### 3.1827 $$\int \frac{1}{(a+b x) (a c+(b c+a d) x+b d x^2)^3} \, dx$$

Optimal. Leaf size=170 $-\frac{6 b^2 d^2}{(a+b x) (b c-a d)^5}-\frac{10 b^2 d^3 \log (a+b x)}{(b c-a d)^6}+\frac{10 b^2 d^3 \log (c+d x)}{(b c-a d)^6}+\frac{3 b^2 d}{2 (a+b x)^2 (b c-a d)^4}-\frac{b^2}{3 (a+b x)^3 (b c-a d)^3}-\frac{4 b d^3}{(c+d x) (b c-a d)^5}-\frac{d^3}{2 (c+d x)^2 (b c-a d)^4}$

[Out]

-b^2/(3*(b*c - a*d)^3*(a + b*x)^3) + (3*b^2*d)/(2*(b*c - a*d)^4*(a + b*x)^2) - (6*b^2*d^2)/((b*c - a*d)^5*(a +
b*x)) - d^3/(2*(b*c - a*d)^4*(c + d*x)^2) - (4*b*d^3)/((b*c - a*d)^5*(c + d*x)) - (10*b^2*d^3*Log[a + b*x])/(
b*c - a*d)^6 + (10*b^2*d^3*Log[c + d*x])/(b*c - a*d)^6

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Rubi [A]  time = 0.159918, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 44} $-\frac{6 b^2 d^2}{(a+b x) (b c-a d)^5}-\frac{10 b^2 d^3 \log (a+b x)}{(b c-a d)^6}+\frac{10 b^2 d^3 \log (c+d x)}{(b c-a d)^6}+\frac{3 b^2 d}{2 (a+b x)^2 (b c-a d)^4}-\frac{b^2}{3 (a+b x)^3 (b c-a d)^3}-\frac{4 b d^3}{(c+d x) (b c-a d)^5}-\frac{d^3}{2 (c+d x)^2 (b c-a d)^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^3),x]

[Out]

-b^2/(3*(b*c - a*d)^3*(a + b*x)^3) + (3*b^2*d)/(2*(b*c - a*d)^4*(a + b*x)^2) - (6*b^2*d^2)/((b*c - a*d)^5*(a +
b*x)) - d^3/(2*(b*c - a*d)^4*(c + d*x)^2) - (4*b*d^3)/((b*c - a*d)^5*(c + d*x)) - (10*b^2*d^3*Log[a + b*x])/(
b*c - a*d)^6 + (10*b^2*d^3*Log[c + d*x])/(b*c - a*d)^6

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+b x) \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^4 (c+d x)^3} \, dx\\ &=\int \left (\frac{b^3}{(b c-a d)^3 (a+b x)^4}-\frac{3 b^3 d}{(b c-a d)^4 (a+b x)^3}+\frac{6 b^3 d^2}{(b c-a d)^5 (a+b x)^2}-\frac{10 b^3 d^3}{(b c-a d)^6 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)^3}+\frac{4 b d^4}{(b c-a d)^5 (c+d x)^2}+\frac{10 b^2 d^4}{(b c-a d)^6 (c+d x)}\right ) \, dx\\ &=-\frac{b^2}{3 (b c-a d)^3 (a+b x)^3}+\frac{3 b^2 d}{2 (b c-a d)^4 (a+b x)^2}-\frac{6 b^2 d^2}{(b c-a d)^5 (a+b x)}-\frac{d^3}{2 (b c-a d)^4 (c+d x)^2}-\frac{4 b d^3}{(b c-a d)^5 (c+d x)}-\frac{10 b^2 d^3 \log (a+b x)}{(b c-a d)^6}+\frac{10 b^2 d^3 \log (c+d x)}{(b c-a d)^6}\\ \end{align*}

Mathematica [A]  time = 0.211589, size = 154, normalized size = 0.91 $-\frac{\frac{36 b^2 d^2 (b c-a d)}{a+b x}-\frac{9 b^2 d (b c-a d)^2}{(a+b x)^2}+\frac{2 b^2 (b c-a d)^3}{(a+b x)^3}+60 b^2 d^3 \log (a+b x)+\frac{24 b d^3 (b c-a d)}{c+d x}+\frac{3 d^3 (b c-a d)^2}{(c+d x)^2}-60 b^2 d^3 \log (c+d x)}{6 (b c-a d)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^3),x]

