+(-5{\mathrm{e}}^{17}-2)x^2-2{\mathrm{e}}^{17}x+{\mathrm{e}}^{17})}{x^4}-x{\mathrm{e}}^{-17}(6{\mathrm{e}}^{17}+2)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-17)*(exp(17)*(6*x + 10*x^2 - 8*x^3 - 6*x^5 + 2*x^6 - 4) + 4*x^2 - 2*x^3 - 2*x^5))/x^5,x)

[Out]

x^2 + (exp(-17)*(exp(17) - 2*x*exp(17) - x^2*(5*exp(17) + 2) + x^3*(8*exp(17) + 2)))/x^4 - x*exp(-17)*(6*exp(1
7) + 2)

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sympy [B]  time = 0.48, size = 54, normalized size = 2.84 $$\begin{array}{c}\frac{x^2{e}^{17}+x(-6e^{17}-2)+\frac{x^3(2+8e^{17})+x^2(-5e^{17}-2)-2x{e}^{17}+e^{17}}{x^4}}{e^{17}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**6-6*x**5-8*x**3+10*x**2+6*x-4)*exp(17)-2*x**5-2*x**3+4*x**2)/x**5/exp(17),x)

[Out]

(x**2*exp(17) + x*(-6*exp(17) - 2) + (x**3*(2 + 8*exp(17)) + x**2*(-5*exp(17) - 2) - 2*x*exp(17) + exp(17))/x*
*4)*exp(-17)

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