3.1328 \(\int \frac{(c+d x)^{10}}{(a+b x)^{17}} \, dx\)

Optimal. Leaf size=182 \[ \frac{d^5 (c+d x)^{11}}{48048 (a+b x)^{11} (b c-a d)^6}-\frac{d^4 (c+d x)^{11}}{4368 (a+b x)^{12} (b c-a d)^5}+\frac{d^3 (c+d x)^{11}}{728 (a+b x)^{13} (b c-a d)^4}-\frac{d^2 (c+d x)^{11}}{168 (a+b x)^{14} (b c-a d)^3}+\frac{d (c+d x)^{11}}{48 (a+b x)^{15} (b c-a d)^2}-\frac{(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)} \]

[Out]

-(c + d*x)^11/(16*(b*c - a*d)*(a + b*x)^16) + (d*(c + d*x)^11)/(48*(b*c - a*d)^2*(a + b*x)^15) - (d^2*(c + d*x
)^11)/(168*(b*c - a*d)^3*(a + b*x)^14) + (d^3*(c + d*x)^11)/(728*(b*c - a*d)^4*(a + b*x)^13) - (d^4*(c + d*x)^
11)/(4368*(b*c - a*d)^5*(a + b*x)^12) + (d^5*(c + d*x)^11)/(48048*(b*c - a*d)^6*(a + b*x)^11)

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Rubi [A]  time = 0.062674, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{d^5 (c+d x)^{11}}{48048 (a+b x)^{11} (b c-a d)^6}-\frac{d^4 (c+d x)^{11}}{4368 (a+b x)^{12} (b c-a d)^5}+\frac{d^3 (c+d x)^{11}}{728 (a+b x)^{13} (b c-a d)^4}-\frac{d^2 (c+d x)^{11}}{168 (a+b x)^{14} (b c-a d)^3}+\frac{d (c+d x)^{11}}{48 (a+b x)^{15} (b c-a d)^2}-\frac{(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^17,x]

[Out]

-(c + d*x)^11/(16*(b*c - a*d)*(a + b*x)^16) + (d*(c + d*x)^11)/(48*(b*c - a*d)^2*(a + b*x)^15) - (d^2*(c + d*x
)^11)/(168*(b*c - a*d)^3*(a + b*x)^14) + (d^3*(c + d*x)^11)/(728*(b*c - a*d)^4*(a + b*x)^13) - (d^4*(c + d*x)^
11)/(4368*(b*c - a*d)^5*(a + b*x)^12) + (d^5*(c + d*x)^11)/(48048*(b*c - a*d)^6*(a + b*x)^11)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(c+d x)^{10}}{(a+b x)^{17}} \, dx &=-\frac{(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}-\frac{(5 d) \int \frac{(c+d x)^{10}}{(a+b x)^{16}} \, dx}{16 (b c-a d)}\\ &=-\frac{(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac{d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}+\frac{d^2 \int \frac{(c+d x)^{10}}{(a+b x)^{15}} \, dx}{12 (b c-a d)^2}\\ &=-\frac{(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac{d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac{d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}-\frac{d^3 \int \frac{(c+d x)^{10}}{(a+b x)^{14}} \, dx}{56 (b c-a d)^3}\\ &=-\frac{(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac{d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac{d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}+\frac{d^3 (c+d x)^{11}}{728 (b c-a d)^4 (a+b x)^{13}}+\frac{d^4 \int \frac{(c+d x)^{10}}{(a+b x)^{13}} \, dx}{364 (b c-a d)^4}\\ &=-\frac{(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac{d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac{d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}+\frac{d^3 (c+d x)^{11}}{728 (b c-a d)^4 (a+b x)^{13}}-\frac{d^4 (c+d x)^{11}}{4368 (b c-a d)^5 (a+b x)^{12}}-\frac{d^5 \int \frac{(c+d x)^{10}}{(a+b x)^{12}} \, dx}{4368 (b c-a d)^5}\\ &=-\frac{(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac{d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac{d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}+\frac{d^3 (c+d x)^{11}}{728 (b c-a d)^4 (a+b x)^{13}}-\frac{d^4 (c+d x)^{11}}{4368 (b c-a d)^5 (a+b x)^{12}}+\frac{d^5 (c+d x)^{11}}{48048 (b c-a d)^6 (a+b x)^{11}}\\ \end{align*}

