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 = 0.286844, 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, Rules used = {43} \[ -\frac{10 d^9 (b c-a d)}{b^{11} (a+b x)}-\frac{45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac{40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac{105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac{252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac{35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac{120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac{45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac{10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac{(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}+\frac{d^{10} \log (a+b x)}{b^{11}} \]

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

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)^{11}} \, dx &=\int \left (\frac{(b c-a d)^{10}}{b^{10} (a+b x)^{11}}+\frac{10 d (b c-a d)^9}{b^{10} (a+b x)^{10}}+\frac{45 d^2 (b c-a d)^8}{b^{10} (a+b x)^9}+\frac{120 d^3 (b c-a d)^7}{b^{10} (a+b x)^8}+\frac{210 d^4 (b c-a d)^6}{b^{10} (a+b x)^7}+\frac{252 d^5 (b c-a d)^5}{b^{10} (a+b x)^6}+\frac{210 d^6 (b c-a d)^4}{b^{10} (a+b x)^5}+\frac{120 d^7 (b c-a d)^3}{b^{10} (a+b x)^4}+\frac{45 d^8 (b c-a d)^2}{b^{10} (a+b x)^3}+\frac{10 d^9 (b c-a d)}{b^{10} (a+b x)^2}+\frac{d^{10}}{b^{10} (a+b x)}\right ) \, dx\\ &=-\frac{(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}-\frac{10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac{45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac{120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac{35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac{252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac{105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac{40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac{45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac{10 d^9 (b c-a d)}{b^{11} (a+b x)}+\frac{d^{10} \log (a+b x)}{b^{11}}\\ \end{align*}

Mathematica [B]  time = 0.358232, size = 591, normalized size = 2.18 \[ \frac{d^{10} \log (a+b x)}{b^{11}}-\frac{(b c-a d) \left (a^2 b^7 d^2 \left (49275 c^5 d^2 x^2+154080 c^4 d^3 x^3+326340 c^3 d^4 x^4+497448 c^2 d^5 x^5+9550 c^6 d x+847 c^7+573300 c d^6 x^6+554400 d^7 x^7\right )+a^3 b^6 d^3 \left (71955 c^4 d^2 x^2+229680 c^3 d^3 x^3+502740 c^2 d^4 x^4+13750 c^5 d x+1207 c^6+814968 c d^5 x^5+1102500 d^6 x^6\right )+a^4 b^5 d^4 \left (100305 c^3 d^2 x^2+330480 c^2 d^3 x^3+18790 c^4 d x+1627 c^5+767340 c d^4 x^4+1450008 d^5 x^5\right )+a^5 b^4 d^5 \left (138105 c^2 d^2 x^2+25090 c^3 d x+2131 c^4+481680 c d^3 x^3+1296540 d^4 x^4\right )+a^6 b^3 d^6 \left (33490 c^2 d x+2761 c^3+194805 c d^2 x^2+784080 d^3 x^3\right )+a^7 b^2 d^7 \left (3601 c^2+46090 c d x+308205 d^2 x^2\right )+a^8 b d^8 (4861 c+71290 d x)+7381 a^9 d^9+a b^8 d \left (30375 c^6 d^2 x^2+93600 c^5 d^3 x^3+194040 c^4 d^4 x^4+285768 c^3 d^5 x^5+308700 c^2 d^6 x^6+5950 c^7 d x+532 c^8+252000 c d^7 x^7+170100 d^8 x^8\right )+b^9 \left (14175 c^7 d^2 x^2+43200 c^6 d^3 x^3+88200 c^5 d^4 x^4+127008 c^4 d^5 x^5+132300 c^3 d^6 x^6+100800 c^2 d^7 x^7+2800 c^8 d x+252 c^9+56700 c d^8 x^8+25200 d^9 x^9\right )\right )}{2520 b^{11} (a+b x)^{10}} \]

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 + 194805*c*d^2*x^2 + 784080*d^3*x^3) + a^5*b^4*d^5*(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) + a^4*b^5*d^4*(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) + a^3*b^6*d^3*(1207*c^6 + 1
3750*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) + 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 + 49744
8*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 + 936
00*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 + 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)))/(2520*b^11*(a + b*x)^10)
+ (d^10*Log[a + b*x])/b^11

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Maple [B]  time = 0.012, size = 1271, normalized size = 4.7 \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)^11,x)

[Out]

40*d^8/b^9/(b*x+a)^9*a^7*c^2+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+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^5-21/b^5/(b*x+a)^10*c^6*d^4*a^4+12/b^4/(b*x+a
)^10*a^3*c^7*d^3-9/2/b^3/(b*x+a)^10*c^8*d^2*a^2+1/b^2/(b*x+a)^10*c^9*d*a+12/b^8/(b*x+a)^10*c^3*d^7*a^7-21/b^7/
(b*x+a)^10*a^6*c^4*d^6+126/5/b^6/(b*x+a)^10*a^5*c^5*d^5+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+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^3-10*d^9/b^10/(b*x+a
)^9*a^8*c+10*d^2/b^3/(b*x+a)^9*a*c^8-120*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^3-45/8*d^10/b^11/(b*x+a)^8*a^8-45/8*d^2/b^3/(b*x+a)^8*c^8-35*d^10/b^11/(b*x+a)^6*a^6-35*d^4/b^5/
(b*x+a)^6*c^6-45/2*d^10/b^11/(b*x+a)^2*a^2-45/2*d^8/b^9/(b*x+a)^2*c^2-1/10/b^11/(b*x+a)^10*a^10*d^10+10/b^11*d
^10/(b*x+a)*a-10/b^10*d^9/(b*x+a)*c+120/7*d^10/b^11/(b*x+a)^7*a^7-120/7*d^3/b^4/(b*x+a)^7*c^7+10/9*d^10/b^11/(
b*x+a)^9*a^9-10/9*d/b^2/(b*x+a)^9*c^9+40*d^10/b^11/(b*x+a)^3*a^3-40*d^7/b^8/(b*x+a)^3*c^3+252/5*d^10/b^11/(b*x
+a)^5*a^5-252/5*d^5/b^6/(b*x+a)^5*c^5-105/2*d^10/b^11/(b*x+a)^4*a^4-105/2*d^6/b^7/(b*x+a)^4*c^4-1/10/b/(b*x+a)
^10*c^10+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+d^10*ln(b*x+a)/b^11-280/3*d^7/b^8/(b*x+a
)^9*a^6*c^3+45*d^9/b^10/(b*x+a)^2*a*c-120*d^9/b^10/(b*x+a)^3*a^2*c+120*d^8/b^9/(b*x+a)^3*a*c^2-252*d^9/b^10/(b
*x+a)^5*a^4*c+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+252*d^6/b^7/(b*x+a)^5*a*c^4+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+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

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Maxima [B]  time = 1.19226, size = 1316, normalized size = 4.86 \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)^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 + 5
04*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 - 738
1*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^1
0*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

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Fricas [B]  time = 2.01085, size = 2456, normalized size = 9.06 \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)^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 + 5
04*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 - 738
1*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^1
0*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 + 4
5*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^10*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 + 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)

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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)**11,x)

[Out]

Timed out

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Giac [B]  time = 1.07657, size = 1180, normalized size = 4.35 \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)^11,x, algorithm="giac")

[Out]

d^10*log(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 + 4410
0*(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 + 10584*(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 + 13
5*(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)