3.1317 \(\int \frac{(c+d x)^{10}}{(a+b x)^6} \, dx\)

Optimal. Leaf size=260 \[ \frac{5 d^9 (a+b x)^4 (b c-a d)}{2 b^{11}}+\frac{15 d^8 (a+b x)^3 (b c-a d)^2}{b^{11}}+\frac{60 d^7 (a+b x)^2 (b c-a d)^3}{b^{11}}+\frac{210 d^6 x (b c-a d)^4}{b^{10}}-\frac{210 d^4 (b c-a d)^6}{b^{11} (a+b x)}-\frac{60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac{15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}+\frac{252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}}-\frac{5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac{(b c-a d)^{10}}{5 b^{11} (a+b x)^5}+\frac{d^{10} (a+b x)^5}{5 b^{11}} \]

[Out]

(210*d^6*(b*c - a*d)^4*x)/b^10 - (b*c - a*d)^10/(5*b^11*(a + b*x)^5) - (5*d*(b*c - a*d)^9)/(2*b^11*(a + b*x)^4
) - (15*d^2*(b*c - a*d)^8)/(b^11*(a + b*x)^3) - (60*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^2) - (210*d^4*(b*c - a*
d)^6)/(b^11*(a + b*x)) + (60*d^7*(b*c - a*d)^3*(a + b*x)^2)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^3)/b^11 + (
5*d^9*(b*c - a*d)*(a + b*x)^4)/(2*b^11) + (d^10*(a + b*x)^5)/(5*b^11) + (252*d^5*(b*c - a*d)^5*Log[a + b*x])/b
^11

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Rubi [A]  time = 0.420905, antiderivative size = 260, 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} \[ \frac{5 d^9 (a+b x)^4 (b c-a d)}{2 b^{11}}+\frac{15 d^8 (a+b x)^3 (b c-a d)^2}{b^{11}}+\frac{60 d^7 (a+b x)^2 (b c-a d)^3}{b^{11}}+\frac{210 d^6 x (b c-a d)^4}{b^{10}}-\frac{210 d^4 (b c-a d)^6}{b^{11} (a+b x)}-\frac{60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac{15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}+\frac{252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}}-\frac{5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac{(b c-a d)^{10}}{5 b^{11} (a+b x)^5}+\frac{d^{10} (a+b x)^5}{5 b^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^6,x]

[Out]

(210*d^6*(b*c - a*d)^4*x)/b^10 - (b*c - a*d)^10/(5*b^11*(a + b*x)^5) - (5*d*(b*c - a*d)^9)/(2*b^11*(a + b*x)^4
) - (15*d^2*(b*c - a*d)^8)/(b^11*(a + b*x)^3) - (60*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^2) - (210*d^4*(b*c - a*
d)^6)/(b^11*(a + b*x)) + (60*d^7*(b*c - a*d)^3*(a + b*x)^2)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^3)/b^11 + (
5*d^9*(b*c - a*d)*(a + b*x)^4)/(2*b^11) + (d^10*(a + b*x)^5)/(5*b^11) + (252*d^5*(b*c - a*d)^5*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)^6} \, dx &=\int \left (\frac{210 d^6 (b c-a d)^4}{b^{10}}+\frac{(b c-a d)^{10}}{b^{10} (a+b x)^6}+\frac{10 d (b c-a d)^9}{b^{10} (a+b x)^5}+\frac{45 d^2 (b c-a d)^8}{b^{10} (a+b x)^4}+\frac{120 d^3 (b c-a d)^7}{b^{10} (a+b x)^3}+\frac{210 d^4 (b c-a d)^6}{b^{10} (a+b x)^2}+\frac{252 d^5 (b c-a d)^5}{b^{10} (a+b x)}+\frac{120 d^7 (b c-a d)^3 (a+b x)}{b^{10}}+\frac{45 d^8 (b c-a d)^2 (a+b x)^2}{b^{10}}+\frac{10 d^9 (b c-a d) (a+b x)^3}{b^{10}}+\frac{d^{10} (a+b x)^4}{b^{10}}\right ) \, dx\\ &=\frac{210 d^6 (b c-a d)^4 x}{b^{10}}-\frac{(b c-a d)^{10}}{5 b^{11} (a+b x)^5}-\frac{5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac{15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}-\frac{60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac{210 d^4 (b c-a d)^6}{b^{11} (a+b x)}+\frac{60 d^7 (b c-a d)^3 (a+b x)^2}{b^{11}}+\frac{15 d^8 (b c-a d)^2 (a+b x)^3}{b^{11}}+\frac{5 d^9 (b c-a d) (a+b x)^4}{2 b^{11}}+\frac{d^{10} (a+b x)^5}{5 b^{11}}+\frac{252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}}\\ \end{align*}

