3.1331 \(\int \frac{(c+d x)^{10}}{(a+b x)^{20}} \, dx\)

Optimal. Leaf size=273 \[ -\frac{d^9 (b c-a d)}{b^{11} (a+b x)^{10}}-\frac{45 d^8 (b c-a d)^2}{11 b^{11} (a+b x)^{11}}-\frac{10 d^7 (b c-a d)^3}{b^{11} (a+b x)^{12}}-\frac{210 d^6 (b c-a d)^4}{13 b^{11} (a+b x)^{13}}-\frac{18 d^5 (b c-a d)^5}{b^{11} (a+b x)^{14}}-\frac{14 d^4 (b c-a d)^6}{b^{11} (a+b x)^{15}}-\frac{15 d^3 (b c-a d)^7}{2 b^{11} (a+b x)^{16}}-\frac{45 d^2 (b c-a d)^8}{17 b^{11} (a+b x)^{17}}-\frac{5 d (b c-a d)^9}{9 b^{11} (a+b x)^{18}}-\frac{(b c-a d)^{10}}{19 b^{11} (a+b x)^{19}}-\frac{d^{10}}{9 b^{11} (a+b x)^9} \]

[Out]

-(b*c - a*d)^10/(19*b^11*(a + b*x)^19) - (5*d*(b*c - a*d)^9)/(9*b^11*(a + b*x)^18) - (45*d^2*(b*c - a*d)^8)/(1
7*b^11*(a + b*x)^17) - (15*d^3*(b*c - a*d)^7)/(2*b^11*(a + b*x)^16) - (14*d^4*(b*c - a*d)^6)/(b^11*(a + b*x)^1
5) - (18*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^14) - (210*d^6*(b*c - a*d)^4)/(13*b^11*(a + b*x)^13) - (10*d^7*(b*
c - a*d)^3)/(b^11*(a + b*x)^12) - (45*d^8*(b*c - a*d)^2)/(11*b^11*(a + b*x)^11) - (d^9*(b*c - a*d))/(b^11*(a +
b*x)^10) - d^10/(9*b^11*(a + b*x)^9)

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Rubi [A]  time = 0.284117, antiderivative size = 273, 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} \[ -\frac{d^9 (b c-a d)}{b^{11} (a+b x)^{10}}-\frac{45 d^8 (b c-a d)^2}{11 b^{11} (a+b x)^{11}}-\frac{10 d^7 (b c-a d)^3}{b^{11} (a+b x)^{12}}-\frac{210 d^6 (b c-a d)^4}{13 b^{11} (a+b x)^{13}}-\frac{18 d^5 (b c-a d)^5}{b^{11} (a+b x)^{14}}-\frac{14 d^4 (b c-a d)^6}{b^{11} (a+b x)^{15}}-\frac{15 d^3 (b c-a d)^7}{2 b^{11} (a+b x)^{16}}-\frac{45 d^2 (b c-a d)^8}{17 b^{11} (a+b x)^{17}}-\frac{5 d (b c-a d)^9}{9 b^{11} (a+b x)^{18}}-\frac{(b c-a d)^{10}}{19 b^{11} (a+b x)^{19}}-\frac{d^{10}}{9 b^{11} (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^20,x]

[Out]

-(b*c - a*d)^10/(19*b^11*(a + b*x)^19) - (5*d*(b*c - a*d)^9)/(9*b^11*(a + b*x)^18) - (45*d^2*(b*c - a*d)^8)/(1
7*b^11*(a + b*x)^17) - (15*d^3*(b*c - a*d)^7)/(2*b^11*(a + b*x)^16) - (14*d^4*(b*c - a*d)^6)/(b^11*(a + b*x)^1
5) - (18*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^14) - (210*d^6*(b*c - a*d)^4)/(13*b^11*(a + b*x)^13) - (10*d^7*(b*
c - a*d)^3)/(b^11*(a + b*x)^12) - (45*d^8*(b*c - a*d)^2)/(11*b^11*(a + b*x)^11) - (d^9*(b*c - a*d))/(b^11*(a +
b*x)^10) - d^10/(9*b^11*(a + b*x)^9)

