3.84.30 \(\int \frac {2552-8633 x+11437 x^2-9906 x^3+5150 x^4-1033 x^5+x^6}{677-4010 x+9103 x^2-9876 x^3+5135 x^4-1030 x^5+x^6} \, dx\)

Optimal. Leaf size=28 \[ x-3 \left (-\log (2)+\log \left (-1-4 \left (16-\frac {3}{(1-x)^2}\right )^2+x\right )\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 38, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 2, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2074, 1587} \begin {gather*} -3 \log \left (-x^5+1029 x^4-4106 x^3+5770 x^2-3333 x+677\right )+x+12 \log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2552 - 8633*x + 11437*x^2 - 9906*x^3 + 5150*x^4 - 1033*x^5 + x^6)/(677 - 4010*x + 9103*x^2 - 9876*x^3 + 5
135*x^4 - 1030*x^5 + x^6),x]

[Out]

x + 12*Log[1 - x] - 3*Log[677 - 3333*x + 5770*x^2 - 4106*x^3 + 1029*x^4 - x^5]

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 (1+\frac {12}{-1+x}-\frac {3 \left (3333-11540 x+12318 x^2-4116 x^3+5 x^4\right )}{-677+3333 x-5770 x^2+4106 x^3-1029 x^4+x^5}\right ) \, dx\\ &=x+12 \log (1-x)-3 \int \frac {3333-11540 x+12318 x^2-4116 x^3+5 x^4}{-677+3333 x-5770 x^2+4106 x^3-1029 x^4+x^5} \, dx\\ &=x+12 \log (1-x)-3 \log \left (677-3333 x+5770 x^2-4106 x^3+1029 x^4-x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.36 \begin {gather*} x+12 \log (1-x)-3 \log \left (677-3333 x+5770 x^2-4106 x^3+1029 x^4-x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2552 - 8633*x + 11437*x^2 - 9906*x^3 + 5150*x^4 - 1033*x^5 + x^6)/(677 - 4010*x + 9103*x^2 - 9876*x
^3 + 5135*x^4 - 1030*x^5 + x^6),x]

[Out]

x + 12*Log[1 - x] - 3*Log[677 - 3333*x + 5770*x^2 - 4106*x^3 + 1029*x^4 - x^5]

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fricas [A]  time = 0.73, size = 34, normalized size = 1.21 \begin {gather*} x - 3 \, \log \left (x^{5} - 1029 \, x^{4} + 4106 \, x^{3} - 5770 \, x^{2} + 3333 \, x - 677\right ) + 12 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1033*x^5+5150*x^4-9906*x^3+11437*x^2-8633*x+2552)/(x^6-1030*x^5+5135*x^4-9876*x^3+9103*x^2-4010
*x+677),x, algorithm="fricas")

[Out]

x - 3*log(x^5 - 1029*x^4 + 4106*x^3 - 5770*x^2 + 3333*x - 677) + 12*log(x - 1)

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giac [A]  time = 0.13, size = 36, normalized size = 1.29 \begin {gather*} x - 3 \, \log \left ({\left | x^{5} - 1029 \, x^{4} + 4106 \, x^{3} - 5770 \, x^{2} + 3333 \, x - 677 \right |}\right ) + 12 \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1033*x^5+5150*x^4-9906*x^3+11437*x^2-8633*x+2552)/(x^6-1030*x^5+5135*x^4-9876*x^3+9103*x^2-4010
*x+677),x, algorithm="giac")

[Out]

x - 3*log(abs(x^5 - 1029*x^4 + 4106*x^3 - 5770*x^2 + 3333*x - 677)) + 12*log(abs(x - 1))

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maple [A]  time = 0.04, size = 35, normalized size = 1.25

method	result	size
default	\(x -3 \ln \left (x^{5}-1029 x^{4}+4106 x^{3}-5770 x^{2}+3333 x -677\right )+12 \ln \left (x -1\right )\)	\(35\)
norman	\(x -3 \ln \left (x^{5}-1029 x^{4}+4106 x^{3}-5770 x^{2}+3333 x -677\right )+12 \ln \left (x -1\right )\)	\(35\)
risch	\(x -3 \ln \left (x^{5}-1029 x^{4}+4106 x^{3}-5770 x^{2}+3333 x -677\right )+12 \ln \left (x -1\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1033*x^5+5150*x^4-9906*x^3+11437*x^2-8633*x+2552)/(x^6-1030*x^5+5135*x^4-9876*x^3+9103*x^2-4010*x+677
),x,method=_RETURNVERBOSE)

[Out]

x-3*ln(x^5-1029*x^4+4106*x^3-5770*x^2+3333*x-677)+12*ln(x-1)

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maxima [A]  time = 0.35, size = 34, normalized size = 1.21 \begin {gather*} x - 3 \, \log \left (x^{5} - 1029 \, x^{4} + 4106 \, x^{3} - 5770 \, x^{2} + 3333 \, x - 677\right ) + 12 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1033*x^5+5150*x^4-9906*x^3+11437*x^2-8633*x+2552)/(x^6-1030*x^5+5135*x^4-9876*x^3+9103*x^2-4010
*x+677),x, algorithm="maxima")

[Out]

x - 3*log(x^5 - 1029*x^4 + 4106*x^3 - 5770*x^2 + 3333*x - 677) + 12*log(x - 1)

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mupad [B]  time = 5.16, size = 34, normalized size = 1.21 \begin {gather*} x-3\,\ln \left (x^5-1029\,x^4+4106\,x^3-5770\,x^2+3333\,x-677\right )+12\,\ln \left (x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((11437*x^2 - 8633*x - 9906*x^3 + 5150*x^4 - 1033*x^5 + x^6 + 2552)/(9103*x^2 - 4010*x - 9876*x^3 + 5135*x^
4 - 1030*x^5 + x^6 + 677),x)

[Out]

x - 3*log(3333*x - 5770*x^2 + 4106*x^3 - 1029*x^4 + x^5 - 677) + 12*log(x - 1)

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sympy [A]  time = 0.14, size = 34, normalized size = 1.21 \begin {gather*} x + 12 \log {\left (x - 1 \right )} - 3 \log {\left (x^{5} - 1029 x^{4} + 4106 x^{3} - 5770 x^{2} + 3333 x - 677 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1033*x**5+5150*x**4-9906*x**3+11437*x**2-8633*x+2552)/(x**6-1030*x**5+5135*x**4-9876*x**3+9103
*x**2-4010*x+677),x)

[Out]

x + 12*log(x - 1) - 3*log(x**5 - 1029*x**4 + 4106*x**3 - 5770*x**2 + 3333*x - 677)

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