3.1289 \(\int \frac{(c+d x)^7}{(a+b x)^7} \, dx\)

Optimal. Leaf size=186 \[ \frac{7 d^6 (b c-a d) \log (a+b x)}{b^8}-\frac{21 d^5 (b c-a d)^2}{b^8 (a+b x)}-\frac{35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac{35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac{21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac{7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac{(b c-a d)^7}{6 b^8 (a+b x)^6}+\frac{d^7 x}{b^7} \]

[Out]

(d^7*x)/b^7 - (b*c - a*d)^7/(6*b^8*(a + b*x)^6) - (7*d*(b*c - a*d)^6)/(5*b^8*(a
+ b*x)^5) - (21*d^2*(b*c - a*d)^5)/(4*b^8*(a + b*x)^4) - (35*d^3*(b*c - a*d)^4)/
(3*b^8*(a + b*x)^3) - (35*d^4*(b*c - a*d)^3)/(2*b^8*(a + b*x)^2) - (21*d^5*(b*c
- a*d)^2)/(b^8*(a + b*x)) + (7*d^6*(b*c - a*d)*Log[a + b*x])/b^8

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Rubi [A]  time = 0.424064, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{7 d^6 (b c-a d) \log (a+b x)}{b^8}-\frac{21 d^5 (b c-a d)^2}{b^8 (a+b x)}-\frac{35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac{35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac{21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac{7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac{(b c-a d)^7}{6 b^8 (a+b x)^6}+\frac{d^7 x}{b^7} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^7/(a + b*x)^7,x]

[Out]

(d^7*x)/b^7 - (b*c - a*d)^7/(6*b^8*(a + b*x)^6) - (7*d*(b*c - a*d)^6)/(5*b^8*(a
+ b*x)^5) - (21*d^2*(b*c - a*d)^5)/(4*b^8*(a + b*x)^4) - (35*d^3*(b*c - a*d)^4)/
(3*b^8*(a + b*x)^3) - (35*d^4*(b*c - a*d)^3)/(2*b^8*(a + b*x)^2) - (21*d^5*(b*c
- a*d)^2)/(b^8*(a + b*x)) + (7*d^6*(b*c - a*d)*Log[a + b*x])/b^8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{7} \int \frac{1}{b^{7}}\, dx - \frac{7 d^{6} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{8}} - \frac{21 d^{5} \left (a d - b c\right )^{2}}{b^{8} \left (a + b x\right )} + \frac{35 d^{4} \left (a d - b c\right )^{3}}{2 b^{8} \left (a + b x\right )^{2}} - \frac{35 d^{3} \left (a d - b c\right )^{4}}{3 b^{8} \left (a + b x\right )^{3}} + \frac{21 d^{2} \left (a d - b c\right )^{5}}{4 b^{8} \left (a + b x\right )^{4}} - \frac{7 d \left (a d - b c\right )^{6}}{5 b^{8} \left (a + b x\right )^{5}} + \frac{\left (a d - b c\right )^{7}}{6 b^{8} \left (a + b x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**7/(b*x+a)**7,x)

[Out]

d**7*Integral(b**(-7), x) - 7*d**6*(a*d - b*c)*log(a + b*x)/b**8 - 21*d**5*(a*d
- b*c)**2/(b**8*(a + b*x)) + 35*d**4*(a*d - b*c)**3/(2*b**8*(a + b*x)**2) - 35*d
**3*(a*d - b*c)**4/(3*b**8*(a + b*x)**3) + 21*d**2*(a*d - b*c)**5/(4*b**8*(a + b
*x)**4) - 7*d*(a*d - b*c)**6/(5*b**8*(a + b*x)**5) + (a*d - b*c)**7/(6*b**8*(a +
b*x)**6)

