\(\int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx\) [64]

Optimal result

Integrand size = 20, antiderivative size = 396 \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=-\frac {3 a b^2 (c+d x)^2}{f}+\frac {b^3 (c+d x)^2}{2 f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f} \] Output:

-3*a*b^2*(d*x+c)^2/f+1/2*b^3*(d*x+c)^2/f+1/3*a^3*(d*x+c)^3/d-a^2*b*(d*x+c) 
^3/d+a*b^2*(d*x+c)^3/d-1/3*b^3*(d*x+c)^3/d+6*a*b^2*d*(d*x+c)*ln(1+exp(2*f* 
x+2*e))/f^2+3*a^2*b*(d*x+c)^2*ln(1+exp(2*f*x+2*e))/f+b^3*(d*x+c)^2*ln(1+ex 
p(2*f*x+2*e))/f+b^3*d^2*ln(cosh(f*x+e))/f^3+3*a*b^2*d^2*polylog(2,-exp(2*f 
*x+2*e))/f^3+3*a^2*b*d*(d*x+c)*polylog(2,-exp(2*f*x+2*e))/f^2+b^3*d*(d*x+c 
)*polylog(2,-exp(2*f*x+2*e))/f^2-3/2*a^2*b*d^2*polylog(3,-exp(2*f*x+2*e))/ 
f^3-1/2*b^3*d^2*polylog(3,-exp(2*f*x+2*e))/f^3-b^3*d*(d*x+c)*tanh(f*x+e)/f 
^2-3*a*b^2*(d*x+c)^2*tanh(f*x+e)/f-1/2*b^3*(d*x+c)^2*tanh(f*x+e)^2/f
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1163\) vs. \(2(396)=792\).

Time = 7.17 (sec) , antiderivative size = 1163, normalized size of antiderivative = 2.94 \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx =\text {Too large to display} \] Input:

Integrate[(c + d*x)^2*(a + b*Tanh[e + f*x])^3,x]
 

Output:

(b*(-4*E^(2*e)*f*x*(9*a*b*d*f*(2*c + d*x) + 3*a^2*f^2*(3*c^2 + 3*c*d*x + d 
^2*x^2) + b^2*(3*c^2*f^2 + 3*c*d*f^2*x + d^2*(3 + f^2*x^2))) + 6*(1 + E^(2 
*e))*(6*a*b*d*f*(c + d*x) + 3*a^2*f^2*(c + d*x)^2 + b^2*(c^2*f^2 + 2*c*d*f 
^2*x + d^2*(1 + f^2*x^2)))*Log[1 + E^(2*(e + f*x))] + 6*d*(1 + E^(2*e))*(3 
*a*b*d + 3*a^2*f*(c + d*x) + b^2*f*(c + d*x))*PolyLog[2, -E^(2*(e + f*x))] 
 - 3*(3*a^2 + b^2)*d^2*(1 + E^(2*e))*PolyLog[3, -E^(2*(e + f*x))]))/(6*(1 
+ E^(2*e))*f^3) + (Sech[e]*Sech[e + f*x]^2*(6*b^3*c^2*f*Cosh[e] + 12*b^3*c 
*d*f*x*Cosh[e] + 6*a^3*c^2*f^2*x*Cosh[e] + 18*a*b^2*c^2*f^2*x*Cosh[e] + 6* 
b^3*d^2*f*x^2*Cosh[e] + 6*a^3*c*d*f^2*x^2*Cosh[e] + 18*a*b^2*c*d*f^2*x^2*C 
osh[e] + 2*a^3*d^2*f^2*x^3*Cosh[e] + 6*a*b^2*d^2*f^2*x^3*Cosh[e] + 3*a^3*c 
^2*f^2*x*Cosh[e + 2*f*x] + 9*a*b^2*c^2*f^2*x*Cosh[e + 2*f*x] + 3*a^3*c*d*f 
^2*x^2*Cosh[e + 2*f*x] + 9*a*b^2*c*d*f^2*x^2*Cosh[e + 2*f*x] + a^3*d^2*f^2 
*x^3*Cosh[e + 2*f*x] + 3*a*b^2*d^2*f^2*x^3*Cosh[e + 2*f*x] + 3*a^3*c^2*f^2 
*x*Cosh[3*e + 2*f*x] + 9*a*b^2*c^2*f^2*x*Cosh[3*e + 2*f*x] + 3*a^3*c*d*f^2 
*x^2*Cosh[3*e + 2*f*x] + 9*a*b^2*c*d*f^2*x^2*Cosh[3*e + 2*f*x] + a^3*d^2*f 
^2*x^3*Cosh[3*e + 2*f*x] + 3*a*b^2*d^2*f^2*x^3*Cosh[3*e + 2*f*x] + 6*b^3*c 
*d*Sinh[e] + 18*a*b^2*c^2*f*Sinh[e] + 6*b^3*d^2*x*Sinh[e] + 36*a*b^2*c*d*f 
*x*Sinh[e] + 18*a^2*b*c^2*f^2*x*Sinh[e] + 6*b^3*c^2*f^2*x*Sinh[e] + 18*a*b 
^2*d^2*f*x^2*Sinh[e] + 18*a^2*b*c*d*f^2*x^2*Sinh[e] + 6*b^3*c*d*f^2*x^2*Si 
nh[e] + 6*a^2*b*d^2*f^2*x^3*Sinh[e] + 2*b^3*d^2*f^2*x^3*Sinh[e] - 6*b^3...
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4205, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^2 (a-i b \tan (i e+i f x))^3dx\)

