3.1333 \(\int \frac{(c+d x)^{10}}{(a+b x)^{22}} \, dx\)

Optimal. Leaf size=279 \[ -\frac{5 d^9 (b c-a d)}{6 b^{11} (a+b x)^{12}}-\frac{45 d^8 (b c-a d)^2}{13 b^{11} (a+b x)^{13}}-\frac{60 d^7 (b c-a d)^3}{7 b^{11} (a+b x)^{14}}-\frac{14 d^6 (b c-a d)^4}{b^{11} (a+b x)^{15}}-\frac{63 d^5 (b c-a d)^5}{4 b^{11} (a+b x)^{16}}-\frac{210 d^4 (b c-a d)^6}{17 b^{11} (a+b x)^{17}}-\frac{20 d^3 (b c-a d)^7}{3 b^{11} (a+b x)^{18}}-\frac{45 d^2 (b c-a d)^8}{19 b^{11} (a+b x)^{19}}-\frac{d (b c-a d)^9}{2 b^{11} (a+b x)^{20}}-\frac{(b c-a d)^{10}}{21 b^{11} (a+b x)^{21}}-\frac{d^{10}}{11 b^{11} (a+b x)^{11}} \]

[Out]

-(b*c - a*d)^10/(21*b^11*(a + b*x)^21) - (d*(b*c - a*d)^9)/(2*b^11*(a + b*x)^20) - (45*d^2*(b*c - a*d)^8)/(19*
b^11*(a + b*x)^19) - (20*d^3*(b*c - a*d)^7)/(3*b^11*(a + b*x)^18) - (210*d^4*(b*c - a*d)^6)/(17*b^11*(a + b*x)
^17) - (63*d^5*(b*c - a*d)^5)/(4*b^11*(a + b*x)^16) - (14*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^15) - (60*d^7*(b*
c - a*d)^3)/(7*b^11*(a + b*x)^14) - (45*d^8*(b*c - a*d)^2)/(13*b^11*(a + b*x)^13) - (5*d^9*(b*c - a*d))/(6*b^1
1*(a + b*x)^12) - d^10/(11*b^11*(a + b*x)^11)

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Rubi [A]  time = 0.270568, 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} \[ -\frac{5 d^9 (b c-a d)}{6 b^{11} (a+b x)^{12}}-\frac{45 d^8 (b c-a d)^2}{13 b^{11} (a+b x)^{13}}-\frac{60 d^7 (b c-a d)^3}{7 b^{11} (a+b x)^{14}}-\frac{14 d^6 (b c-a d)^4}{b^{11} (a+b x)^{15}}-\frac{63 d^5 (b c-a d)^5}{4 b^{11} (a+b x)^{16}}-\frac{210 d^4 (b c-a d)^6}{17 b^{11} (a+b x)^{17}}-\frac{20 d^3 (b c-a d)^7}{3 b^{11} (a+b x)^{18}}-\frac{45 d^2 (b c-a d)^8}{19 b^{11} (a+b x)^{19}}-\frac{d (b c-a d)^9}{2 b^{11} (a+b x)^{20}}-\frac{(b c-a d)^{10}}{21 b^{11} (a+b x)^{21}}-\frac{d^{10}}{11 b^{11} (a+b x)^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^22,x]

[Out]

-(b*c - a*d)^10/(21*b^11*(a + b*x)^21) - (d*(b*c - a*d)^9)/(2*b^11*(a + b*x)^20) - (45*d^2*(b*c - a*d)^8)/(19*
b^11*(a + b*x)^19) - (20*d^3*(b*c - a*d)^7)/(3*b^11*(a + b*x)^18) - (210*d^4*(b*c - a*d)^6)/(17*b^11*(a + b*x)
^17) - (63*d^5*(b*c - a*d)^5)/(4*b^11*(a + b*x)^16) - (14*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^15) - (60*d^7*(b*
c - a*d)^3)/(7*b^11*(a + b*x)^14) - (45*d^8*(b*c - a*d)^2)/(13*b^11*(a + b*x)^13) - (5*d^9*(b*c - a*d))/(6*b^1
1*(a + b*x)^12) - d^10/(11*b^11*(a + b*x)^11)

