3.1537 \(\int \frac{1}{(a-b x) (a+b x) (c+d x)^3} \, dx\)
Optimal. Leaf size=161 \[ \frac{2 b^2 c d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )^2}+\frac{d}{2 (c+d x)^2 \left (b^2 c^2-a^2 d^2\right )}-\frac{b^2 d \left (a^2 d^2+3 b^2 c^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac{b^2 \log (a-b x)}{2 a (a d+b c)^3}+\frac{b^2 \log (a+b x)}{2 a (b c-a d)^3} \]
[Out]
d/(2*(b^2*c^2 - a^2*d^2)*(c + d*x)^2) + (2*b^2*c*d)/((b^2*c^2 - a^2*d^2)^2*(c + d*x)) - (b^2*Log[a - b*x])/(2*
a*(b*c + a*d)^3) + (b^2*Log[a + b*x])/(2*a*(b*c - a*d)^3) - (b^2*d*(3*b^2*c^2 + a^2*d^2)*Log[c + d*x])/(b^2*c^
2 - a^2*d^2)^3
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Rubi [A] time = 0.152564, antiderivative size = 161, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used =
{72} \[ \frac{2 b^2 c d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )^2}+\frac{d}{2 (c+d x)^2 \left (b^2 c^2-a^2 d^2\right )}-\frac{b^2 d \left (a^2 d^2+3 b^2 c^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac{b^2 \log (a-b x)}{2 a (a d+b c)^3}+\frac{b^2 \log (a+b x)}{2 a (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
Int[1/((a - b*x)*(a + b*x)*(c + d*x)^3),x]
[Out]
d/(2*(b^2*c^2 - a^2*d^2)*(c + d*x)^2) + (2*b^2*c*d)/((b^2*c^2 - a^2*d^2)^2*(c + d*x)) - (b^2*Log[a - b*x])/(2*
a*(b*c + a*d)^3) + (b^2*Log[a + b*x])/(2*a*(b*c - a*d)^3) - (b^2*d*(3*b^2*c^2 + a^2*d^2)*Log[c + d*x])/(b^2*c^
2 - a^2*d^2)^3
Rule 72
Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Rubi steps
\begin{align*} \int \frac{1}{(a-b x) (a+b x) (c+d x)^3} \, dx &=\int \left (\frac{b^3}{2 a (b c+a d)^3 (a-b x)}-\frac{b^3}{2 a (-b c+a d)^3 (a+b x)}-\frac{d^2}{\left (b^2 c^2-a^2 d^2\right ) (c+d x)^3}-\frac{2 b^2 c d^2}{\left (b^2 c^2-a^2 d^2\right )^2 (c+d x)^2}-\frac{d^2 \left (3 b^4 c^2+a^2 b^2 d^2\right )}{\left (b^2 c^2-a^2 d^2\right )^3 (c+d x)}\right ) \, dx\\ &=\frac{d}{2 \left (b^2 c^2-a^2 d^2\right ) (c+d x)^2}+\frac{2 b^2 c d}{\left (b^2 c^2-a^2 d^2\right )^2 (c+d x)}-\frac{b^2 \log (a-b x)}{2 a (b c+a d)^3}+\frac{b^2 \log (a+b x)}{2 a (b c-a d)^3}-\frac{b^2 d \left (3 b^2 c^2+a^2 d^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.292268, size = 147, normalized size = 0.91 \[ \frac{1}{2} \left (\frac{d \left (\frac{\left (b^2 c^2-a^2 d^2\right ) \left (b^2 c (5 c+4 d x)-a^2 d^2\right )}{(c+d x)^2}-2 \left (a^2 b^2 d^2+3 b^4 c^2\right ) \log (c+d x)\right )}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac{b^2 \log (a-b x)}{a (a d+b c)^3}-\frac{b^2 \log (a+b x)}{a (a d-b c)^3}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a - b*x)*(a + b*x)*(c + d*x)^3),x]
