3.1798 \(\int \frac{1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx\)

Optimal. Leaf size=130 \[ \frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}+\frac{d^3}{(a+b x) (b c-a d)^4}-\frac{d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{d}{3 (a+b x)^3 (b c-a d)^2}-\frac{1}{4 (a+b x)^4 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.242531, antiderivative size = 130, 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 \[ \frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}+\frac{d^3}{(a+b x) (b c-a d)^4}-\frac{d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{d}{3 (a+b x)^3 (b c-a d)^2}-\frac{1}{4 (a+b x)^4 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^4*(a*c + (b*c + a*d)*x + b*d*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 51.4974, size = 109, normalized size = 0.84 \[ - \frac{d^{4} \log{\left (a + b x \right )}}{\left (a d - b c\right )^{5}} + \frac{d^{4} \log{\left (c + d x \right )}}{\left (a d - b c\right )^{5}} + \frac{d^{3}}{\left (a + b x\right ) \left (a d - b c\right )^{4}} + \frac{d^{2}}{2 \left (a + b x\right )^{2} \left (a d - b c\right )^{3}} + \frac{d}{3 \left (a + b x\right )^{3} \left (a d - b c\right )^{2}} + \frac{1}{4 \left (a + b x\right )^{4} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.0896276, size = 130, normalized size = 1. \[ \frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}+\frac{d^3}{(a+b x) (b c-a d)^4}-\frac{d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{d}{3 (a+b x)^3 (b c-a d)^2}+\frac{1}{4 (a+b x)^4 (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^4*(a*c + (b*c + a*d)*x + b*d*x^2)),x]

[Out]

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

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Maple [A]  time = 0.018, size = 125, normalized size = 1. \[{\frac{1}{ \left ( 4\,ad-4\,bc \right ) \left ( bx+a \right ) ^{4}}}+{\frac{d}{3\, \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{3}}}+{\frac{{d}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}+{\frac{{d}^{3}}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}-{\frac{{d}^{4}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{5}}}+{\frac{{d}^{4}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^4/(a*c+(a*d+b*c)*x+x^2*b*d),x)

[Out]

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

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Maxima [A]  time = 0.790529, size = 753, normalized size = 5.79 \[ \frac{d^{4} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{d^{4} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac{12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{12 \,{\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} +{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \,{\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \,{\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^4),x, algorithm="maxima")

[Out]

d^4*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*
d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - d^4*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*
a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) + 1/12*(12*b^3*d
^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2
- 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/(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 + (b^8*c^4 -
4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^4 + 4*(a*b^
7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^3
+ 6*(a^2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*
b^2*d^4)*x^2 + 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*
c*d^3 + a^7*b*d^4)*x)

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Fricas [A]  time = 0.215416, size = 887, normalized size = 6.82 \[ -\frac{3 \, b^{4} c^{4} - 16 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} - 48 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4} - 12 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} x - 12 \,{\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (d x + c\right )}{12 \,{\left (a^{4} b^{5} c^{5} - 5 \, a^{5} b^{4} c^{4} d + 10 \, a^{6} b^{3} c^{3} d^{2} - 10 \, a^{7} b^{2} c^{2} d^{3} + 5 \, a^{8} b c d^{4} - a^{9} d^{5} +{\left (b^{9} c^{5} - 5 \, a b^{8} c^{4} d + 10 \, a^{2} b^{7} c^{3} d^{2} - 10 \, a^{3} b^{6} c^{2} d^{3} + 5 \, a^{4} b^{5} c d^{4} - a^{5} b^{4} d^{5}\right )} x^{4} + 4 \,{\left (a b^{8} c^{5} - 5 \, a^{2} b^{7} c^{4} d + 10 \, a^{3} b^{6} c^{3} d^{2} - 10 \, a^{4} b^{5} c^{2} d^{3} + 5 \, a^{5} b^{4} c d^{4} - a^{6} b^{3} d^{5}\right )} x^{3} + 6 \,{\left (a^{2} b^{7} c^{5} - 5 \, a^{3} b^{6} c^{4} d + 10 \, a^{4} b^{5} c^{3} d^{2} - 10 \, a^{5} b^{4} c^{2} d^{3} + 5 \, a^{6} b^{3} c d^{4} - a^{7} b^{2} d^{5}\right )} x^{2} + 4 \,{\left (a^{3} b^{6} c^{5} - 5 \, a^{4} b^{5} c^{4} d + 10 \, a^{5} b^{4} c^{3} d^{2} - 10 \, a^{6} b^{3} c^{2} d^{3} + 5 \, a^{7} b^{2} c d^{4} - a^{8} b d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^4),x, algorithm="fricas")

