3.50.32 \(\int \frac {3-48 x+32 x^2}{3 x-19 x^2+16 x^3} \, dx\)

Optimal. Leaf size=16 \[ \log \left (\frac {25 (3-16 x)^2 x}{-1+x}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1594, 1628} \begin {gather*} 2 \log (3-16 x)-\log (1-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 48*x + 32*x^2)/(3*x - 19*x^2 + 16*x^3),x]

[Out]

2*Log[3 - 16*x] - Log[1 - x] + Log[x]

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]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3-48 x+32 x^2}{x \left (3-19 x+16 x^2\right )} \, dx\\ &=\int \left (\frac {1}{1-x}+\frac {1}{x}+\frac {32}{-3+16 x}\right ) \, dx\\ &=2 \log (3-16 x)-\log (1-x)+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.19 \begin {gather*} 2 \log (3-16 x)-\log (1-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 48*x + 32*x^2)/(3*x - 19*x^2 + 16*x^3),x]

[Out]

2*Log[3 - 16*x] - Log[1 - x] + Log[x]

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fricas [A]  time = 0.55, size = 17, normalized size = 1.06 \begin {gather*} 2 \, \log \left (16 \, x - 3\right ) - \log \left (x - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^2-48*x+3)/(16*x^3-19*x^2+3*x),x, algorithm="fricas")

[Out]

2*log(16*x - 3) - log(x - 1) + log(x)

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giac [A]  time = 0.14, size = 20, normalized size = 1.25 \begin {gather*} 2 \, \log \left ({\left | 16 \, x - 3 \right |}\right ) - \log \left ({\left | x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^2-48*x+3)/(16*x^3-19*x^2+3*x),x, algorithm="giac")

[Out]

2*log(abs(16*x - 3)) - log(abs(x - 1)) + log(abs(x))

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maple [A]  time = 0.03, size = 18, normalized size = 1.12

method	result	size
default	\(2 \ln \left (16 x -3\right )-\ln \left (x -1\right )+\ln \relax (x )\)	\(18\)
norman	\(2 \ln \left (16 x -3\right )-\ln \left (x -1\right )+\ln \relax (x )\)	\(18\)
risch	\(2 \ln \left (16 x -3\right )-\ln \left (x -1\right )+\ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x^2-48*x+3)/(16*x^3-19*x^2+3*x),x,method=_RETURNVERBOSE)

[Out]

2*ln(16*x-3)-ln(x-1)+ln(x)

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maxima [A]  time = 0.35, size = 17, normalized size = 1.06 \begin {gather*} 2 \, \log \left (16 \, x - 3\right ) - \log \left (x - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^2-48*x+3)/(16*x^3-19*x^2+3*x),x, algorithm="maxima")

[Out]

2*log(16*x - 3) - log(x - 1) + log(x)

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mupad [B]  time = 0.11, size = 21, normalized size = 1.31 \begin {gather*} 2\,\ln \left (x-\frac {3}{16}\right )-2\,\mathrm {atanh}\left (\frac {4563}{9088\,\left (\frac {71\,x}{64}+\frac {27}{128}\right )}-\frac {98}{71}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x^2 - 48*x + 3)/(3*x - 19*x^2 + 16*x^3),x)

[Out]

2*log(x - 3/16) - 2*atanh(4563/(9088*((71*x)/64 + 27/128)) - 98/71)

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sympy [A]  time = 0.14, size = 15, normalized size = 0.94 \begin {gather*} \log {\relax (x )} - \log {\left (x - 1 \right )} + 2 \log {\left (x - \frac {3}{16} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x**2-48*x+3)/(16*x**3-19*x**2+3*x),x)

[Out]

log(x) - log(x - 1) + 2*log(x - 3/16)

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