3.89.23 \(\int \frac {-16+x^2-4 e^{e^9} x^2-192 x^3-96 x^4}{4 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {4}{x}+\frac {x}{4}-(3+x) \left (e^{e^9}+8 x^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {6, 12, 14} \begin {gather*} -8 x^3-24 x^2+\frac {1}{4} \left (1-4 e^{e^9}\right ) x+\frac {4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16 + x^2 - 4*E^E^9*x^2 - 192*x^3 - 96*x^4)/(4*x^2),x]

[Out]

4/x + ((1 - 4*E^E^9)*x)/4 - 24*x^2 - 8*x^3

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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} &=\int \frac {-16+\left (1-4 e^{e^9}\right ) x^2-192 x^3-96 x^4}{4 x^2} \, dx\\ &=\frac {1}{4} \int \frac {-16+\left (1-4 e^{e^9}\right ) x^2-192 x^3-96 x^4}{x^2} \, dx\\ &=\frac {1}{4} \int \left (1-4 e^{e^9}-\frac {16}{x^2}-192 x-96 x^2\right ) \, dx\\ &=\frac {4}{x}+\frac {1}{4} \left (1-4 e^{e^9}\right ) x-24 x^2-8 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.07 \begin {gather*} \frac {4}{x}+\frac {x}{4}-e^{e^9} x-24 x^2-8 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 + x^2 - 4*E^E^9*x^2 - 192*x^3 - 96*x^4)/(4*x^2),x]

[Out]

4/x + x/4 - E^E^9*x - 24*x^2 - 8*x^3

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fricas [A]  time = 0.49, size = 30, normalized size = 1.11 \begin {gather*} -\frac {32 \, x^{4} + 96 \, x^{3} + 4 \, x^{2} e^{\left (e^{9}\right )} - x^{2} - 16}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x^2*exp(exp(9))-96*x^4-192*x^3+x^2-16)/x^2,x, algorithm="fricas")

[Out]

-1/4*(32*x^4 + 96*x^3 + 4*x^2*e^(e^9) - x^2 - 16)/x

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giac [A]  time = 0.21, size = 25, normalized size = 0.93 \begin {gather*} -8 \, x^{3} - 24 \, x^{2} - x e^{\left (e^{9}\right )} + \frac {1}{4} \, x + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x^2*exp(exp(9))-96*x^4-192*x^3+x^2-16)/x^2,x, algorithm="giac")

[Out]

-8*x^3 - 24*x^2 - x*e^(e^9) + 1/4*x + 4/x

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maple [A]  time = 0.06, size = 26, normalized size = 0.96

method	result	size
default	\(-8 x^{3}-24 x^{2}-x \,{\mathrm e}^{{\mathrm e}^{9}}+\frac {x}{4}+\frac {4}{x}\)	\(26\)
risch	\(-8 x^{3}-24 x^{2}-x \,{\mathrm e}^{{\mathrm e}^{9}}+\frac {x}{4}+\frac {4}{x}\)	\(26\)
norman	\(\frac {4+\left (-{\mathrm e}^{{\mathrm e}^{9}}+\frac {1}{4}\right ) x^{2}-24 x^{3}-8 x^{4}}{x}\)	\(28\)
gosper	\(-\frac {32 x^{4}+4 x^{2} {\mathrm e}^{{\mathrm e}^{9}}+96 x^{3}-x^{2}-16}{4 x}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-4*x^2*exp(exp(9))-96*x^4-192*x^3+x^2-16)/x^2,x,method=_RETURNVERBOSE)

[Out]

-8*x^3-24*x^2-x*exp(exp(9))+1/4*x+4/x

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maxima [A]  time = 0.36, size = 26, normalized size = 0.96 \begin {gather*} -8 \, x^{3} - 24 \, x^{2} - \frac {1}{4} \, x {\left (4 \, e^{\left (e^{9}\right )} - 1\right )} + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x^2*exp(exp(9))-96*x^4-192*x^3+x^2-16)/x^2,x, algorithm="maxima")

[Out]

-8*x^3 - 24*x^2 - 1/4*x*(4*e^(e^9) - 1) + 4/x

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mupad [B]  time = 0.05, size = 24, normalized size = 0.89 \begin {gather*} \frac {4}{x}-x\,\left ({\mathrm {e}}^{{\mathrm {e}}^9}-\frac {1}{4}\right )-24\,x^2-8\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*exp(exp(9)) - x^2/4 + 48*x^3 + 24*x^4 + 4)/x^2,x)

[Out]

4/x - x*(exp(exp(9)) - 1/4) - 24*x^2 - 8*x^3

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sympy [A]  time = 0.09, size = 24, normalized size = 0.89 \begin {gather*} - 8 x^{3} - 24 x^{2} - \frac {x \left (-1 + 4 e^{e^{9}}\right )}{4} + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x**2*exp(exp(9))-96*x**4-192*x**3+x**2-16)/x**2,x)

[Out]

-8*x**3 - 24*x**2 - x*(-1 + 4*exp(exp(9)))/4 + 4/x

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