3.1332 \(\int \frac{(c+d x)^{10}}{(a+b x)^{21}} \, dx\)

Optimal. Leaf size=279 \[ -\frac{10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac{15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac{120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac{15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac{84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac{105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac{120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac{5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac{10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac{(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac{d^{10}}{10 b^{11} (a+b x)^{10}} \]

[Out]

-(b*c - a*d)^10/(20*b^11*(a + b*x)^20) - (10*d*(b*c - a*d)^9)/(19*b^11*(a + b*x)^19) - (5*d^2*(b*c - a*d)^8)/(
2*b^11*(a + b*x)^18) - (120*d^3*(b*c - a*d)^7)/(17*b^11*(a + b*x)^17) - (105*d^4*(b*c - a*d)^6)/(8*b^11*(a + b
*x)^16) - (84*d^5*(b*c - a*d)^5)/(5*b^11*(a + b*x)^15) - (15*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^14) - (120*d^7
*(b*c - a*d)^3)/(13*b^11*(a + b*x)^13) - (15*d^8*(b*c - a*d)^2)/(4*b^11*(a + b*x)^12) - (10*d^9*(b*c - a*d))/(
11*b^11*(a + b*x)^11) - d^10/(10*b^11*(a + b*x)^10)

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Rubi [A]  time = 0.272436, antiderivative size = 279, 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, Rules used = {43} \[ -\frac{10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac{15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac{120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac{15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac{84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac{105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac{120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac{5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac{10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac{(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac{d^{10}}{10 b^{11} (a+b x)^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^21,x]

[Out]

-(b*c - a*d)^10/(20*b^11*(a + b*x)^20) - (10*d*(b*c - a*d)^9)/(19*b^11*(a + b*x)^19) - (5*d^2*(b*c - a*d)^8)/(
2*b^11*(a + b*x)^18) - (120*d^3*(b*c - a*d)^7)/(17*b^11*(a + b*x)^17) - (105*d^4*(b*c - a*d)^6)/(8*b^11*(a + b
*x)^16) - (84*d^5*(b*c - a*d)^5)/(5*b^11*(a + b*x)^15) - (15*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^14) - (120*d^7
*(b*c - a*d)^3)/(13*b^11*(a + b*x)^13) - (15*d^8*(b*c - a*d)^2)/(4*b^11*(a + b*x)^12) - (10*d^9*(b*c - a*d))/(
11*b^11*(a + b*x)^11) - d^10/(10*b^11*(a + b*x)^10)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(c+d x)^{10}}{(a+b x)^{21}} \, dx &=\int \left (\frac{(b c-a d)^{10}}{b^{10} (a+b x)^{21}}+\frac{10 d (b c-a d)^9}{b^{10} (a+b x)^{20}}+\frac{45 d^2 (b c-a d)^8}{b^{10} (a+b x)^{19}}+\frac{120 d^3 (b c-a d)^7}{b^{10} (a+b x)^{18}}+\frac{210 d^4 (b c-a d)^6}{b^{10} (a+b x)^{17}}+\frac{252 d^5 (b c-a d)^5}{b^{10} (a+b x)^{16}}+\frac{210 d^6 (b c-a d)^4}{b^{10} (a+b x)^{15}}+\frac{120 d^7 (b c-a d)^3}{b^{10} (a+b x)^{14}}+\frac{45 d^8 (b c-a d)^2}{b^{10} (a+b x)^{13}}+\frac{10 d^9 (b c-a d)}{b^{10} (a+b x)^{12}}+\frac{d^{10}}{b^{10} (a+b x)^{11}}\right ) \, dx\\ &=-\frac{(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac{10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac{5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac{120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac{105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac{84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac{15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac{120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac{15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac{10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac{d^{10}}{10 b^{11} (a+b x)^{10}}\\ \end{align*}