Mathematica [A]  time = 0.222811, size = 305, normalized size = 1.17 \[ \frac{10 b^3 d^8 x^3 \left (7 a^2 d^2-20 a b c d+15 b^2 c^2\right )+10 b^2 d^7 x^2 \left (105 a^2 b c d^2-28 a^3 d^3-135 a b^2 c^2 d+60 b^3 c^3\right )+10 b d^6 x \left (945 a^2 b^2 c^2 d^2-560 a^3 b c d^3+126 a^4 d^4-720 a b^3 c^3 d+210 b^4 c^4\right )+5 b^4 d^9 x^4 (5 b c-3 a d)-\frac{2100 d^4 (b c-a d)^6}{a+b x}+\frac{600 d^3 (a d-b c)^7}{(a+b x)^2}-\frac{150 d^2 (b c-a d)^8}{(a+b x)^3}+2520 d^5 (b c-a d)^5 \log (a+b x)+\frac{25 d (a d-b c)^9}{(a+b x)^4}-\frac{2 (b c-a d)^{10}}{(a+b x)^5}+2 b^5 d^{10} x^5}{10 b^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^6,x]

[Out]

(10*b*d^6*(210*b^4*c^4 - 720*a*b^3*c^3*d + 945*a^2*b^2*c^2*d^2 - 560*a^3*b*c*d^3 + 126*a^4*d^4)*x + 10*b^2*d^7
*(60*b^3*c^3 - 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 28*a^3*d^3)*x^2 + 10*b^3*d^8*(15*b^2*c^2 - 20*a*b*c*d + 7*a
^2*d^2)*x^3 + 5*b^4*d^9*(5*b*c - 3*a*d)*x^4 + 2*b^5*d^10*x^5 - (2*(b*c - a*d)^10)/(a + b*x)^5 + (25*d*(-(b*c)
+ a*d)^9)/(a + b*x)^4 - (150*d^2*(b*c - a*d)^8)/(a + b*x)^3 + (600*d^3*(-(b*c) + a*d)^7)/(a + b*x)^2 - (2100*d
^4*(b*c - a*d)^6)/(a + b*x) + 2520*d^5*(b*c - a*d)^5*Log[a + b*x])/(10*b^11)

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Maple [B]  time = 0.02, size = 1199, normalized size = 4.6 \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)^6,x)

[Out]