Mathematica [B]  time = 0.269272, size = 694, normalized size = 3.81 \[ -\frac{3 a^2 b^8 d^2 \left (18480 c^6 d^2 x^2+47040 c^5 d^3 x^3+76440 c^4 d^4 x^4+81536 c^3 d^5 x^5+56056 c^2 d^6 x^6+4224 c^7 d x+429 c^8+22880 c d^7 x^7+4290 d^8 x^8\right )+8 a^3 b^7 d^3 \left (3780 c^5 d^2 x^2+8820 c^4 d^3 x^3+12740 c^3 d^4 x^4+11466 c^2 d^5 x^5+924 c^6 d x+99 c^7+6006 c d^6 x^6+1430 d^7 x^7\right )+14 a^4 b^6 d^4 \left (1080 c^4 d^2 x^2+2240 c^3 d^3 x^3+2730 c^2 d^4 x^4+288 c^5 d x+33 c^6+1872 c d^5 x^5+572 d^6 x^6\right )+84 a^5 b^5 d^5 \left (80 c^3 d^2 x^2+140 c^2 d^3 x^3+24 c^4 d x+3 c^5+130 c d^4 x^4+52 d^5 x^5\right )+14 a^6 b^4 d^6 \left (180 c^2 d^2 x^2+64 c^3 d x+9 c^4+240 c d^3 x^3+130 d^4 x^4\right )+8 a^7 b^3 d^7 \left (42 c^2 d x+7 c^3+90 c d^2 x^2+70 d^3 x^3\right )+3 a^8 b^2 d^8 \left (7 c^2+32 c d x+40 d^2 x^2\right )+2 a^9 b d^9 (3 c+8 d x)+a^{10} d^{10}+2 a b^9 d \left (47520 c^7 d^2 x^2+129360 c^6 d^3 x^3+229320 c^5 d^4 x^4+275184 c^4 d^5 x^5+224224 c^3 d^6 x^6+120120 c^2 d^7 x^7+10296 c^8 d x+1001 c^9+38610 c d^8 x^8+5720 d^9 x^9\right )+b^{10} \left (154440 c^8 d^2 x^2+443520 c^7 d^3 x^3+840840 c^6 d^4 x^4+1100736 c^5 d^5 x^5+1009008 c^4 d^6 x^6+640640 c^3 d^7 x^7+270270 c^2 d^8 x^8+32032 c^9 d x+3003 c^{10}+68640 c d^9 x^9+8008 d^{10} x^{10}\right )}{48048 b^{11} (a+b x)^{16}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^17,x]

[Out]

