\(\int \frac {(a+b x^2)^{3/2}}{x^4 (c+d x)^4} \, dx\) [1112]

Optimal result

Integrand size = 22, antiderivative size = 362 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx=-\frac {a \sqrt {a+b x^2}}{3 c^4 x^3}+\frac {2 a d \sqrt {a+b x^2}}{c^5 x^2}-\frac {2 \left (2 b c^2+15 a d^2\right ) \sqrt {a+b x^2}}{3 c^6 x}-\frac {d \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}{3 c^4 (c+d x)^3}-\frac {d \left (5 b c^2+12 a d^2\right ) \sqrt {a+b x^2}}{6 c^5 (c+d x)^2}-\frac {d \left (11 b^2 c^4+68 a b c^2 d^2+60 a^2 d^4\right ) \sqrt {a+b x^2}}{6 c^6 \left (b c^2+a d^2\right ) (c+d x)}-\frac {\left (2 b^3 c^6+33 a b^2 c^4 d^2+72 a^2 b c^2 d^4+40 a^3 d^6\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 c^7 \left (b c^2+a d^2\right )^{3/2}}+\frac {2 \sqrt {a} d \left (3 b c^2+10 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^7} \] Output:

-1/3*a*(b*x^2+a)^(1/2)/c^4/x^3+2*a*d*(b*x^2+a)^(1/2)/c^5/x^2-2/3*(15*a*d^2 
+2*b*c^2)*(b*x^2+a)^(1/2)/c^6/x-1/3*d*(a*d^2+b*c^2)*(b*x^2+a)^(1/2)/c^4/(d 
*x+c)^3-1/6*d*(12*a*d^2+5*b*c^2)*(b*x^2+a)^(1/2)/c^5/(d*x+c)^2-1/6*d*(60*a 
^2*d^4+68*a*b*c^2*d^2+11*b^2*c^4)*(b*x^2+a)^(1/2)/c^6/(a*d^2+b*c^2)/(d*x+c 
)-1/2*(40*a^3*d^6+72*a^2*b*c^2*d^4+33*a*b^2*c^4*d^2+2*b^3*c^6)*arctanh((-b 
*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/c^7/(a*d^2+b*c^2)^(3/2)+2*a 
^(1/2)*d*(10*a*d^2+3*b*c^2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/c^7
 

Mathematica [A] (verified)

Time = 10.87 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx=-\frac {c \sqrt {a+b x^2} \left (\frac {2 a c^2}{x^3}-\frac {12 a c d}{x^2}+\frac {8 b c^2+60 a d^2}{x}+\frac {2 c^2 d \left (b c^2+a d^2\right )}{(c+d x)^3}+\frac {c d \left (5 b c^2+12 a d^2\right )}{(c+d x)^2}+\frac {d \left (11 b^2 c^4+68 a b c^2 d^2+60 a^2 d^4\right )}{\left (b c^2+a d^2\right ) (c+d x)}\right )+12 \sqrt {a} d \left (3 b c^2+10 a d^2\right ) \log (x)-\frac {3 \left (2 b^3 c^6+33 a b^2 c^4 d^2+72 a^2 b c^2 d^4+40 a^3 d^6\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{3/2}}-12 \sqrt {a} d \left (3 b c^2+10 a d^2\right ) \log \left (a+\sqrt {a} \sqrt {a+b x^2}\right )+\frac {3 \left (2 b^3 c^6+33 a b^2 c^4 d^2+72 a^2 b c^2 d^4+40 a^3 d^6\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{3/2}}}{6 c^7} \] Input:

Integrate[(a + b*x^2)^(3/2)/(x^4*(c + d*x)^4),x]
 

Output:

-1/6*(c*Sqrt[a + b*x^2]*((2*a*c^2)/x^3 - (12*a*c*d)/x^2 + (8*b*c^2 + 60*a* 
d^2)/x + (2*c^2*d*(b*c^2 + a*d^2))/(c + d*x)^3 + (c*d*(5*b*c^2 + 12*a*d^2) 
)/(c + d*x)^2 + (d*(11*b^2*c^4 + 68*a*b*c^2*d^2 + 60*a^2*d^4))/((b*c^2 + a 
*d^2)*(c + d*x))) + 12*Sqrt[a]*d*(3*b*c^2 + 10*a*d^2)*Log[x] - (3*(2*b^3*c 
^6 + 33*a*b^2*c^4*d^2 + 72*a^2*b*c^2*d^4 + 40*a^3*d^6)*Log[c + d*x])/(b*c^ 
2 + a*d^2)^(3/2) - 12*Sqrt[a]*d*(3*b*c^2 + 10*a*d^2)*Log[a + Sqrt[a]*Sqrt[ 
a + b*x^2]] + (3*(2*b^3*c^6 + 33*a*b^2*c^4*d^2 + 72*a^2*b*c^2*d^4 + 40*a^3 
*d^6)*Log[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(b*c^2 + a*d 
^2)^(3/2))/c^7
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(855\) vs. \(2(362)=724\).

