3.54.67 \(\int \frac {-12+12 x-36 x^2+48 x^3}{-x+x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{4} \left (1+48 \left (x+\log \left (\frac {3}{25} e^{2 x^2} (1-x) x\right )\right )\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 21, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1593, 1620} \begin {gather*} 24 x^2+12 x+12 \log (1-x)+12 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 + 12*x - 36*x^2 + 48*x^3)/(-x + x^2),x]

[Out]

12*x + 24*x^2 + 12*Log[1 - x] + 12*Log[x]

Rule 1593

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

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-12+12 x-36 x^2+48 x^3}{(-1+x) x} \, dx\\ &=\int \left (12+\frac {12}{-1+x}+\frac {12}{x}+48 x\right ) \, dx\\ &=12 x+24 x^2+12 \log (1-x)+12 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.61 \begin {gather*} 12 \left (x+2 x^2+\log (1-x)+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 + 12*x - 36*x^2 + 48*x^3)/(-x + x^2),x]

[Out]

12*(x + 2*x^2 + Log[1 - x] + Log[x])

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fricas [A]  time = 0.62, size = 19, normalized size = 0.68 \begin {gather*} 24 \, x^{2} + 12 \, x + 12 \, \log \left (x^{2} - x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^3-36*x^2+12*x-12)/(x^2-x),x, algorithm="fricas")

[Out]

24*x^2 + 12*x + 12*log(x^2 - x)

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giac [A]  time = 0.11, size = 21, normalized size = 0.75 \begin {gather*} 24 \, x^{2} + 12 \, x + 12 \, \log \left ({\left | x - 1 \right |}\right ) + 12 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^3-36*x^2+12*x-12)/(x^2-x),x, algorithm="giac")

[Out]

24*x^2 + 12*x + 12*log(abs(x - 1)) + 12*log(abs(x))

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maple [A]  time = 0.21, size = 20, normalized size = 0.71

method	result	size
default	\(24 x^{2}+12 x +12 \ln \left (x -1\right )+12 \ln \relax (x )\)	\(20\)
norman	\(24 x^{2}+12 x +12 \ln \left (x -1\right )+12 \ln \relax (x )\)	\(20\)
risch	\(24 x^{2}+12 x +12 \ln \left (x^{2}-x \right )\)	\(20\)
meijerg	\(12 \ln \relax (x )+12 i \pi +8 x \left (6+3 x \right )-36 x +12 \ln \left (1-x \right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((48*x^3-36*x^2+12*x-12)/(x^2-x),x,method=_RETURNVERBOSE)

[Out]

24*x^2+12*x+12*ln(x-1)+12*ln(x)

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maxima [A]  time = 0.35, size = 19, normalized size = 0.68 \begin {gather*} 24 \, x^{2} + 12 \, x + 12 \, \log \left (x - 1\right ) + 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^3-36*x^2+12*x-12)/(x^2-x),x, algorithm="maxima")

[Out]

24*x^2 + 12*x + 12*log(x - 1) + 12*log(x)

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mupad [B]  time = 0.06, size = 17, normalized size = 0.61 \begin {gather*} 12\,x+12\,\ln \left (x\,\left (x-1\right )\right )+24\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x - 36*x^2 + 48*x^3 - 12)/(x - x^2),x)

[Out]

12*x + 12*log(x*(x - 1)) + 24*x^2

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sympy [A]  time = 0.08, size = 15, normalized size = 0.54 \begin {gather*} 24 x^{2} + 12 x + 12 \log {\left (x^{2} - x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x**3-36*x**2+12*x-12)/(x**2-x),x)

[Out]

24*x**2 + 12*x + 12*log(x**2 - x)

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