3.556 \(\int \frac{243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{2} \log (3-2 x) \]

[Out]

-Log[3 - 2*x]/2

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Rubi [A]  time = 0.0133504, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {1586, 31} \[ -\frac{1}{2} \log (3-2 x) \]

Antiderivative was successfully verified.

[In]

Int[(243 + 162*x + 108*x^2 + 72*x^3 + 48*x^4 + 32*x^5)/(729 - 64*x^6),x]

[Out]

-Log[3 - 2*x]/2

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{729-64 x^6} \, dx &=\int \frac{1}{3-2 x} \, dx\\ &=-\frac{1}{2} \log (3-2 x)\\ \end{align*}

Mathematica [A]  time = 0.0010264, size = 10, normalized size = 1. \[ -\frac{1}{2} \log (3-2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(243 + 162*x + 108*x^2 + 72*x^3 + 48*x^4 + 32*x^5)/(729 - 64*x^6),x]

[Out]

-Log[3 - 2*x]/2

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Maple [A]  time = 0.001, size = 9, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( -3+2\,x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x^5+48*x^4+72*x^3+108*x^2+162*x+243)/(-64*x^6+729),x)

[Out]

-1/2*ln(-3+2*x)

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Maxima [A]  time = 0.920725, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{2} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^5+48*x^4+72*x^3+108*x^2+162*x+243)/(-64*x^6+729),x, algorithm="maxima")

[Out]

-1/2*log(2*x - 3)

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Fricas [A]  time = 1.35855, size = 26, normalized size = 2.6 \begin{align*} -\frac{1}{2} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^5+48*x^4+72*x^3+108*x^2+162*x+243)/(-64*x^6+729),x, algorithm="fricas")

[Out]

-1/2*log(2*x - 3)

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Sympy [A]  time = 0.068114, size = 8, normalized size = 0.8 \begin{align*} - \frac{\log{\left (2 x - 3 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x**5+48*x**4+72*x**3+108*x**2+162*x+243)/(-64*x**6+729),x)

[Out]

-log(2*x - 3)/2

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Giac [A]  time = 1.05218, size = 12, normalized size = 1.2 \begin{align*} -\frac{1}{2} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^5+48*x^4+72*x^3+108*x^2+162*x+243)/(-64*x^6+729),x, algorithm="giac")

[Out]

-1/2*log(abs(2*x - 3))