3.23.36 \(\int \frac {-9-x+60 x^2+4 x^3}{-x+4 x^3} \, dx\)

Optimal. Leaf size=20 \[ -5+x+3 \left (2+\log \left (2 x^3 \left (1-4 x^2\right )\right )\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1593, 1802, 260} \begin {gather*} 3 \log \left (1-4 x^2\right )+x+9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - x + 60*x^2 + 4*x^3)/(-x + 4*x^3),x]

[Out]

x + 9*Log[x] + 3*Log[1 - 4*x^2]

Rule 260

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

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 1802

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9-x+60 x^2+4 x^3}{x \left (-1+4 x^2\right )} \, dx\\ &=\int \left (1+\frac {9}{x}+\frac {24 x}{-1+4 x^2}\right ) \, dx\\ &=x+9 \log (x)+24 \int \frac {x}{-1+4 x^2} \, dx\\ &=x+9 \log (x)+3 \log \left (1-4 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} x+9 \log (x)+3 \log \left (1-4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - x + 60*x^2 + 4*x^3)/(-x + 4*x^3),x]

[Out]

x + 9*Log[x] + 3*Log[1 - 4*x^2]

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fricas [A]  time = 0.83, size = 16, normalized size = 0.80 \begin {gather*} x + 3 \, \log \left (4 \, x^{2} - 1\right ) + 9 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+60*x^2-x-9)/(4*x^3-x),x, algorithm="fricas")

[Out]

x + 3*log(4*x^2 - 1) + 9*log(x)

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giac [A]  time = 0.18, size = 25, normalized size = 1.25 \begin {gather*} x + 3 \, \log \left ({\left | 2 \, x + 1 \right |}\right ) + 3 \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + 9 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+60*x^2-x-9)/(4*x^3-x),x, algorithm="giac")

[Out]

x + 3*log(abs(2*x + 1)) + 3*log(abs(2*x - 1)) + 9*log(abs(x))

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maple [A]  time = 0.42, size = 17, normalized size = 0.85

method	result	size
risch	\(x +9 \ln \relax (x )+3 \ln \left (4 x^{2}-1\right )\)	\(17\)
default	\(x +3 \ln \left (2 x +1\right )+9 \ln \relax (x )+3 \ln \left (2 x -1\right )\)	\(23\)
norman	\(x +3 \ln \left (2 x +1\right )+9 \ln \relax (x )+3 \ln \left (2 x -1\right )\)	\(23\)
meijerg	\(3 \ln \left (-4 x^{2}+1\right )+9 \ln \relax (x )+9 \ln \relax (2)+\frac {9 i \pi }{2}-\frac {i \left (4 i x -2 i \arctanh \left (2 x \right )\right )}{4}+\frac {\arctanh \left (2 x \right )}{2}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3+60*x^2-x-9)/(4*x^3-x),x,method=_RETURNVERBOSE)

[Out]

x+9*ln(x)+3*ln(4*x^2-1)

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maxima [A]  time = 0.68, size = 22, normalized size = 1.10 \begin {gather*} x + 3 \, \log \left (2 \, x + 1\right ) + 3 \, \log \left (2 \, x - 1\right ) + 9 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+60*x^2-x-9)/(4*x^3-x),x, algorithm="maxima")

[Out]

x + 3*log(2*x + 1) + 3*log(2*x - 1) + 9*log(x)

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mupad [B]  time = 0.05, size = 14, normalized size = 0.70 \begin {gather*} x+3\,\ln \left (x^2-\frac {1}{4}\right )+9\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 60*x^2 - 4*x^3 + 9)/(x - 4*x^3),x)

[Out]

x + 3*log(x^2 - 1/4) + 9*log(x)

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sympy [A]  time = 0.09, size = 15, normalized size = 0.75 \begin {gather*} x + 9 \log {\relax (x )} + 3 \log {\left (4 x^{2} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3+60*x**2-x-9)/(4*x**3-x),x)

[Out]

x + 9*log(x) + 3*log(4*x**2 - 1)

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