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)^{22}} \, dx &=\int \left (\frac{(b c-a d)^{10}}{b^{10} (a+b x)^{22}}+\frac{10 d (b c-a d)^9}{b^{10} (a+b x)^{21}}+\frac{45 d^2 (b c-a d)^8}{b^{10} (a+b x)^{20}}+\frac{120 d^3 (b c-a d)^7}{b^{10} (a+b x)^{19}}+\frac{210 d^4 (b c-a d)^6}{b^{10} (a+b x)^{18}}+\frac{252 d^5 (b c-a d)^5}{b^{10} (a+b x)^{17}}+\frac{210 d^6 (b c-a d)^4}{b^{10} (a+b x)^{16}}+\frac{120 d^7 (b c-a d)^3}{b^{10} (a+b x)^{15}}+\frac{45 d^8 (b c-a d)^2}{b^{10} (a+b x)^{14}}+\frac{10 d^9 (b c-a d)}{b^{10} (a+b x)^{13}}+\frac{d^{10}}{b^{10} (a+b x)^{12}}\right ) \, dx\\ &=-\frac{(b c-a d)^{10}}{21 b^{11} (a+b x)^{21}}-\frac{d (b c-a d)^9}{2 b^{11} (a+b x)^{20}}-\frac{45 d^2 (b c-a d)^8}{19 b^{11} (a+b x)^{19}}-\frac{20 d^3 (b c-a d)^7}{3 b^{11} (a+b x)^{18}}-\frac{210 d^4 (b c-a d)^6}{17 b^{11} (a+b x)^{17}}-\frac{63 d^5 (b c-a d)^5}{4 b^{11} (a+b x)^{16}}-\frac{14 d^6 (b c-a d)^4}{b^{11} (a+b x)^{15}}-\frac{60 d^7 (b c-a d)^3}{7 b^{11} (a+b x)^{14}}-\frac{45 d^8 (b c-a d)^2}{13 b^{11} (a+b x)^{13}}-\frac{5 d^9 (b c-a d)}{6 b^{11} (a+b x)^{12}}-\frac{d^{10}}{11 b^{11} (a+b x)^{11}}\\ \end{align*}

Mathematica [B]  time = 0.287279, size = 692, normalized size = 2.48 \[ -\frac{3 a^2 b^8 d^2 \left (560560 c^6 d^2 x^2+1331330 c^5 d^3 x^3+1996995 c^4 d^4 x^4+1939938 c^3 d^5 x^5+1193808 c^2 d^6 x^6+136136 c^7 d x+14586 c^8+426360 c d^7 x^7+67830 d^8 x^8\right )+2 a^3 b^7 d^3 \left (315315 c^5 d^2 x^2+665665 c^4 d^3 x^3+855855 c^3 d^4 x^4+671517 c^2 d^5 x^5+84084 c^6 d x+9724 c^7+298452 c d^6 x^6+58140 d^7 x^7\right )+7 a^4 b^6 d^4 \left (30030 c^4 d^2 x^2+54340 c^3 d^3 x^3+56430 c^2 d^4 x^4+9009 c^5 d x+1144 c^6+31977 c d^5 x^5+7752 d^6 x^6\right )+21 a^5 b^5 d^5 \left (2860 c^3 d^2 x^2+4180 c^2 d^3 x^3+1001 c^4 d x+143 c^5+3135 c d^4 x^4+969 d^5 x^5\right )+7 a^6 b^4 d^6 \left (1980 c^2 d^2 x^2+858 c^3 d x+143 c^4+2090 c d^3 x^3+855 d^4 x^4\right )+2 a^7 b^3 d^7 \left (693 c^2 d x+143 c^3+1155 c d^2 x^2+665 d^3 x^3\right )+3 a^8 b^2 d^8 \left (22 c^2+77 c d x+70 d^2 x^2\right )+a^9 b d^9 (11 c+21 d x)+a^{10} d^{10}+a b^9 d \left (4084080 c^7 d^2 x^2+10650640 c^6 d^3 x^3+17972955 c^5 d^4 x^4+20369349 c^4 d^5 x^5+15519504 c^3 d^6 x^6+7674480 c^2 d^7 x^7+918918 c^8 d x+92378 c^9+2238390 c d^8 x^8+293930 d^9 x^9\right )+b^{10} \left (9189180 c^8 d^2 x^2+25865840 c^7 d^3 x^3+47927880 c^6 d^4 x^4+61108047 c^5 d^5 x^5+54318264 c^4 d^6 x^6+33256080 c^3 d^7 x^7+13430340 c^2 d^8 x^8+1939938 c^9 d x+184756 c^{10}+3233230 c d^9 x^9+352716 d^{10} x^{10}\right )}{3879876 b^{11} (a+b x)^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^22,x]

