3.103 \(\int \frac{-2+5 x^3}{-27+18 x^2-8 x^3+x^4} \, dx\)

Optimal. Leaf size=41 \[ \frac{407}{16 (3-x)}-\frac{133}{8 (3-x)^2}+\frac{313}{64} \log (3-x)+\frac{7}{64} \log (x+1) \]

[Out]

-133/(8*(3 - x)^2) + 407/(16*(3 - x)) + (313*Log[3 - x])/64 + (7*Log[1 + x])/64

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Rubi [A]  time = 0.0370383, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2074} \[ \frac{407}{16 (3-x)}-\frac{133}{8 (3-x)^2}+\frac{313}{64} \log (3-x)+\frac{7}{64} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(-2 + 5*x^3)/(-27 + 18*x^2 - 8*x^3 + x^4),x]

[Out]

-133/(8*(3 - x)^2) + 407/(16*(3 - x)) + (313*Log[3 - x])/64 + (7*Log[1 + x])/64

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{align*} \int \frac{-2+5 x^3}{-27+18 x^2-8 x^3+x^4} \, dx &=\int \left (\frac{133}{4 (-3+x)^3}+\frac{407}{16 (-3+x)^2}+\frac{313}{64 (-3+x)}+\frac{7}{64 (1+x)}\right ) \, dx\\ &=-\frac{133}{8 (3-x)^2}+\frac{407}{16 (3-x)}+\frac{313}{64} \log (3-x)+\frac{7}{64} \log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.019543, size = 37, normalized size = 0.9 \[ -\frac{407}{16 (x-3)}-\frac{133}{8 (x-3)^2}+\frac{313}{64} \log (3-x)+\frac{7}{64} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 5*x^3)/(-27 + 18*x^2 - 8*x^3 + x^4),x]

[Out]

-133/(8*(-3 + x)^2) - 407/(16*(-3 + x)) + (313*Log[3 - x])/64 + (7*Log[1 + x])/64

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Maple [A]  time = 0.007, size = 28, normalized size = 0.7 \begin{align*}{\frac{7\,\ln \left ( 1+x \right ) }{64}}-{\frac{133}{8\, \left ( -3+x \right ) ^{2}}}-{\frac{407}{-48+16\,x}}+{\frac{313\,\ln \left ( -3+x \right ) }{64}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^3-2)/(x^4-8*x^3+18*x^2-27),x)

[Out]

7/64*ln(1+x)-133/8/(-3+x)^2-407/16/(-3+x)+313/64*ln(-3+x)

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Maxima [A]  time = 0.933254, size = 41, normalized size = 1. \begin{align*} -\frac{407 \, x - 955}{16 \,{\left (x^{2} - 6 \, x + 9\right )}} + \frac{7}{64} \, \log \left (x + 1\right ) + \frac{313}{64} \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^3-2)/(x^4-8*x^3+18*x^2-27),x, algorithm="maxima")

[Out]

-1/16*(407*x - 955)/(x^2 - 6*x + 9) + 7/64*log(x + 1) + 313/64*log(x - 3)

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Fricas [A]  time = 2.08478, size = 138, normalized size = 3.37 \begin{align*} \frac{7 \,{\left (x^{2} - 6 \, x + 9\right )} \log \left (x + 1\right ) + 313 \,{\left (x^{2} - 6 \, x + 9\right )} \log \left (x - 3\right ) - 1628 \, x + 3820}{64 \,{\left (x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^3-2)/(x^4-8*x^3+18*x^2-27),x, algorithm="fricas")

[Out]

1/64*(7*(x^2 - 6*x + 9)*log(x + 1) + 313*(x^2 - 6*x + 9)*log(x - 3) - 1628*x + 3820)/(x^2 - 6*x + 9)

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Sympy [A]  time = 0.123652, size = 31, normalized size = 0.76 \begin{align*} - \frac{407 x - 955}{16 x^{2} - 96 x + 144} + \frac{313 \log{\left (x - 3 \right )}}{64} + \frac{7 \log{\left (x + 1 \right )}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**3-2)/(x**4-8*x**3+18*x**2-27),x)

[Out]

-(407*x - 955)/(16*x**2 - 96*x + 144) + 313*log(x - 3)/64 + 7*log(x + 1)/64

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Giac [A]  time = 1.05388, size = 36, normalized size = 0.88 \begin{align*} -\frac{407 \, x - 955}{16 \,{\left (x - 3\right )}^{2}} + \frac{7}{64} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{313}{64} \, \log \left ({\left | x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^3-2)/(x^4-8*x^3+18*x^2-27),x, algorithm="giac")

[Out]

-1/16*(407*x - 955)/(x - 3)^2 + 7/64*log(abs(x + 1)) + 313/64*log(abs(x - 3))