3.6.55 \(\int \frac {243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}{729-64 x^6} \, dx\) [555]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 = {1600, 31} \begin {gather*} \frac {1}{2} \log (2 x+3) \end {gather*}

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 31

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

Rule 1600

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]

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*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log (3+2 x) \end {gather*}

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 = 0.44, size = 9, normalized size = 0.90

method	result	size
default	\(\frac {\ln \left (2 x +3\right )}{2}\)	\(9\)
norman	\(\frac {\ln \left (2 x +3\right )}{2}\)	\(9\)
risch	\(\frac {\ln \left (2 x +3\right )}{2}\)	\(9\)
meijerg	\(-\frac {x \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{12 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {\ln \left (1-\frac {64 x^{6}}{729}\right )}{12}-\frac {x^{5} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{12 \left (x^{6}\right )^{\frac {5}{6}}}+\frac {x^{4} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{12 \left (x^{6}\right )^{\frac {2}{3}}}+\frac {\arctanh \left (\frac {8 x^{3}}{27}\right )}{6}+\frac {x^{2} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{12 \left (x^{6}\right )^{\frac {1}{3}}}\)	\(399\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \log \left (2 \, x + 3\right ) \end {gather*}

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 = 0.37, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \log \left (2 \, x + 3\right ) \end {gather*}

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 = 0.01, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (2 x + 3 \right )}}{2} \end {gather*}

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 = 0.67, size = 9, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.06, size = 6, normalized size = 0.60 \begin {gather*} \frac {\ln \left (x+\frac {3}{2}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3/2)/2

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