[Out]
(-((b^2*Log[a - b*x])/(a*(b*c + a*d)^3)) - (b^2*Log[a + b*x])/(a*(-(b*c) + a*d)^3) + (d*(((b^2*c^2 - a^2*d^2)*
(-(a^2*d^2) + b^2*c*(5*c + 4*d*x)))/(c + d*x)^2 - 2*(3*b^4*c^2 + a^2*b^2*d^2)*Log[c + d*x]))/(b^2*c^2 - a^2*d^
2)^3)/2
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Maple [A] time = 0.011, size = 182, normalized size = 1.1 \begin{align*} -{\frac{d}{ \left ( 2\,ad+2\,bc \right ) \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}+{\frac{{d}^{3}{b}^{2}\ln \left ( dx+c \right ){a}^{2}}{ \left ( ad+bc \right ) ^{3} \left ( ad-bc \right ) ^{3}}}+3\,{\frac{d{b}^{4}\ln \left ( dx+c \right ){c}^{2}}{ \left ( ad+bc \right ) ^{3} \left ( ad-bc \right ) ^{3}}}+2\,{\frac{{b}^{2}dc}{ \left ( ad+bc \right ) ^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-{\frac{{b}^{2}\ln \left ( bx+a \right ) }{2\,a \left ( ad-bc \right ) ^{3}}}-{\frac{{b}^{2}\ln \left ( bx-a \right ) }{2\, \left ( ad+bc \right ) ^{3}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(-b*x+a)/(b*x+a)/(d*x+c)^3,x)
[Out]
-1/2*d/(a*d+b*c)/(a*d-b*c)/(d*x+c)^2+d^3*b^2/(a*d+b*c)^3/(a*d-b*c)^3*ln(d*x+c)*a^2+3*d*b^4/(a*d+b*c)^3/(a*d-b*
c)^3*ln(d*x+c)*c^2+2*d*b^2*c/(a*d+b*c)^2/(a*d-b*c)^2/(d*x+c)-1/2/a*b^2/(a*d-b*c)^3*ln(b*x+a)-1/2*b^2/(a*d+b*c)
^3/a*ln(b*x-a)
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Maxima [B] time = 1.10842, size = 424, normalized size = 2.63 \begin{align*} \frac{b^{2} \log \left (b x + a\right )}{2 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} - \frac{b^{2} \log \left (b x - a\right )}{2 \,{\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )}} - \frac{{\left (3 \, b^{4} c^{2} d + a^{2} b^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{6} c^{6} - 3 \, a^{2} b^{4} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{4} - a^{6} d^{6}} + \frac{4 \, b^{2} c d^{2} x + 5 \, b^{2} c^{2} d - a^{2} d^{3}}{2 \,{\left (b^{4} c^{6} - 2 \, a^{2} b^{2} c^{4} d^{2} + a^{4} c^{2} d^{4} +{\left (b^{4} c^{4} d^{2} - 2 \, a^{2} b^{2} c^{2} d^{4} + a^{4} d^{6}\right )} x^{2} + 2 \,{\left (b^{4} c^{5} d - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{4} c d^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-b*x+a)/(b*x+a)/(d*x+c)^3,x, algorithm="maxima")
[Out]
1/2*b^2*log(b*x + a)/(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3) - 1/2*b^2*log(b*x - a)/(a*b^3*c^3
+ 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 + a^4*d^3) - (3*b^4*c^2*d + a^2*b^2*d^3)*log(d*x + c)/(b^6*c^6 - 3*a^2*b^4*
c^4*d^2 + 3*a^4*b^2*c^2*d^4 - a^6*d^6) + 1/2*(4*b^2*c*d^2*x + 5*b^2*c^2*d - a^2*d^3)/(b^4*c^6 - 2*a^2*b^2*c^4*