[Out]

-(a^10*d^10 + a^9*b*d^9*(11*c + 21*d*x) + 3*a^8*b^2*d^8*(22*c^2 + 77*c*d*x + 70*d^2*x^2) + 2*a^7*b^3*d^7*(143*
c^3 + 693*c^2*d*x + 1155*c*d^2*x^2 + 665*d^3*x^3) + 7*a^6*b^4*d^6*(143*c^4 + 858*c^3*d*x + 1980*c^2*d^2*x^2 +
2090*c*d^3*x^3 + 855*d^4*x^4) + 21*a^5*b^5*d^5*(143*c^5 + 1001*c^4*d*x + 2860*c^3*d^2*x^2 + 4180*c^2*d^3*x^3 +
3135*c*d^4*x^4 + 969*d^5*x^5) + 7*a^4*b^6*d^4*(1144*c^6 + 9009*c^5*d*x + 30030*c^4*d^2*x^2 + 54340*c^3*d^3*x^
3 + 56430*c^2*d^4*x^4 + 31977*c*d^5*x^5 + 7752*d^6*x^6) + 2*a^3*b^7*d^3*(9724*c^7 + 84084*c^6*d*x + 315315*c^5
*d^2*x^2 + 665665*c^4*d^3*x^3 + 855855*c^3*d^4*x^4 + 671517*c^2*d^5*x^5 + 298452*c*d^6*x^6 + 58140*d^7*x^7) +
3*a^2*b^8*d^2*(14586*c^8 + 136136*c^7*d*x + 560560*c^6*d^2*x^2 + 1331330*c^5*d^3*x^3 + 1996995*c^4*d^4*x^4 + 1
939938*c^3*d^5*x^5 + 1193808*c^2*d^6*x^6 + 426360*c*d^7*x^7 + 67830*d^8*x^8) + a*b^9*d*(92378*c^9 + 918918*c^8
*d*x + 4084080*c^7*d^2*x^2 + 10650640*c^6*d^3*x^3 + 17972955*c^5*d^4*x^4 + 20369349*c^4*d^5*x^5 + 15519504*c^3
*d^6*x^6 + 7674480*c^2*d^7*x^7 + 2238390*c*d^8*x^8 + 293930*d^9*x^9) + b^10*(184756*c^10 + 1939938*c^9*d*x + 9
189180*c^8*d^2*x^2 + 25865840*c^7*d^3*x^3 + 47927880*c^6*d^4*x^4 + 61108047*c^5*d^5*x^5 + 54318264*c^4*d^6*x^6
+ 33256080*c^3*d^7*x^7 + 13430340*c^2*d^8*x^8 + 3233230*c*d^9*x^9 + 352716*d^10*x^10))/(3879876*b^11*(a + b*x
)^21)

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Maple [B]  time = 0.009, 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)^22,x)

[Out]

1/2*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)^20+60/7*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)^14+63/4*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)^16-1/11*d^10/b^11/(b*x+a)^11-210/17*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)^17-45/13*d^8*(a^2*d^2-2*a*b*c*d+b
^2*c^2)/b^11/(b*x+a)^13-14*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)^15
-45/19*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)^19+20/3*d^3*(a^7*d^7-7*a^6*b*c*d^6+21*a^5*b^2*c^2*d^5-3
5*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)^18+5/6*d^9*(a*d-b*
c)/b^11/(b*x+a)^12-1/21*(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^6-2
52*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)^21

