3.1322 \(\int \frac{(c+d x)^{10}}{(a+b x)^{11}} \, dx\)

Optimal. Leaf size=271 \[ -\frac{10 d^9 (b c-a d)}{b^{11} (a+b x)}-\frac{45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac{40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac{105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac{252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac{35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac{120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac{45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac{10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac{(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}+\frac{d^{10} \log (a+b x)}{b^{11}} \]

[Out]

-(b*c - a*d)^10/(10*b^11*(a + b*x)^10) - (10*d*(b*c - a*d)^9)/(9*b^11*(a + b*x)^
9) - (45*d^2*(b*c - a*d)^8)/(8*b^11*(a + b*x)^8) - (120*d^3*(b*c - a*d)^7)/(7*b^
11*(a + b*x)^7) - (35*d^4*(b*c - a*d)^6)/(b^11*(a + b*x)^6) - (252*d^5*(b*c - a*
d)^5)/(5*b^11*(a + b*x)^5) - (105*d^6*(b*c - a*d)^4)/(2*b^11*(a + b*x)^4) - (40*
d^7*(b*c - a*d)^3)/(b^11*(a + b*x)^3) - (45*d^8*(b*c - a*d)^2)/(2*b^11*(a + b*x)
^2) - (10*d^9*(b*c - a*d))/(b^11*(a + b*x)) + (d^10*Log[a + b*x])/b^11

_______________________________________________________________________________________

Rubi [A]  time = 0.796309, antiderivative size = 271, 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 \[ -\frac{10 d^9 (b c-a d)}{b^{11} (a+b x)}-\frac{45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac{40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac{105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac{252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac{35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac{120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac{45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac{10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac{(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}+\frac{d^{10} \log (a+b x)}{b^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^10/(a + b*x)^11,x]

[Out]

-(b*c - a*d)^10/(10*b^11*(a + b*x)^10) - (10*d*(b*c - a*d)^9)/(9*b^11*(a + b*x)^
9) - (45*d^2*(b*c - a*d)^8)/(8*b^11*(a + b*x)^8) - (120*d^3*(b*c - a*d)^7)/(7*b^
11*(a + b*x)^7) - (35*d^4*(b*c - a*d)^6)/(b^11*(a + b*x)^6) - (252*d^5*(b*c - a*
d)^5)/(5*b^11*(a + b*x)^5) - (105*d^6*(b*c - a*d)^4)/(2*b^11*(a + b*x)^4) - (40*
d^7*(b*c - a*d)^3)/(b^11*(a + b*x)^3) - (45*d^8*(b*c - a*d)^2)/(2*b^11*(a + b*x)
^2) - (10*d^9*(b*c - a*d))/(b^11*(a + b*x)) + (d^10*Log[a + b*x])/b^11

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 129.327, size = 252, normalized size = 0.93 \[ \frac{d^{10} \log{\left (a + b x \right )}}{b^{11}} + \frac{10 d^{9} \left (a d - b c\right )}{b^{11} \left (a + b x\right )} - \frac{45 d^{8} \left (a d - b c\right )^{2}}{2 b^{11} \left (a + b x\right )^{2}} + \frac{40 d^{7} \left (a d - b c\right )^{3}}{b^{11} \left (a + b x\right )^{3}} - \frac{105 d^{6} \left (a d - b c\right )^{4}}{2 b^{11} \left (a + b x\right )^{4}} + \frac{252 d^{5} \left (a d - b c\right )^{5}}{5 b^{11} \left (a + b x\right )^{5}} - \frac{35 d^{4} \left (a d - b c\right )^{6}}{b^{11} \left (a + b x\right )^{6}} + \frac{120 d^{3} \left (a d - b c\right )^{7}}{7 b^{11} \left (a + b x\right )^{7}} - \frac{45 d^{2} \left (a d - b c\right )^{8}}{8 b^{11} \left (a + b x\right )^{8}} + \frac{10 d \left (a d - b c\right )^{9}}{9 b^{11} \left (a + b x\right )^{9}} - \frac{\left (a d - b c\right )^{10}}{10 b^{11} \left (a + b x\right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**11,x)