[Out]

-((2*b^2*(b*c - a*d)^3)/(a + b*x)^3 - (9*b^2*d*(b*c - a*d)^2)/(a + b*x)^2 + (36*b^2*d^2*(b*c - a*d))/(a + b*x)
+ (3*d^3*(b*c - a*d)^2)/(c + d*x)^2 + (24*b*d^3*(b*c - a*d))/(c + d*x) + 60*b^2*d^3*Log[a + b*x] - 60*b^2*d^3
*Log[c + d*x])/(6*(b*c - a*d)^6)

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Maple [A]  time = 0.054, size = 165, normalized size = 1. \begin{align*} -{\frac{{d}^{3}}{2\, \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) ^{2}}}+10\,{\frac{{d}^{3}{b}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{6}}}+4\,{\frac{{d}^{3}b}{ \left ( ad-bc \right ) ^{5} \left ( dx+c \right ) }}+{\frac{{b}^{2}}{3\, \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{3}}}-10\,{\frac{{d}^{3}{b}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{6}}}+6\,{\frac{{b}^{2}{d}^{2}}{ \left ( ad-bc \right ) ^{5} \left ( bx+a \right ) }}+{\frac{3\,{b}^{2}d}{2\, \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)

[Out]

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

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Maxima [B]  time = 1.35181, size = 1200, normalized size = 7.06 \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)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")

[Out]

-10*b^2*d^3*log(b*x + a)/(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d
^4 - 6*a^5*b*c*d^5 + a^6*d^6) + 10*b^2*d^3*log(d*x + c)/(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3
*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6) - 1/6*(60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*
d + 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 +
23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11*a*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/(
a^3*b^5*c^7 - 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5 + (b^8
*c^5*d^2 - 5*a*b^7*c^4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*a^3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^3*d^7)*x^5 + (2
*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*d^3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6
- 3*a^6*b^2*d^7)*x^4 + (b^8*c^7 + a*b^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25*a^4*b^4*c^3*d^4 -
a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 - 3*a^7*b*d^7)*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25*
a^4*b^4*c^4*d^3 - 35*a^5*b^3*c^3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*x^2 + (3*a^2*b^6*c^7 - 13*a
^3*b^5*c^6*d + 20*a^4*b^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^8*c*d^6)*x)

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Fricas [B]  time = 1.72821, size = 2313, normalized size = 13.61 \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)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")

[Out]

-1/6*(2*b^5*c^5 - 15*a*b^4*c^4*d + 60*a^2*b^3*c^3*d^2 - 20*a^3*b^2*c^2*d^3 - 30*a^4*b*c*d^4 + 3*a^5*d^5 + 60*(
b^5*c*d^4 - a*b^4*d^5)*x^4 + 30*(3*b^5*c^2*d^3 + 2*a*b^4*c*d^4 - 5*a^2*b^3*d^5)*x^3 + 10*(2*b^5*c^3*d^2 + 21*a
*b^4*c^2*d^3 - 12*a^2*b^3*c*d^4 - 11*a^3*b^2*d^5)*x^2 - 5*(b^5*c^4*d - 12*a*b^4*c^3*d^2 - 24*a^2*b^3*c^2*d^3 +
32*a^3*b^2*c*d^4 + 3*a^4*b*d^5)*x + 60*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^
5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^
2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a) - 60*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*
d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^
5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(d*x + c))/(a^3*b^6*c^8 - 6*a^4*b^5*c^7*d + 15*a^5*b^4*c^
6*d^2 - 20*a^6*b^3*c^5*d^3 + 15*a^7*b^2*c^4*d^4 - 6*a^8*b*c^3*d^5 + a^9*c^2*d^6 + (b^9*c^6*d^2 - 6*a*b^8*c^5*d
^3 + 15*a^2*b^7*c^4*d^4 - 20*a^3*b^6*c^3*d^5 + 15*a^4*b^5*c^2*d^6 - 6*a^5*b^4*c*d^7 + a^6*b^3*d^8)*x^5 + (2*b^
9*c^7*d - 9*a*b^8*c^6*d^2 + 12*a^2*b^7*c^5*d^3 + 5*a^3*b^6*c^4*d^4 - 30*a^4*b^5*c^3*d^5 + 33*a^5*b^4*c^2*d^6 -
16*a^6*b^3*c*d^7 + 3*a^7*b^2*d^8)*x^4 + (b^9*c^8 - 18*a^2*b^7*c^6*d^2 + 52*a^3*b^6*c^5*d^3 - 60*a^4*b^5*c^4*d
^4 + 24*a^5*b^4*c^3*d^5 + 10*a^6*b^3*c^2*d^6 - 12*a^7*b^2*c*d^7 + 3*a^8*b*d^8)*x^3 + (3*a*b^8*c^8 - 12*a^2*b^7
*c^7*d + 10*a^3*b^6*c^6*d^2 + 24*a^4*b^5*c^5*d^3 - 60*a^5*b^4*c^4*d^4 + 52*a^6*b^3*c^3*d^5 - 18*a^7*b^2*c^2*d^
6 + a^9*d^8)*x^2 + (3*a^2*b^7*c^8 - 16*a^3*b^6*c^7*d + 33*a^4*b^5*c^6*d^2 - 30*a^5*b^4*c^5*d^3 + 5*a^6*b^3*c^4
*d^4 + 12*a^7*b^2*c^3*d^5 - 9*a^8*b*c^2*d^6 + 2*a^9*c*d^7)*x)