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Maxima [B]  time = 1.38266, size = 1465, normalized size = 5.25 \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)^22,x, algorithm="maxima")

[Out]

-1/3879876*(352716*b^10*d^10*x^10 + 184756*b^10*c^10 + 92378*a*b^9*c^9*d + 43758*a^2*b^8*c^8*d^2 + 19448*a^3*b
^7*c^7*d^3 + 8008*a^4*b^6*c^6*d^4 + 3003*a^5*b^5*c^5*d^5 + 1001*a^6*b^4*c^4*d^6 + 286*a^7*b^3*c^3*d^7 + 66*a^8
*b^2*c^2*d^8 + 11*a^9*b*c*d^9 + a^10*d^10 + 293930*(11*b^10*c*d^9 + a*b^9*d^10)*x^9 + 203490*(66*b^10*c^2*d^8
+ 11*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 116280*(286*b^10*c^3*d^7 + 66*a*b^9*c^2*d^8 + 11*a^2*b^8*c*d^9 + a^3*b^
7*d^10)*x^7 + 54264*(1001*b^10*c^4*d^6 + 286*a*b^9*c^3*d^7 + 66*a^2*b^8*c^2*d^8 + 11*a^3*b^7*c*d^9 + a^4*b^6*d
^10)*x^6 + 20349*(3003*b^10*c^5*d^5 + 1001*a*b^9*c^4*d^6 + 286*a^2*b^8*c^3*d^7 + 66*a^3*b^7*c^2*d^8 + 11*a^4*b
^6*c*d^9 + a^5*b^5*d^10)*x^5 + 5985*(8008*b^10*c^6*d^4 + 3003*a*b^9*c^5*d^5 + 1001*a^2*b^8*c^4*d^6 + 286*a^3*b
^7*c^3*d^7 + 66*a^4*b^6*c^2*d^8 + 11*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1330*(19448*b^10*c^7*d^3 + 8008*a*b^9
*c^6*d^4 + 3003*a^2*b^8*c^5*d^5 + 1001*a^3*b^7*c^4*d^6 + 286*a^4*b^6*c^3*d^7 + 66*a^5*b^5*c^2*d^8 + 11*a^6*b^4
*c*d^9 + a^7*b^3*d^10)*x^3 + 210*(43758*b^10*c^8*d^2 + 19448*a*b^9*c^7*d^3 + 8008*a^2*b^8*c^6*d^4 + 3003*a^3*b
^7*c^5*d^5 + 1001*a^4*b^6*c^4*d^6 + 286*a^5*b^5*c^3*d^7 + 66*a^6*b^4*c^2*d^8 + 11*a^7*b^3*c*d^9 + a^8*b^2*d^10
)*x^2 + 21*(92378*b^10*c^9*d + 43758*a*b^9*c^8*d^2 + 19448*a^2*b^8*c^7*d^3 + 8008*a^3*b^7*c^6*d^4 + 3003*a^4*b
^6*c^5*d^5 + 1001*a^5*b^5*c^4*d^6 + 286*a^6*b^4*c^3*d^7 + 66*a^7*b^3*c^2*d^8 + 11*a^8*b^2*c*d^9 + a^9*b*d^10)*
x)/(b^32*x^21 + 21*a*b^31*x^20 + 210*a^2*b^30*x^19 + 1330*a^3*b^29*x^18 + 5985*a^4*b^28*x^17 + 20349*a^5*b^27*
x^16 + 54264*a^6*b^26*x^15 + 116280*a^7*b^25*x^14 + 203490*a^8*b^24*x^13 + 293930*a^9*b^23*x^12 + 352716*a^10*
b^22*x^11 + 352716*a^11*b^21*x^10 + 293930*a^12*b^20*x^9 + 203490*a^13*b^19*x^8 + 116280*a^14*b^18*x^7 + 54264
*a^15*b^17*x^6 + 20349*a^16*b^16*x^5 + 5985*a^17*b^15*x^4 + 1330*a^18*b^14*x^3 + 210*a^19*b^13*x^2 + 21*a^20*b
^12*x + a^21*b^11)

