3.50.33 \(\int \frac {9+60 e-66 x-47 x^2-8 x^3}{9+6 x+x^2} \, dx\)

Optimal. Leaf size=18 \[ -9+x+4 x \left (-x+\frac {5 e}{3+x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 1850} \begin {gather*} -4 x^2+x-\frac {60 e}{x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9 + 60*E - 66*x - 47*x^2 - 8*x^3)/(9 + 6*x + x^2),x]

[Out]

x - 4*x^2 - (60*E)/(3 + 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 {9+60 e-66 x-47 x^2-8 x^3}{(3+x)^2} \, dx\\ &=\int \left (1-8 x+\frac {60 e}{(3+x)^2}\right ) \, dx\\ &=x-4 x^2-\frac {60 e}{3+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.83 \begin {gather*} x-4 x^2-\frac {60 e}{3+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + 60*E - 66*x - 47*x^2 - 8*x^3)/(9 + 6*x + x^2),x]

[Out]

x - 4*x^2 - (60*E)/(3 + x)

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fricas [A]  time = 1.14, size = 25, normalized size = 1.39 \begin {gather*} -\frac {4 \, x^{3} + 11 \, x^{2} - 3 \, x + 60 \, e}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((60*exp(1)-8*x^3-47*x^2-66*x+9)/(x^2+6*x+9),x, algorithm="fricas")

[Out]

-(4*x^3 + 11*x^2 - 3*x + 60*e)/(x + 3)

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giac [A]  time = 0.12, size = 16, normalized size = 0.89 \begin {gather*} -4 \, x^{2} + x - \frac {60 \, e}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((60*exp(1)-8*x^3-47*x^2-66*x+9)/(x^2+6*x+9),x, algorithm="giac")

[Out]

-4*x^2 + x - 60*e/(x + 3)

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maple [A]  time = 0.18, size = 17, normalized size = 0.94

method	result	size
default	\(-4 x^{2}+x -\frac {60 \,{\mathrm e}}{3+x}\)	\(17\)
risch	\(-4 x^{2}+x -\frac {60 \,{\mathrm e}}{3+x}\)	\(17\)
norman	\(\frac {-4 x^{3}-11 x^{2}-9-60 \,{\mathrm e}}{3+x}\)	\(23\)
gosper	\(-\frac {4 x^{3}+11 x^{2}+60 \,{\mathrm e}+9}{3+x}\)	\(24\)
meijerg	\(\frac {23 x}{1+\frac {x}{3}}+\frac {20 \,{\mathrm e} x}{3 \left (1+\frac {x}{3}\right )}+\frac {6 x \left (-\frac {2}{9} x^{2}+2 x +12\right )}{1+\frac {x}{3}}-\frac {47 x \left (x +6\right )}{3 \left (1+\frac {x}{3}\right )}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((60*exp(1)-8*x^3-47*x^2-66*x+9)/(x^2+6*x+9),x,method=_RETURNVERBOSE)

[Out]

-4*x^2+x-60/(3+x)*exp(1)

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maxima [A]  time = 0.37, size = 16, normalized size = 0.89 \begin {gather*} -4 \, x^{2} + x - \frac {60 \, e}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((60*exp(1)-8*x^3-47*x^2-66*x+9)/(x^2+6*x+9),x, algorithm="maxima")

[Out]

-4*x^2 + x - 60*e/(x + 3)

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mupad [B]  time = 0.11, size = 16, normalized size = 0.89 \begin {gather*} x-4\,x^2-\frac {60\,\mathrm {e}}{x+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(66*x - 60*exp(1) + 47*x^2 + 8*x^3 - 9)/(6*x + x^2 + 9),x)

[Out]

x - 4*x^2 - (60*exp(1))/(x + 3)

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sympy [A]  time = 0.11, size = 14, normalized size = 0.78 \begin {gather*} - 4 x^{2} + x - \frac {60 e}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((60*exp(1)-8*x**3-47*x**2-66*x+9)/(x**2+6*x+9),x)

[Out]

-4*x**2 + x - 60*E/(x + 3)

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