-(a^10*d^10 + 2*a^9*b*d^9*(3*c + 8*d*x) + 3*a^8*b^2*d^8*(7*c^2 + 32*c*d*x + 40*d^2*x^2) + 8*a^7*b^3*d^7*(7*c^3
+ 42*c^2*d*x + 90*c*d^2*x^2 + 70*d^3*x^3) + 14*a^6*b^4*d^6*(9*c^4 + 64*c^3*d*x + 180*c^2*d^2*x^2 + 240*c*d^3*
x^3 + 130*d^4*x^4) + 84*a^5*b^5*d^5*(3*c^5 + 24*c^4*d*x + 80*c^3*d^2*x^2 + 140*c^2*d^3*x^3 + 130*c*d^4*x^4 + 5
2*d^5*x^5) + 14*a^4*b^6*d^4*(33*c^6 + 288*c^5*d*x + 1080*c^4*d^2*x^2 + 2240*c^3*d^3*x^3 + 2730*c^2*d^4*x^4 + 1
872*c*d^5*x^5 + 572*d^6*x^6) + 8*a^3*b^7*d^3*(99*c^7 + 924*c^6*d*x + 3780*c^5*d^2*x^2 + 8820*c^4*d^3*x^3 + 127
40*c^3*d^4*x^4 + 11466*c^2*d^5*x^5 + 6006*c*d^6*x^6 + 1430*d^7*x^7) + 3*a^2*b^8*d^2*(429*c^8 + 4224*c^7*d*x +
18480*c^6*d^2*x^2 + 47040*c^5*d^3*x^3 + 76440*c^4*d^4*x^4 + 81536*c^3*d^5*x^5 + 56056*c^2*d^6*x^6 + 22880*c*d^
7*x^7 + 4290*d^8*x^8) + 2*a*b^9*d*(1001*c^9 + 10296*c^8*d*x + 47520*c^7*d^2*x^2 + 129360*c^6*d^3*x^3 + 229320*
c^5*d^4*x^4 + 275184*c^4*d^5*x^5 + 224224*c^3*d^6*x^6 + 120120*c^2*d^7*x^7 + 38610*c*d^8*x^8 + 5720*d^9*x^9) +
b^10*(3003*c^10 + 32032*c^9*d*x + 154440*c^8*d^2*x^2 + 443520*c^7*d^3*x^3 + 840840*c^6*d^4*x^4 + 1100736*c^5*
d^5*x^5 + 1009008*c^4*d^6*x^6 + 640640*c^3*d^7*x^7 + 270270*c^2*d^8*x^8 + 68640*c*d^9*x^9 + 8008*d^10*x^10))/(
48048*b^11*(a + b*x)^16)

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Maple [B]  time = 0.009, size = 867, normalized size = 4.8 \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)^17,x)

[Out]

-21*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)^10-45/8*d^8*(a^2*d^2-2*a*
b*c*d+b^2*c^2)/b^11/(b*x+a)^8-1/6*d^10/b^11/(b*x+a)^6-45/14*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)^14
-1/16*(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)^16+252/11*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)^11+120/13*
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)^13+2/3*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)^15-35/2
*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)^12+40/3*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)^9+10/7*d^9*(a*d-b*c)/b^11/(b
*x+a)^7

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Maxima [B]  time = 1.26298, size = 1391, normalized size = 7.64 \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)^17,x, algorithm="maxima")

[Out]

-1/48048*(8008*b^10*d^10*x^10 + 3003*b^10*c^10 + 2002*a*b^9*c^9*d + 1287*a^2*b^8*c^8*d^2 + 792*a^3*b^7*c^7*d^3
+ 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c^5*d^5 + 126*a^6*b^4*c^4*d^6 + 56*a^7*b^3*c^3*d^7 + 21*a^8*b^2*c^2*d^8 +
6*a^9*b*c*d^9 + a^10*d^10 + 11440*(6*b^10*c*d^9 + a*b^9*d^10)*x^9 + 12870*(21*b^10*c^2*d^8 + 6*a*b^9*c*d^9 +
a^2*b^8*d^10)*x^8 + 11440*(56*b^10*c^3*d^7 + 21*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 8008*(12
6*b^10*c^4*d^6 + 56*a*b^9*c^3*d^7 + 21*a^2*b^8*c^2*d^8 + 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 4368*(252*b^10*
c^5*d^5 + 126*a*b^9*c^4*d^6 + 56*a^2*b^8*c^3*d^7 + 21*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 +
1820*(462*b^10*c^6*d^4 + 252*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4*d^6 + 56*a^3*b^7*c^3*d^7 + 21*a^4*b^6*c^2*d^8 + 6
*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 560*(792*b^10*c^7*d^3 + 462*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 + 126*a^3
*b^7*c^4*d^6 + 56*a^4*b^6*c^3*d^7 + 21*a^5*b^5*c^2*d^8 + 6*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 120*(1287*b^10*
c^8*d^2 + 792*a*b^9*c^7*d^3 + 462*a^2*b^8*c^6*d^4 + 252*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 + 56*a^5*b^5*c^3
*d^7 + 21*a^6*b^4*c^2*d^8 + 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 16*(2002*b^10*c^9*d + 1287*a*b^9*c^8*d^2 + 7
92*a^2*b^8*c^7*d^3 + 462*a^3*b^7*c^6*d^4 + 252*a^4*b^6*c^5*d^5 + 126*a^5*b^5*c^4*d^6 + 56*a^6*b^4*c^3*d^7 + 21
*a^7*b^3*c^2*d^8 + 6*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^27*x^16 + 16*a*b^26*x^15 + 120*a^2*b^25*x^14 + 560*a^3*
b^24*x^13 + 1820*a^4*b^23*x^12 + 4368*a^5*b^22*x^11 + 8008*a^6*b^21*x^10 + 11440*a^7*b^20*x^9 + 12870*a^8*b^19
*x^8 + 11440*a^9*b^18*x^7 + 8008*a^10*b^17*x^6 + 4368*a^11*b^16*x^5 + 1820*a^12*b^15*x^4 + 560*a^13*b^14*x^3 +
120*a^14*b^13*x^2 + 16*a^15*b^12*x + a^16*b^11)