Mathematica [B]  time = 0.270999, size = 692, normalized size = 2.48 \[ -\frac{5 a^2 b^8 d^2 \left (190190 c^6 d^2 x^2+456456 c^5 d^3 x^3+692835 c^4 d^4 x^4+682176 c^3 d^5 x^5+426360 c^2 d^6 x^6+45760 c^7 d x+4862 c^8+155040 c d^7 x^7+25194 d^8 x^8\right )+20 a^3 b^7 d^3 \left (19019 c^5 d^2 x^2+40755 c^4 d^3 x^3+53295 c^3 d^4 x^4+42636 c^2 d^5 x^5+5005 c^6 d x+572 c^7+19380 c d^6 x^6+3876 d^7 x^7\right )+5 a^4 b^6 d^4 \left (27170 c^4 d^2 x^2+50160 c^3 d^3 x^3+53295 c^2 d^4 x^4+8008 c^5 d x+1001 c^6+31008 c d^5 x^5+7752 d^6 x^6\right )+2 a^5 b^5 d^5 \left (20900 c^3 d^2 x^2+31350 c^2 d^3 x^3+7150 c^4 d x+1001 c^5+24225 c d^4 x^4+7752 d^5 x^5\right )+5 a^6 b^4 d^6 \left (2090 c^2 d^2 x^2+880 c^3 d x+143 c^4+2280 c d^3 x^3+969 d^4 x^4\right )+20 a^7 b^3 d^7 \left (55 c^2 d x+11 c^3+95 c d^2 x^2+57 d^3 x^3\right )+5 a^8 b^2 d^8 \left (11 c^2+40 c d x+38 d^2 x^2\right )+10 a^9 b d^9 (c+2 d x)+a^{10} d^{10}+10 a b^9 d \left (217360 c^7 d^2 x^2+570570 c^6 d^3 x^3+969969 c^5 d^4 x^4+1108536 c^4 d^5 x^5+852720 c^3 d^6 x^6+426360 c^2 d^7 x^7+48620 c^8 d x+4862 c^9+125970 c d^8 x^8+16796 d^9 x^9\right )+b^{10} \left (4618900 c^8 d^2 x^2+13041600 c^7 d^3 x^3+24249225 c^6 d^4 x^4+31039008 c^5 d^5 x^5+27713400 c^4 d^6 x^6+17054400 c^3 d^7 x^7+6928350 c^2 d^8 x^8+972400 c^9 d x+92378 c^{10}+1679600 c d^9 x^9+184756 d^{10} x^{10}\right )}{1847560 b^{11} (a+b x)^{20}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^21,x]

[Out]

-(a^10*d^10 + 10*a^9*b*d^9*(c + 2*d*x) + 5*a^8*b^2*d^8*(11*c^2 + 40*c*d*x + 38*d^2*x^2) + 20*a^7*b^3*d^7*(11*c
^3 + 55*c^2*d*x + 95*c*d^2*x^2 + 57*d^3*x^3) + 5*a^6*b^4*d^6*(143*c^4 + 880*c^3*d*x + 2090*c^2*d^2*x^2 + 2280*
c*d^3*x^3 + 969*d^4*x^4) + 2*a^5*b^5*d^5*(1001*c^5 + 7150*c^4*d*x + 20900*c^3*d^2*x^2 + 31350*c^2*d^3*x^3 + 24
225*c*d^4*x^4 + 7752*d^5*x^5) + 5*a^4*b^6*d^4*(1001*c^6 + 8008*c^5*d*x + 27170*c^4*d^2*x^2 + 50160*c^3*d^3*x^3
+ 53295*c^2*d^4*x^4 + 31008*c*d^5*x^5 + 7752*d^6*x^6) + 20*a^3*b^7*d^3*(572*c^7 + 5005*c^6*d*x + 19019*c^5*d^
2*x^2 + 40755*c^4*d^3*x^3 + 53295*c^3*d^4*x^4 + 42636*c^2*d^5*x^5 + 19380*c*d^6*x^6 + 3876*d^7*x^7) + 5*a^2*b^
8*d^2*(4862*c^8 + 45760*c^7*d*x + 190190*c^6*d^2*x^2 + 456456*c^5*d^3*x^3 + 692835*c^4*d^4*x^4 + 682176*c^3*d^
5*x^5 + 426360*c^2*d^6*x^6 + 155040*c*d^7*x^7 + 25194*d^8*x^8) + 10*a*b^9*d*(4862*c^9 + 48620*c^8*d*x + 217360
*c^7*d^2*x^2 + 570570*c^6*d^3*x^3 + 969969*c^5*d^4*x^4 + 1108536*c^4*d^5*x^5 + 852720*c^3*d^6*x^6 + 426360*c^2
*d^7*x^7 + 125970*c*d^8*x^8 + 16796*d^9*x^9) + b^10*(92378*c^10 + 972400*c^9*d*x + 4618900*c^8*d^2*x^2 + 13041
600*c^7*d^3*x^3 + 24249225*c^6*d^4*x^4 + 31039008*c^5*d^5*x^5 + 27713400*c^4*d^6*x^6 + 17054400*c^3*d^7*x^7 +
6928350*c^2*d^8*x^8 + 1679600*c*d^9*x^9 + 184756*d^10*x^10))/(1847560*b^11*(a + b*x)^20)