\(\Big \downarrow \) 4205

\(\displaystyle \int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \tanh (e+f x)+3 a b^2 (c+d x)^2 \tanh ^2(e+f x)+b^3 (c+d x)^2 \tanh ^3(e+f x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {3 a b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a b^2 (c+d x)^3}{d}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tanh (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b^3 (c+d x)^2 \tanh ^2(e+f x)}{2 f}+\frac {b^3 (c+d x)^2}{2 f}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \log (\cosh (e+f x))}{f^3}\)

Input:

Int[(c + d*x)^2*(a + b*Tanh[e + f*x])^3,x]
 

Output:

(-3*a*b^2*(c + d*x)^2)/f + (b^3*(c + d*x)^2)/(2*f) + (a^3*(c + d*x)^3)/(3* 
d) - (a^2*b*(c + d*x)^3)/d + (a*b^2*(c + d*x)^3)/d - (b^3*(c + d*x)^3)/(3* 
d) + (6*a*b^2*d*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f^2 + (3*a^2*b*(c + d* 
x)^2*Log[1 + E^(2*(e + f*x))])/f + (b^3*(c + d*x)^2*Log[1 + E^(2*(e + f*x) 
)])/f + (b^3*d^2*Log[Cosh[e + f*x]])/f^3 + (3*a*b^2*d^2*PolyLog[2, -E^(2*( 
e + f*x))])/f^3 + (3*a^2*b*d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 + 
 (b^3*d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (3*a^2*b*d^2*PolyLog 
[3, -E^(2*(e + f*x))])/(2*f^3) - (b^3*d^2*PolyLog[3, -E^(2*(e + f*x))])/(2 
*f^3) - (b^3*d*(c + d*x)*Tanh[e + f*x])/f^2 - (3*a*b^2*(c + d*x)^2*Tanh[e 
+ f*x])/f - (b^3*(c + d*x)^2*Tanh[e + f*x]^2)/(2*f)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4205

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1065\) vs. \(2(384)=768\).