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Mathematica [B]  time = 0.577239, size = 390, normalized size = 2.1 \[ -\frac{669 a^7 d^7+3 a^6 b d^6 (1198 d x-343 c)+3 a^5 b^2 d^5 \left (70 c^2-1918 c d x+2575 d^2 x^2\right )+5 a^4 b^3 d^4 \left (14 c^3+252 c^2 d x-2625 c d^2 x^2+1640 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+84 c^3 d x+630 c^2 d^2 x^2-3080 c d^3 x^3+810 d^4 x^4\right )+3 a^2 b^5 d^2 \left (7 c^5+70 c^4 d x+350 c^3 d^2 x^2+1400 c^2 d^3 x^3-3150 c d^4 x^4+120 d^5 x^5\right )+a b^6 d \left (14 c^6+126 c^5 d x+525 c^4 d^2 x^2+1400 c^3 d^3 x^3+3150 c^2 d^4 x^4-2520 c d^5 x^5-360 d^6 x^6\right )+420 d^6 (a+b x)^6 (a d-b c) \log (a+b x)+b^7 \left (10 c^7+84 c^6 d x+315 c^5 d^2 x^2+700 c^4 d^3 x^3+1050 c^3 d^4 x^4+1260 c^2 d^5 x^5-60 d^7 x^7\right )}{60 b^8 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^7/(a + b*x)^7,x]

[Out]

-(669*a^7*d^7 + 3*a^6*b*d^6*(-343*c + 1198*d*x) + 3*a^5*b^2*d^5*(70*c^2 - 1918*c
*d*x + 2575*d^2*x^2) + 5*a^4*b^3*d^4*(14*c^3 + 252*c^2*d*x - 2625*c*d^2*x^2 + 16
40*d^3*x^3) + 5*a^3*b^4*d^3*(7*c^4 + 84*c^3*d*x + 630*c^2*d^2*x^2 - 3080*c*d^3*x
^3 + 810*d^4*x^4) + 3*a^2*b^5*d^2*(7*c^5 + 70*c^4*d*x + 350*c^3*d^2*x^2 + 1400*c
^2*d^3*x^3 - 3150*c*d^4*x^4 + 120*d^5*x^5) + a*b^6*d*(14*c^6 + 126*c^5*d*x + 525
*c^4*d^2*x^2 + 1400*c^3*d^3*x^3 + 3150*c^2*d^4*x^4 - 2520*c*d^5*x^5 - 360*d^6*x^
6) + b^7*(10*c^7 + 84*c^6*d*x + 315*c^5*d^2*x^2 + 700*c^4*d^3*x^3 + 1050*c^3*d^4
*x^4 + 1260*c^2*d^5*x^5 - 60*d^7*x^7) + 420*d^6*(-(b*c) + a*d)*(a + b*x)^6*Log[a
+ b*x])/(60*b^8*(a + b*x)^6)

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Maple [B]  time = 0.019, size = 666, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^7/(b*x+a)^7,x)

[Out]

d^7*x/b^7+35/2/b^8*d^7/(b*x+a)^2*a^3-35/2/b^5*d^4/(b*x+a)^2*c^3-21/b^8*d^7/(b*x+
a)*a^2-21/b^6*d^5/(b*x+a)*c^2+7/b^7*d^6*ln(b*x+a)*c+1/6/b^8/(b*x+a)^6*a^7*d^7+21
/4/b^8*d^7/(b*x+a)^4*a^5-21/4/b^3*d^2/(b*x+a)^4*c^5-35/3/b^8*d^7/(b*x+a)^3*a^4-3
5/3/b^4*d^3/(b*x+a)^3*c^4-7/5/b^8*d^7/(b*x+a)^5*a^6-7/5/b^2*d/(b*x+a)^5*c^6-7/b^
8*d^7*ln(b*x+a)*a-1/6/b/(b*x+a)^6*c^7+28/b^5*d^4/(b*x+a)^5*a^3*c^3-21/b^4*d^3/(b
*x+a)^5*a^2*c^4+42/5/b^3*d^2/(b*x+a)^5*a*c^5-21/b^6*d^5/(b*x+a)^5*a^4*c^2+105/4/
b^4*d^3/(b*x+a)^4*a*c^4+140/3/b^7*d^6/(b*x+a)^3*a^3*c+42/5/b^7*d^6/(b*x+a)^5*a^5
*c-35/6/b^5/(b*x+a)^6*a^4*c^3*d^4+35/6/b^4/(b*x+a)^6*a^3*c^4*d^3-7/2/b^3/(b*x+a)
^6*a^2*c^5*d^2+7/6/b^2/(b*x+a)^6*a*c^6*d-105/4/b^7*d^6/(b*x+a)^4*a^4*c+105/2/b^6
*d^5/(b*x+a)^4*a^3*c^2-105/2/b^5*d^4/(b*x+a)^4*a^2*c^3-105/2/b^7*d^6/(b*x+a)^2*a
^2*c+105/2/b^6*d^5/(b*x+a)^2*a*c^2+42/b^7*d^6/(b*x+a)*a*c-7/6/b^7/(b*x+a)^6*a^6*
c*d^6+7/2/b^6/(b*x+a)^6*a^5*c^2*d^5-70/b^6*d^5/(b*x+a)^3*a^2*c^2+140/3/b^5*d^4/(
b*x+a)^3*a*c^3

