3.262 \(\int \frac{x^4}{(a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=164 \[ -\frac{a^4}{b^2 (a+b x) (b c-a d)^3}-\frac{a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac{c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac{c^4}{2 d^3 (c+d x)^2 (b c-a d)^2}+\frac{2 c^3 (b c-2 a d)}{d^3 (c+d x) (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.353745, antiderivative size = 164, 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 \[ -\frac{a^4}{b^2 (a+b x) (b c-a d)^3}-\frac{a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac{c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac{c^4}{2 d^3 (c+d x)^2 (b c-a d)^2}+\frac{2 c^3 (b c-2 a d)}{d^3 (c+d x) (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 63.3011, size = 150, normalized size = 0.91 \[ \frac{a^{4}}{b^{2} \left (a + b x\right ) \left (a d - b c\right )^{3}} + \frac{a^{3} \left (a d - 4 b c\right ) \log{\left (a + b x \right )}}{b^{2} \left (a d - b c\right )^{4}} - \frac{c^{4}}{2 d^{3} \left (c + d x\right )^{2} \left (a d - b c\right )^{2}} + \frac{2 c^{3} \left (2 a d - b c\right )}{d^{3} \left (c + d x\right ) \left (a d - b c\right )^{3}} + \frac{c^{2} \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{d^{3} \left (a d - b c\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.396927, size = 162, normalized size = 0.99 \[ -\frac{a^4}{b^2 (a+b x) (b c-a d)^3}+\frac{a^3 (a d-4 b c) \log (a+b x)}{b^2 (b c-a d)^4}+\frac{c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac{c^4}{2 d^3 (c+d x)^2 (b c-a d)^2}-\frac{2 c^3 (b c-2 a d)}{d^3 (c+d x) (a d-b c)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 231, normalized size = 1.4 \[ -{\frac{{c}^{4}}{2\,{d}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+6\,{\frac{{c}^{2}\ln \left ( dx+c \right ){a}^{2}}{ \left ( ad-bc \right ) ^{4}d}}-4\,{\frac{{c}^{3}\ln \left ( dx+c \right ) ab}{ \left ( ad-bc \right ) ^{4}{d}^{2}}}+{\frac{{c}^{4}\ln \left ( dx+c \right ){b}^{2}}{ \left ( ad-bc \right ) ^{4}{d}^{3}}}+4\,{\frac{{c}^{3}a}{{d}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-2\,{\frac{{c}^{4}b}{{d}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+{\frac{{a}^{4}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}{b}^{2}}}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}b}}+{\frac{{a}^{4}}{{b}^{2} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^2/(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.38679, size = 699, normalized size = 4.26 \[ -\frac{{\left (4 \, a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}} + \frac{{\left (b^{2} c^{4} - 4 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}} + \frac{3 \, a b^{3} c^{5} - 7 \, a^{2} b^{2} c^{4} d - 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (2 \, b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} - a^{4} d^{5}\right )} x^{2} +{\left (3 \, b^{4} c^{5} - 3 \, a b^{3} c^{4} d - 8 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{4} c d^{4}\right )} x}{2 \,{\left (a b^{5} c^{5} d^{3} - 3 \, a^{2} b^{4} c^{4} d^{4} + 3 \, a^{3} b^{3} c^{3} d^{5} - a^{4} b^{2} c^{2} d^{6} +{\left (b^{6} c^{3} d^{5} - 3 \, a b^{5} c^{2} d^{6} + 3 \, a^{2} b^{4} c d^{7} - a^{3} b^{3} d^{8}\right )} x^{3} +{\left (2 \, b^{6} c^{4} d^{4} - 5 \, a b^{5} c^{3} d^{5} + 3 \, a^{2} b^{4} c^{2} d^{6} + a^{3} b^{3} c d^{7} - a^{4} b^{2} d^{8}\right )} x^{2} +{\left (b^{6} c^{5} d^{3} - a b^{5} c^{4} d^{4} - 3 \, a^{2} b^{4} c^{3} d^{5} + 5 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.247689, size = 1076, normalized size = 6.56 \[ \frac{3 \, a b^{4} c^{6} - 10 \, a^{2} b^{3} c^{5} d + 7 \, a^{3} b^{2} c^{4} d^{2} - 2 \, a^{4} b c^{3} d^{3} + 2 \, a^{5} c^{2} d^{4} + 2 \,{\left (2 \, b^{5} c^{5} d - 6 \, a b^{4} c^{4} d^{2} + 4 \, a^{2} b^{3} c^{3} d^{3} - a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} +{\left (3 \, b^{5} c^{6} - 6 \, a b^{4} c^{5} d - 5 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 4 \, a^{4} b c^{2} d^{4} + 4 \, a^{5} c d^{5}\right )} x - 2 \,{\left (4 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4} +{\left (4 \, a^{3} b^{2} c d^{5} - a^{4} b d^{6}\right )} x^{3} +{\left (8 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - a^{5} d^{6}\right )} x^{2} +{\left (4 \, a^{3} b^{2} c^{3} d^{3} + 7 \, a^{4} b