\(\int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx\) [6223]

Optimal result

Integrand size = 47, antiderivative size = 25 \[ \int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx=\log \left (4 x+\frac {x}{3+\frac {x}{25+x^2+x (2+x)}}\right ) \]

[Out]

ln(x/(x/(x^2+x*(2+x)+25)+3)+4*x)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2099, 642} \[ \int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx=-\log \left (6 x^2+7 x+75\right )+\log \left (26 x^2+30 x+325\right )+\log (x) \]

[In]

Int[(24375 + 4500*x + 4110*x^2 + 364*x^3 + 156*x^4)/(24375*x + 4525*x^2 + 4110*x^3 + 362*x^4 + 156*x^5),x]

[Out]

Log[x] - Log[75 + 7*x + 6*x^2] + Log[325 + 30*x + 26*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 {-7-12 x}{75+7 x+6 x^2}+\frac {2 (15+26 x)}{325+30 x+26 x^2}\right ) \, dx \\ & = \log (x)+2 \int \frac {15+26 x}{325+30 x+26 x^2} \, dx+\int \frac {-7-12 x}{75+7 x+6 x^2} \, dx \\ & = \log (x)-\log \left (75+7 x+6 x^2\right )+\log \left (325+30 x+26 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx=\log (x)-\log \left (75+7 x+6 x^2\right )+\log \left (325+30 x+26 x^2\right ) \]

[In]

Integrate[(24375 + 4500*x + 4110*x^2 + 364*x^3 + 156*x^4)/(24375*x + 4525*x^2 + 4110*x^3 + 362*x^4 + 156*x^5),
x]

[Out]

Log[x] - Log[75 + 7*x + 6*x^2] + Log[325 + 30*x + 26*x^2]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(\ln \left (x \right )-\ln \left (x^{2}+\frac {7}{6} x +\frac {25}{2}\right )+\ln \left (x^{2}+\frac {15}{13} x +\frac {25}{2}\right )\)	\(24\)
default	\(-\ln \left (6 x^{2}+7 x +75\right )+\ln \left (26 x^{2}+30 x +325\right )+\ln \left (x \right )\)	\(28\)
norman	\(-\ln \left (6 x^{2}+7 x +75\right )+\ln \left (26 x^{2}+30 x +325\right )+\ln \left (x \right )\)	\(28\)
risch	\(-\ln \left (6 x^{2}+7 x +75\right )+\ln \left (26 x^{3}+30 x^{2}+325 x \right )\)	\(30\)

[In]

int((156*x^4+364*x^3+4110*x^2+4500*x+24375)/(156*x^5+362*x^4+4110*x^3+4525*x^2+24375*x),x,method=_RETURNVERBOS
E)

[Out]

ln(x)-ln(x^2+7/6*x+25/2)+ln(x^2+15/13*x+25/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx=\log \left (26 \, x^{3} + 30 \, x^{2} + 325 \, x\right ) - \log \left (6 \, x^{2} + 7 \, x + 75\right ) \]

[In]

integrate((156*x^4+364*x^3+4110*x^2+4500*x+24375)/(156*x^5+362*x^4+4110*x^3+4525*x^2+24375*x),x, algorithm="fr
icas")

[Out]

log(26*x^3 + 30*x^2 + 325*x) - log(6*x^2 + 7*x + 75)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx=- \log {\left (6 x^{2} + 7 x + 75 \right )} + \log {\left (26 x^{3} + 30 x^{2} + 325 x \right )} \]

[In]

integrate((156*x**4+364*x**3+4110*x**2+4500*x+24375)/(156*x**5+362*x**4+4110*x**3+4525*x**2+24375*x),x)

[Out]

-log(6*x**2 + 7*x + 75) + log(26*x**3 + 30*x**2 + 325*x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx=\log \left (26 \, x^{2} + 30 \, x + 325\right ) - \log \left (6 \, x^{2} + 7 \, x + 75\right ) + \log \left (x\right ) \]

[In]

integrate((156*x^4+364*x^3+4110*x^2+4500*x+24375)/(156*x^5+362*x^4+4110*x^3+4525*x^2+24375*x),x, algorithm="ma
xima")

[Out]

log(26*x^2 + 30*x + 325) - log(6*x^2 + 7*x + 75) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx=\log \left (26 \, x^{2} + 30 \, x + 325\right ) - \log \left (6 \, x^{2} + 7 \, x + 75\right ) + \log \left ({\left | x \right |}\right ) \]

[In]

integrate((156*x^4+364*x^3+4110*x^2+4500*x+24375)/(156*x^5+362*x^4+4110*x^3+4525*x^2+24375*x),x, algorithm="gi
ac")

[Out]

log(26*x^2 + 30*x + 325) - log(6*x^2 + 7*x + 75) + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {24375+4500 x+4110 x^2+364 x^3+156 x^4}{24375 x+4525 x^2+4110 x^3+362 x^4+156 x^5} \, dx=\ln \left (x\,\left (26\,x^2+30\,x+325\right )\right )-\ln \left (x^2+\frac {7\,x}{6}+\frac {25}{2}\right ) \]

[In]

int((4500*x + 4110*x^2 + 364*x^3 + 156*x^4 + 24375)/(24375*x + 4525*x^2 + 4110*x^3 + 362*x^4 + 156*x^5),x)

[Out]

log(x*(30*x + 26*x^2 + 325)) - log((7*x)/6 + x^2 + 25/2)