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Maxima [A]  time = 1.44381, size = 697, normalized size = 3.75 \[ \frac{d^{7} x}{b^{7}} - \frac{10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \,{\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \,{\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \,{\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \,{\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \,{\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \,{\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac{7 \,{\left (b c d^{6} - a d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^7,x, algorithm="maxima")

[Out]

d^7*x/b^7 - 1/60*(10*b^7*c^7 + 14*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*
c^4*d^3 + 70*a^4*b^3*c^3*d^4 + 210*a^5*b^2*c^2*d^5 - 1029*a^6*b*c*d^6 + 669*a^7*
d^7 + 1260*(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1050*(b^7*c^3*d^4 +
3*a*b^6*c^2*d^5 - 9*a^2*b^5*c*d^6 + 5*a^3*b^4*d^7)*x^4 + 700*(b^7*c^4*d^3 + 2*a
*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 22*a^3*b^4*c*d^6 + 13*a^4*b^3*d^7)*x^3 + 105*
(3*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 - 125
*a^4*b^3*c*d^6 + 77*a^5*b^2*d^7)*x^2 + 42*(2*b^7*c^6*d + 3*a*b^6*c^5*d^2 + 5*a^2
*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 + 30*a^4*b^3*c^2*d^5 - 137*a^5*b^2*c*d^6 + 87*
a^6*b*d^7)*x)/(b^14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 20*a^3*b^11*x^3 + 15*
a^4*b^10*x^2 + 6*a^5*b^9*x + a^6*b^8) + 7*(b*c*d^6 - a*d^7)*log(b*x + a)/b^8

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Fricas [A]  time = 0.225912, size = 934, normalized size = 5.02 \[ \frac{60 \, b^{7} d^{7} x^{7} + 360 \, a b^{6} d^{7} x^{6} - 10 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 70 \, a^{4} b^{3} c^{3} d^{4} - 210 \, a^{5} b^{2} c^{2} d^{5} + 1029 \, a^{6} b c d^{6} - 669 \, a^{7} d^{7} - 180 \,{\left (7 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + 2 \, a^{2} b^{5} d^{7}\right )} x^{5} - 150 \,{\left (7 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} - 63 \, a^{2} b^{5} c d^{6} + 27 \, a^{3} b^{4} d^{7}\right )} x^{4} - 100 \,{\left (7 \, b^{7} c^{4} d^{3} + 14 \, a b^{6} c^{3} d^{4} + 42 \, a^{2} b^{5} c^{2} d^{5} - 154 \, a^{3} b^{4} c d^{6} + 82 \, a^{4} b^{3} d^{7}\right )} x^{3} - 15 \,{\left (21 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 70 \, a^{2} b^{5} c^{3} d^{4} + 210 \, a^{3} b^{4} c^{2} d^{5} - 875 \, a^{4} b^{3} c d^{6} + 515 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \,{\left (14 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 70 \, a^{3} b^{4} c^{3} d^{4} + 210 \, a^{4} b^{3} c^{2} d^{5} - 959 \, a^{5} b^{2} c d^{6} + 599 \, a^{6} b d^{7}\right )} x + 420 \,{\left (a^{6} b c d^{6} - a^{7} d^{7} +{\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 6 \,{\left (a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 15 \,{\left (a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 20 \,{\left (a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 15 \,{\left (a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 6 \,{\left (a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{60 \,{\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^7,x, algorithm="fricas")