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Fricas [B]  time = 1.96428, size = 2552, normalized size = 9.15 \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)^22,x, algorithm="fricas")

[Out]

-1/3879876*(352716*b^10*d^10*x^10 + 184756*b^10*c^10 + 92378*a*b^9*c^9*d + 43758*a^2*b^8*c^8*d^2 + 19448*a^3*b
^7*c^7*d^3 + 8008*a^4*b^6*c^6*d^4 + 3003*a^5*b^5*c^5*d^5 + 1001*a^6*b^4*c^4*d^6 + 286*a^7*b^3*c^3*d^7 + 66*a^8
*b^2*c^2*d^8 + 11*a^9*b*c*d^9 + a^10*d^10 + 293930*(11*b^10*c*d^9 + a*b^9*d^10)*x^9 + 203490*(66*b^10*c^2*d^8
+ 11*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 116280*(286*b^10*c^3*d^7 + 66*a*b^9*c^2*d^8 + 11*a^2*b^8*c*d^9 + a^3*b^
7*d^10)*x^7 + 54264*(1001*b^10*c^4*d^6 + 286*a*b^9*c^3*d^7 + 66*a^2*b^8*c^2*d^8 + 11*a^3*b^7*c*d^9 + a^4*b^6*d
^10)*x^6 + 20349*(3003*b^10*c^5*d^5 + 1001*a*b^9*c^4*d^6 + 286*a^2*b^8*c^3*d^7 + 66*a^3*b^7*c^2*d^8 + 11*a^4*b
^6*c*d^9 + a^5*b^5*d^10)*x^5 + 5985*(8008*b^10*c^6*d^4 + 3003*a*b^9*c^5*d^5 + 1001*a^2*b^8*c^4*d^6 + 286*a^3*b
^7*c^3*d^7 + 66*a^4*b^6*c^2*d^8 + 11*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1330*(19448*b^10*c^7*d^3 + 8008*a*b^9
*c^6*d^4 + 3003*a^2*b^8*c^5*d^5 + 1001*a^3*b^7*c^4*d^6 + 286*a^4*b^6*c^3*d^7 + 66*a^5*b^5*c^2*d^8 + 11*a^6*b^4
*c*d^9 + a^7*b^3*d^10)*x^3 + 210*(43758*b^10*c^8*d^2 + 19448*a*b^9*c^7*d^3 + 8008*a^2*b^8*c^6*d^4 + 3003*a^3*b
^7*c^5*d^5 + 1001*a^4*b^6*c^4*d^6 + 286*a^5*b^5*c^3*d^7 + 66*a^6*b^4*c^2*d^8 + 11*a^7*b^3*c*d^9 + a^8*b^2*d^10
)*x^2 + 21*(92378*b^10*c^9*d + 43758*a*b^9*c^8*d^2 + 19448*a^2*b^8*c^7*d^3 + 8008*a^3*b^7*c^6*d^4 + 3003*a^4*b
^6*c^5*d^5 + 1001*a^5*b^5*c^4*d^6 + 286*a^6*b^4*c^3*d^7 + 66*a^7*b^3*c^2*d^8 + 11*a^8*b^2*c*d^9 + a^9*b*d^10)*
x)/(b^32*x^21 + 21*a*b^31*x^20 + 210*a^2*b^30*x^19 + 1330*a^3*b^29*x^18 + 5985*a^4*b^28*x^17 + 20349*a^5*b^27*
x^16 + 54264*a^6*b^26*x^15 + 116280*a^7*b^25*x^14 + 203490*a^8*b^24*x^13 + 293930*a^9*b^23*x^12 + 352716*a^10*
b^22*x^11 + 352716*a^11*b^21*x^10 + 293930*a^12*b^20*x^9 + 203490*a^13*b^19*x^8 + 116280*a^14*b^18*x^7 + 54264
*a^15*b^17*x^6 + 20349*a^16*b^16*x^5 + 5985*a^17*b^15*x^4 + 1330*a^18*b^14*x^3 + 210*a^19*b^13*x^2 + 21*a^20*b
^12*x + a^21*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)**22,x)