Time = 1.50 (sec) , antiderivative size = 1066, normalized size of antiderivative = 2.69

method	result	size
risch	\(\text {Expression too large to display}\)	\(1066\)

Input:

int((d*x+c)^2*(a+b*tanh(f*x+e))^3,x,method=_RETURNVERBOSE)
 

Output:

3*a*b^2*d^2*polylog(2,-exp(2*f*x+2*e))/f^3-3/2*a^2*b*d^2*polylog(3,-exp(2* 
f*x+2*e))/f^3-1/2*b^3*d^2*polylog(3,-exp(2*f*x+2*e))/f^3-3*d*a^2*b*c*x^2+3 
*d*a*b^2*c*x^2+3*a^2*b*c^2*x+3*a*b^2*c^2*x-4/f*b^3*c*d*e*x-d^2*a^2*b*x^3+d 
^2*a*b^2*x^3+d*a^3*c*x^2-d*b^3*c*x^2+a^3*c^2*x+b^3*c^2*x+1/d*a^2*b*c^3+1/d 
*a*b^2*c^3-12/f^2*b^2*a*d^2*e*x+6/f^2*b*d^2*a^2*e^2*x-6/f^2*b*d*c*a^2*e^2- 
6/f^3*b*e^2*d^2*a^2*ln(exp(f*x+e))-12/f^2*b^2*a*c*d*ln(exp(f*x+e))+6/f^2*b 
^2*a*c*d*ln(1+exp(2*f*x+2*e))+4/f^2*b^3*e*c*d*ln(exp(f*x+e))+2/f*b^3*c*d*l 
n(1+exp(2*f*x+2*e))*x+6/f^2*b^2*a*d^2*ln(1+exp(2*f*x+2*e))*x+3/f*b*d^2*a^2 
*ln(1+exp(2*f*x+2*e))*x^2+3/f^2*b*d^2*a^2*polylog(2,-exp(2*f*x+2*e))*x+12/ 
f^3*b^2*e*a*d^2*ln(exp(f*x+e))+3/f^2*b*d*c*a^2*polylog(2,-exp(2*f*x+2*e))+ 
4/3/f^3*b^3*d^2*e^3-2/f*b^3*c^2*ln(exp(f*x+e))+1/f*b^3*c^2*ln(1+exp(2*f*x+ 
2*e))-2/f^3*b^3*d^2*ln(exp(f*x+e))+1/f^3*b^3*d^2*ln(1+exp(2*f*x+2*e))+2*b^ 
2*(3*a*d^2*f*x^2*exp(2*f*x+2*e)+b*d^2*f*x^2*exp(2*f*x+2*e)+6*a*c*d*f*x*exp 
(2*f*x+2*e)+2*b*c*d*f*x*exp(2*f*x+2*e)+3*a*c^2*f*exp(2*f*x+2*e)+3*a*d^2*f* 
x^2+b*c^2*f*exp(2*f*x+2*e)+b*d^2*x*exp(2*f*x+2*e)+6*a*c*d*f*x+b*c*d*exp(2* 
f*x+2*e)+3*a*c^2*f+b*d^2*x+b*c*d)/f^2/(1+exp(2*f*x+2*e))^2-12/f*b*d*c*a^2* 
e*x+12/f^2*b*e*d*c*a^2*ln(exp(f*x+e))+6/f*b*d*c*a^2*ln(1+exp(2*f*x+2*e))*x 
+1/3*d^2*a^3*x^3-1/3*d^2*b^3*x^3+1/3/d*a^3*c^3+1/3/d*b^3*c^3-6/f*b^2*a*d^2 
*x^2-2/f^2*b^3*c*d*e^2+1/f^2*b^3*d^2*polylog(2,-exp(2*f*x+2*e))*x+1/f^2*b^ 
3*c*d*polylog(2,-exp(2*f*x+2*e))-6/f*b*a^2*c^2*ln(exp(f*x+e))+3/f*b*a^2...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 7298, normalized size of antiderivative = 18.43 \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^2*(a+b*tanh(f*x+e))^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\int \left (a + b \tanh {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{2}\, dx \] Input:

integrate((d*x+c)**2*(a+b*tanh(f*x+e))**3,x)
 

Output:

Integral((a + b*tanh(e + f*x))**3*(c + d*x)**2, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 869 vs. \(2 (381) = 762\).