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Sympy [B]  time = 7.18453, size = 1217, normalized size = 7.16 \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)/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

10*b**2*d**3*log(x + (-10*a**7*b**2*d**10/(a*d - b*c)**6 + 70*a**6*b**3*c*d**9/(a*d - b*c)**6 - 210*a**5*b**4*
c**2*d**8/(a*d - b*c)**6 + 350*a**4*b**5*c**3*d**7/(a*d - b*c)**6 - 350*a**3*b**6*c**4*d**6/(a*d - b*c)**6 + 2
10*a**2*b**7*c**5*d**5/(a*d - b*c)**6 - 70*a*b**8*c**6*d**4/(a*d - b*c)**6 + 10*a*b**2*d**4 + 10*b**9*c**7*d**
3/(a*d - b*c)**6 + 10*b**3*c*d**3)/(20*b**3*d**4))/(a*d - b*c)**6 - 10*b**2*d**3*log(x + (10*a**7*b**2*d**10/(
a*d - b*c)**6 - 70*a**6*b**3*c*d**9/(a*d - b*c)**6 + 210*a**5*b**4*c**2*d**8/(a*d - b*c)**6 - 350*a**4*b**5*c*
*3*d**7/(a*d - b*c)**6 + 350*a**3*b**6*c**4*d**6/(a*d - b*c)**6 - 210*a**2*b**7*c**5*d**5/(a*d - b*c)**6 + 70*
a*b**8*c**6*d**4/(a*d - b*c)**6 + 10*a*b**2*d**4 - 10*b**9*c**7*d**3/(a*d - b*c)**6 + 10*b**3*c*d**3)/(20*b**3
*d**4))/(a*d - b*c)**6 + (-3*a**4*d**4 + 27*a**3*b*c*d**3 + 47*a**2*b**2*c**2*d**2 - 13*a*b**3*c**3*d + 2*b**4
*c**4 + 60*b**4*d**4*x**4 + x**3*(150*a*b**3*d**4 + 90*b**4*c*d**3) + x**2*(110*a**2*b**2*d**4 + 230*a*b**3*c*
d**3 + 20*b**4*c**2*d**2) + x*(15*a**3*b*d**4 + 175*a**2*b**2*c*d**3 + 55*a*b**3*c**2*d**2 - 5*b**4*c**3*d))/(
6*a**8*c**2*d**5 - 30*a**7*b*c**3*d**4 + 60*a**6*b**2*c**4*d**3 - 60*a**5*b**3*c**5*d**2 + 30*a**4*b**4*c**6*d
- 6*a**3*b**5*c**7 + x**5*(6*a**5*b**3*d**7 - 30*a**4*b**4*c*d**6 + 60*a**3*b**5*c**2*d**5 - 60*a**2*b**6*c**
3*d**4 + 30*a*b**7*c**4*d**3 - 6*b**8*c**5*d**2) + x**4*(18*a**6*b**2*d**7 - 78*a**5*b**3*c*d**6 + 120*a**4*b*
*4*c**2*d**5 - 60*a**3*b**5*c**3*d**4 - 30*a**2*b**6*c**4*d**3 + 42*a*b**7*c**5*d**2 - 12*b**8*c**6*d) + x**3*
(18*a**7*b*d**7 - 54*a**6*b**2*c*d**6 + 6*a**5*b**3*c**2*d**5 + 150*a**4*b**4*c**3*d**4 - 210*a**3*b**5*c**4*d
**3 + 102*a**2*b**6*c**5*d**2 - 6*a*b**7*c**6*d - 6*b**8*c**7) + x**2*(6*a**8*d**7 + 6*a**7*b*c*d**6 - 102*a**
6*b**2*c**2*d**5 + 210*a**5*b**3*c**3*d**4 - 150*a**4*b**4*c**4*d**3 - 6*a**3*b**5*c**5*d**2 + 54*a**2*b**6*c*
*6*d - 18*a*b**7*c**7) + x*(12*a**8*c*d**6 - 42*a**7*b*c**2*d**5 + 30*a**6*b**2*c**3*d**4 + 60*a**5*b**3*c**4*
d**3 - 120*a**4*b**4*c**5*d**2 + 78*a**3*b**5*c**6*d - 18*a**2*b**6*c**7))

