3.93.4 \(\int \frac {-10+74 x-32 x^2+18 x^3-28 x^4-70 x^5+50 x^6}{2 x-11 x^2+7 x^3-23 x^4+81 x^5-85 x^6+25 x^7} \, dx\)

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

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Rubi [A]  time = 0.16, antiderivative size = 37, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2074, 1587} \begin {gather*} 2 \log \left (5 x^4-6 x^3+x^2+1\right )-2 \log (1-5 x)+\log (2-x)-5 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10 + 74*x - 32*x^2 + 18*x^3 - 28*x^4 - 70*x^5 + 50*x^6)/(2*x - 11*x^2 + 7*x^3 - 23*x^4 + 81*x^5 - 85*x^6
+ 25*x^7),x]

[Out]

-2*Log[1 - 5*x] + Log[2 - x] - 5*Log[x] + 2*Log[1 + x^2 - 6*x^3 + 5*x^4]

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}{-2+x}-\frac {5}{x}-\frac {10}{-1+5 x}+\frac {4 x \left (1-9 x+10 x^2\right )}{1+x^2-6 x^3+5 x^4}\right ) \, dx\\ &=-2 \log (1-5 x)+\log (2-x)-5 \log (x)+4 \int \frac {x \left (1-9 x+10 x^2\right )}{1+x^2-6 x^3+5 x^4} \, dx\\ &=-2 \log (1-5 x)+\log (2-x)-5 \log (x)+2 \log \left (1+x^2-6 x^3+5 x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 1.30 \begin {gather*} 2 \left (-\log (1-5 x)+\frac {1}{2} \log (2-x)-\frac {5 \log (x)}{2}+\log \left (1+x^2-6 x^3+5 x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 74*x - 32*x^2 + 18*x^3 - 28*x^4 - 70*x^5 + 50*x^6)/(2*x - 11*x^2 + 7*x^3 - 23*x^4 + 81*x^5 -
85*x^6 + 25*x^7),x]

[Out]

2*(-Log[1 - 5*x] + Log[2 - x]/2 - (5*Log[x])/2 + Log[1 + x^2 - 6*x^3 + 5*x^4])

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fricas [A]  time = 0.57, size = 35, normalized size = 1.06 \begin {gather*} 2 \, \log \left (5 \, x^{4} - 6 \, x^{3} + x^{2} + 1\right ) - 2 \, \log \left (5 \, x - 1\right ) + \log \left (x - 2\right ) - 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^6-70*x^5-28*x^4+18*x^3-32*x^2+74*x-10)/(25*x^7-85*x^6+81*x^5-23*x^4+7*x^3-11*x^2+2*x),x, algor
ithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 38, normalized size = 1.15 \begin {gather*} 2 \, \log \left (5 \, x^{4} - 6 \, x^{3} + x^{2} + 1\right ) - 2 \, \log \left ({\left | 5 \, x - 1 \right |}\right ) + \log \left ({\left | x - 2 \right |}\right ) - 5 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^6-70*x^5-28*x^4+18*x^3-32*x^2+74*x-10)/(25*x^7-85*x^6+81*x^5-23*x^4+7*x^3-11*x^2+2*x),x, algor
ithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 36, normalized size = 1.09

method	result	size
default	\(-5 \ln \relax (x )+\ln \left (x -2\right )-2 \ln \left (5 x -1\right )+2 \ln \left (5 x^{4}-6 x^{3}+x^{2}+1\right )\)	\(36\)
norman	\(-5 \ln \relax (x )+\ln \left (x -2\right )-2 \ln \left (5 x -1\right )+2 \ln \left (5 x^{4}-6 x^{3}+x^{2}+1\right )\)	\(36\)
risch	\(-5 \ln \relax (x )+\ln \left (x -2\right )-2 \ln \left (5 x -1\right )+2 \ln \left (5 x^{4}-6 x^{3}+x^{2}+1\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x^6-70*x^5-28*x^4+18*x^3-32*x^2+74*x-10)/(25*x^7-85*x^6+81*x^5-23*x^4+7*x^3-11*x^2+2*x),x,method=_RETU
RNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 35, normalized size = 1.06 \begin {gather*} 2 \, \log \left (5 \, x^{4} - 6 \, x^{3} + x^{2} + 1\right ) - 2 \, \log \left (5 \, x - 1\right ) + \log \left (x - 2\right ) - 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^6-70*x^5-28*x^4+18*x^3-32*x^2+74*x-10)/(25*x^7-85*x^6+81*x^5-23*x^4+7*x^3-11*x^2+2*x),x, algor
ithm="maxima")

[Out]

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

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mupad [B]  time = 0.16, size = 33, normalized size = 1.00 \begin {gather*} 2\,\ln \left (5\,x^4-6\,x^3+x^2+1\right )+\ln \left (x-2\right )-2\,\ln \left (x-\frac {1}{5}\right )-5\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x^2 - 74*x - 18*x^3 + 28*x^4 + 70*x^5 - 50*x^6 + 10)/(2*x - 11*x^2 + 7*x^3 - 23*x^4 + 81*x^5 - 85*x^6
+ 25*x^7),x)

[Out]

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

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sympy [A]  time = 0.23, size = 36, normalized size = 1.09 \begin {gather*} - 5 \log {\relax (x )} + \log {\left (x - 2 \right )} - 2 \log {\left (x - \frac {1}{5} \right )} + 2 \log {\left (5 x^{4} - 6 x^{3} + x^{2} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x**6-70*x**5-28*x**4+18*x**3-32*x**2+74*x-10)/(25*x**7-85*x**6+81*x**5-23*x**4+7*x**3-11*x**2+2*
x),x)

[Out]

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

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