3.1295 \(\int \frac{(c+d x)^7}{(a+b x)^{13}} \, dx\)

Optimal. Leaf size=151 \[ -\frac{d^4 (c+d x)^8}{3960 (a+b x)^8 (b c-a d)^5}+\frac{d^3 (c+d x)^8}{495 (a+b x)^9 (b c-a d)^4}-\frac{d^2 (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^3}+\frac{d (c+d x)^8}{33 (a+b x)^{11} (b c-a d)^2}-\frac{(c+d x)^8}{12 (a+b x)^{12} (b c-a d)} \]

[Out]

-(c + d*x)^8/(12*(b*c - a*d)*(a + b*x)^12) + (d*(c + d*x)^8)/(33*(b*c - a*d)^2*(
a + b*x)^11) - (d^2*(c + d*x)^8)/(110*(b*c - a*d)^3*(a + b*x)^10) + (d^3*(c + d*
x)^8)/(495*(b*c - a*d)^4*(a + b*x)^9) - (d^4*(c + d*x)^8)/(3960*(b*c - a*d)^5*(a
+ b*x)^8)

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Rubi [A]  time = 0.136383, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{d^4 (c+d x)^8}{3960 (a+b x)^8 (b c-a d)^5}+\frac{d^3 (c+d x)^8}{495 (a+b x)^9 (b c-a d)^4}-\frac{d^2 (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^3}+\frac{d (c+d x)^8}{33 (a+b x)^{11} (b c-a d)^2}-\frac{(c+d x)^8}{12 (a+b x)^{12} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^7/(a + b*x)^13,x]

[Out]

-(c + d*x)^8/(12*(b*c - a*d)*(a + b*x)^12) + (d*(c + d*x)^8)/(33*(b*c - a*d)^2*(
a + b*x)^11) - (d^2*(c + d*x)^8)/(110*(b*c - a*d)^3*(a + b*x)^10) + (d^3*(c + d*
x)^8)/(495*(b*c - a*d)^4*(a + b*x)^9) - (d^4*(c + d*x)^8)/(3960*(b*c - a*d)^5*(a
+ b*x)^8)

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Rubi in Sympy [A]  time = 31.8089, size = 128, normalized size = 0.85 \[ \frac{d^{4} \left (c + d x\right )^{8}}{3960 \left (a + b x\right )^{8} \left (a d - b c\right )^{5}} + \frac{d^{3} \left (c + d x\right )^{8}}{495 \left (a + b x\right )^{9} \left (a d - b c\right )^{4}} + \frac{d^{2} \left (c + d x\right )^{8}}{110 \left (a + b x\right )^{10} \left (a d - b c\right )^{3}} + \frac{d \left (c + d x\right )^{8}}{33 \left (a + b x\right )^{11} \left (a d - b c\right )^{2}} + \frac{\left (c + d x\right )^{8}}{12 \left (a + b x\right )^{12} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**7/(b*x+a)**13,x)

[Out]

d**4*(c + d*x)**8/(3960*(a + b*x)**8*(a*d - b*c)**5) + d**3*(c + d*x)**8/(495*(a
+ b*x)**9*(a*d - b*c)**4) + d**2*(c + d*x)**8/(110*(a + b*x)**10*(a*d - b*c)**3
) + d*(c + d*x)**8/(33*(a + b*x)**11*(a*d - b*c)**2) + (c + d*x)**8/(12*(a + b*x
)**12*(a*d - b*c))

