3.1.62 \(\int \frac {3-3 x+30 x^2+160 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\) [62]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {1601} \begin {gather*} \frac {1}{8} \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right ) \end {gather*}

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 1601

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]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; 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*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{8} \log \left (9+24 x-12 x^2+80 x^3+320 x^4\right ) \end {gather*}

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 = 0.01, size = 24, normalized size = 0.96

method	result	size
default	\(\frac {\ln \left (320 x^{4}+80 x^{3}-12 x^{2}+24 x +9\right )}{8}\)	\(24\)
norman	\(\frac {\ln \left (320 x^{4}+80 x^{3}-12 x^{2}+24 x +9\right )}{8}\)	\(24\)
risch	\(\frac {\ln \left (320 x^{4}+80 x^{3}-12 x^{2}+24 x +9\right )}{8}\)	\(24\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{8} \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \end {gather*}

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 = 0.97, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{8} \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \end {gather*}

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 = 0.03, size = 22, normalized size = 0.88 \begin {gather*} \frac {\log {\left (320 x^{4} + 80 x^{3} - 12 x^{2} + 24 x + 9 \right )}}{8} \end {gather*}

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 = 1.23, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{8} \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \end {gather*}

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)

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Mupad [B]
time = 0.07, size = 23, normalized size = 0.92 \begin {gather*} \frac {\ln \left (320\,x^4+80\,x^3-12\,x^2+24\,x+9\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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