[Out]

d**10*log(a + b*x)/b**11 + 10*d**9*(a*d - b*c)/(b**11*(a + b*x)) - 45*d**8*(a*d
- b*c)**2/(2*b**11*(a + b*x)**2) + 40*d**7*(a*d - b*c)**3/(b**11*(a + b*x)**3) -
105*d**6*(a*d - b*c)**4/(2*b**11*(a + b*x)**4) + 252*d**5*(a*d - b*c)**5/(5*b**
11*(a + b*x)**5) - 35*d**4*(a*d - b*c)**6/(b**11*(a + b*x)**6) + 120*d**3*(a*d -
b*c)**7/(7*b**11*(a + b*x)**7) - 45*d**2*(a*d - b*c)**8/(8*b**11*(a + b*x)**8)
+ 10*d*(a*d - b*c)**9/(9*b**11*(a + b*x)**9) - (a*d - b*c)**10/(10*b**11*(a + b*
x)**10)

_______________________________________________________________________________________

Mathematica [B]  time = 0.811004, size = 591, normalized size = 2.18 \[ \frac{d^{10} \log (a+b x)}{b^{11}}-\frac{(b c-a d) \left (7381 a^9 d^9+a^8 b d^8 (4861 c+71290 d x)+a^7 b^2 d^7 \left (3601 c^2+46090 c d x+308205 d^2 x^2\right )+a^6 b^3 d^6 \left (2761 c^3+33490 c^2 d x+194805 c d^2 x^2+784080 d^3 x^3\right )+a^5 b^4 d^5 \left (2131 c^4+25090 c^3 d x+138105 c^2 d^2 x^2+481680 c d^3 x^3+1296540 d^4 x^4\right )+a^4 b^5 d^4 \left (1627 c^5+18790 c^4 d x+100305 c^3 d^2 x^2+330480 c^2 d^3 x^3+767340 c d^4 x^4+1450008 d^5 x^5\right )+a^3 b^6 d^3 \left (1207 c^6+13750 c^5 d x+71955 c^4 d^2 x^2+229680 c^3 d^3 x^3+502740 c^2 d^4 x^4+814968 c d^5 x^5+1102500 d^6 x^6\right )+a^2 b^7 d^2 \left (847 c^7+9550 c^6 d x+49275 c^5 d^2 x^2+154080 c^4 d^3 x^3+326340 c^3 d^4 x^4+497448 c^2 d^5 x^5+573300 c d^6 x^6+554400 d^7 x^7\right )+a b^8 d \left (532 c^8+5950 c^7 d x+30375 c^6 d^2 x^2+93600 c^5 d^3 x^3+194040 c^4 d^4 x^4+285768 c^3 d^5 x^5+308700 c^2 d^6 x^6+252000 c d^7 x^7+170100 d^8 x^8\right )+b^9 \left (252 c^9+2800 c^8 d x+14175 c^7 d^2 x^2+43200 c^6 d^3 x^3+88200 c^5 d^4 x^4+127008 c^4 d^5 x^5+132300 c^3 d^6 x^6+100800 c^2 d^7 x^7+56700 c d^8 x^8+25200 d^9 x^9\right )\right )}{2520 b^{11} (a+b x)^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^10/(a + b*x)^11,x]

[Out]

-((b*c - a*d)*(7381*a^9*d^9 + a^8*b*d^8*(4861*c + 71290*d*x) + a^7*b^2*d^7*(3601
*c^2 + 46090*c*d*x + 308205*d^2*x^2) + a^6*b^3*d^6*(2761*c^3 + 33490*c^2*d*x + 1
94805*c*d^2*x^2 + 784080*d^3*x^3) + a^5*b^4*d^5*(2131*c^4 + 25090*c^3*d*x + 1381
05*c^2*d^2*x^2 + 481680*c*d^3*x^3 + 1296540*d^4*x^4) + a^4*b^5*d^4*(1627*c^5 + 1
8790*c^4*d*x + 100305*c^3*d^2*x^2 + 330480*c^2*d^3*x^3 + 767340*c*d^4*x^4 + 1450
008*d^5*x^5) + a^3*b^6*d^3*(1207*c^6 + 13750*c^5*d*x + 71955*c^4*d^2*x^2 + 22968
0*c^3*d^3*x^3 + 502740*c^2*d^4*x^4 + 814968*c*d^5*x^5 + 1102500*d^6*x^6) + a^2*b
^7*d^2*(847*c^7 + 9550*c^6*d*x + 49275*c^5*d^2*x^2 + 154080*c^4*d^3*x^3 + 326340
*c^3*d^4*x^4 + 497448*c^2*d^5*x^5 + 573300*c*d^6*x^6 + 554400*d^7*x^7) + a*b^8*d
*(532*c^8 + 5950*c^7*d*x + 30375*c^6*d^2*x^2 + 93600*c^5*d^3*x^3 + 194040*c^4*d^
4*x^4 + 285768*c^3*d^5*x^5 + 308700*c^2*d^6*x^6 + 252000*c*d^7*x^7 + 170100*d^8*
x^8) + b^9*(252*c^9 + 2800*c^8*d*x + 14175*c^7*d^2*x^2 + 43200*c^6*d^3*x^3 + 882
00*c^5*d^4*x^4 + 127008*c^4*d^5*x^5 + 132300*c^3*d^6*x^6 + 100800*c^2*d^7*x^7 +
56700*c*d^8*x^8 + 25200*d^9*x^9)))/(2520*b^11*(a + b*x)^10) + (d^10*Log[a + b*x]
)/b^11