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Mathematica [B]  time = 0.29688, size = 371, normalized size = 2.46 \[ -\frac{a^7 d^7+a^6 b d^6 (5 c+12 d x)+3 a^5 b^2 d^5 \left (5 c^2+20 c d x+22 d^2 x^2\right )+5 a^4 b^3 d^4 \left (7 c^3+36 c^2 d x+66 c d^2 x^2+44 d^3 x^3\right )+5 a^3 b^4 d^3 \left (14 c^4+84 c^3 d x+198 c^2 d^2 x^2+220 c d^3 x^3+99 d^4 x^4\right )+3 a^2 b^5 d^2 \left (42 c^5+280 c^4 d x+770 c^3 d^2 x^2+1100 c^2 d^3 x^3+825 c d^4 x^4+264 d^5 x^5\right )+a b^6 d \left (210 c^6+1512 c^5 d x+4620 c^4 d^2 x^2+7700 c^3 d^3 x^3+7425 c^2 d^4 x^4+3960 c d^5 x^5+924 d^6 x^6\right )+b^7 \left (330 c^7+2520 c^6 d x+8316 c^5 d^2 x^2+15400 c^4 d^3 x^3+17325 c^3 d^4 x^4+11880 c^2 d^5 x^5+4620 c d^6 x^6+792 d^7 x^7\right )}{3960 b^8 (a+b x)^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^7/(a + b*x)^13,x]

[Out]

-(a^7*d^7 + a^6*b*d^6*(5*c + 12*d*x) + 3*a^5*b^2*d^5*(5*c^2 + 20*c*d*x + 22*d^2*
x^2) + 5*a^4*b^3*d^4*(7*c^3 + 36*c^2*d*x + 66*c*d^2*x^2 + 44*d^3*x^3) + 5*a^3*b^
4*d^3*(14*c^4 + 84*c^3*d*x + 198*c^2*d^2*x^2 + 220*c*d^3*x^3 + 99*d^4*x^4) + 3*a
^2*b^5*d^2*(42*c^5 + 280*c^4*d*x + 770*c^3*d^2*x^2 + 1100*c^2*d^3*x^3 + 825*c*d^
4*x^4 + 264*d^5*x^5) + a*b^6*d*(210*c^6 + 1512*c^5*d*x + 4620*c^4*d^2*x^2 + 7700
*c^3*d^3*x^3 + 7425*c^2*d^4*x^4 + 3960*c*d^5*x^5 + 924*d^6*x^6) + b^7*(330*c^7 +
2520*c^6*d*x + 8316*c^5*d^2*x^2 + 15400*c^4*d^3*x^3 + 17325*c^3*d^4*x^4 + 11880
*c^2*d^5*x^5 + 4620*c*d^6*x^6 + 792*d^7*x^7))/(3960*b^8*(a + b*x)^12)

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Maple [B]  time = 0.013, size = 464, normalized size = 3.1 \[ -{\frac{{d}^{7}}{5\,{b}^{8} \left ( bx+a \right ) ^{5}}}+{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{6\,{b}^{8} \left ( bx+a \right ) ^{6}}}-{\frac{-{a}^{7}{d}^{7}+7\,c{d}^{6}{a}^{6}b-21\,{a}^{5}{c}^{2}{d}^{5}{b}^{2}+35\,{a}^{4}{b}^{3}{c}^{3}{d}^{4}-35\,{a}^{3}{b}^{4}{c}^{4}{d}^{3}+21\,{a}^{2}{c}^{5}{d}^{2}{b}^{5}-7\,a{b}^{6}{c}^{6}d+{c}^{7}{b}^{7}}{12\,{b}^{8} \left ( bx+a \right ) ^{12}}}-3\,{\frac{{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{{b}^{8} \left ( bx+a \right ) ^{7}}}+{\frac{35\,{d}^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}+{\frac{21\,{d}^{2} \left ({a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5} \right ) }{10\,{b}^{8} \left ( bx+a \right ) ^{10}}}-{\frac{7\,d \left ({a}^{6}{d}^{6}-6\,{a}^{5}bc{d}^{5}+15\,{a}^{4}{b}^{2}{c}^{2}{d}^{4}-20\,{a}^{3}{b}^{3}{c}^{3}{d}^{3}+15\,{a}^{2}{b}^{4}{c}^{4}{d}^{2}-6\,a{b}^{5}{c}^{5}d+{b}^{6}{c}^{6} \right ) }{11\,{b}^{8} \left ( bx+a \right ) ^{11}}}-{\frac{35\,{d}^{3} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{9\,{b}^{8} \left ( bx+a \right ) ^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^7/(b*x+a)^13,x)

