3.74.80 \(\int \frac {-1+3 x-9 x^2-2 x^3+4 x^4}{x-2 x^2+x^3} \, dx\)

Optimal. Leaf size=23 \[ (4+x) \left (-3+x+\frac {x^2}{-1+x}\right )-\log (2 x) \]

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Rubi [A]  time = 0.05, antiderivative size = 22, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1594, 27, 1620} \begin {gather*} 2 x^2+6 x-\frac {5}{1-x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 3*x - 9*x^2 - 2*x^3 + 4*x^4)/(x - 2*x^2 + x^3),x]

[Out]

-5/(1 - x) + 6*x + 2*x^2 - 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 {-1+3 x-9 x^2-2 x^3+4 x^4}{x \left (1-2 x+x^2\right )} \, dx\\ &=\int \frac {-1+3 x-9 x^2-2 x^3+4 x^4}{(-1+x)^2 x} \, dx\\ &=\int \left (6-\frac {5}{(-1+x)^2}-\frac {1}{x}+4 x\right ) \, dx\\ &=-\frac {5}{1-x}+6 x+2 x^2-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.87 \begin {gather*} \frac {5}{-1+x}+6 x+2 x^2-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 3*x - 9*x^2 - 2*x^3 + 4*x^4)/(x - 2*x^2 + x^3),x]

[Out]

5/(-1 + x) + 6*x + 2*x^2 - Log[x]

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fricas [A]  time = 0.97, size = 28, normalized size = 1.22 \begin {gather*} \frac {2 \, x^{3} + 4 \, x^{2} - {\left (x - 1\right )} \log \relax (x) - 6 \, x + 5}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4-2*x^3-9*x^2+3*x-1)/(x^3-2*x^2+x),x, algorithm="fricas")

[Out]

(2*x^3 + 4*x^2 - (x - 1)*log(x) - 6*x + 5)/(x - 1)

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giac [A]  time = 0.19, size = 21, normalized size = 0.91 \begin {gather*} 2 \, x^{2} + 6 \, x + \frac {5}{x - 1} - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4-2*x^3-9*x^2+3*x-1)/(x^3-2*x^2+x),x, algorithm="giac")

[Out]

2*x^2 + 6*x + 5/(x - 1) - log(abs(x))

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maple [A]  time = 0.03, size = 21, normalized size = 0.91

method	result	size
default	\(2 x^{2}+6 x -\ln \relax (x )+\frac {5}{x -1}\)	\(21\)
risch	\(2 x^{2}+6 x -\ln \relax (x )+\frac {5}{x -1}\)	\(21\)
norman	\(\frac {2 x^{3}+4 x^{2}-1}{x -1}-\ln \relax (x )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^4-2*x^3-9*x^2+3*x-1)/(x^3-2*x^2+x),x,method=_RETURNVERBOSE)

[Out]

2*x^2+6*x-ln(x)+5/(x-1)

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maxima [A]  time = 0.44, size = 20, normalized size = 0.87 \begin {gather*} 2 \, x^{2} + 6 \, x + \frac {5}{x - 1} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4-2*x^3-9*x^2+3*x-1)/(x^3-2*x^2+x),x, algorithm="maxima")

[Out]

2*x^2 + 6*x + 5/(x - 1) - log(x)

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mupad [B]  time = 0.07, size = 20, normalized size = 0.87 \begin {gather*} 6\,x-\ln \relax (x)+\frac {5}{x-1}+2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*x^2 - 3*x + 2*x^3 - 4*x^4 + 1)/(x - 2*x^2 + x^3),x)

[Out]

6*x - log(x) + 5/(x - 1) + 2*x^2

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sympy [A]  time = 0.08, size = 15, normalized size = 0.65 \begin {gather*} 2 x^{2} + 6 x - \log {\relax (x )} + \frac {5}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**4-2*x**3-9*x**2+3*x-1)/(x**3-2*x**2+x),x)

[Out]

2*x**2 + 6*x - log(x) + 5/(x - 1)

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