_______________________________________________________________________________________

Maple [B]  time = 0.018, size = 1271, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^11,x)

[Out]

10/9*d^10/b^11/(b*x+a)^9*a^9-10/9*d/b^2/(b*x+a)^9*c^9+10/b^11*d^10/(b*x+a)*a-10/
b^10*d^9/(b*x+a)*c+40*d^10/b^11/(b*x+a)^3*a^3-40*d^7/b^8/(b*x+a)^3*c^3-45/2*d^10
/b^11/(b*x+a)^2*a^2-45/2*d^8/b^9/(b*x+a)^2*c^2-45/8*d^10/b^11/(b*x+a)^8*a^8-45/8
*d^2/b^3/(b*x+a)^8*c^8-1/10/b^11/(b*x+a)^10*a^10*d^10-105/2*d^10/b^11/(b*x+a)^4*
a^4-105/2*d^6/b^7/(b*x+a)^4*c^4+252/5*d^10/b^11/(b*x+a)^5*a^5-252/5*d^5/b^6/(b*x
+a)^5*c^5-35*d^10/b^11/(b*x+a)^6*a^6-35*d^4/b^5/(b*x+a)^6*c^6+120/7*d^10/b^11/(b
*x+a)^7*a^7-120/7*d^3/b^4/(b*x+a)^7*c^7-9/2/b^3/(b*x+a)^10*c^8*d^2*a^2+1/b^2/(b*
x+a)^10*c^9*d*a-252*d^9/b^10/(b*x+a)^5*a^4*c+252*d^6/b^7/(b*x+a)^5*a*c^4+210*d^9
/b^10/(b*x+a)^6*a^5*c-525*d^8/b^9/(b*x+a)^6*a^4*c^2+700*d^7/b^8/(b*x+a)^6*a^3*c^
3-525*d^6/b^7/(b*x+a)^6*a^2*c^4+210*d^5/b^6/(b*x+a)^6*a*c^5-120*d^9/b^10/(b*x+a)
^7*a^6*c+360*d^8/b^9/(b*x+a)^7*a^5*c^2-600*d^7/b^8/(b*x+a)^7*a^4*c^3-10*d^9/b^10
/(b*x+a)^9*a^8*c+40*d^8/b^9/(b*x+a)^9*a^7*c^2-1/10/b/(b*x+a)^10*c^10+600*d^6/b^7
/(b*x+a)^7*a^3*c^4-360*d^5/b^6/(b*x+a)^7*a^2*c^5+120*d^4/b^5/(b*x+a)^7*a*c^6+210
*d^9/b^10/(b*x+a)^4*a^3*c-315*d^8/b^9/(b*x+a)^4*a^2*c^2+210*d^7/b^8/(b*x+a)^4*a*
c^3-120*d^9/b^10/(b*x+a)^3*a^2*c+120*d^8/b^9/(b*x+a)^3*a*c^2+45*d^9/b^10/(b*x+a)
^2*a*c+45*d^9/b^10/(b*x+a)^8*a^7*c-315/2*d^8/b^9/(b*x+a)^8*a^6*c^2+315*d^7/b^8/(
b*x+a)^8*a^5*c^3-1575/4*d^6/b^7/(b*x+a)^8*a^4*c^4+315*d^5/b^6/(b*x+a)^8*a^3*c^5-
315/2*d^4/b^5/(b*x+a)^8*a^2*c^6+45*d^3/b^4/(b*x+a)^8*a*c^7+1/b^10/(b*x+a)^10*c*d
^9*a^9-9/2/b^9/(b*x+a)^10*c^2*d^8*a^8+d^10*ln(b*x+a)/b^11+12/b^8/(b*x+a)^10*c^3*
d^7*a^7-21/b^7/(b*x+a)^10*c^4*d^6*a^6-280/3*d^7/b^8/(b*x+a)^9*a^6*c^3+140*d^6/b^
7/(b*x+a)^9*a^5*c^4-140*d^5/b^6/(b*x+a)^9*a^4*c^5+280/3*d^4/b^5/(b*x+a)^9*a^3*c^
6-40*d^3/b^4/(b*x+a)^9*a^2*c^7+10*d^2/b^3/(b*x+a)^9*a*c^8+504*d^8/b^9/(b*x+a)^5*
a^3*c^2-504*d^7/b^8/(b*x+a)^5*a^2*c^3+126/5/b^6/(b*x+a)^10*a^5*c^5*d^5-21/b^5/(b
*x+a)^10*a^4*c^6*d^4+12/b^4/(b*x+a)^10*c^7*d^3*a^3