[Out]

-1/5*d^7/b^8/(b*x+a)^5+7/6*d^6*(a*d-b*c)/b^8/(b*x+a)^6-1/12*(-a^7*d^7+7*a^6*b*c*
d^6-21*a^5*b^2*c^2*d^5+35*a^4*b^3*c^3*d^4-35*a^3*b^4*c^4*d^3+21*a^2*b^5*c^5*d^2-
7*a*b^6*c^6*d+b^7*c^7)/b^8/(b*x+a)^12-3*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^8/(b*x
+a)^7+35/8*d^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^8/(b*x+a)^8+21/10
*d^2*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-
b^5*c^5)/b^8/(b*x+a)^10-7/11*d*(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*
b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)/b^8/(b*x+a)^11-35/9*d^3*(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^8/(b*x+a)^9

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Maxima [A]  time = 1.40713, size = 784, normalized size = 5.19 \[ -\frac{792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \,{\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \,{\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \,{\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \,{\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \,{\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \,{\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \,{\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^13,x, algorithm="maxima")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 +
70*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 +
a^7*d^7 + 924*(5*b^7*c*d^6 + a*b^6*d^7)*x^6 + 792*(15*b^7*c^2*d^5 + 5*a*b^6*c*d^
6 + a^2*b^5*d^7)*x^5 + 495*(35*b^7*c^3*d^4 + 15*a*b^6*c^2*d^5 + 5*a^2*b^5*c*d^6
+ a^3*b^4*d^7)*x^4 + 220*(70*b^7*c^4*d^3 + 35*a*b^6*c^3*d^4 + 15*a^2*b^5*c^2*d^5
+ 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 66*(126*b^7*c^5*d^2 + 70*a*b^6*c^4*d^3 +
35*a^2*b^5*c^3*d^4 + 15*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 +
12*(210*b^7*c^6*d + 126*a*b^6*c^5*d^2 + 70*a^2*b^5*c^4*d^3 + 35*a^3*b^4*c^3*d^4
+ 15*a^4*b^3*c^2*d^5 + 5*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^20*x^12 + 12*a*b^19*x^
11 + 66*a^2*b^18*x^10 + 220*a^3*b^17*x^9 + 495*a^4*b^16*x^8 + 792*a^5*b^15*x^7 +
924*a^6*b^14*x^6 + 792*a^7*b^13*x^5 + 495*a^8*b^12*x^4 + 220*a^9*b^11*x^3 + 66*
a^10*b^10*x^2 + 12*a^11*b^9*x + a^12*b^8)