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Maple [B]  time = 0.01, size = 867, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^21,x)

[Out]

-1/10*d^10/b^11/(b*x+a)^10-1/20*(a^10*d^10-10*a^9*b*c*d^9+45*a^8*b^2*c^2*d^8-120*a^7*b^3*c^3*d^7+210*a^6*b^4*c
^4*d^6-252*a^5*b^5*c^5*d^5+210*a^4*b^6*c^6*d^4-120*a^3*b^7*c^7*d^3+45*a^2*b^8*c^8*d^2-10*a*b^9*c^9*d+b^10*c^10
)/b^11/(b*x+a)^20-15*d^6*(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^11/(b*x+a)^14-105/8
*d^4*(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^
11/(b*x+a)^16+10/11*d^9*(a*d-b*c)/b^11/(b*x+a)^11+120/17*d^3*(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^11/(b*x+a)^17+120/13*d^7*(a^3*d^3-3
*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^11/(b*x+a)^13+84/5*d^5*(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^11/(b*x+a)^15+10/19*d*(a^9*d^9-9*a^8*b*c*d^8+36*a^7*b^2*c^2*d^7-84*a^6*b^
3*c^3*d^6+126*a^5*b^4*c^4*d^5-126*a^4*b^5*c^5*d^4+84*a^3*b^6*c^6*d^3-36*a^2*b^7*c^7*d^2+9*a*b^8*c^8*d-b^9*c^9)
/b^11/(b*x+a)^19-5/2*d^2*(a^8*d^8-8*a^7*b*c*d^7+28*a^6*b^2*c^2*d^6-56*a^5*b^3*c^3*d^5+70*a^4*b^4*c^4*d^4-56*a^
3*b^5*c^5*d^3+28*a^2*b^6*c^6*d^2-8*a*b^7*c^7*d+b^8*c^8)/b^11/(b*x+a)^18-15/4*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b
^11/(b*x+a)^12

