3.23.73 \(\int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx\)

Optimal. Leaf size=27 \[ 4-\log \left (-1+\frac {1}{x}+\frac {3 x^2}{\left (4-x-2 x^2\right )^2}\right ) \]

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Rubi [A]  time = 0.19, antiderivative size = 39, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 3, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2074, 628, 1587} \begin {gather*} 2 \log \left (-2 x^2-x+4\right )-\log \left (-4 x^5+22 x^3-7 x^2-24 x+16\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(64 - 48*x - 84*x^2 + 23*x^3 + 42*x^4 - 24*x^5 - 8*x^6)/(64*x - 112*x^2 - 36*x^3 + 143*x^4 - 8*x^5 - 60*x^
6 + 4*x^7 + 8*x^8),x]

[Out]

Log[x] + 2*Log[4 - x - 2*x^2] - Log[16 - 24*x - 7*x^2 + 22*x^3 - 4*x^5]

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 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{x}+\frac {2 (1+4 x)}{-4+x+2 x^2}-\frac {2 \left (12+7 x-33 x^2+10 x^4\right )}{-16+24 x+7 x^2-22 x^3+4 x^5}\right ) \, dx\\ &=\log (x)+2 \int \frac {1+4 x}{-4+x+2 x^2} \, dx-2 \int \frac {12+7 x-33 x^2+10 x^4}{-16+24 x+7 x^2-22 x^3+4 x^5} \, dx\\ &=\log (x)+2 \log \left (4-x-2 x^2\right )-\log \left (16-24 x-7 x^2+22 x^3-4 x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 39, normalized size = 1.44 \begin {gather*} \log (x)+2 \log \left (4-x-2 x^2\right )-\log \left (16-24 x-7 x^2+22 x^3-4 x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64 - 48*x - 84*x^2 + 23*x^3 + 42*x^4 - 24*x^5 - 8*x^6)/(64*x - 112*x^2 - 36*x^3 + 143*x^4 - 8*x^5 -
60*x^6 + 4*x^7 + 8*x^8),x]

[Out]

Log[x] + 2*Log[4 - x - 2*x^2] - Log[16 - 24*x - 7*x^2 + 22*x^3 - 4*x^5]

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fricas [A]  time = 0.82, size = 37, normalized size = 1.37 \begin {gather*} -\log \left (4 \, x^{5} - 22 \, x^{3} + 7 \, x^{2} + 24 \, x - 16\right ) + 2 \, \log \left (2 \, x^{2} + x - 4\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^6-24*x^5+42*x^4+23*x^3-84*x^2-48*x+64)/(8*x^8+4*x^7-60*x^6-8*x^5+143*x^4-36*x^3-112*x^2+64*x),
x, algorithm="fricas")

[Out]

-log(4*x^5 - 22*x^3 + 7*x^2 + 24*x - 16) + 2*log(2*x^2 + x - 4) + log(x)

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giac [A]  time = 0.25, size = 40, normalized size = 1.48 \begin {gather*} -\log \left ({\left | 4 \, x^{5} - 22 \, x^{3} + 7 \, x^{2} + 24 \, x - 16 \right |}\right ) + 2 \, \log \left ({\left | 2 \, x^{2} + x - 4 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^6-24*x^5+42*x^4+23*x^3-84*x^2-48*x+64)/(8*x^8+4*x^7-60*x^6-8*x^5+143*x^4-36*x^3-112*x^2+64*x),
x, algorithm="giac")

[Out]

-log(abs(4*x^5 - 22*x^3 + 7*x^2 + 24*x - 16)) + 2*log(abs(2*x^2 + x - 4)) + log(abs(x))

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maple [A]  time = 0.05, size = 38, normalized size = 1.41

method	result	size
default	\(\ln \relax (x )-\ln \left (4 x^{5}-22 x^{3}+7 x^{2}+24 x -16\right )+2 \ln \left (2 x^{2}+x -4\right )\)	\(38\)
norman	\(\ln \relax (x )-\ln \left (4 x^{5}-22 x^{3}+7 x^{2}+24 x -16\right )+2 \ln \left (2 x^{2}+x -4\right )\)	\(38\)
risch	\(\ln \relax (x )-\ln \left (4 x^{5}-22 x^{3}+7 x^{2}+24 x -16\right )+2 \ln \left (2 x^{2}+x -4\right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^6-24*x^5+42*x^4+23*x^3-84*x^2-48*x+64)/(8*x^8+4*x^7-60*x^6-8*x^5+143*x^4-36*x^3-112*x^2+64*x),x,meth
od=_RETURNVERBOSE)

[Out]

ln(x)-ln(4*x^5-22*x^3+7*x^2+24*x-16)+2*ln(2*x^2+x-4)

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maxima [A]  time = 0.46, size = 37, normalized size = 1.37 \begin {gather*} -\log \left (4 \, x^{5} - 22 \, x^{3} + 7 \, x^{2} + 24 \, x - 16\right ) + 2 \, \log \left (2 \, x^{2} + x - 4\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^6-24*x^5+42*x^4+23*x^3-84*x^2-48*x+64)/(8*x^8+4*x^7-60*x^6-8*x^5+143*x^4-36*x^3-112*x^2+64*x),
x, algorithm="maxima")

[Out]

-log(4*x^5 - 22*x^3 + 7*x^2 + 24*x - 16) + 2*log(2*x^2 + x - 4) + log(x)

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mupad [B]  time = 0.16, size = 35, normalized size = 1.30 \begin {gather*} 2\,\ln \left (2\,x^2+x-4\right )+\ln \relax (x)-\ln \left (x^5-\frac {11\,x^3}{2}+\frac {7\,x^2}{4}+6\,x-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*x + 84*x^2 - 23*x^3 - 42*x^4 + 24*x^5 + 8*x^6 - 64)/(64*x - 112*x^2 - 36*x^3 + 143*x^4 - 8*x^5 - 60*x
^6 + 4*x^7 + 8*x^8),x)

[Out]

2*log(x + 2*x^2 - 4) + log(x) - log(6*x + (7*x^2)/4 - (11*x^3)/2 + x^5 - 4)

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sympy [A]  time = 0.17, size = 36, normalized size = 1.33 \begin {gather*} \log {\relax (x )} + 2 \log {\left (2 x^{2} + x - 4 \right )} - \log {\left (4 x^{5} - 22 x^{3} + 7 x^{2} + 24 x - 16 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**6-24*x**5+42*x**4+23*x**3-84*x**2-48*x+64)/(8*x**8+4*x**7-60*x**6-8*x**5+143*x**4-36*x**3-112
*x**2+64*x),x)

[Out]

log(x) + 2*log(2*x**2 + x - 4) - log(4*x**5 - 22*x**3 + 7*x**2 + 24*x - 16)

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