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Fricas [A]  time = 0.228861, size = 784, normalized size = 5.19 \[ -\frac{792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \,{\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \,{\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \,{\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \,{\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \,{\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \,{\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \,{\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^13,x, algorithm="fricas")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 +
70*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 +
a^7*d^7 + 924*(5*b^7*c*d^6 + a*b^6*d^7)*x^6 + 792*(15*b^7*c^2*d^5 + 5*a*b^6*c*d^
6 + a^2*b^5*d^7)*x^5 + 495*(35*b^7*c^3*d^4 + 15*a*b^6*c^2*d^5 + 5*a^2*b^5*c*d^6
+ a^3*b^4*d^7)*x^4 + 220*(70*b^7*c^4*d^3 + 35*a*b^6*c^3*d^4 + 15*a^2*b^5*c^2*d^5
+ 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 66*(126*b^7*c^5*d^2 + 70*a*b^6*c^4*d^3 +
35*a^2*b^5*c^3*d^4 + 15*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 +
12*(210*b^7*c^6*d + 126*a*b^6*c^5*d^2 + 70*a^2*b^5*c^4*d^3 + 35*a^3*b^4*c^3*d^4
+ 15*a^4*b^3*c^2*d^5 + 5*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^20*x^12 + 12*a*b^19*x^
11 + 66*a^2*b^18*x^10 + 220*a^3*b^17*x^9 + 495*a^4*b^16*x^8 + 792*a^5*b^15*x^7 +
924*a^6*b^14*x^6 + 792*a^7*b^13*x^5 + 495*a^8*b^12*x^4 + 220*a^9*b^11*x^3 + 66*
a^10*b^10*x^2 + 12*a^11*b^9*x + a^12*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)**13,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228662, size = 670, normalized size = 4.44 \[ -\frac{792 \, b^{7} d^{7} x^{7} + 4620 \, b^{7} c d^{6} x^{6} + 924 \, a b^{6} d^{7} x^{6} + 11880 \, b^{7} c^{2} d^{5} x^{5} + 3960 \, a b^{6} c d^{6} x^{5} + 792 \, a^{2} b^{5} d^{7} x^{5} + 17325 \, b^{7} c^{3} d^{4} x^{4} + 7425 \, a b^{6} c^{2} d^{5} x^{4} + 2475 \, a^{2} b^{5} c d^{6} x^{4} + 495 \, a^{3} b^{4} d^{7} x^{4} + 15400 \, b^{7} c^{4} d^{3} x^{3} + 7700 \, a b^{6} c^{3} d^{4} x^{3} + 3300 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1100 \, a^{3} b^{4} c d^{6} x^{3} + 220 \, a^{4} b^{3} d^{7} x^{3} + 8316 \, b^{7} c^{5} d^{2} x^{2} + 4620 \, a b^{6} c^{4} d^{3} x^{2} + 2310 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 990 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 330 \, a^{4} b^{3} c d^{6} x^{2} + 66 \, a^{5} b^{2} d^{7} x^{2} + 2520 \, b^{7} c^{6} d x + 1512 \, a b^{6} c^{5} d^{2} x + 840 \, a^{2} b^{5} c^{4} d^{3} x + 420 \, a^{3} b^{4} c^{3} d^{4} x + 180 \, a^{4} b^{3} c^{2} d^{5} x + 60 \, a^{5} b^{2} c d^{6} x + 12 \, a^{6} b d^{7} x + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7}}{3960 \,{\left (b x + a\right )}^{12} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^13,x, algorithm="giac")

[Out]

-1/3960*(792*b^7*d^7*x^7 + 4620*b^7*c*d^6*x^6 + 924*a*b^6*d^7*x^6 + 11880*b^7*c^
2*d^5*x^5 + 3960*a*b^6*c*d^6*x^5 + 792*a^2*b^5*d^7*x^5 + 17325*b^7*c^3*d^4*x^4 +
7425*a*b^6*c^2*d^5*x^4 + 2475*a^2*b^5*c*d^6*x^4 + 495*a^3*b^4*d^7*x^4 + 15400*b
^7*c^4*d^3*x^3 + 7700*a*b^6*c^3*d^4*x^3 + 3300*a^2*b^5*c^2*d^5*x^3 + 1100*a^3*b^
4*c*d^6*x^3 + 220*a^4*b^3*d^7*x^3 + 8316*b^7*c^5*d^2*x^2 + 4620*a*b^6*c^4*d^3*x^
2 + 2310*a^2*b^5*c^3*d^4*x^2 + 990*a^3*b^4*c^2*d^5*x^2 + 330*a^4*b^3*c*d^6*x^2 +
66*a^5*b^2*d^7*x^2 + 2520*b^7*c^6*d*x + 1512*a*b^6*c^5*d^2*x + 840*a^2*b^5*c^4*
d^3*x + 420*a^3*b^4*c^3*d^4*x + 180*a^4*b^3*c^2*d^5*x + 60*a^5*b^2*c*d^6*x + 12*
a^6*b*d^7*x + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c
^4*d^3 + 35*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + a^7*d^7)/((b*
x + a)^12*b^8)