[Out]

-1/12*(3*b^4*c^4 - 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 - 48*a^3*b*c*d^3 + 25*a^4
*d^4 - 12*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(b^4*c^2*d^2 - 8*a*b^3*c*d^3 + 7*a^2*b
^2*d^4)*x^2 - 4*(b^4*c^3*d - 6*a*b^3*c^2*d^2 + 18*a^2*b^2*c*d^3 - 13*a^3*b*d^4)*
x - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*
d^4)*log(b*x + a) + 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^
3*b*d^4*x + a^4*d^4)*log(d*x + c))/(a^4*b^5*c^5 - 5*a^5*b^4*c^4*d + 10*a^6*b^3*c
^3*d^2 - 10*a^7*b^2*c^2*d^3 + 5*a^8*b*c*d^4 - a^9*d^5 + (b^9*c^5 - 5*a*b^8*c^4*d
+ 10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5)*x^4
+ 4*(a*b^8*c^5 - 5*a^2*b^7*c^4*d + 10*a^3*b^6*c^3*d^2 - 10*a^4*b^5*c^2*d^3 + 5*a
^5*b^4*c*d^4 - a^6*b^3*d^5)*x^3 + 6*(a^2*b^7*c^5 - 5*a^3*b^6*c^4*d + 10*a^4*b^5*
c^3*d^2 - 10*a^5*b^4*c^2*d^3 + 5*a^6*b^3*c*d^4 - a^7*b^2*d^5)*x^2 + 4*(a^3*b^6*c
^5 - 5*a^4*b^5*c^4*d + 10*a^5*b^4*c^3*d^2 - 10*a^6*b^3*c^2*d^3 + 5*a^7*b^2*c*d^4
- a^8*b*d^5)*x)

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Sympy [A]  time = 10.2323, size = 802, normalized size = 6.17 \[ \frac{d^{4} \log{\left (x + \frac{- \frac{a^{6} d^{10}}{\left (a d - b c\right )^{5}} + \frac{6 a^{5} b c d^{9}}{\left (a d - b c\right )^{5}} - \frac{15 a^{4} b^{2} c^{2} d^{8}}{\left (a d - b c\right )^{5}} + \frac{20 a^{3} b^{3} c^{3} d^{7}}{\left (a d - b c\right )^{5}} - \frac{15 a^{2} b^{4} c^{4} d^{6}}{\left (a d - b c\right )^{5}} + \frac{6 a b^{5} c^{5} d^{5}}{\left (a d - b c\right )^{5}} + a d^{5} - \frac{b^{6} c^{6} d^{4}}{\left (a d - b c\right )^{5}} + b c d^{4}}{2 b d^{5}} \right )}}{\left (a d - b c\right )^{5}} - \frac{d^{4} \log{\left (x + \frac{\frac{a^{6} d^{10}}{\left (a d - b c\right )^{5}} - \frac{6 a^{5} b c d^{9}}{\left (a d - b c\right )^{5}} + \frac{15 a^{4} b^{2} c^{2} d^{8}}{\left (a d - b c\right )^{5}} - \frac{20 a^{3} b^{3} c^{3} d^{7}}{\left (a d - b c\right )^{5}} + \frac{15 a^{2} b^{4} c^{4} d^{6}}{\left (a d - b c\right )^{5}} - \frac{6 a b^{5} c^{5} d^{5}}{\left (a d - b c\right )^{5}} + a d^{5} + \frac{b^{6} c^{6} d^{4}}{\left (a d - b c\right )^{5}} + b c d^{4}}{2 b d^{5}} \right )}}{\left (a d - b c\right )^{5}} + \frac{25 a^{3} d^{3} - 23 a^{2} b c d^{2} + 13 a b^{2} c^{2} d - 3 b^{3} c^{3} + 12 b^{3} d^{3} x^{3} + x^{2} \left (42 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x \left (52 a^{2} b d^{3} - 20 a b^{2} c d^{2} + 4 b^{3} c^{2} d\right )}{12 a^{8} d^{4} - 48 a^{7} b c d^{3} + 72 a^{6} b^{2} c^{2} d^{2} - 48 a^{5} b^{3} c^{3} d + 12 a^{4} b^{4} c^{4} + x^{4} \left (12 a^{4} b^{4} d^{4} - 48 a^{3} b^{5} c d^{3} + 72 a^{2} b^{6} c^{2} d^{2} - 48 a b^{7} c^{3} d + 12 b^{8} c^{4}\right ) + x^{3} \left (48 a^{5} b^{3} d^{4} - 192 a^{4} b^{4} c d^{3} + 288 a^{3} b^{5} c^{2} d^{2} - 192 a^{2} b^{6} c^{3} d + 48 a b^{7} c^{4}\right ) + x^{2} \left (72 a^{6} b^{2} d^{4} - 288 a^{5} b^{3} c d^{3} + 432 a^{4} b^{4} c^{2} d^{2} - 288 a^{3} b^{5} c^{3} d + 72 a^{2} b^{6} c^{4}\right ) + x \left (48 a^{7} b d^{4} - 192 a^{6} b^{2} c d^{3} + 288 a^{5} b^{3} c^{2} d^{2} - 192 a^{4} b^{4} c^{3} d + 48 a^{3} b^{5} c^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2),x)

