3.31.59 \(\int \frac {-9-e x^2-24 x^5+9 x^{10}}{e x^2} \, dx\)

Optimal. Leaf size=20 \[ -x+\frac {x \left (-\frac {3}{x}+x^4\right )^2}{e} \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.35, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {12, 14} \begin {gather*} \frac {x^9}{e}-\frac {6 x^4}{e}-x+\frac {9}{e x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - E*x^2 - 24*x^5 + 9*x^10)/(E*x^2),x]

[Out]

9/(E*x) - x - (6*x^4)/E + x^9/E

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 {-9-e x^2-24 x^5+9 x^{10}}{x^2} \, dx}{e}\\ &=\frac {\int \left (-e-\frac {9}{x^2}-24 x^3+9 x^8\right ) \, dx}{e}\\ &=\frac {9}{e x}-x-\frac {6 x^4}{e}+\frac {x^9}{e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 1.20 \begin {gather*} -\frac {-\frac {9}{x}+e x+6 x^4-x^9}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - E*x^2 - 24*x^5 + 9*x^10)/(E*x^2),x]

[Out]

-((-9/x + E*x + 6*x^4 - x^9)/E)

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fricas [A]  time = 0.62, size = 23, normalized size = 1.15 \begin {gather*} \frac {{\left (x^{10} - 6 \, x^{5} - x^{2} e + 9\right )} e^{\left (-1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(1)+9*x^10-24*x^5-9)/x^2/exp(1),x, algorithm="fricas")

[Out]

(x^10 - 6*x^5 - x^2*e + 9)*e^(-1)/x

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giac [A]  time = 0.19, size = 22, normalized size = 1.10 \begin {gather*} {\left (x^{9} - 6 \, x^{4} - x e + \frac {9}{x}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(1)+9*x^10-24*x^5-9)/x^2/exp(1),x, algorithm="giac")

[Out]

(x^9 - 6*x^4 - x*e + 9/x)*e^(-1)

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maple [A]  time = 0.06, size = 25, normalized size = 1.25

method	result	size
default	\({\mathrm e}^{-1} \left (x^{9}-6 x^{4}-x \,{\mathrm e}+\frac {9}{x}\right )\)	\(25\)
risch	\({\mathrm e}^{-1} x^{9}-6 x^{4} {\mathrm e}^{-1}-x +\frac {9 \,{\mathrm e}^{-1}}{x}\)	\(25\)
gosper	\(-\frac {\left (-x^{10}-9+6 x^{5}+x^{2} {\mathrm e}\right ) {\mathrm e}^{-1}}{x}\)	\(28\)
norman	\(\frac {{\mathrm e}^{-1} x^{10}-x^{2}+9 \,{\mathrm e}^{-1}-6 \,{\mathrm e}^{-1} x^{5}}{x}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2*exp(1)+9*x^10-24*x^5-9)/x^2/exp(1),x,method=_RETURNVERBOSE)

[Out]

1/exp(1)*(x^9-6*x^4-x*exp(1)+9/x)

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maxima [A]  time = 0.38, size = 22, normalized size = 1.10 \begin {gather*} {\left (x^{9} - 6 \, x^{4} - x e + \frac {9}{x}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(1)+9*x^10-24*x^5-9)/x^2/exp(1),x, algorithm="maxima")

[Out]

(x^9 - 6*x^4 - x*e + 9/x)*e^(-1)

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mupad [B]  time = 1.79, size = 24, normalized size = 1.20 \begin {gather*} \frac {9\,{\mathrm {e}}^{-1}}{x}-x-6\,x^4\,{\mathrm {e}}^{-1}+x^9\,{\mathrm {e}}^{-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-1)*(x^2*exp(1) + 24*x^5 - 9*x^10 + 9))/x^2,x)

[Out]

(9*exp(-1))/x - x - 6*x^4*exp(-1) + x^9*exp(-1)

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sympy [A]  time = 0.07, size = 19, normalized size = 0.95 \begin {gather*} \frac {x^{9} - 6 x^{4} - e x + \frac {9}{x}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2*exp(1)+9*x**10-24*x**5-9)/x**2/exp(1),x)

[Out]

(x**9 - 6*x**4 - E*x + 9/x)*exp(-1)

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