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

Mathematica [B]  time = 0.279299, size = 692, normalized size = 2.53 \[ -\frac{9 a^2 b^8 d^2 \left (57057 c^6 d^2 x^2+138567 c^5 d^3 x^3+213180 c^4 d^4 x^4+213180 c^3 d^5 x^5+135660 c^2 d^6 x^6+13585 c^7 d x+1430 c^8+50388 c d^7 x^7+8398 d^8 x^8\right )+3 a^3 b^7 d^3 \left (73359 c^5 d^2 x^2+159885 c^4 d^3 x^3+213180 c^3 d^4 x^4+174420 c^2 d^5 x^5+19019 c^6 d x+2145 c^7+81396 c d^6 x^6+16796 d^7 x^7\right )+3 a^4 b^6 d^4 \left (28215 c^4 d^2 x^2+53295 c^3 d^3 x^3+58140 c^2 d^4 x^4+8151 c^5 d x+1001 c^6+34884 c d^5 x^5+9044 d^6 x^6\right )+9 a^5 b^5 d^5 \left (3135 c^3 d^2 x^2+4845 c^2 d^3 x^3+1045 c^4 d x+143 c^5+3876 c d^4 x^4+1292 d^5 x^5\right )+3 a^6 b^4 d^6 \left (2565 c^2 d^2 x^2+1045 c^3 d x+165 c^4+2907 c d^3 x^3+1292 d^4 x^4\right )+3 a^7 b^3 d^7 \left (285 c^2 d x+55 c^3+513 c d^2 x^2+323 d^3 x^3\right )+9 a^8 b^2 d^8 \left (5 c^2+19 c d x+19 d^2 x^2\right )+a^9 b d^9 (9 c+19 d x)+a^{10} d^{10}+a b^9 d \left (1100385 c^7 d^2 x^2+2909907 c^6 d^3 x^3+4988412 c^5 d^4 x^4+5755860 c^4 d^5 x^5+4476780 c^3 d^6 x^6+2267460 c^2 d^7 x^7+244530 c^8 d x+24310 c^9+680238 c d^8 x^8+92378 d^9 x^9\right )+b^{10} \left (2200770 c^8 d^2 x^2+6235515 c^7 d^3 x^3+11639628 c^6 d^4 x^4+14965236 c^5 d^5 x^5+13430340 c^4 d^6 x^6+8314020 c^3 d^7 x^7+3401190 c^2 d^8 x^8+461890 c^9 d x+43758 c^{10}+831402 c d^9 x^9+92378 d^{10} x^{10}\right )}{831402 b^{11} (a+b x)^{19}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^20,x]

[Out]

-(a^10*d^10 + a^9*b*d^9*(9*c + 19*d*x) + 9*a^8*b^2*d^8*(5*c^2 + 19*c*d*x + 19*d^2*x^2) + 3*a^7*b^3*d^7*(55*c^3
+ 285*c^2*d*x + 513*c*d^2*x^2 + 323*d^3*x^3) + 3*a^6*b^4*d^6*(165*c^4 + 1045*c^3*d*x + 2565*c^2*d^2*x^2 + 290
7*c*d^3*x^3 + 1292*d^4*x^4) + 9*a^5*b^5*d^5*(143*c^5 + 1045*c^4*d*x + 3135*c^3*d^2*x^2 + 4845*c^2*d^3*x^3 + 38
76*c*d^4*x^4 + 1292*d^5*x^5) + 3*a^4*b^6*d^4*(1001*c^6 + 8151*c^5*d*x + 28215*c^4*d^2*x^2 + 53295*c^3*d^3*x^3
+ 58140*c^2*d^4*x^4 + 34884*c*d^5*x^5 + 9044*d^6*x^6) + 3*a^3*b^7*d^3*(2145*c^7 + 19019*c^6*d*x + 73359*c^5*d^
2*x^2 + 159885*c^4*d^3*x^3 + 213180*c^3*d^4*x^4 + 174420*c^2*d^5*x^5 + 81396*c*d^6*x^6 + 16796*d^7*x^7) + 9*a^
2*b^8*d^2*(1430*c^8 + 13585*c^7*d*x + 57057*c^6*d^2*x^2 + 138567*c^5*d^3*x^3 + 213180*c^4*d^4*x^4 + 213180*c^3
*d^5*x^5 + 135660*c^2*d^6*x^6 + 50388*c*d^7*x^7 + 8398*d^8*x^8) + a*b^9*d*(24310*c^9 + 244530*c^8*d*x + 110038
5*c^7*d^2*x^2 + 2909907*c^6*d^3*x^3 + 4988412*c^5*d^4*x^4 + 5755860*c^4*d^5*x^5 + 4476780*c^3*d^6*x^6 + 226746
0*c^2*d^7*x^7 + 680238*c*d^8*x^8 + 92378*d^9*x^9) + b^10*(43758*c^10 + 461890*c^9*d*x + 2200770*c^8*d^2*x^2 +
6235515*c^7*d^3*x^3 + 11639628*c^6*d^4*x^4 + 14965236*c^5*d^5*x^5 + 13430340*c^4*d^6*x^6 + 8314020*c^3*d^7*x^7
+ 3401190*c^2*d^8*x^8 + 831402*c*d^9*x^9 + 92378*d^10*x^10))/(831402*b^11*(a + b*x)^19)

