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

Optimal. Leaf size=228 \[ -\frac{3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}+\frac{b^5}{a^3 (a+b x) (b c-a d)^3}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}-\frac{1}{2 a^2 c^3 x^2}+\frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]

[Out]

-1/(2*a^2*c^3*x^2) + (2*b*c + 3*a*d)/(a^3*c^4*x) + b^5/(a^3*(b*c - a*d)^3*(a + b
*x)) + d^4/(2*c^3*(b*c - a*d)^2*(c + d*x)^2) + (d^4*(5*b*c - 3*a*d))/(c^4*(b*c -
a*d)^3*(c + d*x)) + (3*(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*Log[x])/(a^4*c^5) - (3
*b^5*(b*c - 2*a*d)*Log[a + b*x])/(a^4*(b*c - a*d)^4) - (3*d^4*(5*b^2*c^2 - 6*a*b
*c*d + 2*a^2*d^2)*Log[c + d*x])/(c^5*(b*c - a*d)^4)

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Rubi [A]  time = 0.61653, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}+\frac{b^5}{a^3 (a+b x) (b c-a d)^3}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}-\frac{1}{2 a^2 c^3 x^2}+\frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a^2*c^3*x^2) + (2*b*c + 3*a*d)/(a^3*c^4*x) + b^5/(a^3*(b*c - a*d)^3*(a + b
*x)) + d^4/(2*c^3*(b*c - a*d)^2*(c + d*x)^2) + (d^4*(5*b*c - 3*a*d))/(c^4*(b*c -
a*d)^3*(c + d*x)) + (3*(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*Log[x])/(a^4*c^5) - (3
*b^5*(b*c - 2*a*d)*Log[a + b*x])/(a^4*(b*c - a*d)^4) - (3*d^4*(5*b^2*c^2 - 6*a*b
*c*d + 2*a^2*d^2)*Log[c + d*x])/(c^5*(b*c - a*d)^4)

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Rubi in Sympy [A]  time = 154.082, size = 223, normalized size = 0.98 \[ \frac{d^{4}}{2 c^{3} \left (c + d x\right )^{2} \left (a d - b c\right )^{2}} + \frac{d^{4} \left (3 a d - 5 b c\right )}{c^{4} \left (c + d x\right ) \left (a d - b c\right )^{3}} - \frac{3 d^{4} \left (2 a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{c^{5} \left (a d - b c\right )^{4}} - \frac{1}{2 a^{2} c^{3} x^{2}} - \frac{b^{5}}{a^{3} \left (a + b x\right ) \left (a d - b c\right )^{3}} + \frac{3 a d + 2 b c}{a^{3} c^{4} x} + \frac{3 b^{5} \left (2 a d - b c\right ) \log{\left (a + b x \right )}}{a^{4} \left (a d - b c\right )^{4}} + \frac{3 \left (2 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \log{\left (x \right )}}{a^{4} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.557149, size = 230, normalized size = 1.01 \[ \frac{3 b^5 (2 a d-b c) \log (a+b x)}{a^4 (b c-a d)^4}-\frac{b^5}{a^3 (a+b x) (a d-b c)^3}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}-\frac{1}{2 a^2 c^3 x^2}+\frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a^2*c^3*x^2) + (2*b*c + 3*a*d)/(a^3*c^4*x) - b^5/(a^3*(-(b*c) + a*d)^3*(a
+ b*x)) + d^4/(2*c^3*(b*c - a*d)^2*(c + d*x)^2) + (d^4*(5*b*c - 3*a*d))/(c^4*(b*
c - a*d)^3*(c + d*x)) + (3*(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*Log[x])/(a^4*c^5) +
(3*b^5*(-(b*c) + 2*a*d)*Log[a + b*x])/(a^4*(b*c - a*d)^4) - (3*d^4*(5*b^2*c^2 -
6*a*b*c*d + 2*a^2*d^2)*Log[c + d*x])/(c^5*(b*c - a*d)^4)