Time = 2.11 (sec) , antiderivative size = 855, normalized size of antiderivative = 2.36, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx\)

\(\Big \downarrow \) 617

\(\displaystyle \int \left (\frac {20 d^4 \left (a+b x^2\right )^{3/2}}{c^7 (c+d x)}-\frac {20 d^3 \left (a+b x^2\right )^{3/2}}{c^7 x}+\frac {10 d^4 \left (a+b x^2\right )^{3/2}}{c^6 (c+d x)^2}+\frac {10 d^2 \left (a+b x^2\right )^{3/2}}{c^6 x^2}+\frac {4 d^4 \left (a+b x^2\right )^{3/2}}{c^5 (c+d x)^3}-\frac {4 d \left (a+b x^2\right )^{3/2}}{c^5 x^3}+\frac {d^4 \left (a+b x^2\right )^{3/2}}{c^4 (c+d x)^4}+\frac {\left (a+b x^2\right )^{3/2}}{c^4 x^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {10 \left (b x^2+a\right )^{3/2} d^3}{c^6 (c+d x)}-\frac {2 \left (b x^2+a\right )^{3/2} d^3}{c^5 (c+d x)^2}-\frac {\left (b x^2+a\right )^{3/2} d^3}{3 c^4 (c+d x)^3}+\frac {20 a^{3/2} \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d^3}{c^7}-\frac {20 a \sqrt {b x^2+a} d^3}{c^7}-\frac {10 \left (b x^2+a\right )^{3/2} d^2}{c^6 x}+\frac {15 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^2}{c^6}+\frac {15 b x \sqrt {b x^2+a} d^2}{c^6}+\frac {2 \left (b x^2+a\right )^{3/2} d}{c^5 x^2}+\frac {6 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d}{c^5}-\frac {15 b (2 c-d x) \sqrt {b x^2+a} d}{c^6}+\frac {6 b (2 c+d x) \sqrt {b x^2+a} d}{c^5 (c+d x)}+\frac {10 \left (2 \left (b c^2+a d^2\right )-b c d x\right ) \sqrt {b x^2+a} d}{c^7}-\frac {b \left (c \left (2 b c^2+a d^2\right )+d \left (3 b c^2+2 a d^2\right ) x\right ) \sqrt {b x^2+a} d}{2 c^4 \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {6 b \sqrt {b x^2+a} d}{c^5}-\frac {\left (b x^2+a\right )^{3/2}}{3 c^4 x^3}+\frac {15 \sqrt {b} \left (2 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{c^6}-\frac {10 \sqrt {b} \left (2 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{c^6}-\frac {10 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{c^4}-\frac {20 \left (b c^2+a d^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right )}{c^7}-\frac {6 b \left (2 b c^2+a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right )}{c^5 \sqrt {b c^2+a d^2}}+\frac {b^2 \left (2 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right )}{2 c^3 \left (b c^2+a d^2\right )^{3/2}}+\frac {30 b \sqrt {b c^2+a d^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right )}{c^5}-\frac {b \sqrt {b x^2+a}}{c^4 x}\)

Input:

Int[(a + b*x^2)^(3/2)/(x^4*(c + d*x)^4),x]
 

Output:

(-6*b*d*Sqrt[a + b*x^2])/c^5 - (20*a*d^3*Sqrt[a + b*x^2])/c^7 - (b*Sqrt[a 
+ b*x^2])/(c^4*x) + (15*b*d^2*x*Sqrt[a + b*x^2])/c^6 - (15*b*d*(2*c - d*x) 
*Sqrt[a + b*x^2])/c^6 + (6*b*d*(2*c + d*x)*Sqrt[a + b*x^2])/(c^5*(c + d*x) 
) + (10*d*(2*(b*c^2 + a*d^2) - b*c*d*x)*Sqrt[a + b*x^2])/c^7 - (b*d*(c*(2* 
b*c^2 + a*d^2) + d*(3*b*c^2 + 2*a*d^2)*x)*Sqrt[a + b*x^2])/(2*c^4*(b*c^2 + 
 a*d^2)*(c + d*x)^2) - (a + b*x^2)^(3/2)/(3*c^4*x^3) + (2*d*(a + b*x^2)^(3 
/2))/(c^5*x^2) - (10*d^2*(a + b*x^2)^(3/2))/(c^6*x) - (d^3*(a + b*x^2)^(3/ 
2))/(3*c^4*(c + d*x)^3) - (2*d^3*(a + b*x^2)^(3/2))/(c^5*(c + d*x)^2) - (1 
0*d^3*(a + b*x^2)^(3/2))/(c^6*(c + d*x)) - (10*b^(3/2)*ArcTanh[(Sqrt[b]*x) 
/Sqrt[a + b*x^2]])/c^4 + (15*a*Sqrt[b]*d^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b* 
x^2]])/c^6 + (15*Sqrt[b]*(2*b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b* 
x^2]])/c^6 - (10*Sqrt[b]*(2*b*c^2 + 3*a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + 
b*x^2]])/c^6 + (30*b*Sqrt[b*c^2 + a*d^2]*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 
 + a*d^2]*Sqrt[a + b*x^2])])/c^5 - (20*(b*c^2 + a*d^2)^(3/2)*ArcTanh[(a*d 
- b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/c^7 - (6*b*(2*b*c^2 + a*d 
^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c^5*Sqr 
t[b*c^2 + a*d^2]) + (b^2*(2*b*c^2 + 3*a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b 
*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(2*c^3*(b*c^2 + a*d^2)^(3/2)) + (6*Sqrt[a 
]*b*d*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/c^5 + (20*a^(3/2)*d^3*ArcTanh[Sqrt 
[a + b*x^2]/Sqrt[a]])/c^7
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1711\) vs. \(2(326)=652\).