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Maple [B]  time = 0.009, size = 866, normalized size = 3.2 \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)^20,x)

[Out]

d^9*(a*d-b*c)/b^11/(b*x+a)^10+18*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)^14+15/2*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)^16-45/11*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*
x+a)^11-45/17*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)^17-210/13*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)^13-14*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)^15-1/19*(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)^19+5/9*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)^18+10*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)^12-1/9*d^10/b^11/(b*x+a)^9

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Maxima [B]  time = 1.3383, size = 1435, normalized size = 5.26 \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)^20,x, algorithm="maxima")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^7*c
^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b^2*
c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10 + 92378*(9*b^10*c*d^9 + a*b^9*d^10)*x^9 + 75582*(45*b^10*c^2*d^8 + 9*a*b^9
*c*d^9 + a^2*b^8*d^10)*x^8 + 50388*(165*b^10*c^3*d^7 + 45*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7
+ 27132*(495*b^10*c^4*d^6 + 165*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 + 9*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 116
28*(1287*b^10*c^5*d^5 + 495*a*b^9*c^4*d^6 + 165*a^2*b^8*c^3*d^7 + 45*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 + a^5*b
^5*d^10)*x^5 + 3876*(3003*b^10*c^6*d^4 + 1287*a*b^9*c^5*d^5 + 495*a^2*b^8*c^4*d^6 + 165*a^3*b^7*c^3*d^7 + 45*a
^4*b^6*c^2*d^8 + 9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 969*(6435*b^10*c^7*d^3 + 3003*a*b^9*c^6*d^4 + 1287*a^2*
b^8*c^5*d^5 + 495*a^3*b^7*c^4*d^6 + 165*a^4*b^6*c^3*d^7 + 45*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 + a^7*b^3*d^10)
*x^3 + 171*(12870*b^10*c^8*d^2 + 6435*a*b^9*c^7*d^3 + 3003*a^2*b^8*c^6*d^4 + 1287*a^3*b^7*c^5*d^5 + 495*a^4*b^
6*c^4*d^6 + 165*a^5*b^5*c^3*d^7 + 45*a^6*b^4*c^2*d^8 + 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 19*(24310*b^10*c^
9*d + 12870*a*b^9*c^8*d^2 + 6435*a^2*b^8*c^7*d^3 + 3003*a^3*b^7*c^6*d^4 + 1287*a^4*b^6*c^5*d^5 + 495*a^5*b^5*c
^4*d^6 + 165*a^6*b^4*c^3*d^7 + 45*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^30*x^19 + 19*a*b^29*x^
18 + 171*a^2*b^28*x^17 + 969*a^3*b^27*x^16 + 3876*a^4*b^26*x^15 + 11628*a^5*b^25*x^14 + 27132*a^6*b^24*x^13 +
50388*a^7*b^23*x^12 + 75582*a^8*b^22*x^11 + 92378*a^9*b^21*x^10 + 92378*a^10*b^20*x^9 + 75582*a^11*b^19*x^8 +
50388*a^12*b^18*x^7 + 27132*a^13*b^17*x^6 + 11628*a^14*b^16*x^5 + 3876*a^15*b^15*x^4 + 969*a^16*b^14*x^3 + 171
*a^17*b^13*x^2 + 19*a^18*b^12*x + a^19*b^11)

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Fricas [B]  time = 1.95746, size = 2438, normalized size = 8.93 \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)^20,x, algorithm="fricas")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^7*c
^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b^2*
c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10 + 92378*(9*b^10*c*d^9 + a*b^9*d^10)*x^9 + 75582*(45*b^10*c^2*d^8 + 9*a*b^9
*c*d^9 + a^2*b^8*d^10)*x^8 + 50388*(165*b^10*c^3*d^7 + 45*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7
+ 27132*(495*b^10*c^4*d^6 + 165*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 + 9*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 116
28*(1287*b^10*c^5*d^5 + 495*a*b^9*c^4*d^6 + 165*a^2*b^8*c^3*d^7 + 45*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 + a^5*b
^5*d^10)*x^5 + 3876*(3003*b^10*c^6*d^4 + 1287*a*b^9*c^5*d^5 + 495*a^2*b^8*c^4*d^6 + 165*a^3*b^7*c^3*d^7 + 45*a
^4*b^6*c^2*d^8 + 9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 969*(6435*b^10*c^7*d^3 + 3003*a*b^9*c^6*d^4 + 1287*a^2*
b^8*c^5*d^5 + 495*a^3*b^7*c^4*d^6 + 165*a^4*b^6*c^3*d^7 + 45*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 + a^7*b^3*d^10)
*x^3 + 171*(12870*b^10*c^8*d^2 + 6435*a*b^9*c^7*d^3 + 3003*a^2*b^8*c^6*d^4 + 1287*a^3*b^7*c^5*d^5 + 495*a^4*b^
6*c^4*d^6 + 165*a^5*b^5*c^3*d^7 + 45*a^6*b^4*c^2*d^8 + 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 19*(24310*b^10*c^
9*d + 12870*a*b^9*c^8*d^2 + 6435*a^2*b^8*c^7*d^3 + 3003*a^3*b^7*c^6*d^4 + 1287*a^4*b^6*c^5*d^5 + 495*a^5*b^5*c
^4*d^6 + 165*a^6*b^4*c^3*d^7 + 45*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^30*x^19 + 19*a*b^29*x^
18 + 171*a^2*b^28*x^17 + 969*a^3*b^27*x^16 + 3876*a^4*b^26*x^15 + 11628*a^5*b^25*x^14 + 27132*a^6*b^24*x^13 +
50388*a^7*b^23*x^12 + 75582*a^8*b^22*x^11 + 92378*a^9*b^21*x^10 + 92378*a^10*b^20*x^9 + 75582*a^11*b^19*x^8 +
50388*a^12*b^18*x^7 + 27132*a^13*b^17*x^6 + 11628*a^14*b^16*x^5 + 3876*a^15*b^15*x^4 + 969*a^16*b^14*x^3 + 171
*a^17*b^13*x^2 + 19*a^18*b^12*x + a^19*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)**20,x)