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Fricas [B]  time = 1.83617, size = 2295, normalized size = 12.61 \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)^17,x, algorithm="fricas")

[Out]

-1/48048*(8008*b^10*d^10*x^10 + 3003*b^10*c^10 + 2002*a*b^9*c^9*d + 1287*a^2*b^8*c^8*d^2 + 792*a^3*b^7*c^7*d^3
+ 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c^5*d^5 + 126*a^6*b^4*c^4*d^6 + 56*a^7*b^3*c^3*d^7 + 21*a^8*b^2*c^2*d^8 +
6*a^9*b*c*d^9 + a^10*d^10 + 11440*(6*b^10*c*d^9 + a*b^9*d^10)*x^9 + 12870*(21*b^10*c^2*d^8 + 6*a*b^9*c*d^9 +
a^2*b^8*d^10)*x^8 + 11440*(56*b^10*c^3*d^7 + 21*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 8008*(12
6*b^10*c^4*d^6 + 56*a*b^9*c^3*d^7 + 21*a^2*b^8*c^2*d^8 + 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 4368*(252*b^10*
c^5*d^5 + 126*a*b^9*c^4*d^6 + 56*a^2*b^8*c^3*d^7 + 21*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 +
1820*(462*b^10*c^6*d^4 + 252*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4*d^6 + 56*a^3*b^7*c^3*d^7 + 21*a^4*b^6*c^2*d^8 + 6
*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 560*(792*b^10*c^7*d^3 + 462*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 + 126*a^3
*b^7*c^4*d^6 + 56*a^4*b^6*c^3*d^7 + 21*a^5*b^5*c^2*d^8 + 6*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 120*(1287*b^10*
c^8*d^2 + 792*a*b^9*c^7*d^3 + 462*a^2*b^8*c^6*d^4 + 252*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 + 56*a^5*b^5*c^3
*d^7 + 21*a^6*b^4*c^2*d^8 + 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 16*(2002*b^10*c^9*d + 1287*a*b^9*c^8*d^2 + 7
92*a^2*b^8*c^7*d^3 + 462*a^3*b^7*c^6*d^4 + 252*a^4*b^6*c^5*d^5 + 126*a^5*b^5*c^4*d^6 + 56*a^6*b^4*c^3*d^7 + 21
*a^7*b^3*c^2*d^8 + 6*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^27*x^16 + 16*a*b^26*x^15 + 120*a^2*b^25*x^14 + 560*a^3*
b^24*x^13 + 1820*a^4*b^23*x^12 + 4368*a^5*b^22*x^11 + 8008*a^6*b^21*x^10 + 11440*a^7*b^20*x^9 + 12870*a^8*b^19
*x^8 + 11440*a^9*b^18*x^7 + 8008*a^10*b^17*x^6 + 4368*a^11*b^16*x^5 + 1820*a^12*b^15*x^4 + 560*a^13*b^14*x^3 +
120*a^14*b^13*x^2 + 16*a^15*b^12*x + a^16*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)**17,x)