d^2 + a^4*c^2*d^4 + (b^4*c^4*d^2 - 2*a^2*b^2*c^2*d^4 + a^4*d^6)*x^2 + 2*(b^4*c^5*d - 2*a^2*b^2*c^3*d^3 + a^4*c
*d^5)*x)
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Fricas [B] time = 21.8681, size = 1175, normalized size = 7.3 \begin{align*} \frac{5 \, a b^{4} c^{4} d - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5} + 4 \,{\left (a b^{4} c^{3} d^{2} - a^{3} b^{2} c d^{4}\right )} x +{\left (b^{5} c^{5} + 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + a^{3} b^{2} c^{2} d^{3} +{\left (b^{5} c^{3} d^{2} + 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \,{\left (b^{5} c^{4} d + 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x + a\right ) -{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} +{\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \,{\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x - a\right ) - 2 \,{\left (3 \, a b^{4} c^{4} d + a^{3} b^{2} c^{2} d^{3} +{\left (3 \, a b^{4} c^{2} d^{3} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \,{\left (3 \, a b^{4} c^{3} d^{2} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{6} c^{8} - 3 \, a^{3} b^{4} c^{6} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{4} - a^{7} c^{2} d^{6} +{\left (a b^{6} c^{6} d^{2} - 3 \, a^{3} b^{4} c^{4} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{6} - a^{7} d^{8}\right )} x^{2} + 2 \,{\left (a b^{6} c^{7} d - 3 \, a^{3} b^{4} c^{5} d^{3} + 3 \, a^{5} b^{2} c^{3} d^{5} - a^{7} c d^{7}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-b*x+a)/(b*x+a)/(d*x+c)^3,x, algorithm="fricas")
[Out]
1/2*(5*a*b^4*c^4*d - 6*a^3*b^2*c^2*d^3 + a^5*d^5 + 4*(a*b^4*c^3*d^2 - a^3*b^2*c*d^4)*x + (b^5*c^5 + 3*a*b^4*c^
4*d + 3*a^2*b^3*c^3*d^2 + a^3*b^2*c^2*d^3 + (b^5*c^3*d^2 + 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^
2 + 2*(b^5*c^4*d + 3*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4)*x)*log(b*x + a) - (b^5*c^5 - 3*a*b^4*c
^4*d + 3*a^2*b^3*c^3*d^2 - a^3*b^2*c^2*d^3 + (b^5*c^3*d^2 - 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 - a^3*b^2*d^5)*x
^2 + 2*(b^5*c^4*d - 3*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 - a^3*b^2*c*d^4)*x)*log(b*x - a) - 2*(3*a*b^4*c^4*d +
a^3*b^2*c^2*d^3 + (3*a*b^4*c^2*d^3 + a^3*b^2*d^5)*x^2 + 2*(3*a*b^4*c^3*d^2 + a^3*b^2*c*d^4)*x)*log(d*x + c))/(
a*b^6*c^8 - 3*a^3*b^4*c^6*d^2 + 3*a^5*b^2*c^4*d^4 - a^7*c^2*d^6 + (a*b^6*c^6*d^2 - 3*a^3*b^4*c^4*d^4 + 3*a^5*b
^2*c^2*d^6 - a^7*d^8)*x^2 + 2*(a*b^6*c^7*d - 3*a^3*b^4*c^5*d^3 + 3*a^5*b^2*c^3*d^5 - a^7*c*d^7)*x)
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Sympy [B] time = 98.7421, size = 2152, normalized size = 13.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-b*x+a)/(b*x+a)/(d*x+c)**3,x)
