3.88.45 \(\int \frac {-8-8 x-4 x^3+10 x^4+12 x^5-8 x^6+6 x^8}{x^3} \, dx\)

Optimal. Leaf size=29 \[ -i \pi +\left (2+\frac {2}{x}-x+x^3\right )^2-\log (3-e) \]

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {14} \begin {gather*} x^6-2 x^4+4 x^3+5 x^2+\frac {4}{x^2}-4 x+\frac {8}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 - 8*x - 4*x^3 + 10*x^4 + 12*x^5 - 8*x^6 + 6*x^8)/x^3,x]

[Out]

4/x^2 + 8/x - 4*x + 5*x^2 + 4*x^3 - 2*x^4 + x^6

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 {8}{x^3}-\frac {8}{x^2}+10 x+12 x^2-8 x^3+6 x^5\right ) \, dx\\ &=\frac {4}{x^2}+\frac {8}{x}-4 x+5 x^2+4 x^3-2 x^4+x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.10 \begin {gather*} \frac {4}{x^2}+\frac {8}{x}-4 x+5 x^2+4 x^3-2 x^4+x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 - 8*x - 4*x^3 + 10*x^4 + 12*x^5 - 8*x^6 + 6*x^8)/x^3,x]

[Out]

4/x^2 + 8/x - 4*x + 5*x^2 + 4*x^3 - 2*x^4 + x^6

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fricas [A]  time = 0.43, size = 32, normalized size = 1.10 \begin {gather*} \frac {x^{8} - 2 \, x^{6} + 4 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} + 8 \, x + 4}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^8-8*x^6+12*x^5+10*x^4-4*x^3-8*x-8)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.13, size = 32, normalized size = 1.10 \begin {gather*} x^{6} - 2 \, x^{4} + 4 \, x^{3} + 5 \, x^{2} - 4 \, x + \frac {4 \, {\left (2 \, x + 1\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^8-8*x^6+12*x^5+10*x^4-4*x^3-8*x-8)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 31, normalized size = 1.07

method	result	size
gosper	\(\frac {\left (x^{6}+2 x^{5}+5 x^{2}+6 x +2\right ) \left (x^{2}-2 x +2\right )}{x^{2}}\)	\(31\)
risch	\(x^{6}-2 x^{4}+4 x^{3}+5 x^{2}-4 x +\frac {8 x +4}{x^{2}}\)	\(32\)
default	\(x^{6}-2 x^{4}+4 x^{3}+5 x^{2}-4 x +\frac {4}{x^{2}}+\frac {8}{x}\)	\(33\)
norman	\(\frac {x^{8}-2 x^{6}+4 x^{5}+5 x^{4}-4 x^{3}+8 x +4}{x^{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^8-8*x^6+12*x^5+10*x^4-4*x^3-8*x-8)/x^3,x,method=_RETURNVERBOSE)

[Out]

(x^6+2*x^5+5*x^2+6*x+2)*(x^2-2*x+2)/x^2

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maxima [A]  time = 0.34, size = 32, normalized size = 1.10 \begin {gather*} x^{6} - 2 \, x^{4} + 4 \, x^{3} + 5 \, x^{2} - 4 \, x + \frac {4 \, {\left (2 \, x + 1\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^8-8*x^6+12*x^5+10*x^4-4*x^3-8*x-8)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 31, normalized size = 1.07 \begin {gather*} \frac {8\,x+4}{x^2}-4\,x+5\,x^2+4\,x^3-2\,x^4+x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x + 4*x^3 - 10*x^4 - 12*x^5 + 8*x^6 - 6*x^8 + 8)/x^3,x)

[Out]

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

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sympy [A]  time = 0.07, size = 29, normalized size = 1.00 \begin {gather*} x^{6} - 2 x^{4} + 4 x^{3} + 5 x^{2} - 4 x + \frac {8 x + 4}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x**8-8*x**6+12*x**5+10*x**4-4*x**3-8*x-8)/x**3,x)

[Out]

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

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