Time = 0.24 (sec) , antiderivative size = 869, normalized size of antiderivative = 2.19 \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^2*(a+b*tanh(f*x+e))^3,x, algorithm="maxima")
 

Output:

1/3*a^3*d^2*x^3 + a^3*c*d*x^2 + b^3*c^2*(x + e/f + log(e^(-2*f*x - 2*e) + 
1)/f + 2*e^(-2*f*x - 2*e)/(f*(2*e^(-2*f*x - 2*e) + e^(-4*f*x - 4*e) + 1))) 
 + a^3*c^2*x + 3*a^2*b*c^2*log(cosh(f*x + e))/f + 1/3*(18*a*b^2*c^2*f + 6* 
b^3*c*d + (3*a^2*b*d^2*f^2 + 3*a*b^2*d^2*f^2 + b^3*d^2*f^2)*x^3 + 3*(3*a^2 
*b*c*d*f^2 + b^3*c*d*f^2 + 3*(c*d*f^2 + 2*d^2*f)*a*b^2)*x^2 + 3*(2*b^3*d^2 
 + 3*(c^2*f^2 + 4*c*d*f)*a*b^2)*x + (9*a*b^2*c^2*f^2*x*e^(4*e) + (3*a^2*b* 
d^2*f^2*e^(4*e) + 3*a*b^2*d^2*f^2*e^(4*e) + b^3*d^2*f^2*e^(4*e))*x^3 + 3*( 
3*a^2*b*c*d*f^2*e^(4*e) + 3*a*b^2*c*d*f^2*e^(4*e) + b^3*c*d*f^2*e^(4*e))*x 
^2)*e^(4*f*x) + 2*(9*a*b^2*c^2*f*e^(2*e) + 3*b^3*c*d*e^(2*e) + (3*a^2*b*d^ 
2*f^2*e^(2*e) + 3*a*b^2*d^2*f^2*e^(2*e) + b^3*d^2*f^2*e^(2*e))*x^3 + 3*(3* 
a^2*b*c*d*f^2*e^(2*e) + 3*(c*d*f^2*e^(2*e) + d^2*f*e^(2*e))*a*b^2 + (c*d*f 
^2*e^(2*e) + d^2*f*e^(2*e))*b^3)*x^2 + 3*(3*(c^2*f^2*e^(2*e) + 2*c*d*f*e^( 
2*e))*a*b^2 + (2*c*d*f*e^(2*e) + d^2*e^(2*e))*b^3)*x)*e^(2*f*x))/(f^2*e^(4 
*f*x + 4*e) + 2*f^2*e^(2*f*x + 2*e) + f^2) - 2*(6*a*b^2*c*d*f + b^3*d^2)*x 
/f^2 + 1/2*(3*a^2*b*d^2 + b^3*d^2)*(2*f^2*x^2*log(e^(2*f*x + 2*e) + 1) + 2 
*f*x*dilog(-e^(2*f*x + 2*e)) - polylog(3, -e^(2*f*x + 2*e)))/f^3 + (3*a^2* 
b*c*d*f + b^3*c*d*f + 3*a*b^2*d^2)*(2*f*x*log(e^(2*f*x + 2*e) + 1) + dilog 
(-e^(2*f*x + 2*e)))/f^3 + (6*a*b^2*c*d*f + b^3*d^2)*log(e^(2*f*x + 2*e) + 
1)/f^3 - 2/3*((3*a^2*b*d^2 + b^3*d^2)*f^3*x^3 + 3*(3*a^2*b*c*d*f + b^3*c*d 
*f + 3*a*b^2*d^2)*f^2*x^2)/f^3
 

