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

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

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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 {gather*} \frac {1}{x^4}-\frac {2}{x^3}+x^2-\frac {5+\frac {2}{e^{17}}}{x^2}-2 \left (3+\frac {1}{e^{17}}\right ) x+\frac {2 \left (4+\frac {1}{e^{17}}\right )}{x} \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {4 x^2-2 x^3-2 x^5+e^{17} \left (-4+6 x+10 x^2-8 x^3-6 x^5+2 x^6\right )}{x^5} \, dx}{e^{17}}\\ &=\frac {\int \left (-2 \left (1+3 e^{17}\right )-\frac {4 e^{17}}{x^5}+\frac {6 e^{17}}{x^4}+\frac {2 \left (2+5 e^{17}\right )}{x^3}-\frac {2 \left (1+4 e^{17}\right )}{x^2}+2 e^{17} x\right ) \, dx}{e^{17}}\\ &=\frac {1}{x^4}-\frac {2}{x^3}-\frac {5+\frac {2}{e^{17}}}{x^2}+\frac {2 \left (4+\frac {1}{e^{17}}\right )}{x}-2 \left (3+\frac {1}{e^{17}}\right ) x+x^2\\ \end {aligned} \end {gather*}

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

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 {gather*} -\frac {{\left (2 \, x^{5} - 2 \, x^{3} + 2 \, x^{2} - {\left (x^{6} - 6 \, x^{5} + 8 \, x^{3} - 5 \, x^{2} - 2 \, x + 1\right )} e^{17}\right )} e^{\left (-17\right )}}{x^{4}} \end {gather*}

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 {gather*} {\left (x^{2} e^{17} - 6 \, x e^{17} - 2 \, x + \frac {8 \, x^{3} e^{17} + 2 \, x^{3} - 5 \, x^{2} e^{17} - 2 \, x^{2} - 2 \, x e^{17} + e^{17}}{x^{4}}\right )} e^{\left (-17\right )} \end {gather*}

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	\(x^{2}-6 x -2 \,{\mathrm e}^{-17} x +\frac {{\mathrm e}^{-17} \left (\left (8 \,{\mathrm e}^{17}+2\right ) x^{3}+\left (-5 \,{\mathrm e}^{17}-2\right ) x^{2}-2 \,{\mathrm e}^{17} x +{\mathrm e}^{17}\right )}{x^{4}}\)	\(47\)
default	\({\mathrm e}^{-17} \left ({\mathrm e}^{17} x^{2}-6 \,{\mathrm e}^{17} x -2 x +\frac {{\mathrm e}^{17}}{x^{4}}-\frac {5 \,{\mathrm e}^{17}+2}{x^{2}}-\frac {2 \left (-4 \,{\mathrm e}^{17}-1\right )}{x}-\frac {2 \,{\mathrm e}^{17}}{x^{3}}\right )\)	\(56\)
norman	\(\frac {1+x^{6}-2 x -2 \left (3 \,{\mathrm e}^{17}+1\right ) {\mathrm e}^{-17} x^{5}+2 \left (4 \,{\mathrm e}^{17}+1\right ) {\mathrm e}^{-17} x^{3}-\left (5 \,{\mathrm e}^{17}+2\right ) {\mathrm e}^{-17} x^{2}}{x^{4}}\)	\(58\)
gosper	\(\frac {\left (x^{6} {\mathrm e}^{17}-6 x^{5} {\mathrm e}^{17}-2 x^{5}+8 \,{\mathrm e}^{17} x^{3}-5 \,{\mathrm e}^{17} x^{2}+2 x^{3}-2 \,{\mathrm e}^{17} x -2 x^{2}+{\mathrm e}^{17}\right ) {\mathrm e}^{-17}}{x^{4}}\)	\(59\)

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 {gather*} {\left (x^{2} e^{17} - 2 \, x {\left (3 \, e^{17} + 1\right )} + \frac {2 \, x^{3} {\left (4 \, e^{17} + 1\right )} - x^{2} {\left (5 \, e^{17} + 2\right )} - 2 \, x e^{17} + e^{17}}{x^{4}}\right )} e^{\left (-17\right )} \end {gather*}

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 {gather*} x^2+\frac {{\mathrm {e}}^{-17}\,\left (\left (8\,{\mathrm {e}}^{17}+2\right )\,x^3+\left (-5\,{\mathrm {e}}^{17}-2\right )\,x^2-2\,{\mathrm {e}}^{17}\,x+{\mathrm {e}}^{17}\right )}{x^4}-x\,{\mathrm {e}}^{-17}\,\left (6\,{\mathrm {e}}^{17}+2\right ) \end {gather*}

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 {gather*} \frac {x^{2} e^{17} + x \left (- 6 e^{17} - 2\right ) + \frac {x^{3} \left (2 + 8 e^{17}\right ) + x^{2} \left (- 5 e^{17} - 2\right ) - 2 x e^{17} + e^{17}}{x^{4}}}{e^{17}} \end {gather*}

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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