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Giac [B]  time = 1.2078, size = 618, normalized size = 3.64 \begin{align*} -\frac{10 \, b^{3} d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7} c^{6} - 6 \, a b^{6} c^{5} d + 15 \, a^{2} b^{5} c^{4} d^{2} - 20 \, a^{3} b^{4} c^{3} d^{3} + 15 \, a^{4} b^{3} c^{2} d^{4} - 6 \, a^{5} b^{2} c d^{5} + a^{6} b d^{6}} + \frac{10 \, b^{2} d^{4} \log \left ({\left | d x + c \right |}\right )}{b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} - 20 \, a^{3} b^{3} c^{3} d^{4} + 15 \, a^{4} b^{2} c^{2} d^{5} - 6 \, a^{5} b c d^{6} + a^{6} d^{7}} - \frac{2 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 60 \, a^{2} b^{3} c^{3} d^{2} - 20 \, a^{3} b^{2} c^{2} d^{3} - 30 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5} + 60 \,{\left (b^{5} c d^{4} - a b^{4} d^{5}\right )} x^{4} + 30 \,{\left (3 \, b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} - 5 \, a^{2} b^{3} d^{5}\right )} x^{3} + 10 \,{\left (2 \, b^{5} c^{3} d^{2} + 21 \, a b^{4} c^{2} d^{3} - 12 \, a^{2} b^{3} c d^{4} - 11 \, a^{3} b^{2} d^{5}\right )} x^{2} - 5 \,{\left (b^{5} c^{4} d - 12 \, a b^{4} c^{3} d^{2} - 24 \, a^{2} b^{3} c^{2} d^{3} + 32 \, a^{3} b^{2} c d^{4} + 3 \, a^{4} b d^{5}\right )} x}{6 \,{\left (b c - a d\right )}^{6}{\left (b x + a\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)/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")

[Out]

-10*b^3*d^3*log(abs(b*x + a))/(b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*
c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6) + 10*b^2*d^4*log(abs(d*x + c))/(b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^
4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7) - 1/6*(2*b^5*c^5 - 15*a*b^4*c^4
*d + 60*a^2*b^3*c^3*d^2 - 20*a^3*b^2*c^2*d^3 - 30*a^4*b*c*d^4 + 3*a^5*d^5 + 60*(b^5*c*d^4 - a*b^4*d^5)*x^4 + 3
0*(3*b^5*c^2*d^3 + 2*a*b^4*c*d^4 - 5*a^2*b^3*d^5)*x^3 + 10*(2*b^5*c^3*d^2 + 21*a*b^4*c^2*d^3 - 12*a^2*b^3*c*d^
4 - 11*a^3*b^2*d^5)*x^2 - 5*(b^5*c^4*d - 12*a*b^4*c^3*d^2 - 24*a^2*b^3*c^2*d^3 + 32*a^3*b^2*c*d^4 + 3*a^4*b*d^
5)*x)/((b*c - a*d)^6*(b*x + a)^3*(d*x + c)^2)