_______________________________________________________________________________________

Maxima [A]  time = 1.42912, size = 1316, normalized size = 4.86 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^11,x, algorithm="maxima")

[Out]

-1/2520*(252*b^10*c^10 + 280*a*b^9*c^9*d + 315*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7
*d^3 + 420*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 + 840*a^7
*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 2520*a^9*b*c*d^9 - 7381*a^10*d^10 + 25200*
(b^10*c*d^9 - a*b^9*d^10)*x^9 + 56700*(b^10*c^2*d^8 + 2*a*b^9*c*d^9 - 3*a^2*b^8*
d^10)*x^8 + 50400*(2*b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 - 11*a^3*b
^7*d^10)*x^7 + 44100*(3*b^10*c^4*d^6 + 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 12*
a^3*b^7*c*d^9 - 25*a^4*b^6*d^10)*x^6 + 10584*(12*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6
+ 20*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 + 60*a^4*b^6*c*d^9 - 137*a^5*b^5*d^10
)*x^5 + 8820*(10*b^10*c^6*d^4 + 12*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 20*a^3*b
^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 + 60*a^5*b^5*c*d^9 - 147*a^6*b^4*d^10)*x^4 + 720
*(60*b^10*c^7*d^3 + 70*a*b^9*c^6*d^4 + 84*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6
+ 140*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 + 420*a^6*b^4*c*d^9 - 1089*a^7*b^3*d
^10)*x^3 + 135*(105*b^10*c^8*d^2 + 120*a*b^9*c^7*d^3 + 140*a^2*b^8*c^6*d^4 + 168
*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7 + 420*a^6*b^4*c^2*d
^8 + 840*a^7*b^3*c*d^9 - 2283*a^8*b^2*d^10)*x^2 + 10*(280*b^10*c^9*d + 315*a*b^9
*c^8*d^2 + 360*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 + 630
*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c^2*d^8 + 2520*a^8*b^2*c*d
^9 - 7129*a^9*b*d^10)*x)/(b^21*x^10 + 10*a*b^20*x^9 + 45*a^2*b^19*x^8 + 120*a^3*
b^18*x^7 + 210*a^4*b^17*x^6 + 252*a^5*b^16*x^5 + 210*a^6*b^15*x^4 + 120*a^7*b^14
*x^3 + 45*a^8*b^13*x^2 + 10*a^9*b^12*x + a^10*b^11) + d^10*log(b*x + a)/b^11

_______________________________________________________________________________________

Fricas [A]  time = 0.232779, size = 1494, normalized size = 5.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^11,x, algorithm="fricas")

[Out]

