3.62 \(\int \frac{3-3 x+30 x^2+160 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{8} \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right ) \]

[Out]

Log[9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4]/8

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Rubi [A]  time = 0.0226836, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {1587} \[ \frac{1}{8} \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 - 3*x + 30*x^2 + 160*x^3)/(9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4),x]

[Out]

Log[9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4]/8

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]

Rubi steps

\begin{align*} \int \frac{3-3 x+30 x^2+160 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx &=\frac{1}{8} \log \left (9+24 x-12 x^2+80 x^3+320 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0081255, size = 25, normalized size = 1. \[ \frac{1}{8} \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 3*x + 30*x^2 + 160*x^3)/(9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4),x]

[Out]

Log[9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4]/8

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Maple [A]  time = 0.001, size = 24, normalized size = 1. \begin{align*}{\frac{\ln \left ( 320\,{x}^{4}+80\,{x}^{3}-12\,{x}^{2}+24\,x+9 \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((160*x^3+30*x^2-3*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x)

[Out]

1/8*ln(320*x^4+80*x^3-12*x^2+24*x+9)

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Maxima [A]  time = 0.936682, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{8} \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*x^3+30*x^2-3*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="maxima")

[Out]

1/8*log(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9)

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Fricas [A]  time = 2.30158, size = 63, normalized size = 2.52 \begin{align*} \frac{1}{8} \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*x^3+30*x^2-3*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="fricas")

[Out]

1/8*log(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9)

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Sympy [A]  time = 0.097344, size = 22, normalized size = 0.88 \begin{align*} \frac{\log{\left (320 x^{4} + 80 x^{3} - 12 x^{2} + 24 x + 9 \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*x**3+30*x**2-3*x+3)/(320*x**4+80*x**3-12*x**2+24*x+9),x)

[Out]

log(320*x**4 + 80*x**3 - 12*x**2 + 24*x + 9)/8

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Giac [A]  time = 1.07841, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{8} \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((160*x^3+30*x^2-3*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="giac")

[Out]

1/8*log(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9)