Giac [F]

\[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\int { {\left (d x + c\right )}^{2} {\left (b \tanh \left (f x + e\right ) + a\right )}^{3} \,d x } \] Input:

integrate((d*x+c)^2*(a+b*tanh(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^2*(b*tanh(f*x + e) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\int {\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \] Input:

int((a + b*tanh(e + f*x))^3*(c + d*x)^2,x)
 

Output:

int((a + b*tanh(e + f*x))^3*(c + d*x)^2, x)
 

Reduce [F]

\[ \int (c+d x)^2 (a+b \tanh (e+f x))^3 \, dx=\text {too large to display} \] Input:

int((d*x+c)^2*(a+b*tanh(f*x+e))^3,x)
 

Output:

( - 288*e**(4*e + 4*f*x)*int(x**2/(e**(6*e + 6*f*x) + 3*e**(4*e + 4*f*x) + 
 3*e**(2*e + 2*f*x) + 1),x)*a**2*b*d**2*f**3 - 96*e**(4*e + 4*f*x)*int(x** 
2/(e**(6*e + 6*f*x) + 3*e**(4*e + 4*f*x) + 3*e**(2*e + 2*f*x) + 1),x)*b**3 
*d**2*f**3 - 576*e**(4*e + 4*f*x)*int(x/(e**(6*e + 6*f*x) + 3*e**(4*e + 4* 
f*x) + 3*e**(2*e + 2*f*x) + 1),x)*a**2*b*c*d*f**3 - 432*e**(4*e + 4*f*x)*i 
nt(x/(e**(6*e + 6*f*x) + 3*e**(4*e + 4*f*x) + 3*e**(2*e + 2*f*x) + 1),x)*a 
**2*b*d**2*f**2 - 576*e**(4*e + 4*f*x)*int(x/(e**(6*e + 6*f*x) + 3*e**(4*e 
 + 4*f*x) + 3*e**(2*e + 2*f*x) + 1),x)*a*b**2*d**2*f**2 - 192*e**(4*e + 4* 
f*x)*int(x/(e**(6*e + 6*f*x) + 3*e**(4*e + 4*f*x) + 3*e**(2*e + 2*f*x) + 1 
),x)*b**3*c*d*f**3 - 144*e**(4*e + 4*f*x)*int(x/(e**(6*e + 6*f*x) + 3*e**( 
4*e + 4*f*x) + 3*e**(2*e + 2*f*x) + 1),x)*b**3*d**2*f**2 + 144*e**(4*e + 4 
*f*x)*log(e**(2*e + 2*f*x) + 1)*a**2*b*c**2*f**2 + 216*e**(4*e + 4*f*x)*lo 
g(e**(2*e + 2*f*x) + 1)*a**2*b*c*d*f + 126*e**(4*e + 4*f*x)*log(e**(2*e + 
2*f*x) + 1)*a**2*b*d**2 + 288*e**(4*e + 4*f*x)*log(e**(2*e + 2*f*x) + 1)*a 
*b**2*c*d*f + 216*e**(4*e + 4*f*x)*log(e**(2*e + 2*f*x) + 1)*a*b**2*d**2 + 
 48*e**(4*e + 4*f*x)*log(e**(2*e + 2*f*x) + 1)*b**3*c**2*f**2 + 72*e**(4*e 
 + 4*f*x)*log(e**(2*e + 2*f*x) + 1)*b**3*c*d*f + 90*e**(4*e + 4*f*x)*log(e 
**(2*e + 2*f*x) + 1)*b**3*d**2 + 48*e**(4*e + 4*f*x)*a**3*c**2*f**3*x + 48 
*e**(4*e + 4*f*x)*a**3*c*d*f**3*x**2 + 16*e**(4*e + 4*f*x)*a**3*d**2*f**3* 
x**3 - 144*e**(4*e + 4*f*x)*a**2*b*c**2*f**3*x + 144*e**(4*e + 4*f*x)*a...