c^{2} d^{4} - 2 \, a^{5} c d^{5}\right )} x\right )} \log \left (b x + a\right ) + 2 \,{\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} +{\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4}\right )} x^{3} +{\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} + 6 \, a^{3} b^{2} c^{2} d^{4}\right )} x^{2} +{\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 12 \, a^{3} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{6} c^{6} d^{3} - 4 \, a^{2} b^{5} c^{5} d^{4} + 6 \, a^{3} b^{4} c^{4} d^{5} - 4 \, a^{4} b^{3} c^{3} d^{6} + a^{5} b^{2} c^{2} d^{7} +{\left (b^{7} c^{4} d^{5} - 4 \, a b^{6} c^{3} d^{6} + 6 \, a^{2} b^{5} c^{2} d^{7} - 4 \, a^{3} b^{4} c d^{8} + a^{4} b^{3} d^{9}\right )} x^{3} +{\left (2 \, b^{7} c^{5} d^{4} - 7 \, a b^{6} c^{4} d^{5} + 8 \, a^{2} b^{5} c^{3} d^{6} - 2 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8} + a^{5} b^{2} d^{9}\right )} x^{2} +{\left (b^{7} c^{6} d^{3} - 2 \, a b^{6} c^{5} d^{4} - 2 \, a^{2} b^{5} c^{4} d^{5} + 8 \, a^{3} b^{4} c^{3} d^{6} - 7 \, a^{4} b^{3} c^{2} d^{7} + 2 \, a^{5} b^{2} c d^{8}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 27.159, size = 1083, normalized size = 6.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*(a*d - 4*b*c)*log(x + (a**8*d**7*(a*d - 4*b*c)/(b*(a*d - b*c)**4) - 5*a**7*
c*d**6*(a*d - 4*b*c)/(a*d - b*c)**4 + 10*a**6*b*c**2*d**5*(a*d - 4*b*c)/(a*d - b
*c)**4 - 10*a**5*b**2*c**3*d**4*(a*d - 4*b*c)/(a*d - b*c)**4 + 5*a**4*b**3*c**4*
d**3*(a*d - 4*b*c)/(a*d - b*c)**4 + a**4*c*d**3 - a**3*b**4*c**5*d**2*(a*d - 4*b
*c)/(a*d - b*c)**4 - 10*a**3*b*c**2*d**2 + 4*a**2*b**2*c**3*d - a*b**3*c**4)/(a*
*4*d**4 - 4*a**3*b*c*d**3 - 6*a**2*b**2*c**2*d**2 + 4*a*b**3*c**3*d - b**4*c**4)
)/(b**2*(a*d - b*c)**4) + c**2*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)*log(x + (a*
*5*b*c**2*d**4*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**4 - 5*a**4*b**
2*c**3*d**3*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**4 + a**4*c*d**3 +
10*a**3*b**3*c**4*d**2*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**4 - 1
0*a**3*b*c**2*d**2 - 10*a**2*b**4*c**5*d*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(
a*d - b*c)**4 + 4*a**2*b**2*c**3*d + 5*a*b**5*c**6*(6*a**2*d**2 - 4*a*b*c*d + b*
*2*c**2)/(a*d - b*c)**4 - a*b**3*c**4 - b**6*c**7*(6*a**2*d**2 - 4*a*b*c*d + b**
2*c**2)/(d*(a*d - b*c)**4))/(a**4*d**4 - 4*a**3*b*c*d**3 - 6*a**2*b**2*c**2*d**2
+ 4*a*b**3*c**3*d - b**4*c**4))/(d**3*(a*d - b*c)**4) + (2*a**4*c**2*d**3 + 7*a
**2*b**2*c**4*d - 3*a*b**3*c**5 + x**2*(2*a**4*d**5 + 8*a*b**3*c**3*d**2 - 4*b**
4*c**4*d) + x*(4*a**4*c*d**4 + 8*a**2*b**2*c**3*d**2 + 3*a*b**3*c**4*d - 3*b**4*
c**5))/(2*a**4*b**2*c**2*d**6 - 6*a**3*b**3*c**3*d**5 + 6*a**2*b**4*c**4*d**4 -
2*a*b**5*c**5*d**3 + x**3*(2*a**3*b**3*d**8 - 6*a**2*b**4*c*d**7 + 6*a*b**5*c**2
*d**6 - 2*b**6*c**3*d**5) + x**2*(2*a**4*b**2*d**8 - 2*a**3*b**3*c*d**7 - 6*a**2
*b**4*c**2*d**6 + 10*a*b**5*c**3*d**5 - 4*b**6*c**4*d**4) + x*(4*a**4*b**2*c*d**
7 - 10*a**3*b**3*c**2*d**6 + 6*a**2*b**4*c**3*d**5 + 2*a*b**5*c**4*d**4 - 2*b**6
*c**5*d**3))

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GIAC/XCAS [A]  time = 0.332874, size = 419, normalized size = 2.55 \[ -\frac{a^{4} b^{3}}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )}{\left (b x + a\right )}} + \frac{{\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 6 \, a^{2} b c^{2} d^{2}\right )}{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}} - \frac{{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{2} d^{3}} - \frac{3 \, b^{2} c^{4} d^{2} - 8 \, a b c^{3} d^{3} + \frac{2 \,{\left (b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 4 \, a^{2} b^{2} c^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \,{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a^4*b^3/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*(b*x + a)) +
(b^3*c^4 - 4*a*b^2*c^3*d + 6*a^2*b*c^2*d^2)*ln(abs(b*c/(b*x + a) - a*d/(b*x + a
) + d))/(b^5*c^4*d^3 - 4*a*b^4*c^3*d^4 + 6*a^2*b^3*c^2*d^5 - 4*a^3*b^2*c*d^6 + a
^4*b*d^7) - ln(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^2*d^3) - 1/2*(3*b^2*c^4*d^2
- 8*a*b*c^3*d^3 + 2*(b^4*c^5*d - 5*a*b^3*c^4*d^2 + 4*a^2*b^2*c^3*d^3)/((b*x + a
)*b))/((b*c - a*d)^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^3)