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

Optimal. Leaf size=23 \[ -\frac {6 e^4 x}{3-x}+\log \left (5 e^4 x^3\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 0.74, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {6, 1594, 27, 893} \begin {gather*} 3 \log (x)-\frac {18 e^4}{3-x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*E^4)/(3 - x) + 3*Log[x]

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 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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.65 \begin {gather*} 3 \left (\frac {6 e^4}{-3+x}+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*((6*E^4)/(-3 + x) + Log[x])

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fricas [A]  time = 1.54, size = 18, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left ({\left (x - 3\right )} \log \relax (x) + 6 \, e^{4}\right )}}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*((x - 3)*log(x) + 6*e^4)/(x - 3)

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giac [A]  time = 0.24, size = 15, normalized size = 0.65 \begin {gather*} \frac {18 \, e^{4}}{x - 3} + 3 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18*e^4/(x - 3) + 3*log(abs(x))

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maple [A]  time = 0.04, size = 15, normalized size = 0.65

method	result	size
default	\(3 \ln \relax (x )+\frac {18 \,{\mathrm e}^{4}}{x -3}\)	\(15\)
norman	\(3 \ln \relax (x )+\frac {18 \,{\mathrm e}^{4}}{x -3}\)	\(15\)
risch	\(3 \ln \relax (x )+\frac {18 \,{\mathrm e}^{4}}{x -3}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*ln(x)+18*exp(4)/(x-3)

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maxima [A]  time = 0.46, size = 14, normalized size = 0.61 \begin {gather*} \frac {18 \, e^{4}}{x - 3} + 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18*e^4/(x - 3) + 3*log(x)

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mupad [B]  time = 0.05, size = 14, normalized size = 0.61 \begin {gather*} 3\,\ln \relax (x)+\frac {18\,{\mathrm {e}}^4}{x-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x) + (18*exp(4))/(x - 3)

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sympy [A]  time = 0.15, size = 12, normalized size = 0.52 \begin {gather*} 3 \log {\relax (x )} + \frac {18 e^{4}}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x) + 18*exp(4)/(x - 3)

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