3.18.57 \(\int \frac {-2 e^4+4 x^2-8000 x^3+4800 x^5-960 x^7+64 x^9}{x^2} \, dx\)

Optimal. Leaf size=26 \[ 4 \left (7+\frac {e^4}{2 x}+x+2 \left (5-x^2\right )^4\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {14} \begin {gather*} 8 x^8-160 x^6+1200 x^4-4000 x^2+4 x+\frac {2 e^4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*E^4 + 4*x^2 - 8000*x^3 + 4800*x^5 - 960*x^7 + 64*x^9)/x^2,x]

[Out]

(2*E^4)/x + 4*x - 4000*x^2 + 1200*x^4 - 160*x^6 + 8*x^8

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} &=\int \left (4-\frac {2 e^4}{x^2}-8000 x+4800 x^3-960 x^5+64 x^7\right ) \, dx\\ &=\frac {2 e^4}{x}+4 x-4000 x^2+1200 x^4-160 x^6+8 x^8\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.23 \begin {gather*} \frac {2 e^4}{x}+4 x-4000 x^2+1200 x^4-160 x^6+8 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^4 + 4*x^2 - 8000*x^3 + 4800*x^5 - 960*x^7 + 64*x^9)/x^2,x]

[Out]

(2*E^4)/x + 4*x - 4000*x^2 + 1200*x^4 - 160*x^6 + 8*x^8

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fricas [A]  time = 1.02, size = 33, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (4 \, x^{9} - 80 \, x^{7} + 600 \, x^{5} - 2000 \, x^{3} + 2 \, x^{2} + e^{4}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(4)+64*x^9-960*x^7+4800*x^5-8000*x^3+4*x^2)/x^2,x, algorithm="fricas")

[Out]

2*(4*x^9 - 80*x^7 + 600*x^5 - 2000*x^3 + 2*x^2 + e^4)/x

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giac [A]  time = 0.28, size = 31, normalized size = 1.19 \begin {gather*} 8 \, x^{8} - 160 \, x^{6} + 1200 \, x^{4} - 4000 \, x^{2} + 4 \, x + \frac {2 \, e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(4)+64*x^9-960*x^7+4800*x^5-8000*x^3+4*x^2)/x^2,x, algorithm="giac")

[Out]

8*x^8 - 160*x^6 + 1200*x^4 - 4000*x^2 + 4*x + 2*e^4/x

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maple [A]  time = 0.04, size = 32, normalized size = 1.23

method	result	size
default	\(8 x^{8}-160 x^{6}+1200 x^{4}-4000 x^{2}+4 x +\frac {2 \,{\mathrm e}^{4}}{x}\)	\(32\)
risch	\(8 x^{8}-160 x^{6}+1200 x^{4}-4000 x^{2}+4 x +\frac {2 \,{\mathrm e}^{4}}{x}\)	\(32\)
gosper	\(\frac {4 x^{2}-4000 x^{3}+1200 x^{5}-160 x^{7}+8 x^{9}+2 \,{\mathrm e}^{4}}{x}\)	\(34\)
norman	\(\frac {4 x^{2}-4000 x^{3}+1200 x^{5}-160 x^{7}+8 x^{9}+2 \,{\mathrm e}^{4}}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*exp(4)+64*x^9-960*x^7+4800*x^5-8000*x^3+4*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

8*x^8-160*x^6+1200*x^4-4000*x^2+4*x+2*exp(4)/x

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maxima [A]  time = 0.69, size = 31, normalized size = 1.19 \begin {gather*} 8 \, x^{8} - 160 \, x^{6} + 1200 \, x^{4} - 4000 \, x^{2} + 4 \, x + \frac {2 \, e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(4)+64*x^9-960*x^7+4800*x^5-8000*x^3+4*x^2)/x^2,x, algorithm="maxima")

[Out]

8*x^8 - 160*x^6 + 1200*x^4 - 4000*x^2 + 4*x + 2*e^4/x

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mupad [B]  time = 0.04, size = 31, normalized size = 1.19 \begin {gather*} 4\,x+\frac {2\,{\mathrm {e}}^4}{x}-4000\,x^2+1200\,x^4-160\,x^6+8\,x^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp(4) - 4*x^2 + 8000*x^3 - 4800*x^5 + 960*x^7 - 64*x^9)/x^2,x)

[Out]

4*x + (2*exp(4))/x - 4000*x^2 + 1200*x^4 - 160*x^6 + 8*x^8

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sympy [A]  time = 0.09, size = 29, normalized size = 1.12 \begin {gather*} 8 x^{8} - 160 x^{6} + 1200 x^{4} - 4000 x^{2} + 4 x + \frac {2 e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(4)+64*x**9-960*x**7+4800*x**5-8000*x**3+4*x**2)/x**2,x)

[Out]

8*x**8 - 160*x**6 + 1200*x**4 - 4000*x**2 + 4*x + 2*exp(4)/x

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