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

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

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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} $$\begin{array}{c}-8x^3-24x^2+\frac{1}{4}(1-4e^{e^9})x+\frac{4}{x}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-16+(1-4e^{e^9})x^2-192x^3-96x^4}{4x^2}dx\\ & =\frac{1}{4}\int \frac{-16+(1-4e^{e^9})x^2-192x^3-96x^4}{x^2}dx\\ & =\frac{1}{4}\int (1-4e^{e^9}-\frac{16}{x^2}-192x-96x^2)dx\\ & =\frac{4}{x}+\frac{1}{4}(1-4e^{e^9})x-24x^2-8x^3\end{array}\end{array}$$

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

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 $$\begin{array}{c}-\frac{32x^4+96x^3+4x^2{e}^{\left(e^9\right)}-x^2-16}{4x}\end{array}$$

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 $$\begin{array}{c}-8x^3-24x^2-x{e}^{\left(e^9\right)}+\frac{1}{4}x+\frac{4}{x}\end{array}$$

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	$-8x^3-24x^2-x{\mathrm{e}}^{{\mathrm{e}}^9}+\frac{x}{4}+\frac{4}{x}$	$26$
risch	$-8x^3-24x^2-x{\mathrm{e}}^{{\mathrm{e}}^9}+\frac{x}{4}+\frac{4}{x}$	$26$
norman	$\frac{4+(-{\mathrm{e}}^{{\mathrm{e}}^9}+\frac{1}{4})x^2-24x^3-8x^4}{x}$	$28$
gosper	$-\frac{32x^4+4x^2{\mathrm{e}}^{{\mathrm{e}}^9}+96x^3-x^2-16}{4x}$	$31$

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 $$\begin{array}{c}-8x^3-24x^2-\frac{1}{4}x(4e^{\left(e^9\right)}-1)+\frac{4}{x}\end{array}$$

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 $$\begin{array}{c}\frac{4}{x}-x({\mathrm{e}}^{{\mathrm{e}}^9}-\frac{1}{4})-24x^2-8x^3\end{array}$$

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 $$\begin{array}{c}-8x^3-24x^2-\frac{x(-1+4e^{e^9})}{4}+\frac{4}{x}\end{array}$$

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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