3.44.88 \(\int \frac {-3+9 x-2 x^2+(9-2 x) \log (2)}{3 x+3 \log (2)} \, dx\)

Optimal. Leaf size=28 \[ -3+3 x+\frac {1}{3} \left (-x^2+\log \left (2 e^2\right )\right )-\log (x+\log (2)) \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {1850} \begin {gather*} -\frac {x^2}{3}+3 x-\log (x+\log (2)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 9*x - 2*x^2 + (9 - 2*x)*Log[2])/(3*x + 3*Log[2]),x]

[Out]

3*x - x^2/3 - Log[x + Log[2]]

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 \left (3-\frac {2 x}{3}+\frac {1}{-x-\log (2)}\right ) \, dx\\ &=3 x-\frac {x^2}{3}-\log (x+\log (2))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 30, normalized size = 1.07 \begin {gather*} \frac {1}{27} \left (81 x-9 x^2+\log ^2(8)-27 \log \left (\frac {1}{8} (3 x+\log (8))\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 9*x - 2*x^2 + (9 - 2*x)*Log[2])/(3*x + 3*Log[2]),x]

[Out]

(81*x - 9*x^2 + Log[8]^2 - 27*Log[(3*x + Log[8])/8])/27

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fricas [A]  time = 0.63, size = 16, normalized size = 0.57 \begin {gather*} -\frac {1}{3} \, x^{2} + 3 \, x - \log \left (x + \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+9)*log(2)-2*x^2+9*x-3)/(3*log(2)+3*x),x, algorithm="fricas")

[Out]

-1/3*x^2 + 3*x - log(x + log(2))

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giac [A]  time = 0.14, size = 17, normalized size = 0.61 \begin {gather*} -\frac {1}{3} \, x^{2} + 3 \, x - \log \left ({\left | x + \log \relax (2) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+9)*log(2)-2*x^2+9*x-3)/(3*log(2)+3*x),x, algorithm="giac")

[Out]

-1/3*x^2 + 3*x - log(abs(x + log(2)))

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maple [A]  time = 0.12, size = 17, normalized size = 0.61

method	result	size
default	\(-\frac {x^{2}}{3}+3 x -\ln \left (\ln \relax (2)+x \right )\)	\(17\)
risch	\(-\frac {x^{2}}{3}+3 x -\ln \left (\ln \relax (2)+x \right )\)	\(17\)
norman	\(3 x -\frac {x^{2}}{3}-\ln \left (3 \ln \relax (2)+3 x \right )\)	\(21\)
meijerg	\(3 \ln \relax (2) \ln \left (1+\frac {x}{\ln \relax (2)}\right )-\ln \left (1+\frac {x}{\ln \relax (2)}\right )-\frac {2 \ln \relax (2)^{2} \left (-\frac {x \left (-\frac {3 x}{\ln \relax (2)}+6\right )}{6 \ln \relax (2)}+\ln \left (1+\frac {x}{\ln \relax (2)}\right )\right )}{3}+\left (-\frac {2 \ln \relax (2)}{3}+3\right ) \ln \relax (2) \left (\frac {x}{\ln \relax (2)}-\ln \left (1+\frac {x}{\ln \relax (2)}\right )\right )\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x+9)*ln(2)-2*x^2+9*x-3)/(3*ln(2)+3*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 16, normalized size = 0.57 \begin {gather*} -\frac {1}{3} \, x^{2} + 3 \, x - \log \left (x + \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+9)*log(2)-2*x^2+9*x-3)/(3*log(2)+3*x),x, algorithm="maxima")

[Out]

-1/3*x^2 + 3*x - log(x + log(2))

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mupad [B]  time = 0.07, size = 16, normalized size = 0.57 \begin {gather*} 3\,x-\ln \left (x+\ln \relax (2)\right )-\frac {x^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2)*(2*x - 9) - 9*x + 2*x^2 + 3)/(3*x + 3*log(2)),x)

[Out]

3*x - log(x + log(2)) - x^2/3

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sympy [A]  time = 0.11, size = 14, normalized size = 0.50 \begin {gather*} - \frac {x^{2}}{3} + 3 x - \log {\left (x + \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+9)*ln(2)-2*x**2+9*x-3)/(3*ln(2)+3*x),x)

[Out]

-x**2/3 + 3*x - log(x + log(2))

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