[Out]

1/60*(60*b^7*d^7*x^7 + 360*a*b^6*d^7*x^6 - 10*b^7*c^7 - 14*a*b^6*c^6*d - 21*a^2*
b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 - 70*a^4*b^3*c^3*d^4 - 210*a^5*b^2*c^2*d^5 + 10
29*a^6*b*c*d^6 - 669*a^7*d^7 - 180*(7*b^7*c^2*d^5 - 14*a*b^6*c*d^6 + 2*a^2*b^5*d
^7)*x^5 - 150*(7*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 - 63*a^2*b^5*c*d^6 + 27*a^3*b^4*
d^7)*x^4 - 100*(7*b^7*c^4*d^3 + 14*a*b^6*c^3*d^4 + 42*a^2*b^5*c^2*d^5 - 154*a^3*
b^4*c*d^6 + 82*a^4*b^3*d^7)*x^3 - 15*(21*b^7*c^5*d^2 + 35*a*b^6*c^4*d^3 + 70*a^2
*b^5*c^3*d^4 + 210*a^3*b^4*c^2*d^5 - 875*a^4*b^3*c*d^6 + 515*a^5*b^2*d^7)*x^2 -
6*(14*b^7*c^6*d + 21*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 70*a^3*b^4*c^3*d^4 + 2
10*a^4*b^3*c^2*d^5 - 959*a^5*b^2*c*d^6 + 599*a^6*b*d^7)*x + 420*(a^6*b*c*d^6 - a
^7*d^7 + (b^7*c*d^6 - a*b^6*d^7)*x^6 + 6*(a*b^6*c*d^6 - a^2*b^5*d^7)*x^5 + 15*(a
^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 20*(a^3*b^4*c*d^6 - a^4*b^3*d^7)*x^3 + 15*(a^4
*b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 6*(a^5*b^2*c*d^6 - a^6*b*d^7)*x)*log(b*x + a))/(
b^14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 20*a^3*b^11*x^3 + 15*a^4*b^10*x^2 +
6*a^5*b^9*x + a^6*b^8)

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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((d*x+c)**7/(b*x+a)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22422, size = 620, normalized size = 3.33 \[ \frac{d^{7} x}{b^{7}} + \frac{7 \,{\left (b c d^{6} - a d^{7}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{8}} - \frac{10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \,{\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \,{\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \,{\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \,{\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \,{\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \,{\left (b x + a\right )}^{6} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^7,x, algorithm="giac")

[Out]

d^7*x/b^7 + 7*(b*c*d^6 - a*d^7)*ln(abs(b*x + a))/b^8 - 1/60*(10*b^7*c^7 + 14*a*b
^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 70*a^4*b^3*c^3*d^4 + 210*a^
5*b^2*c^2*d^5 - 1029*a^6*b*c*d^6 + 669*a^7*d^7 + 1260*(b^7*c^2*d^5 - 2*a*b^6*c*d
^6 + a^2*b^5*d^7)*x^5 + 1050*(b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 - 9*a^2*b^5*c*d^6 +
5*a^3*b^4*d^7)*x^4 + 700*(b^7*c^4*d^3 + 2*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 22
*a^3*b^4*c*d^6 + 13*a^4*b^3*d^7)*x^3 + 105*(3*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 10
*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 - 125*a^4*b^3*c*d^6 + 77*a^5*b^2*d^7)*x^2
+ 42*(2*b^7*c^6*d + 3*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 + 3
0*a^4*b^3*c^2*d^5 - 137*a^5*b^2*c*d^6 + 87*a^6*b*d^7)*x)/((b*x + a)^6*b^8)