\(\int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx\) [3307]

Optimal result

Integrand size = 47, antiderivative size = 27 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\log \left (\frac {x \left (-2-\frac {16}{3 \left (5+\frac {4}{x}+3 x\right )}\right )}{12 e^5}\right ) \]

[Out]

ln(1/12*(-2-16/3/(4/x+5+3*x))/exp(5)*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2099, 642} \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=-\log \left (3 x^2+5 x+4\right )+\log \left (9 x^2+23 x+12\right )+\log (x) \]

[In]

Int[(48 + 184*x + 187*x^2 + 90*x^3 + 27*x^4)/(48*x + 152*x^2 + 187*x^3 + 114*x^4 + 27*x^5),x]

[Out]

Log[x] - Log[4 + 5*x + 3*x^2] + Log[12 + 23*x + 9*x^2]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2099

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*} \text {integral}& = \int \left (\frac {1}{x}+\frac {-5-6 x}{4+5 x+3 x^2}+\frac {23+18 x}{12+23 x+9 x^2}\right ) \, dx \\ & = \log (x)+\int \frac {-5-6 x}{4+5 x+3 x^2} \, dx+\int \frac {23+18 x}{12+23 x+9 x^2} \, dx \\ & = \log (x)-\log \left (4+5 x+3 x^2\right )+\log \left (12+23 x+9 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\log (x)-\log \left (4+5 x+3 x^2\right )+\log \left (12+23 x+9 x^2\right ) \]

[In]

Integrate[(48 + 184*x + 187*x^2 + 90*x^3 + 27*x^4)/(48*x + 152*x^2 + 187*x^3 + 114*x^4 + 27*x^5),x]

[Out]

Log[x] - Log[4 + 5*x + 3*x^2] + Log[12 + 23*x + 9*x^2]

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

method	result	size
parallelrisch	\(\ln \left (x \right )-\ln \left (x^{2}+\frac {5}{3} x +\frac {4}{3}\right )+\ln \left (x^{2}+\frac {23}{9} x +\frac {4}{3}\right )\)	\(24\)
default	\(-\ln \left (3 x^{2}+5 x +4\right )+\ln \left (9 x^{2}+23 x +12\right )+\ln \left (x \right )\)	\(28\)
norman	\(-\ln \left (3 x^{2}+5 x +4\right )+\ln \left (9 x^{2}+23 x +12\right )+\ln \left (x \right )\)	\(28\)
risch	\(-\ln \left (3 x^{2}+5 x +4\right )+\ln \left (9 x^{3}+23 x^{2}+12 x \right )\)	\(30\)

[In]

int((27*x^4+90*x^3+187*x^2+184*x+48)/(27*x^5+114*x^4+187*x^3+152*x^2+48*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(x^2+5/3*x+4/3)+ln(x^2+23/9*x+4/3)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\log \left (9 \, x^{3} + 23 \, x^{2} + 12 \, x\right ) - \log \left (3 \, x^{2} + 5 \, x + 4\right ) \]

[In]

integrate((27*x^4+90*x^3+187*x^2+184*x+48)/(27*x^5+114*x^4+187*x^3+152*x^2+48*x),x, algorithm="fricas")

[Out]

log(9*x^3 + 23*x^2 + 12*x) - log(3*x^2 + 5*x + 4)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=- \log {\left (3 x^{2} + 5 x + 4 \right )} + \log {\left (9 x^{3} + 23 x^{2} + 12 x \right )} \]

[In]

integrate((27*x**4+90*x**3+187*x**2+184*x+48)/(27*x**5+114*x**4+187*x**3+152*x**2+48*x),x)

[Out]

-log(3*x**2 + 5*x + 4) + log(9*x**3 + 23*x**2 + 12*x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\log \left (9 \, x^{2} + 23 \, x + 12\right ) - \log \left (3 \, x^{2} + 5 \, x + 4\right ) + \log \left (x\right ) \]

[In]

integrate((27*x^4+90*x^3+187*x^2+184*x+48)/(27*x^5+114*x^4+187*x^3+152*x^2+48*x),x, algorithm="maxima")

[Out]

log(9*x^2 + 23*x + 12) - log(3*x^2 + 5*x + 4) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=-\log \left (3 \, x^{2} + 5 \, x + 4\right ) + \log \left ({\left | 9 \, x^{2} + 23 \, x + 12 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]

[In]

integrate((27*x^4+90*x^3+187*x^2+184*x+48)/(27*x^5+114*x^4+187*x^3+152*x^2+48*x),x, algorithm="giac")

[Out]

-log(3*x^2 + 5*x + 4) + log(abs(9*x^2 + 23*x + 12)) + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\ln \left (x\,\left (9\,x^2+23\,x+12\right )\right )-\ln \left (x^2+\frac {5\,x}{3}+\frac {4}{3}\right ) \]

[In]

int((184*x + 187*x^2 + 90*x^3 + 27*x^4 + 48)/(48*x + 152*x^2 + 187*x^3 + 114*x^4 + 27*x^5),x)

[Out]

log(x*(23*x + 9*x^2 + 12)) - log((5*x)/3 + x^2 + 4/3)