Time = 0.62 (sec) , antiderivative size = 1712, normalized size of antiderivative = 4.73

method	result	size
risch	\(\text {Expression too large to display}\)	\(1712\)
default	\(\text {Expression too large to display}\)	\(5621\)

Input:

int((b*x^2+a)^(3/2)/x^4/(d*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*(b*x^2+a)^(1/2)*(30*a*d^2*x^2+4*b*c^2*x^2-6*a*c*d*x+a*c^2)/c^6/x^3+1/ 
c^6*(c^2*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/d^4*(-1/3/(a*d^2+b*c^2)*d^2/(x+c/ 
d)^3*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+5/3*b*c*d/(a*d^ 
2+b*c^2)*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a 
*d^2+b*c^2)/d^2)^(1/2)+3/2*b*c*d/(a*d^2+b*c^2)*(-1/(a*d^2+b*c^2)*d^2/(x+c/ 
d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^ 
2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a 
*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/ 
2))/(x+c/d)))+1/2*b/(a*d^2+b*c^2)*d^2/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d 
^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b 
*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))-2/3*b/(a*d^2+b*c^2)*d^2*( 
-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^ 
2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2) 
/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c 
/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d))))+2*a*(5*a*d^2+3*b*c^2)*(-1/(a*d^2+ 
b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b 
*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c 
/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2 
+b*c^2)/d^2)^(1/2))/(x+c/d)))+2*a^(1/2)/c*d*(10*a*d^2+3*b*c^2)*ln((2*a+2*a 
^(1/2)*(b*x^2+a)^(1/2))/x)+4*c/d*a*(a*d^2+b*c^2)*(-1/2/(a*d^2+b*c^2)*d^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 929 vs. \(2 (327) = 654\).

Time = 1.88 (sec) , antiderivative size = 3785, normalized size of antiderivative = 10.46 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)/x^4/(d*x+c)^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x^{4} \left (c + d x\right )^{4}}\, dx \] Input:

integrate((b*x**2+a)**(3/2)/x**4/(d*x+c)**4,x)
                                                                                    
                                                                                    
 

Output:

Integral((a + b*x**2)**(3/2)/(x**4*(c + d*x)**4), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{4} x^{4}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/x^4/(d*x+c)^4,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x + c)^4*x^4), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(3/2)/x^4/(d*x+c)^4,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^4\,{\left (c+d\,x\right )}^4} \,d x \] Input:

int((a + b*x^2)^(3/2)/(x^4*(c + d*x)^4),x)
 

Output:

int((a + b*x^2)^(3/2)/(x^4*(c + d*x)^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 2966, normalized size of antiderivative = 8.19 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^4} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(3/2)/x^4/(d*x+c)^4,x)
 

Output:

(120*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a* 
d + b*c*x)*a**3*c**3*d**6*x**3 + 360*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b* 
x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*c**2*d**7*x**4 + 360*sqrt( 
a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x) 
*a**3*c*d**8*x**5 + 120*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a* 
d**2 + b*c**2) - a*d + b*c*x)*a**3*d**9*x**6 + 216*sqrt(a*d**2 + b*c**2)*l 
og(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c**5*d**4* 
x**3 + 648*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2 
) - a*d + b*c*x)*a**2*b*c**4*d**5*x**4 + 648*sqrt(a*d**2 + b*c**2)*log(sqr 
t(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c**3*d**6*x**5 + 
 216*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a* 
d + b*c*x)*a**2*b*c**2*d**7*x**6 + 99*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b 
*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**7*d**2*x**3 + 297*sq 
rt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c 
*x)*a*b**2*c**6*d**3*x**4 + 297*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2) 
*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**5*d**4*x**5 + 99*sqrt(a*d* 
*2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b 
**2*c**4*d**5*x**6 + 6*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d 
**2 + b*c**2) - a*d + b*c*x)*b**3*c**9*x**3 + 18*sqrt(a*d**2 + b*c**2)*log 
(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**3*c**8*d*x**4...