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Maxima [B]  time = 1.33424, size = 1450, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="maxima")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*d^2 + 11440*a^3*b^
7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*c^3*d^7 + 55*a^8*b
^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10 + 167960*(10*b^10*c*d^9 + a*b^9*d^10)*x^9 + 125970*(55*b^10*c^2*d^8 +
10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 77520*(220*b^10*c^3*d^7 + 55*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 + a^3*b^7*d
^10)*x^7 + 38760*(715*b^10*c^4*d^6 + 220*a*b^9*c^3*d^7 + 55*a^2*b^8*c^2*d^8 + 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)
*x^6 + 15504*(2002*b^10*c^5*d^5 + 715*a*b^9*c^4*d^6 + 220*a^2*b^8*c^3*d^7 + 55*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*
d^9 + a^5*b^5*d^10)*x^5 + 4845*(5005*b^10*c^6*d^4 + 2002*a*b^9*c^5*d^5 + 715*a^2*b^8*c^4*d^6 + 220*a^3*b^7*c^3
*d^7 + 55*a^4*b^6*c^2*d^8 + 10*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1140*(11440*b^10*c^7*d^3 + 5005*a*b^9*c^6*d
^4 + 2002*a^2*b^8*c^5*d^5 + 715*a^3*b^7*c^4*d^6 + 220*a^4*b^6*c^3*d^7 + 55*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9
+ a^7*b^3*d^10)*x^3 + 190*(24310*b^10*c^8*d^2 + 11440*a*b^9*c^7*d^3 + 5005*a^2*b^8*c^6*d^4 + 2002*a^3*b^7*c^5*
d^5 + 715*a^4*b^6*c^4*d^6 + 220*a^5*b^5*c^3*d^7 + 55*a^6*b^4*c^2*d^8 + 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 +
20*(48620*b^10*c^9*d + 24310*a*b^9*c^8*d^2 + 11440*a^2*b^8*c^7*d^3 + 5005*a^3*b^7*c^6*d^4 + 2002*a^4*b^6*c^5*d
^5 + 715*a^5*b^5*c^4*d^6 + 220*a^6*b^4*c^3*d^7 + 55*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^31*
x^20 + 20*a*b^30*x^19 + 190*a^2*b^29*x^18 + 1140*a^3*b^28*x^17 + 4845*a^4*b^27*x^16 + 15504*a^5*b^26*x^15 + 38
760*a^6*b^25*x^14 + 77520*a^7*b^24*x^13 + 125970*a^8*b^23*x^12 + 167960*a^9*b^22*x^11 + 184756*a^10*b^21*x^10
+ 167960*a^11*b^20*x^9 + 125970*a^12*b^19*x^8 + 77520*a^13*b^18*x^7 + 38760*a^14*b^17*x^6 + 15504*a^15*b^16*x^
5 + 4845*a^16*b^15*x^4 + 1140*a^17*b^14*x^3 + 190*a^18*b^13*x^2 + 20*a^19*b^12*x + a^20*b^11)

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Fricas [B]  time = 1.91626, size = 2504, normalized size = 8.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="fricas")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*d^2 + 11440*a^3*b^
7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*c^3*d^7 + 55*a^8*b
^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10 + 167960*(10*b^10*c*d^9 + a*b^9*d^10)*x^9 + 125970*(55*b^10*c^2*d^8 +
10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 77520*(220*b^10*c^3*d^7 + 55*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 + a^3*b^7*d
^10)*x^7 + 38760*(715*b^10*c^4*d^6 + 220*a*b^9*c^3*d^7 + 55*a^2*b^8*c^2*d^8 + 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)
*x^6 + 15504*(2002*b^10*c^5*d^5 + 715*a*b^9*c^4*d^6 + 220*a^2*b^8*c^3*d^7 + 55*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*
d^9 + a^5*b^5*d^10)*x^5 + 4845*(5005*b^10*c^6*d^4 + 2002*a*b^9*c^5*d^5 + 715*a^2*b^8*c^4*d^6 + 220*a^3*b^7*c^3
*d^7 + 55*a^4*b^6*c^2*d^8 + 10*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1140*(11440*b^10*c^7*d^3 + 5005*a*b^9*c^6*d
^4 + 2002*a^2*b^8*c^5*d^5 + 715*a^3*b^7*c^4*d^6 + 220*a^4*b^6*c^3*d^7 + 55*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9
+ a^7*b^3*d^10)*x^3 + 190*(24310*b^10*c^8*d^2 + 11440*a*b^9*c^7*d^3 + 5005*a^2*b^8*c^6*d^4 + 2002*a^3*b^7*c^5*
d^5 + 715*a^4*b^6*c^4*d^6 + 220*a^5*b^5*c^3*d^7 + 55*a^6*b^4*c^2*d^8 + 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 +
20*(48620*b^10*c^9*d + 24310*a*b^9*c^8*d^2 + 11440*a^2*b^8*c^7*d^3 + 5005*a^3*b^7*c^6*d^4 + 2002*a^4*b^6*c^5*d
^5 + 715*a^5*b^5*c^4*d^6 + 220*a^6*b^4*c^3*d^7 + 55*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^31*
x^20 + 20*a*b^30*x^19 + 190*a^2*b^29*x^18 + 1140*a^3*b^28*x^17 + 4845*a^4*b^27*x^16 + 15504*a^5*b^26*x^15 + 38
760*a^6*b^25*x^14 + 77520*a^7*b^24*x^13 + 125970*a^8*b^23*x^12 + 167960*a^9*b^22*x^11 + 184756*a^10*b^21*x^10
+ 167960*a^11*b^20*x^9 + 125970*a^12*b^19*x^8 + 77520*a^13*b^18*x^7 + 38760*a^14*b^17*x^6 + 15504*a^15*b^16*x^
5 + 4845*a^16*b^15*x^4 + 1140*a^17*b^14*x^3 + 190*a^18*b^13*x^2 + 20*a^19*b^12*x + a^20*b^11)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**10/(b*x+a)**21,x)