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Maple [A]  time = 0.027, size = 307, normalized size = 1.4 \[{\frac{{d}^{4}}{2\,{c}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+3\,{\frac{{d}^{5}a}{{c}^{4} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-5\,{\frac{{d}^{4}b}{{c}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-6\,{\frac{{d}^{6}\ln \left ( dx+c \right ){a}^{2}}{{c}^{5} \left ( ad-bc \right ) ^{4}}}+18\,{\frac{{d}^{5}\ln \left ( dx+c \right ) ab}{{c}^{4} \left ( ad-bc \right ) ^{4}}}-15\,{\frac{{d}^{4}\ln \left ( dx+c \right ){b}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{4}}}-{\frac{1}{2\,{a}^{2}{c}^{3}{x}^{2}}}+3\,{\frac{d}{x{a}^{2}{c}^{4}}}+2\,{\frac{b}{x{a}^{3}{c}^{3}}}+6\,{\frac{\ln \left ( x \right ){d}^{2}}{{a}^{2}{c}^{5}}}+6\,{\frac{b\ln \left ( x \right ) d}{{a}^{3}{c}^{4}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{4}{c}^{3}}}-{\frac{{b}^{5}}{ \left ( ad-bc \right ) ^{3}{a}^{3} \left ( bx+a \right ) }}+6\,{\frac{{b}^{5}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}{a}^{3}}}-3\,{\frac{{b}^{6}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^4/c^3/(a*d-b*c)^2/(d*x+c)^2+3*d^5/c^4/(a*d-b*c)^3/(d*x+c)*a-5*d^4/c^3/(a*d
-b*c)^3/(d*x+c)*b-6*d^6/c^5/(a*d-b*c)^4*ln(d*x+c)*a^2+18*d^5/c^4/(a*d-b*c)^4*ln(
d*x+c)*a*b-15*d^4/c^3/(a*d-b*c)^4*ln(d*x+c)*b^2-1/2/a^2/c^3/x^2+3/x/a^2/c^4*d+2/
x/a^3/c^3*b+6/a^2/c^5*ln(x)*d^2+6/a^3/c^4*ln(x)*b*d+3/a^4/c^3*ln(x)*b^2-b^5/(a*d
-b*c)^3/a^3/(b*x+a)+6*b^5/(a*d-b*c)^4/a^3*ln(b*x+a)*d-3*b^6/(a*d-b*c)^4/a^4*ln(b
*x+a)*c

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Maxima [A]  time = 1.4093, size = 1017, normalized size = 4.46 \[ -\frac{3 \,{\left (b^{6} c - 2 \, a b^{5} d\right )} \log \left (b x + a\right )}{a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4}} - \frac{3 \,{\left (5 \, b^{2} c^{2} d^{4} - 6 \, a b c d^{5} + 2 \, a^{2} d^{6}\right )} \log \left (d x + c\right )}{b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4}} - \frac{a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} - 6 \,{\left (b^{5} c^{4} d^{2} - a b^{4} c^{3} d^{3} - a^{2} b^{3} c^{2} d^{4} + 4 \, a^{3} b^{2} c d^{5} - 2 \, a^{4} b d^{6}\right )} x^{4} - 3 \,{\left (4 \, b^{5} c^{5} d - 3 \, a b^{4} c^{4} d^{2} - 5 \, a^{2} b^{3} c^{3} d^{3} + 10 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - 4 \, a^{5} d^{6}\right )} x^{3} -{\left (6 \, b^{5} c^{6} - 13 \, a^{2} b^{3} c^{4} d^{2} - a^{3} b^{2} c^{3} d^{3} + 32 \, a^{4} b c^{2} d^{4} - 18 \, a^{5} c d^{5}\right )} x^{2} -{\left (3 \, a b^{4} c^{6} - 5 \, a^{2} b^{3} c^{5} d - 3 \, a^{3} b^{2} c^{4} d^{2} + 9 \, a^{4} b c^{3} d^{3} - 4 \, a^{5} c^{2} d^{4}\right )} x}{2 \,{\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{5} +{\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{4} +{\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{3} +{\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x^{2}\right )}} + \frac{3 \,{\left (b^{2} c^{2} + 2 \, a b c d + 2 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{4} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^3*x^3),x, algorithm="maxima")

[Out]

