3.8.31 \(\int \frac {-4065+100 x-134 x^2+4 x^3-x^4}{-17875+5165 x-742 x^2+146 x^3-7 x^4+x^5} \, dx\)

Optimal. Leaf size=30 \[ \log \left (\frac {2}{5 \left (3-x+\frac {4}{4-\frac {1}{20} (5-x) (3+x)}\right )}\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 28, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2074, 628, 1587} \begin {gather*} \log \left (x^2-2 x+65\right )-\log \left (-x^3+5 x^2-71 x+275\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4065 + 100*x - 134*x^2 + 4*x^3 - x^4)/(-17875 + 5165*x - 742*x^2 + 146*x^3 - 7*x^4 + x^5),x]

[Out]

Log[65 - 2*x + x^2] - Log[275 - 71*x + 5*x^2 - x^3]

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 {2 (-1+x)}{65-2 x+x^2}+\frac {-71+10 x-3 x^2}{-275+71 x-5 x^2+x^3}\right ) \, dx\\ &=2 \int \frac {-1+x}{65-2 x+x^2} \, dx+\int \frac {-71+10 x-3 x^2}{-275+71 x-5 x^2+x^3} \, dx\\ &=\log \left (65-2 x+x^2\right )-\log \left (275-71 x+5 x^2-x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.93 \begin {gather*} \log \left (65-2 x+x^2\right )-\log \left (275-71 x+5 x^2-x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4065 + 100*x - 134*x^2 + 4*x^3 - x^4)/(-17875 + 5165*x - 742*x^2 + 146*x^3 - 7*x^4 + x^5),x]

[Out]

Log[65 - 2*x + x^2] - Log[275 - 71*x + 5*x^2 - x^3]

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fricas [A]  time = 0.56, size = 26, normalized size = 0.87 \begin {gather*} -\log \left (x^{3} - 5 \, x^{2} + 71 \, x - 275\right ) + \log \left (x^{2} - 2 \, x + 65\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+4*x^3-134*x^2+100*x-4065)/(x^5-7*x^4+146*x^3-742*x^2+5165*x-17875),x, algorithm="fricas")

[Out]

-log(x^3 - 5*x^2 + 71*x - 275) + log(x^2 - 2*x + 65)

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giac [A]  time = 0.31, size = 27, normalized size = 0.90 \begin {gather*} \log \left (x^{2} - 2 \, x + 65\right ) - \log \left ({\left | x^{3} - 5 \, x^{2} + 71 \, x - 275 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+4*x^3-134*x^2+100*x-4065)/(x^5-7*x^4+146*x^3-742*x^2+5165*x-17875),x, algorithm="giac")

[Out]

log(x^2 - 2*x + 65) - log(abs(x^3 - 5*x^2 + 71*x - 275))

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maple [A]  time = 0.04, size = 27, normalized size = 0.90

method	result	size
default	\(\ln \left (x^{2}-2 x +65\right )-\ln \left (x^{3}-5 x^{2}+71 x -275\right )\)	\(27\)
norman	\(\ln \left (x^{2}-2 x +65\right )-\ln \left (x^{3}-5 x^{2}+71 x -275\right )\)	\(27\)
risch	\(\ln \left (x^{2}-2 x +65\right )-\ln \left (x^{3}-5 x^{2}+71 x -275\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+4*x^3-134*x^2+100*x-4065)/(x^5-7*x^4+146*x^3-742*x^2+5165*x-17875),x,method=_RETURNVERBOSE)

[Out]

ln(x^2-2*x+65)-ln(x^3-5*x^2+71*x-275)

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maxima [A]  time = 0.46, size = 26, normalized size = 0.87 \begin {gather*} -\log \left (x^{3} - 5 \, x^{2} + 71 \, x - 275\right ) + \log \left (x^{2} - 2 \, x + 65\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+4*x^3-134*x^2+100*x-4065)/(x^5-7*x^4+146*x^3-742*x^2+5165*x-17875),x, algorithm="maxima")

[Out]

-log(x^3 - 5*x^2 + 71*x - 275) + log(x^2 - 2*x + 65)

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mupad [B]  time = 0.56, size = 26, normalized size = 0.87 \begin {gather*} \ln \left (x^2-2\,x+65\right )-\ln \left (x^3-5\,x^2+71\,x-275\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(134*x^2 - 100*x - 4*x^3 + x^4 + 4065)/(5165*x - 742*x^2 + 146*x^3 - 7*x^4 + x^5 - 17875),x)

[Out]

log(x^2 - 2*x + 65) - log(71*x - 5*x^2 + x^3 - 275)

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sympy [A]  time = 0.12, size = 24, normalized size = 0.80 \begin {gather*} \log {\left (x^{2} - 2 x + 65 \right )} - \log {\left (x^{3} - 5 x^{2} + 71 x - 275 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4+4*x**3-134*x**2+100*x-4065)/(x**5-7*x**4+146*x**3-742*x**2+5165*x-17875),x)

[Out]

log(x**2 - 2*x + 65) - log(x**3 - 5*x**2 + 71*x - 275)

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