3.9.32 \(\int \frac {-27-18 x-3 x^2+e^3 (48 x+24 x^2)}{9 x^4+6 x^5+x^6} \, dx\)

Optimal. Leaf size=18 \[ -1+\frac {1}{x^3}-\frac {8 e^3}{x^2 (3+x)} \]

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Rubi [A]  time = 0.06, antiderivative size = 36, normalized size of antiderivative = 2.00, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {1594, 27, 1820} \begin {gather*} \frac {1}{x^3}-\frac {8 e^3}{3 x^2}-\frac {8 e^3}{9 (x+3)}+\frac {8 e^3}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-27 - 18*x - 3*x^2 + E^3*(48*x + 24*x^2))/(9*x^4 + 6*x^5 + x^6),x]

[Out]

x^(-3) - (8*E^3)/(3*x^2) + (8*E^3)/(9*x) - (8*E^3)/(9*(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 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1820

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-27-18 x-3 x^2+e^3 \left (48 x+24 x^2\right )}{x^4 \left (9+6 x+x^2\right )} \, dx\\ &=\int \frac {-27-18 x-3 x^2+e^3 \left (48 x+24 x^2\right )}{x^4 (3+x)^2} \, dx\\ &=\int \left (-\frac {3}{x^4}+\frac {16 e^3}{3 x^3}-\frac {8 e^3}{9 x^2}+\frac {8 e^3}{9 (3+x)^2}\right ) \, dx\\ &=\frac {1}{x^3}-\frac {8 e^3}{3 x^2}+\frac {8 e^3}{9 x}-\frac {8 e^3}{9 (3+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {3+x-8 e^3 x}{x^3 (3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-27 - 18*x - 3*x^2 + E^3*(48*x + 24*x^2))/(9*x^4 + 6*x^5 + x^6),x]

[Out]

(3 + x - 8*E^3*x)/(x^3*(3 + x))

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fricas [A]  time = 0.60, size = 23, normalized size = 1.28 \begin {gather*} -\frac {8 \, x e^{3} - x - 3}{x^{4} + 3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^2+48*x)*exp(3)-3*x^2-18*x-27)/(x^6+6*x^5+9*x^4),x, algorithm="fricas")

[Out]

-(8*x*e^3 - x - 3)/(x^4 + 3*x^3)

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giac [A]  time = 0.25, size = 29, normalized size = 1.61 \begin {gather*} -\frac {8 \, e^{3}}{9 \, {\left (x + 3\right )}} + \frac {8 \, x^{2} e^{3} - 24 \, x e^{3} + 9}{9 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^2+48*x)*exp(3)-3*x^2-18*x-27)/(x^6+6*x^5+9*x^4),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 20, normalized size = 1.11

method	result	size
norman	\(\frac {3+\left (-8 \,{\mathrm e}^{3}+1\right ) x}{\left (3+x \right ) x^{3}}\)	\(20\)
risch	\(\frac {3+\left (-8 \,{\mathrm e}^{3}+1\right ) x}{\left (3+x \right ) x^{3}}\)	\(20\)
gosper	\(-\frac {8 x \,{\mathrm e}^{3}-x -3}{x^{3} \left (3+x \right )}\)	\(21\)
default	\(\frac {1}{x^{3}}-\frac {8 \,{\mathrm e}^{3}}{3 x^{2}}+\frac {8 \,{\mathrm e}^{3}}{9 x}-\frac {8 \,{\mathrm e}^{3}}{9 \left (3+x \right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x^2+48*x)*exp(3)-3*x^2-18*x-27)/(x^6+6*x^5+9*x^4),x,method=_RETURNVERBOSE)

[Out]

(3+(-8*exp(3)+1)*x)/(3+x)/x^3

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maxima [A]  time = 0.41, size = 23, normalized size = 1.28 \begin {gather*} -\frac {x {\left (8 \, e^{3} - 1\right )} - 3}{x^{4} + 3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^2+48*x)*exp(3)-3*x^2-18*x-27)/(x^6+6*x^5+9*x^4),x, algorithm="maxima")

[Out]

-(x*(8*e^3 - 1) - 3)/(x^4 + 3*x^3)

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mupad [B]  time = 0.63, size = 23, normalized size = 1.28 \begin {gather*} -\frac {x\,\left (8\,{\mathrm {e}}^3-1\right )-3}{x^4+3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*x - exp(3)*(48*x + 24*x^2) + 3*x^2 + 27)/(9*x^4 + 6*x^5 + x^6),x)

[Out]

-(x*(8*exp(3) - 1) - 3)/(3*x^3 + x^4)

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sympy [A]  time = 0.39, size = 19, normalized size = 1.06 \begin {gather*} - \frac {x \left (-1 + 8 e^{3}\right ) - 3}{x^{4} + 3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x**2+48*x)*exp(3)-3*x**2-18*x-27)/(x**6+6*x**5+9*x**4),x)

[Out]

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

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