-3*(b^6*c - 2*a*b^5*d)*log(b*x + a)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c
^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4) - 3*(5*b^2*c^2*d^4 - 6*a*b*c*d^5 + 2*a^2*d^6)*
log(d*x + c)/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^
4*c^5*d^4) - 1/2*(a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3
- 6*(b^5*c^4*d^2 - a*b^4*c^3*d^3 - a^2*b^3*c^2*d^4 + 4*a^3*b^2*c*d^5 - 2*a^4*b*d
^6)*x^4 - 3*(4*b^5*c^5*d - 3*a*b^4*c^4*d^2 - 5*a^2*b^3*c^3*d^3 + 10*a^3*b^2*c^2*
d^4 + 2*a^4*b*c*d^5 - 4*a^5*d^6)*x^3 - (6*b^5*c^6 - 13*a^2*b^3*c^4*d^2 - a^3*b^2
*c^3*d^3 + 32*a^4*b*c^2*d^4 - 18*a^5*c*d^5)*x^2 - (3*a*b^4*c^6 - 5*a^2*b^3*c^5*d
- 3*a^3*b^2*c^4*d^2 + 9*a^4*b*c^3*d^3 - 4*a^5*c^2*d^4)*x)/((a^3*b^4*c^7*d^2 - 3
*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^5 + (2*a^3*b^4*c^8*d - 5
*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^4 + (a^3*b
^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^
3 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2) + 3*(b^
2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*log(x)/(a^4*c^5)

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Fricas [A]  time = 65.1588, size = 1758, normalized size = 7.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^3*x^3),x, algorithm="fricas")

[Out]

-1/2*(a^3*b^4*c^8 - 4*a^4*b^3*c^7*d + 6*a^5*b^2*c^6*d^2 - 4*a^6*b*c^5*d^3 + a^7*
c^4*d^4 - 6*(a*b^6*c^6*d^2 - 2*a^2*b^5*c^5*d^3 + 5*a^4*b^3*c^3*d^5 - 6*a^5*b^2*c
^2*d^6 + 2*a^6*b*c*d^7)*x^4 - 3*(4*a*b^6*c^7*d - 7*a^2*b^5*c^6*d^2 - 2*a^3*b^4*c
^5*d^3 + 15*a^4*b^3*c^4*d^4 - 8*a^5*b^2*c^3*d^5 - 6*a^6*b*c^2*d^6 + 4*a^7*c*d^7)
*x^3 - (6*a*b^6*c^8 - 6*a^2*b^5*c^7*d - 13*a^3*b^4*c^6*d^2 + 12*a^4*b^3*c^5*d^3
+ 33*a^5*b^2*c^4*d^4 - 50*a^6*b*c^3*d^5 + 18*a^7*c^2*d^6)*x^2 - (3*a^2*b^5*c^8 -
8*a^3*b^4*c^7*d + 2*a^4*b^3*c^6*d^2 + 12*a^5*b^2*c^5*d^3 - 13*a^6*b*c^4*d^4 + 4
*a^7*c^3*d^5)*x + 6*((b^7*c^6*d^2 - 2*a*b^6*c^5*d^3)*x^5 + (2*b^7*c^7*d - 3*a*b^
6*c^6*d^2 - 2*a^2*b^5*c^5*d^3)*x^4 + (b^7*c^8 - 4*a^2*b^5*c^6*d^2)*x^3 + (a*b^6*
c^8 - 2*a^2*b^5*c^7*d)*x^2)*log(b*x + a) + 6*((5*a^4*b^3*c^2*d^6 - 6*a^5*b^2*c*d
^7 + 2*a^6*b*d^8)*x^5 + (10*a^4*b^3*c^3*d^5 - 7*a^5*b^2*c^2*d^6 - 2*a^6*b*c*d^7
+ 2*a^7*d^8)*x^4 + (5*a^4*b^3*c^4*d^4 + 4*a^5*b^2*c^3*d^5 - 10*a^6*b*c^2*d^6 + 4
*a^7*c*d^7)*x^3 + (5*a^5*b^2*c^4*d^4 - 6*a^6*b*c^3*d^5 + 2*a^7*c^2*d^6)*x^2)*log
(d*x + c) - 6*((b^7*c^6*d^2 - 2*a*b^6*c^5*d^3 + 5*a^4*b^3*c^2*d^6 - 6*a^5*b^2*c*
d^7 + 2*a^6*b*d^8)*x^5 + (2*b^7*c^7*d - 3*a*b^6*c^6*d^2 - 2*a^2*b^5*c^5*d^3 + 10
*a^4*b^3*c^3*d^5 - 7*a^5*b^2*c^2*d^6 - 2*a^6*b*c*d^7 + 2*a^7*d^8)*x^4 + (b^7*c^8
- 4*a^2*b^5*c^6*d^2 + 5*a^4*b^3*c^4*d^4 + 4*a^5*b^2*c^3*d^5 - 10*a^6*b*c^2*d^6
+ 4*a^7*c*d^7)*x^3 + (a*b^6*c^8 - 2*a^2*b^5*c^7*d + 5*a^5*b^2*c^4*d^4 - 6*a^6*b*
c^3*d^5 + 2*a^7*c^2*d^6)*x^2)*log(x))/((a^4*b^5*c^9*d^2 - 4*a^5*b^4*c^8*d^3 + 6*
a^6*b^3*c^7*d^4 - 4*a^7*b^2*c^6*d^5 + a^8*b*c^5*d^6)*x^5 + (2*a^4*b^5*c^10*d - 7
*a^5*b^4*c^9*d^2 + 8*a^6*b^3*c^8*d^3 - 2*a^7*b^2*c^7*d^4 - 2*a^8*b*c^6*d^5 + a^9
*c^5*d^6)*x^4 + (a^4*b^5*c^11 - 2*a^5*b^4*c^10*d - 2*a^6*b^3*c^9*d^2 + 8*a^7*b^2
*c^8*d^3 - 7*a^8*b*c^7*d^4 + 2*a^9*c^6*d^5)*x^3 + (a^5*b^4*c^11 - 4*a^6*b^3*c^10
*d + 6*a^7*b^2*c^9*d^2 - 4*a^8*b*c^8*d^3 + a^9*c^7*d^4)*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.309697, size = 1168, normalized size = 5.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^3*x^3),x, algorithm="giac")

