3.53.14 \(\int \frac {100-20 x-59 x^2+10 x^3+8 x^4}{-80 x^2+16 x^3+16 x^4} \, dx\)

Optimal. Leaf size=35 \[ 5+\frac {1}{2} (-2+x)+\frac {1}{4} \left (3+\frac {5}{x}+\frac {1}{4} \log \left (5-x-x^2\right )\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 28, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1594, 1628, 628} \begin {gather*} \frac {1}{16} \log \left (-x^2-x+5\right )+\frac {x}{2}+\frac {5}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(100 - 20*x - 59*x^2 + 10*x^3 + 8*x^4)/(-80*x^2 + 16*x^3 + 16*x^4),x]

[Out]

5/(4*x) + x/2 + Log[5 - x - x^2]/16

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

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 {100-20 x-59 x^2+10 x^3+8 x^4}{x^2 \left (-80+16 x+16 x^2\right )} \, dx\\ &=\int \left (\frac {1}{2}-\frac {5}{4 x^2}+\frac {1+2 x}{16 \left (-5+x+x^2\right )}\right ) \, dx\\ &=\frac {5}{4 x}+\frac {x}{2}+\frac {1}{16} \int \frac {1+2 x}{-5+x+x^2} \, dx\\ &=\frac {5}{4 x}+\frac {x}{2}+\frac {1}{16} \log \left (5-x-x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.69 \begin {gather*} \frac {1}{16} \left (\frac {20}{x}+8 x+\log \left (5-x-x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100 - 20*x - 59*x^2 + 10*x^3 + 8*x^4)/(-80*x^2 + 16*x^3 + 16*x^4),x]

[Out]

(20/x + 8*x + Log[5 - x - x^2])/16

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fricas [A]  time = 0.52, size = 21, normalized size = 0.60 \begin {gather*} \frac {8 \, x^{2} + x \log \left (x^{2} + x - 5\right ) + 20}{16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4+10*x^3-59*x^2-20*x+100)/(16*x^4+16*x^3-80*x^2),x, algorithm="fricas")

[Out]

1/16*(8*x^2 + x*log(x^2 + x - 5) + 20)/x

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giac [A]  time = 1.32, size = 19, normalized size = 0.54 \begin {gather*} \frac {1}{2} \, x + \frac {5}{4 \, x} + \frac {1}{16} \, \log \left ({\left | x^{2} + x - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4+10*x^3-59*x^2-20*x+100)/(16*x^4+16*x^3-80*x^2),x, algorithm="giac")

[Out]

1/2*x + 5/4/x + 1/16*log(abs(x^2 + x - 5))

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maple [A]  time = 0.04, size = 19, normalized size = 0.54

method	result	size
default	\(\frac {x}{2}+\frac {\ln \left (x^{2}+x -5\right )}{16}+\frac {5}{4 x}\)	\(19\)
risch	\(\frac {x}{2}+\frac {\ln \left (x^{2}+x -5\right )}{16}+\frac {5}{4 x}\)	\(19\)
norman	\(\frac {\frac {5}{4}+\frac {x^{2}}{2}}{x}+\frac {\ln \left (x^{2}+x -5\right )}{16}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^4+10*x^3-59*x^2-20*x+100)/(16*x^4+16*x^3-80*x^2),x,method=_RETURNVERBOSE)

[Out]

1/2*x+1/16*ln(x^2+x-5)+5/4/x

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maxima [A]  time = 0.34, size = 18, normalized size = 0.51 \begin {gather*} \frac {1}{2} \, x + \frac {5}{4 \, x} + \frac {1}{16} \, \log \left (x^{2} + x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4+10*x^3-59*x^2-20*x+100)/(16*x^4+16*x^3-80*x^2),x, algorithm="maxima")

[Out]

1/2*x + 5/4/x + 1/16*log(x^2 + x - 5)

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mupad [B]  time = 0.06, size = 18, normalized size = 0.51 \begin {gather*} \frac {x}{2}+\frac {\ln \left (x^2+x-5\right )}{16}+\frac {5}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x^3 - 59*x^2 - 20*x + 8*x^4 + 100)/(16*x^3 - 80*x^2 + 16*x^4),x)

[Out]

x/2 + log(x + x^2 - 5)/16 + 5/(4*x)

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sympy [A]  time = 0.10, size = 17, normalized size = 0.49 \begin {gather*} \frac {x}{2} + \frac {\log {\left (x^{2} + x - 5 \right )}}{16} + \frac {5}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**4+10*x**3-59*x**2-20*x+100)/(16*x**4+16*x**3-80*x**2),x)

[Out]

x/2 + log(x**2 + x - 5)/16 + 5/(4*x)

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