[Out]

Timed out

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Giac [B]  time = 1.07213, size = 1297, normalized size = 4.75 \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)^20,x, algorithm="giac")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 831402*b^10*c*d^9*x^9 + 92378*a*b^9*d^10*x^9 + 3401190*b^10*c^2*d^8*x^8 + 68
0238*a*b^9*c*d^9*x^8 + 75582*a^2*b^8*d^10*x^8 + 8314020*b^10*c^3*d^7*x^7 + 2267460*a*b^9*c^2*d^8*x^7 + 453492*
a^2*b^8*c*d^9*x^7 + 50388*a^3*b^7*d^10*x^7 + 13430340*b^10*c^4*d^6*x^6 + 4476780*a*b^9*c^3*d^7*x^6 + 1220940*a
^2*b^8*c^2*d^8*x^6 + 244188*a^3*b^7*c*d^9*x^6 + 27132*a^4*b^6*d^10*x^6 + 14965236*b^10*c^5*d^5*x^5 + 5755860*a
*b^9*c^4*d^6*x^5 + 1918620*a^2*b^8*c^3*d^7*x^5 + 523260*a^3*b^7*c^2*d^8*x^5 + 104652*a^4*b^6*c*d^9*x^5 + 11628
*a^5*b^5*d^10*x^5 + 11639628*b^10*c^6*d^4*x^4 + 4988412*a*b^9*c^5*d^5*x^4 + 1918620*a^2*b^8*c^4*d^6*x^4 + 6395
40*a^3*b^7*c^3*d^7*x^4 + 174420*a^4*b^6*c^2*d^8*x^4 + 34884*a^5*b^5*c*d^9*x^4 + 3876*a^6*b^4*d^10*x^4 + 623551
5*b^10*c^7*d^3*x^3 + 2909907*a*b^9*c^6*d^4*x^3 + 1247103*a^2*b^8*c^5*d^5*x^3 + 479655*a^3*b^7*c^4*d^6*x^3 + 15
9885*a^4*b^6*c^3*d^7*x^3 + 43605*a^5*b^5*c^2*d^8*x^3 + 8721*a^6*b^4*c*d^9*x^3 + 969*a^7*b^3*d^10*x^3 + 2200770
*b^10*c^8*d^2*x^2 + 1100385*a*b^9*c^7*d^3*x^2 + 513513*a^2*b^8*c^6*d^4*x^2 + 220077*a^3*b^7*c^5*d^5*x^2 + 8464
5*a^4*b^6*c^4*d^6*x^2 + 28215*a^5*b^5*c^3*d^7*x^2 + 7695*a^6*b^4*c^2*d^8*x^2 + 1539*a^7*b^3*c*d^9*x^2 + 171*a^
8*b^2*d^10*x^2 + 461890*b^10*c^9*d*x + 244530*a*b^9*c^8*d^2*x + 122265*a^2*b^8*c^7*d^3*x + 57057*a^3*b^7*c^6*d
^4*x + 24453*a^4*b^6*c^5*d^5*x + 9405*a^5*b^5*c^4*d^6*x + 3135*a^6*b^4*c^3*d^7*x + 855*a^7*b^3*c^2*d^8*x + 171
*a^8*b^2*c*d^9*x + 19*a^9*b*d^10*x + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^
7*c^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b
^2*c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^19*b^11)