3.6.63 \(\int \frac {-721875+991375 x-129420 x^2-1857 x^3+x^4}{721875 x-135125 x^2-1855 x^3+x^4} \, dx\)

Optimal. Leaf size=28 \[ \log \left (\frac {e^x \left (-1+\frac {80}{3+\frac {x}{25}}\right )}{(5-x) x}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2074} \begin {gather*} x-\log (5-x)+\log (1925-x)-\log (x)-\log (x+75) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-721875 + 991375*x - 129420*x^2 - 1857*x^3 + x^4)/(721875*x - 135125*x^2 - 1855*x^3 + x^4),x]

[Out]

x - Log[5 - x] + Log[1925 - x] - Log[x] - Log[75 + 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 {1}{-75-x}+\frac {1}{5-x}+\frac {1}{-1925+x}-\frac {1}{x}\right ) \, dx\\ &=x-\log (5-x)+\log (1925-x)-\log (x)-\log (75+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.89 \begin {gather*} x+\log (1925-x)-\log (x)-\log \left (375-70 x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-721875 + 991375*x - 129420*x^2 - 1857*x^3 + x^4)/(721875*x - 135125*x^2 - 1855*x^3 + x^4),x]

[Out]

x + Log[1925 - x] - Log[x] - Log[375 - 70*x - x^2]

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fricas [A]  time = 0.62, size = 21, normalized size = 0.75 \begin {gather*} x - \log \left (x^{3} + 70 \, x^{2} - 375 \, x\right ) + \log \left (x - 1925\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1857*x^3-129420*x^2+991375*x-721875)/(x^4-1855*x^3-135125*x^2+721875*x),x, algorithm="fricas")

[Out]

x - log(x^3 + 70*x^2 - 375*x) + log(x - 1925)

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giac [A]  time = 0.51, size = 26, normalized size = 0.93 \begin {gather*} x - \log \left ({\left | x + 75 \right |}\right ) - \log \left ({\left | x - 5 \right |}\right ) + \log \left ({\left | x - 1925 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1857*x^3-129420*x^2+991375*x-721875)/(x^4-1855*x^3-135125*x^2+721875*x),x, algorithm="giac")

[Out]

x - log(abs(x + 75)) - log(abs(x - 5)) + log(abs(x - 1925)) - log(abs(x))

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maple [A]  time = 0.06, size = 22, normalized size = 0.79

method	result	size
risch	\(x +\ln \left (x -1925\right )-\ln \left (x^{3}+70 x^{2}-375 x \right )\)	\(22\)
default	\(x -\ln \relax (x )+\ln \left (x -1925\right )-\ln \left (x -5\right )-\ln \left (x +75\right )\)	\(23\)
norman	\(x -\ln \relax (x )+\ln \left (x -1925\right )-\ln \left (x -5\right )-\ln \left (x +75\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1857*x^3-129420*x^2+991375*x-721875)/(x^4-1855*x^3-135125*x^2+721875*x),x,method=_RETURNVERBOSE)

[Out]

x+ln(x-1925)-ln(x^3+70*x^2-375*x)

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maxima [A]  time = 0.54, size = 22, normalized size = 0.79 \begin {gather*} x - \log \left (x + 75\right ) - \log \left (x - 5\right ) + \log \left (x - 1925\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1857*x^3-129420*x^2+991375*x-721875)/(x^4-1855*x^3-135125*x^2+721875*x),x, algorithm="maxima")

[Out]

x - log(x + 75) - log(x - 5) + log(x - 1925) - log(x)

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mupad [B]  time = 0.50, size = 19, normalized size = 0.68 \begin {gather*} x+\ln \left (x-1925\right )-\ln \left (x\,\left (x^2+70\,x-375\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(129420*x^2 - 991375*x + 1857*x^3 - x^4 + 721875)/(721875*x - 135125*x^2 - 1855*x^3 + x^4),x)

[Out]

x + log(x - 1925) - log(x*(70*x + x^2 - 375))

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sympy [A]  time = 0.11, size = 19, normalized size = 0.68 \begin {gather*} x + \log {\left (x - 1925 \right )} - \log {\left (x^{3} + 70 x^{2} - 375 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1857*x**3-129420*x**2+991375*x-721875)/(x**4-1855*x**3-135125*x**2+721875*x),x)

[Out]

x + log(x - 1925) - log(x**3 + 70*x**2 - 375*x)

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