[Out]

Timed out

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Giac [B]  time = 1.06939, 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)^22,x, algorithm="giac")

[Out]

-1/3879876*(352716*b^10*d^10*x^10 + 3233230*b^10*c*d^9*x^9 + 293930*a*b^9*d^10*x^9 + 13430340*b^10*c^2*d^8*x^8
+ 2238390*a*b^9*c*d^9*x^8 + 203490*a^2*b^8*d^10*x^8 + 33256080*b^10*c^3*d^7*x^7 + 7674480*a*b^9*c^2*d^8*x^7 +
1279080*a^2*b^8*c*d^9*x^7 + 116280*a^3*b^7*d^10*x^7 + 54318264*b^10*c^4*d^6*x^6 + 15519504*a*b^9*c^3*d^7*x^6
+ 3581424*a^2*b^8*c^2*d^8*x^6 + 596904*a^3*b^7*c*d^9*x^6 + 54264*a^4*b^6*d^10*x^6 + 61108047*b^10*c^5*d^5*x^5
+ 20369349*a*b^9*c^4*d^6*x^5 + 5819814*a^2*b^8*c^3*d^7*x^5 + 1343034*a^3*b^7*c^2*d^8*x^5 + 223839*a^4*b^6*c*d^
9*x^5 + 20349*a^5*b^5*d^10*x^5 + 47927880*b^10*c^6*d^4*x^4 + 17972955*a*b^9*c^5*d^5*x^4 + 5990985*a^2*b^8*c^4*
d^6*x^4 + 1711710*a^3*b^7*c^3*d^7*x^4 + 395010*a^4*b^6*c^2*d^8*x^4 + 65835*a^5*b^5*c*d^9*x^4 + 5985*a^6*b^4*d^
10*x^4 + 25865840*b^10*c^7*d^3*x^3 + 10650640*a*b^9*c^6*d^4*x^3 + 3993990*a^2*b^8*c^5*d^5*x^3 + 1331330*a^3*b^
7*c^4*d^6*x^3 + 380380*a^4*b^6*c^3*d^7*x^3 + 87780*a^5*b^5*c^2*d^8*x^3 + 14630*a^6*b^4*c*d^9*x^3 + 1330*a^7*b^
3*d^10*x^3 + 9189180*b^10*c^8*d^2*x^2 + 4084080*a*b^9*c^7*d^3*x^2 + 1681680*a^2*b^8*c^6*d^4*x^2 + 630630*a^3*b
^7*c^5*d^5*x^2 + 210210*a^4*b^6*c^4*d^6*x^2 + 60060*a^5*b^5*c^3*d^7*x^2 + 13860*a^6*b^4*c^2*d^8*x^2 + 2310*a^7
*b^3*c*d^9*x^2 + 210*a^8*b^2*d^10*x^2 + 1939938*b^10*c^9*d*x + 918918*a*b^9*c^8*d^2*x + 408408*a^2*b^8*c^7*d^3
*x + 168168*a^3*b^7*c^6*d^4*x + 63063*a^4*b^6*c^5*d^5*x + 21021*a^5*b^5*c^4*d^6*x + 6006*a^6*b^4*c^3*d^7*x + 1
386*a^7*b^3*c^2*d^8*x + 231*a^8*b^2*c*d^9*x + 21*a^9*b*d^10*x + 184756*b^10*c^10 + 92378*a*b^9*c^9*d + 43758*a
^2*b^8*c^8*d^2 + 19448*a^3*b^7*c^7*d^3 + 8008*a^4*b^6*c^6*d^4 + 3003*a^5*b^5*c^5*d^5 + 1001*a^6*b^4*c^4*d^6 +
286*a^7*b^3*c^3*d^7 + 66*a^8*b^2*c^2*d^8 + 11*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^21*b^11)