Maximum of $\frac{Bx}{x^3 + Rx + P}$

Question

I submitted this function to Wolfram Alpha and similar tools, but none can find for me the symbolic expression for maximum value, unless I give specific values to each parameter: $$\frac{Bx}{x^3 + Rx + P}$$

The curve has this aspect with "typical" parameters:

Any suggestion?

Given that the tool can calculate the derivative:

$$\frac{B (P - 2 x^3)}{{P + x (R + x^2)}^2}=0$$

Why can't it find the maximum, which is just $(P - 2x^3)=0$ , i.e. $x^3 = \frac{P}{2}$ ?

I also tried adding parameters, but I get this weird result, where I can see sliders but I can't use them!

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EDIT

Actually the curve I need to study has an additional parameter A:

$$y(x) = \frac{Bx}{Ax^3 + Rx + P}$$

It represents range of a vehicle given:

• $A= \frac1 2 \rho C_d A_r $

  • $\rho = 1.225 \frac{kg}{m^{3-}}$ (Air Density)
  • $C_d$ = Air Drag Coefficient (around 0.3, adimensional)
  • $A_r$ = Frontal Area (Around 2.2 [$m^2$])

• $R = m \cdot g \cdot C_{rr}$

  • $m$ =Mass [kg]
  • $g = 9.81 [\frac{m}{s^{2}}]$ (Gravity Acceleration)
  • $C_{rr}$ = Rolling Friction Coefficient (around 0.010, adimensional)

• $P$ = Power of additional services [Watt]

• $B$ = Battery capacity [Wh]

The starting form of above equation was:

$$ Range = \frac{B}{(\frac1 2 \rho C_d A ) x^2 + (mgC_{rr}) + \frac P x }$$

which can be also written as:

$$ Range = \frac{Bx}{ (\frac1 2 \rho C_d A ) x^3 +(mgC_{rr})x + P }$$

and hence:

$$\frac{Bx}{Ax^3 + Rx + P}$$

The maximum of the function is the maximum range at constant speed.

$x$ represents speed in km/h.

Answer 1

$$f(x)=\frac{Bx}{x^3 + Rx + P}\implies f'(x)=\frac{B \left(P-2 x^3\right)}{\left(x^3 + Rx + P\right)^2}\implies f''(x)=-\frac{2 B \left(P R+6 P x^2+R x^3-3 x^5\right)}{\left(x^3 + Rx + P\right)^3}$$

If $(B,P,R)$ are positive, the $x_*=\sqrt[3]{\frac{P}{2}}$ corresponds to an extremum; the second derivaive test confirms that it is a maximum.

$$f(x_*)=\frac{2^{2/3} B}{3 P^{2/3}+2^{2/3} R}$$

======= EDIT

adapting to edited question...

$$ f'(x) =\frac{B (P - 2 A x^3)}{P + R x + A x^3}^2$$ $$ f''(x) = -\frac{2 B (P R + 6 A P x^2 + A R x^3 - 3 A^2 x^5)}{(P + R x + A x^3)^3}$$

$x_*=\sqrt[3]{\frac{P}{2A}}$