3.27.84 \(\int \frac {-104+52 x-2 x^2-160 x^3+80 x^4-10 x^5}{16 x^2-8 x^3+x^4} \, dx\) [2684]

Optimal. Leaf size=33 \[ e^3+\frac {14+x}{(4-x) x}+\frac {3+x \left (1-5 x^2\right )}{x} \]

[Out]

exp(3)+(3+x*(-5*x^2+1))/x+(x+14)/(4-x)/x

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Rubi [A]
time = 0.04, antiderivative size = 24, normalized size of antiderivative = 0.73, number of steps used = 4, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1608, 27, 1634} \begin {gather*} -5 x^2+\frac {9}{2 (4-x)}+\frac {13}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-104 + 52*x - 2*x^2 - 160*x^3 + 80*x^4 - 10*x^5)/(16*x^2 - 8*x^3 + x^4),x]

[Out]

9/(2*(4 - x)) + 13/(2*x) - 5*x^2

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 1608

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 1634

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 {-104+52 x-2 x^2-160 x^3+80 x^4-10 x^5}{x^2 \left (16-8 x+x^2\right )} \, dx\\ &=\int \frac {-104+52 x-2 x^2-160 x^3+80 x^4-10 x^5}{(-4+x)^2 x^2} \, dx\\ &=\int \left (\frac {9}{2 (-4+x)^2}-\frac {13}{2 x^2}-10 x\right ) \, dx\\ &=\frac {9}{2 (4-x)}+\frac {13}{2 x}-5 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.79 \begin {gather*} -2 \left (\frac {9}{4 (-4+x)}-\frac {13}{4 x}+\frac {5 x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-104 + 52*x - 2*x^2 - 160*x^3 + 80*x^4 - 10*x^5)/(16*x^2 - 8*x^3 + x^4),x]

[Out]

-2*(9/(4*(-4 + x)) - 13/(4*x) + (5*x^2)/2)

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Maple [A]
time = 0.09, size = 19, normalized size = 0.58

method	result	size
default	\(-5 x^{2}+\frac {13}{2 x}-\frac {9}{2 \left (x -4\right )}\)	\(19\)
risch	\(-5 x^{2}+\frac {2 x -26}{\left (x -4\right ) x}\)	\(21\)
norman	\(\frac {-5 x^{4}+20 x^{3}+2 x -26}{\left (x -4\right ) x}\)	\(25\)
gosper	\(-\frac {5 x^{4}-20 x^{3}-2 x +26}{x \left (x -4\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*x^5+80*x^4-160*x^3-2*x^2+52*x-104)/(x^4-8*x^3+16*x^2),x,method=_RETURNVERBOSE)

[Out]

-5*x^2+13/2/x-9/2/(x-4)

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Maxima [A]
time = 0.26, size = 20, normalized size = 0.61 \begin {gather*} -5 \, x^{2} + \frac {2 \, {\left (x - 13\right )}}{x^{2} - 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^5+80*x^4-160*x^3-2*x^2+52*x-104)/(x^4-8*x^3+16*x^2),x, algorithm="maxima")

[Out]

-5*x^2 + 2*(x - 13)/(x^2 - 4*x)

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Fricas [A]
time = 0.33, size = 26, normalized size = 0.79 \begin {gather*} -\frac {5 \, x^{4} - 20 \, x^{3} - 2 \, x + 26}{x^{2} - 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^5+80*x^4-160*x^3-2*x^2+52*x-104)/(x^4-8*x^3+16*x^2),x, algorithm="fricas")

[Out]

-(5*x^4 - 20*x^3 - 2*x + 26)/(x^2 - 4*x)

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Sympy [A]
time = 0.03, size = 17, normalized size = 0.52 \begin {gather*} - 5 x^{2} - \frac {26 - 2 x}{x^{2} - 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x**5+80*x**4-160*x**3-2*x**2+52*x-104)/(x**4-8*x**3+16*x**2),x)

[Out]

-5*x**2 - (26 - 2*x)/(x**2 - 4*x)

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Giac [A]
time = 0.38, size = 20, normalized size = 0.61 \begin {gather*} -5 \, x^{2} + \frac {2 \, {\left (x - 13\right )}}{x^{2} - 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^5+80*x^4-160*x^3-2*x^2+52*x-104)/(x^4-8*x^3+16*x^2),x, algorithm="giac")

[Out]

-5*x^2 + 2*(x - 13)/(x^2 - 4*x)

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Mupad [B]
time = 1.46, size = 20, normalized size = 0.61 \begin {gather*} \frac {2\,x-26}{x\,\left (x-4\right )}-5\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^2 - 52*x + 160*x^3 - 80*x^4 + 10*x^5 + 104)/(16*x^2 - 8*x^3 + x^4),x)

[Out]

(2*x - 26)/(x*(x - 4)) - 5*x^2

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