[Out]
b**2*d*(a**2*d**2 + 3*b**2*c**2)*log(x + (12*a**18*b**4*d**18*(a**2*d**2 + 3*b**2*c**2)**2/((a*d - b*c)**6*(a*
d + b*c)**6) - 24*a**16*b**6*c**2*d**16*(a**2*d**2 + 3*b**2*c**2)**2/((a*d - b*c)**6*(a*d + b*c)**6) - 104*a**
14*b**8*c**4*d**14*(a**2*d**2 + 3*b**2*c**2)**2/((a*d - b*c)**6*(a*d + b*c)**6) - 6*a**14*b**4*d**14*(a**2*d**
2 + 3*b**2*c**2)/((a*d - b*c)**3*(a*d + b*c)**3) + 456*a**12*b**10*c**6*d**12*(a**2*d**2 + 3*b**2*c**2)**2/((a
*d - b*c)**6*(a*d + b*c)**6) + 12*a**12*b**6*c**2*d**12*(a**2*d**2 + 3*b**2*c**2)/((a*d - b*c)**3*(a*d + b*c)*
*3) - 720*a**10*b**12*c**8*d**10*(a**2*d**2 + 3*b**2*c**2)**2/((a*d - b*c)**6*(a*d + b*c)**6) - 2*a**10*b**8*c
**4*d**10*(a**2*d**2 + 3*b**2*c**2)/((a*d - b*c)**3*(a*d + b*c)**3) - 6*a**10*b**4*d**10 + 568*a**8*b**14*c**1
0*d**8*(a**2*d**2 + 3*b**2*c**2)**2/((a*d - b*c)**6*(a*d + b*c)**6) + 8*a**8*b**10*c**6*d**8*(a**2*d**2 + 3*b*
*2*c**2)/((a*d - b*c)**3*(a*d + b*c)**3) - 69*a**8*b**6*c**2*d**8 - 216*a**6*b**16*c**12*d**6*(a**2*d**2 + 3*b
**2*c**2)**2/((a*d - b*c)**6*(a*d + b*c)**6) - 42*a**6*b**12*c**8*d**6*(a**2*d**2 + 3*b**2*c**2)/((a*d - b*c)*
*3*(a*d + b*c)**3) - 236*a**6*b**8*c**4*d**6 + 24*a**4*b**18*c**14*d**4*(a**2*d**2 + 3*b**2*c**2)**2/((a*d - b
*c)**6*(a*d + b*c)**6) + 44*a**4*b**14*c**10*d**4*(a**2*d**2 + 3*b**2*c**2)/((a*d - b*c)**3*(a*d + b*c)**3) -
194*a**4*b**10*c**6*d**4 + 4*a**2*b**20*c**16*d**2*(a**2*d**2 + 3*b**2*c**2)**2/((a*d - b*c)**6*(a*d + b*c)**6
) - 14*a**2*b**16*c**12*d**2*(a**2*d**2 + 3*b**2*c**2)/((a*d - b*c)**3*(a*d + b*c)**3) - 6*a**2*b**12*c**8*d**
2 - b**14*c**10)/(27*a**8*b**6*c*d**9 + 144*a**6*b**8*c**3*d**7 + 270*a**4*b**10*c**5*d**5 + 72*a**2*b**12*c**
7*d**3 - b**14*c**9*d))/((a*d - b*c)**3*(a*d + b*c)**3) + (-a**2*d**3 + 5*b**2*c**2*d + 4*b**2*c*d**2*x)/(2*a*
*4*c**2*d**4 - 4*a**2*b**2*c**4*d**2 + 2*b**4*c**6 + x**2*(2*a**4*d**6 - 4*a**2*b**2*c**2*d**4 + 2*b**4*c**4*d
**2) + x*(4*a**4*c*d**5 - 8*a**2*b**2*c**3*d**3 + 4*b**4*c**5*d)) - b**2*log(x + (3*a**16*b**4*d**16/(a*d + b*
c)**6 - 6*a**14*b**6*c**2*d**14/(a*d + b*c)**6 + 3*a**13*b**4*d**13/(a*d + b*c)**3 - 26*a**12*b**8*c**4*d**12/
(a*d + b*c)**6 - 6*a**11*b**6*c**2*d**11/(a*d + b*c)**3 + 114*a**10*b**10*c**6*d**10/(a*d + b*c)**6 - 6*a**10*
b**4*d**10 + a**9*b**8*c**4*d**9/(a*d + b*c)**3 - 180*a**8*b**12*c**8*d**8/(a*d + b*c)**6 - 69*a**8*b**6*c**2*
d**8 - 4*a**7*b**10*c**6*d**7/(a*d + b*c)**3 + 142*a**6*b**14*c**10*d**6/(a*d + b*c)**6 - 236*a**6*b**8*c**4*d