[Out]

Timed out

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Giac [B]  time = 1.06038, size = 1297, normalized size = 7.13 \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)^17,x, algorithm="giac")

[Out]

-1/48048*(8008*b^10*d^10*x^10 + 68640*b^10*c*d^9*x^9 + 11440*a*b^9*d^10*x^9 + 270270*b^10*c^2*d^8*x^8 + 77220*
a*b^9*c*d^9*x^8 + 12870*a^2*b^8*d^10*x^8 + 640640*b^10*c^3*d^7*x^7 + 240240*a*b^9*c^2*d^8*x^7 + 68640*a^2*b^8*
c*d^9*x^7 + 11440*a^3*b^7*d^10*x^7 + 1009008*b^10*c^4*d^6*x^6 + 448448*a*b^9*c^3*d^7*x^6 + 168168*a^2*b^8*c^2*
d^8*x^6 + 48048*a^3*b^7*c*d^9*x^6 + 8008*a^4*b^6*d^10*x^6 + 1100736*b^10*c^5*d^5*x^5 + 550368*a*b^9*c^4*d^6*x^
5 + 244608*a^2*b^8*c^3*d^7*x^5 + 91728*a^3*b^7*c^2*d^8*x^5 + 26208*a^4*b^6*c*d^9*x^5 + 4368*a^5*b^5*d^10*x^5 +
840840*b^10*c^6*d^4*x^4 + 458640*a*b^9*c^5*d^5*x^4 + 229320*a^2*b^8*c^4*d^6*x^4 + 101920*a^3*b^7*c^3*d^7*x^4
+ 38220*a^4*b^6*c^2*d^8*x^4 + 10920*a^5*b^5*c*d^9*x^4 + 1820*a^6*b^4*d^10*x^4 + 443520*b^10*c^7*d^3*x^3 + 2587
20*a*b^9*c^6*d^4*x^3 + 141120*a^2*b^8*c^5*d^5*x^3 + 70560*a^3*b^7*c^4*d^6*x^3 + 31360*a^4*b^6*c^3*d^7*x^3 + 11
760*a^5*b^5*c^2*d^8*x^3 + 3360*a^6*b^4*c*d^9*x^3 + 560*a^7*b^3*d^10*x^3 + 154440*b^10*c^8*d^2*x^2 + 95040*a*b^
9*c^7*d^3*x^2 + 55440*a^2*b^8*c^6*d^4*x^2 + 30240*a^3*b^7*c^5*d^5*x^2 + 15120*a^4*b^6*c^4*d^6*x^2 + 6720*a^5*b
^5*c^3*d^7*x^2 + 2520*a^6*b^4*c^2*d^8*x^2 + 720*a^7*b^3*c*d^9*x^2 + 120*a^8*b^2*d^10*x^2 + 32032*b^10*c^9*d*x
+ 20592*a*b^9*c^8*d^2*x + 12672*a^2*b^8*c^7*d^3*x + 7392*a^3*b^7*c^6*d^4*x + 4032*a^4*b^6*c^5*d^5*x + 2016*a^5
*b^5*c^4*d^6*x + 896*a^6*b^4*c^3*d^7*x + 336*a^7*b^3*c^2*d^8*x + 96*a^8*b^2*c*d^9*x + 16*a^9*b*d^10*x + 3003*b
^10*c^10 + 2002*a*b^9*c^9*d + 1287*a^2*b^8*c^8*d^2 + 792*a^3*b^7*c^7*d^3 + 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c
^5*d^5 + 126*a^6*b^4*c^4*d^6 + 56*a^7*b^3*c^3*d^7 + 21*a^8*b^2*c^2*d^8 + 6*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)
^16*b^11)