3.72.99 \(\int \frac {24 x^2+16 x^3+8 x^4+2 x^5}{4+4 x+x^2} \, dx\)

Optimal. Leaf size=23 \[ -1+\left (-\frac {4}{-2-x}+\frac {x}{2}\right ) x^3+\log (4) \]

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {27, 1850} \begin {gather*} \frac {x^4}{2}+4 x^2-8 x-\frac {32}{x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24*x^2 + 16*x^3 + 8*x^4 + 2*x^5)/(4 + 4*x + x^2),x]

[Out]

-8*x + 4*x^2 + x^4/2 - 32/(2 + x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {24 x^2+16 x^3+8 x^4+2 x^5}{(2+x)^2} \, dx\\ &=\int \left (-8+8 x+2 x^3+\frac {32}{(2+x)^2}\right ) \, dx\\ &=-8 x+4 x^2+\frac {x^4}{2}-\frac {32}{2+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.17 \begin {gather*} \frac {-224-112 x+8 x^3+2 x^4+x^5}{2 (2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x^2 + 16*x^3 + 8*x^4 + 2*x^5)/(4 + 4*x + x^2),x]

[Out]

(-224 - 112*x + 8*x^3 + 2*x^4 + x^5)/(2*(2 + x))

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fricas [A]  time = 0.71, size = 25, normalized size = 1.09 \begin {gather*} \frac {x^{5} + 2 \, x^{4} + 8 \, x^{3} - 32 \, x - 64}{2 \, {\left (x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+8*x^4+16*x^3+24*x^2)/(x^2+4*x+4),x, algorithm="fricas")

[Out]

1/2*(x^5 + 2*x^4 + 8*x^3 - 32*x - 64)/(x + 2)

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giac [A]  time = 0.15, size = 21, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, x^{4} + 4 \, x^{2} - 8 \, x - \frac {32}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+8*x^4+16*x^3+24*x^2)/(x^2+4*x+4),x, algorithm="giac")

[Out]

1/2*x^4 + 4*x^2 - 8*x - 32/(x + 2)

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maple [A]  time = 0.15, size = 19, normalized size = 0.83

method	result	size
gosper	\(\frac {x^{3} \left (x^{2}+2 x +8\right )}{2 x +4}\)	\(19\)
norman	\(\frac {x^{4}+4 x^{3}+\frac {1}{2} x^{5}}{2+x}\)	\(21\)
default	\(\frac {x^{4}}{2}+4 x^{2}-8 x -\frac {32}{2+x}\)	\(22\)
risch	\(\frac {x^{4}}{2}+4 x^{2}-8 x -\frac {32}{2+x}\)	\(22\)
meijerg	\(-\frac {4 x \left (-\frac {3}{16} x^{4}+\frac {5}{8} x^{3}-\frac {5}{2} x^{2}+15 x +60\right )}{3 \left (1+\frac {x}{2}\right )}+\frac {32 x \left (\frac {5}{8} x^{3}-\frac {5}{2} x^{2}+15 x +60\right )}{15 \left (1+\frac {x}{2}\right )}-\frac {8 x \left (-\frac {1}{2} x^{2}+3 x +12\right )}{1+\frac {x}{2}}+\frac {8 x \left (\frac {3 x}{2}+6\right )}{1+\frac {x}{2}}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^5+8*x^4+16*x^3+24*x^2)/(x^2+4*x+4),x,method=_RETURNVERBOSE)

[Out]

1/2*x^3*(x^2+2*x+8)/(2+x)

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maxima [A]  time = 0.34, size = 21, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, x^{4} + 4 \, x^{2} - 8 \, x - \frac {32}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+8*x^4+16*x^3+24*x^2)/(x^2+4*x+4),x, algorithm="maxima")

[Out]

1/2*x^4 + 4*x^2 - 8*x - 32/(x + 2)

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mupad [B]  time = 0.05, size = 21, normalized size = 0.91 \begin {gather*} 4\,x^2-\frac {32}{x+2}-8\,x+\frac {x^4}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^2 + 16*x^3 + 8*x^4 + 2*x^5)/(4*x + x^2 + 4),x)

[Out]

4*x^2 - 32/(x + 2) - 8*x + x^4/2

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sympy [A]  time = 0.10, size = 17, normalized size = 0.74 \begin {gather*} \frac {x^{4}}{2} + 4 x^{2} - 8 x - \frac {32}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**5+8*x**4+16*x**3+24*x**2)/(x**2+4*x+4),x)

[Out]

x**4/2 + 4*x**2 - 8*x - 32/(x + 2)

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