[Out]

b^11/((a^3*b^9*c^3 - 3*a^4*b^8*c^2*d + 3*a^5*b^7*c*d^2 - a^6*b^6*d^3)*(b*x + a))
+ 3/2*(b^6*c - 2*a*b^5*d)*ln(abs(-b*c/(b*x + a) + a*b*c/(b*x + a)^2 + 2*a*d/(b*
x + a) - a^2*d/(b*x + a)^2 - d))/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*
d^2 - 4*a^7*b*c*d^3 + a^8*d^4) - 3/2*(b^8*c^6 - 2*a*b^7*c^5*d + 10*a^4*b^4*c^2*d
^4 - 12*a^5*b^3*c*d^5 + 4*a^6*b^2*d^6)*ln(abs(-2*a*b^2*c/(b*x + a) + b^2*c - 2*a
*b*d + 2*a^2*b*d/(b*x + a) - b^2*abs(c))/abs(-2*a*b^2*c/(b*x + a) + b^2*c - 2*a*
b*d + 2*a^2*b*d/(b*x + a) + b^2*abs(c)))/((a^4*b^4*c^8 - 4*a^5*b^3*c^7*d + 6*a^6
*b^2*c^6*d^2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4)*b^2*abs(c)) + 1/2*(5*b^6*c^5*d^2 -
14*a*b^5*c^4*d^3 + 6*a^2*b^4*c^3*d^4 + 16*a^3*b^3*c^2*d^5 - 30*a^4*b^2*c*d^6 +
12*a^5*b*d^7 + 2*(5*b^8*c^6*d - 22*a*b^7*c^5*d^2 + 29*a^2*b^6*c^4*d^3 + 4*a^3*b^
5*c^3*d^4 - 47*a^4*b^4*c^2*d^5 + 54*a^5*b^3*c*d^6 - 18*a^6*b^2*d^7)/((b*x + a)*b
) + (5*b^10*c^7 - 36*a*b^9*c^6*d + 87*a^2*b^8*c^5*d^2 - 70*a^3*b^7*c^4*d^3 - 45*
a^4*b^6*c^3*d^4 + 144*a^5*b^5*c^2*d^5 - 126*a^6*b^4*c*d^6 + 36*a^7*b^3*d^7)/((b*
x + a)^2*b^2) - 6*(a*b^11*c^7 - 5*a^2*b^10*c^6*d + 9*a^3*b^9*c^5*d^2 - 5*a^4*b^8
*c^4*d^3 - 5*a^5*b^7*c^3*d^4 + 11*a^6*b^6*c^2*d^5 - 8*a^7*b^5*c*d^6 + 2*a^8*b^4*
d^7)/((b*x + a)^3*b^3))/((b*c - a*d)^4*a^4*(b*c/(b*x + a) - a*b*c/(b*x + a)^2 -
2*a*d/(b*x + a) + a^2*d/(b*x + a)^2 + d)^2*c^4)