[Out]

Timed out

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Giac [B]  time = 1.06218, size = 1297, normalized size = 4.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="giac")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 1679600*b^10*c*d^9*x^9 + 167960*a*b^9*d^10*x^9 + 6928350*b^10*c^2*d^8*x^8
+ 1259700*a*b^9*c*d^9*x^8 + 125970*a^2*b^8*d^10*x^8 + 17054400*b^10*c^3*d^7*x^7 + 4263600*a*b^9*c^2*d^8*x^7 +
775200*a^2*b^8*c*d^9*x^7 + 77520*a^3*b^7*d^10*x^7 + 27713400*b^10*c^4*d^6*x^6 + 8527200*a*b^9*c^3*d^7*x^6 + 21
31800*a^2*b^8*c^2*d^8*x^6 + 387600*a^3*b^7*c*d^9*x^6 + 38760*a^4*b^6*d^10*x^6 + 31039008*b^10*c^5*d^5*x^5 + 11
085360*a*b^9*c^4*d^6*x^5 + 3410880*a^2*b^8*c^3*d^7*x^5 + 852720*a^3*b^7*c^2*d^8*x^5 + 155040*a^4*b^6*c*d^9*x^5
+ 15504*a^5*b^5*d^10*x^5 + 24249225*b^10*c^6*d^4*x^4 + 9699690*a*b^9*c^5*d^5*x^4 + 3464175*a^2*b^8*c^4*d^6*x^
4 + 1065900*a^3*b^7*c^3*d^7*x^4 + 266475*a^4*b^6*c^2*d^8*x^4 + 48450*a^5*b^5*c*d^9*x^4 + 4845*a^6*b^4*d^10*x^4
+ 13041600*b^10*c^7*d^3*x^3 + 5705700*a*b^9*c^6*d^4*x^3 + 2282280*a^2*b^8*c^5*d^5*x^3 + 815100*a^3*b^7*c^4*d^
6*x^3 + 250800*a^4*b^6*c^3*d^7*x^3 + 62700*a^5*b^5*c^2*d^8*x^3 + 11400*a^6*b^4*c*d^9*x^3 + 1140*a^7*b^3*d^10*x
^3 + 4618900*b^10*c^8*d^2*x^2 + 2173600*a*b^9*c^7*d^3*x^2 + 950950*a^2*b^8*c^6*d^4*x^2 + 380380*a^3*b^7*c^5*d^
5*x^2 + 135850*a^4*b^6*c^4*d^6*x^2 + 41800*a^5*b^5*c^3*d^7*x^2 + 10450*a^6*b^4*c^2*d^8*x^2 + 1900*a^7*b^3*c*d^
9*x^2 + 190*a^8*b^2*d^10*x^2 + 972400*b^10*c^9*d*x + 486200*a*b^9*c^8*d^2*x + 228800*a^2*b^8*c^7*d^3*x + 10010
0*a^3*b^7*c^6*d^4*x + 40040*a^4*b^6*c^5*d^5*x + 14300*a^5*b^5*c^4*d^6*x + 4400*a^6*b^4*c^3*d^7*x + 1100*a^7*b^
3*c^2*d^8*x + 200*a^8*b^2*c*d^9*x + 20*a^9*b*d^10*x + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*
d^2 + 11440*a^3*b^7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*
c^3*d^7 + 55*a^8*b^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^20*b^11)