-1/2520*(252*b^10*c^10 + 280*a*b^9*c^9*d + 315*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7
*d^3 + 420*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 + 840*a^7
*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 2520*a^9*b*c*d^9 - 7381*a^10*d^10 + 25200*
(b^10*c*d^9 - a*b^9*d^10)*x^9 + 56700*(b^10*c^2*d^8 + 2*a*b^9*c*d^9 - 3*a^2*b^8*
d^10)*x^8 + 50400*(2*b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 - 11*a^3*b
^7*d^10)*x^7 + 44100*(3*b^10*c^4*d^6 + 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 12*
a^3*b^7*c*d^9 - 25*a^4*b^6*d^10)*x^6 + 10584*(12*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6
+ 20*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 + 60*a^4*b^6*c*d^9 - 137*a^5*b^5*d^10
)*x^5 + 8820*(10*b^10*c^6*d^4 + 12*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 20*a^3*b
^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 + 60*a^5*b^5*c*d^9 - 147*a^6*b^4*d^10)*x^4 + 720
*(60*b^10*c^7*d^3 + 70*a*b^9*c^6*d^4 + 84*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6
+ 140*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 + 420*a^6*b^4*c*d^9 - 1089*a^7*b^3*d
^10)*x^3 + 135*(105*b^10*c^8*d^2 + 120*a*b^9*c^7*d^3 + 140*a^2*b^8*c^6*d^4 + 168
*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7 + 420*a^6*b^4*c^2*d
^8 + 840*a^7*b^3*c*d^9 - 2283*a^8*b^2*d^10)*x^2 + 10*(280*b^10*c^9*d + 315*a*b^9
*c^8*d^2 + 360*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 + 630
*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c^2*d^8 + 2520*a^8*b^2*c*d
^9 - 7129*a^9*b*d^10)*x - 2520*(b^10*d^10*x^10 + 10*a*b^9*d^10*x^9 + 45*a^2*b^8*
d^10*x^8 + 120*a^3*b^7*d^10*x^7 + 210*a^4*b^6*d^10*x^6 + 252*a^5*b^5*d^10*x^5 +
210*a^6*b^4*d^10*x^4 + 120*a^7*b^3*d^10*x^3 + 45*a^8*b^2*d^10*x^2 + 10*a^9*b*d^1
0*x + a^10*d^10)*log(b*x + a))/(b^21*x^10 + 10*a*b^20*x^9 + 45*a^2*b^19*x^8 + 12
0*a^3*b^18*x^7 + 210*a^4*b^17*x^6 + 252*a^5*b^16*x^5 + 210*a^6*b^15*x^4 + 120*a^
7*b^14*x^3 + 45*a^8*b^13*x^2 + 10*a^9*b^12*x + a^10*b^11)

_______________________________________________________________________________________

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)**10/(b*x+a)**11,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223707, size = 1180, normalized size = 4.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^11,x, algorithm="giac")

[Out]

d^10*ln(abs(b*x + a))/b^11 - 1/2520*(25200*(b^9*c*d^9 - a*b^8*d^10)*x^9 + 56700*
(b^9*c^2*d^8 + 2*a*b^8*c*d^9 - 3*a^2*b^7*d^10)*x^8 + 50400*(2*b^9*c^3*d^7 + 3*a*
b^8*c^2*d^8 + 6*a^2*b^7*c*d^9 - 11*a^3*b^6*d^10)*x^7 + 44100*(3*b^9*c^4*d^6 + 4*
a*b^8*c^3*d^7 + 6*a^2*b^7*c^2*d^8 + 12*a^3*b^6*c*d^9 - 25*a^4*b^5*d^10)*x^6 + 10
584*(12*b^9*c^5*d^5 + 15*a*b^8*c^4*d^6 + 20*a^2*b^7*c^3*d^7 + 30*a^3*b^6*c^2*d^8
+ 60*a^4*b^5*c*d^9 - 137*a^5*b^4*d^10)*x^5 + 8820*(10*b^9*c^6*d^4 + 12*a*b^8*c^
5*d^5 + 15*a^2*b^7*c^4*d^6 + 20*a^3*b^6*c^3*d^7 + 30*a^4*b^5*c^2*d^8 + 60*a^5*b^
4*c*d^9 - 147*a^6*b^3*d^10)*x^4 + 720*(60*b^9*c^7*d^3 + 70*a*b^8*c^6*d^4 + 84*a^
2*b^7*c^5*d^5 + 105*a^3*b^6*c^4*d^6 + 140*a^4*b^5*c^3*d^7 + 210*a^5*b^4*c^2*d^8
+ 420*a^6*b^3*c*d^9 - 1089*a^7*b^2*d^10)*x^3 + 135*(105*b^9*c^8*d^2 + 120*a*b^8*
c^7*d^3 + 140*a^2*b^7*c^6*d^4 + 168*a^3*b^6*c^5*d^5 + 210*a^4*b^5*c^4*d^6 + 280*
a^5*b^4*c^3*d^7 + 420*a^6*b^3*c^2*d^8 + 840*a^7*b^2*c*d^9 - 2283*a^8*b*d^10)*x^2
+ 10*(280*b^9*c^9*d + 315*a*b^8*c^8*d^2 + 360*a^2*b^7*c^7*d^3 + 420*a^3*b^6*c^6
*d^4 + 504*a^4*b^5*c^5*d^5 + 630*a^5*b^4*c^4*d^6 + 840*a^6*b^3*c^3*d^7 + 1260*a^
7*b^2*c^2*d^8 + 2520*a^8*b*c*d^9 - 7129*a^9*d^10)*x + (252*b^10*c^10 + 280*a*b^9
*c^9*d + 315*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 + 504*a
^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 + 840*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^
8 + 2520*a^9*b*c*d^9 - 7381*a^10*d^10)/b)/((b*x + a)^10*b^10)