\(\int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx\) [9239]

Optimal result

Integrand size = 35, antiderivative size = 19 \[ \int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx=\log \left (x \left (2-x-9 \left (4-x^2\right )^2\right )\right ) \]

[Out]

ln((2-x-3*(-x^2+4)*(-3*x^2+12))*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1601} \[ \int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx=\log \left (9 x^5-72 x^3+x^2+142 x\right ) \]

[In]

Int[(142 + 2*x - 216*x^2 + 45*x^4)/(142*x + x^2 - 72*x^3 + 9*x^5),x]

[Out]

Log[142*x + x^2 - 72*x^3 + 9*x^5]

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*} \text {integral}& = \log \left (142 x+x^2-72 x^3+9 x^5\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx=\log (x)+\log \left (142+x-72 x^2+9 x^4\right ) \]

[In]

Integrate[(142 + 2*x - 216*x^2 + 45*x^4)/(142*x + x^2 - 72*x^3 + 9*x^5),x]

[Out]

Log[x] + Log[142 + x - 72*x^2 + 9*x^4]

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
default	\(\ln \left (x \left (9 x^{4}-72 x^{2}+x +142\right )\right )\)	\(17\)
derivativedivides	\(\ln \left (9 x^{5}-72 x^{3}+x^{2}+142 x \right )\)	\(19\)
risch	\(\ln \left (9 x^{5}-72 x^{3}+x^{2}+142 x \right )\)	\(19\)
parallelrisch	\(\ln \left (x \right )+\ln \left (-2+x \right )+\ln \left (x^{3}+2 x^{2}-4 x -\frac {71}{9}\right )\)	\(22\)
norman	\(\ln \left (x \right )+\ln \left (-2+x \right )+\ln \left (9 x^{3}+18 x^{2}-36 x -71\right )\)	\(24\)

[In]

int((45*x^4-216*x^2+2*x+142)/(9*x^5-72*x^3+x^2+142*x),x,method=_RETURNVERBOSE)

[Out]

ln(x*(9*x^4-72*x^2+x+142))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx=\log \left (9 \, x^{5} - 72 \, x^{3} + x^{2} + 142 \, x\right ) \]

[In]

integrate((45*x^4-216*x^2+2*x+142)/(9*x^5-72*x^3+x^2+142*x),x, algorithm="fricas")

[Out]

log(9*x^5 - 72*x^3 + x^2 + 142*x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx=\log {\left (9 x^{5} - 72 x^{3} + x^{2} + 142 x \right )} \]

[In]

integrate((45*x**4-216*x**2+2*x+142)/(9*x**5-72*x**3+x**2+142*x),x)

[Out]

log(9*x**5 - 72*x**3 + x**2 + 142*x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx=\log \left (9 \, x^{5} - 72 \, x^{3} + x^{2} + 142 \, x\right ) \]

[In]

integrate((45*x^4-216*x^2+2*x+142)/(9*x^5-72*x^3+x^2+142*x),x, algorithm="maxima")

[Out]

log(9*x^5 - 72*x^3 + x^2 + 142*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx=\log \left ({\left | 9 \, x^{5} - 72 \, x^{3} + x^{2} + 142 \, x \right |}\right ) \]

[In]

integrate((45*x^4-216*x^2+2*x+142)/(9*x^5-72*x^3+x^2+142*x),x, algorithm="giac")

[Out]

log(abs(9*x^5 - 72*x^3 + x^2 + 142*x))

Mupad [B] (verification not implemented)

Time = 13.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {142+2 x-216 x^2+45 x^4}{142 x+x^2-72 x^3+9 x^5} \, dx=\ln \left (x\,\left (9\,x^4-72\,x^2+x+142\right )\right ) \]

[In]

int((2*x - 216*x^2 + 45*x^4 + 142)/(142*x + x^2 - 72*x^3 + 9*x^5),x)

[Out]

log(x*(x - 72*x^2 + 9*x^4 + 142))