[Out]

d**4*log(x + (-a**6*d**10/(a*d - b*c)**5 + 6*a**5*b*c*d**9/(a*d - b*c)**5 - 15*a
**4*b**2*c**2*d**8/(a*d - b*c)**5 + 20*a**3*b**3*c**3*d**7/(a*d - b*c)**5 - 15*a
**2*b**4*c**4*d**6/(a*d - b*c)**5 + 6*a*b**5*c**5*d**5/(a*d - b*c)**5 + a*d**5 -
b**6*c**6*d**4/(a*d - b*c)**5 + b*c*d**4)/(2*b*d**5))/(a*d - b*c)**5 - d**4*log
(x + (a**6*d**10/(a*d - b*c)**5 - 6*a**5*b*c*d**9/(a*d - b*c)**5 + 15*a**4*b**2*
c**2*d**8/(a*d - b*c)**5 - 20*a**3*b**3*c**3*d**7/(a*d - b*c)**5 + 15*a**2*b**4*
c**4*d**6/(a*d - b*c)**5 - 6*a*b**5*c**5*d**5/(a*d - b*c)**5 + a*d**5 + b**6*c**
6*d**4/(a*d - b*c)**5 + b*c*d**4)/(2*b*d**5))/(a*d - b*c)**5 + (25*a**3*d**3 - 2
3*a**2*b*c*d**2 + 13*a*b**2*c**2*d - 3*b**3*c**3 + 12*b**3*d**3*x**3 + x**2*(42*
a*b**2*d**3 - 6*b**3*c*d**2) + x*(52*a**2*b*d**3 - 20*a*b**2*c*d**2 + 4*b**3*c**
2*d))/(12*a**8*d**4 - 48*a**7*b*c*d**3 + 72*a**6*b**2*c**2*d**2 - 48*a**5*b**3*c
**3*d + 12*a**4*b**4*c**4 + x**4*(12*a**4*b**4*d**4 - 48*a**3*b**5*c*d**3 + 72*a
**2*b**6*c**2*d**2 - 48*a*b**7*c**3*d + 12*b**8*c**4) + x**3*(48*a**5*b**3*d**4
- 192*a**4*b**4*c*d**3 + 288*a**3*b**5*c**2*d**2 - 192*a**2*b**6*c**3*d + 48*a*b
**7*c**4) + x**2*(72*a**6*b**2*d**4 - 288*a**5*b**3*c*d**3 + 432*a**4*b**4*c**2*
d**2 - 288*a**3*b**5*c**3*d + 72*a**2*b**6*c**4) + x*(48*a**7*b*d**4 - 192*a**6*
b**2*c*d**3 + 288*a**5*b**3*c**2*d**2 - 192*a**4*b**4*c**3*d + 48*a**3*b**5*c**4
))

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GIAC/XCAS [A]  time = 0.213312, size = 456, normalized size = 3.51 \[ \frac{b d^{4}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac{d^{5}{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac{3 \, b^{4} c^{4} - 16 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} - 48 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4} - 12 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} x}{12 \,{\left (b c - a d\right )}^{5}{\left (b x + a\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^4),x, algorithm="giac")

[Out]

b*d^4*ln(abs(b*x + a))/(b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^
3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5) - d^5*ln(abs(d*x + c))/(b^5*c^5*d - 5*a
*b^4*c^4*d^2 + 10*a^2*b^3*c^3*d^3 - 10*a^3*b^2*c^2*d^4 + 5*a^4*b*c*d^5 - a^5*d^6
) - 1/12*(3*b^4*c^4 - 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 - 48*a^3*b*c*d^3 + 25*
a^4*d^4 - 12*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(b^4*c^2*d^2 - 8*a*b^3*c*d^3 + 7*a^
2*b^2*d^4)*x^2 - 4*(b^4*c^3*d - 6*a*b^3*c^2*d^2 + 18*a^2*b^2*c*d^3 - 13*a^3*b*d^
4)*x)/((b*c - a*d)^5*(b*x + a)^4)