**6 + 21*a**5*b**12*c**8*d**5/(a*d + b*c)**3 - 54*a**4*b**16*c**12*d**4/(a*d + b*c)**6 - 194*a**4*b**10*c**6*d
**4 - 22*a**3*b**14*c**10*d**3/(a*d + b*c)**3 + 6*a**2*b**18*c**14*d**2/(a*d + b*c)**6 - 6*a**2*b**12*c**8*d**
2 + 7*a*b**16*c**12*d/(a*d + b*c)**3 + b**20*c**16/(a*d + b*c)**6 - b**14*c**10)/(27*a**8*b**6*c*d**9 + 144*a*
*6*b**8*c**3*d**7 + 270*a**4*b**10*c**5*d**5 + 72*a**2*b**12*c**7*d**3 - b**14*c**9*d))/(2*a*(a*d + b*c)**3) -
b**2*log(x + (3*a**16*b**4*d**16/(a*d - b*c)**6 - 6*a**14*b**6*c**2*d**14/(a*d - b*c)**6 + 3*a**13*b**4*d**13
/(a*d - b*c)**3 - 26*a**12*b**8*c**4*d**12/(a*d - b*c)**6 - 6*a**11*b**6*c**2*d**11/(a*d - b*c)**3 + 114*a**10
*b**10*c**6*d**10/(a*d - b*c)**6 - 6*a**10*b**4*d**10 + a**9*b**8*c**4*d**9/(a*d - b*c)**3 - 180*a**8*b**12*c*
*8*d**8/(a*d - b*c)**6 - 69*a**8*b**6*c**2*d**8 - 4*a**7*b**10*c**6*d**7/(a*d - b*c)**3 + 142*a**6*b**14*c**10
*d**6/(a*d - b*c)**6 - 236*a**6*b**8*c**4*d**6 + 21*a**5*b**12*c**8*d**5/(a*d - b*c)**3 - 54*a**4*b**16*c**12*
d**4/(a*d - b*c)**6 - 194*a**4*b**10*c**6*d**4 - 22*a**3*b**14*c**10*d**3/(a*d - b*c)**3 + 6*a**2*b**18*c**14*
d**2/(a*d - b*c)**6 - 6*a**2*b**12*c**8*d**2 + 7*a*b**16*c**12*d/(a*d - b*c)**3 + b**20*c**16/(a*d - b*c)**6 -
b**14*c**10)/(27*a**8*b**6*c*d**9 + 144*a**6*b**8*c**3*d**7 + 270*a**4*b**10*c**5*d**5 + 72*a**2*b**12*c**7*d
**3 - b**14*c**9*d))/(2*a*(a*d - b*c)**3)
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Giac [A] time = 1.25549, size = 374, normalized size = 2.32 \begin{align*} \frac{b^{3} \log \left ({\left | b x + a \right |}\right )}{2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )}} - \frac{b^{3} \log \left ({\left | b x - a \right |}\right )}{2 \,{\left (a b^{4} c^{3} + 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )}} - \frac{{\left (3 \, b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{6} c^{6} d - 3 \, a^{2} b^{4} c^{4} d^{3} + 3 \, a^{4} b^{2} c^{2} d^{5} - a^{6} d^{7}} + \frac{5 \, b^{4} c^{4} d - 6 \, a^{2} b^{2} c^{2} d^{3} + a^{4} d^{5} + 4 \,{\left (b^{4} c^{3} d^{2} - a^{2} b^{2} c d^{4}\right )} x}{2 \,{\left (b c + a d\right )}^{3}{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-b*x+a)/(b*x+a)/(d*x+c)^3,x, algorithm="giac")
[Out]
1/2*b^3*log(abs(b*x + a))/(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3) - 1/2*b^3*log(abs(b*x -
a))/(a*b^4*c^3 + 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 + a^4*b*d^3) - (3*b^4*c^2*d^2 + a^2*b^2*d^4)*log(abs(d*x +
c))/(b^6*c^6*d - 3*a^2*b^4*c^4*d^3 + 3*a^4*b^2*c^2*d^5 - a^6*d^7) + 1/2*(5*b^4*c^4*d - 6*a^2*b^2*c^2*d^3 + a^4
*d^5 + 4*(b^4*c^3*d^2 - a^2*b^2*c*d^4)*x)/((b*c + a*d)^3*(b*c - a*d)^3*(d*x + c)^2)