3.92.25 \(\int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx\)

Optimal. Leaf size=25 \[ -3 \left (\frac {5 x}{-2+x}-x^6+(3+\log (2))^2\right )+\log (x) \]

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Rubi [A]  time = 0.05, antiderivative size = 17, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {1594, 27, 1620} \begin {gather*} 3 x^6+\frac {30}{2-x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 26*x + x^2 + 72*x^6 - 72*x^7 + 18*x^8)/(4*x - 4*x^2 + x^3),x]

[Out]

30/(2 - x) + 3*x^6 + Log[x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 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 {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{x \left (4-4 x+x^2\right )} \, dx\\ &=\int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{(-2+x)^2 x} \, dx\\ &=\int \left (\frac {30}{(-2+x)^2}+\frac {1}{x}+18 x^5\right ) \, dx\\ &=\frac {30}{2-x}+3 x^6+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.60 \begin {gather*} -\frac {30}{-2+x}+3 x^6+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 26*x + x^2 + 72*x^6 - 72*x^7 + 18*x^8)/(4*x - 4*x^2 + x^3),x]

[Out]

-30/(-2 + x) + 3*x^6 + Log[x]

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fricas [A]  time = 0.67, size = 24, normalized size = 0.96 \begin {gather*} \frac {3 \, x^{7} - 6 \, x^{6} + {\left (x - 2\right )} \log \relax (x) - 30}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8-72*x^7+72*x^6+x^2+26*x+4)/(x^3-4*x^2+4*x),x, algorithm="fricas")

[Out]

(3*x^7 - 6*x^6 + (x - 2)*log(x) - 30)/(x - 2)

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giac [A]  time = 0.11, size = 16, normalized size = 0.64 \begin {gather*} 3 \, x^{6} - \frac {30}{x - 2} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8-72*x^7+72*x^6+x^2+26*x+4)/(x^3-4*x^2+4*x),x, algorithm="giac")

[Out]

3*x^6 - 30/(x - 2) + log(abs(x))

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maple [A]  time = 0.04, size = 16, normalized size = 0.64

method	result	size
default	\(3 x^{6}+\ln \relax (x )-\frac {30}{x -2}\)	\(16\)
risch	\(3 x^{6}+\ln \relax (x )-\frac {30}{x -2}\)	\(16\)
norman	\(\frac {3 x^{7}-6 x^{6}-30}{x -2}+\ln \relax (x )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x^8-72*x^7+72*x^6+x^2+26*x+4)/(x^3-4*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

3*x^6+ln(x)-30/(x-2)

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maxima [A]  time = 0.35, size = 15, normalized size = 0.60 \begin {gather*} 3 \, x^{6} - \frac {30}{x - 2} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x^8-72*x^7+72*x^6+x^2+26*x+4)/(x^3-4*x^2+4*x),x, algorithm="maxima")

[Out]

3*x^6 - 30/(x - 2) + log(x)

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mupad [B]  time = 0.06, size = 15, normalized size = 0.60 \begin {gather*} \ln \relax (x)-\frac {30}{x-2}+3\,x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((26*x + x^2 + 72*x^6 - 72*x^7 + 18*x^8 + 4)/(4*x - 4*x^2 + x^3),x)

[Out]

log(x) - 30/(x - 2) + 3*x^6

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sympy [A]  time = 0.09, size = 12, normalized size = 0.48 \begin {gather*} 3 x^{6} + \log {\relax (x )} - \frac {30}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x**8-72*x**7+72*x**6+x**2+26*x+4)/(x**3-4*x**2+4*x),x)

[Out]

3*x**6 + log(x) - 30/(x - 2)

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