-2520/b^9*d^8*ln(b*x+a)*a^3*c^2+2520/b^8*d^7*ln(b*x+a)*a^2*c^3-1260/b^7*d^6*ln(b*x+a)*a*c^4-45/2/b^10*d^9/(b*x
+a)^4*a^8*c+90/b^9*d^8/(b*x+a)^4*a^7*c^2-210/b^8*d^7/(b*x+a)^4*a^6*c^3+315/b^7*d^6/(b*x+a)^4*a^5*c^4+2/b^2/(b*
x+a)^5*a*c^9*d+1260/b^10*d^9*ln(b*x+a)*a^4*c-9/b^3/(b*x+a)^5*a^2*c^8*d^2+45/2/b^3*d^2/(b*x+a)^4*a*c^8-1260/b^6
*d^5/(b*x+a)^2*a^2*c^5+420/b^5*d^4/(b*x+a)^2*a*c^6+120/b^10*d^9/(b*x+a)^3*a^7*c-420/b^9*d^8/(b*x+a)^3*a^6*c^2+
840/b^8*d^7/(b*x+a)^3*a^5*c^3-1050/b^7*d^6/(b*x+a)^3*a^4*c^4+840/b^6*d^5/(b*x+a)^3*a^3*c^5-315/b^6*d^5/(b*x+a)
^4*a^4*c^5+210/b^5*d^4/(b*x+a)^4*a^3*c^6-90/b^4*d^3/(b*x+a)^4*a^2*c^7-420/b^5*d^4/(b*x+a)^3*a^2*c^6+120/b^4*d^
3/(b*x+a)^3*a*c^7+1/5*d^10/b^6*x^5-1/5/b/(b*x+a)^5*c^10-3/2*d^10/b^7*x^4*a+5/2*d^9/b^6*x^4*c+7*d^10/b^8*x^3*a^
2+15*d^8/b^6*x^3*c^2-28*d^10/b^9*x^2*a^3+60*d^7/b^6*x^2*c^3+126*d^10/b^10*a^4*x+210*d^6/b^6*c^4*x-210/b^11*d^1
0/(b*x+a)*a^6-210/b^5*d^4/(b*x+a)*c^6+60/b^11*d^10/(b*x+a)^2*a^7-60/b^4*d^3/(b*x+a)^2*c^7-15/b^11*d^10/(b*x+a)
^3*a^8-15/b^3*d^2/(b*x+a)^3*c^8-1/5/b^11/(b*x+a)^5*a^10*d^10-252/b^11*d^10*ln(b*x+a)*a^5+252/b^6*d^5*ln(b*x+a)
*c^5+5/2/b^11*d^10/(b*x+a)^4*a^9-5/2/b^2*d/(b*x+a)^4*c^9-2100/b^8*d^7/(b*x+a)^2*a^4*c^3+2100/b^7*d^6/(b*x+a)^2
*a^3*c^4+24/b^4/(b*x+a)^5*a^3*c^7*d^3-20*d^9/b^7*x^3*a*c+105*d^9/b^8*x^2*a^2*c-135*d^8/b^7*x^2*a*c^2-560*d^9/b
^9*a^3*c*x+945*d^8/b^8*a^2*c^2*x-720*d^7/b^7*a*c^3*x+1260/b^10*d^9/(b*x+a)*a^5*c-3150/b^9*d^8/(b*x+a)*a^4*c^2+
4200/b^8*d^7/(b*x+a)*a^3*c^3-3150/b^7*d^6/(b*x+a)*a^2*c^4+1260/b^6*d^5/(b*x+a)*a*c^5-420/b^10*d^9/(b*x+a)^2*a^
6*c+1260/b^9*d^8/(b*x+a)^2*a^5*c^2+2/b^10/(b*x+a)^5*a^9*c*d^9-9/b^9/(b*x+a)^5*a^8*c^2*d^8+24/b^8/(b*x+a)^5*a^7
*c^3*d^7-42/b^7/(b*x+a)^5*a^6*c^4*d^6+252/5/b^6/(b*x+a)^5*a^5*c^5*d^5-42/b^5/(b*x+a)^5*a^4*c^6*d^4

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Maxima [B]  time = 1.27585, size = 1231, normalized size = 4.73 \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)^6,x, algorithm="maxima")

[Out]

-1/10*(2*b^10*c^10 + 5*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 - 5754*a^5*
b^5*c^5*d^5 + 18270*a^6*b^4*c^4*d^6 - 27540*a^7*b^3*c^3*d^7 + 22290*a^8*b^2*c^2*d^8 - 9395*a^9*b*c*d^9 + 1627*
a^10*d^10 + 2100*(b^10*c^6*d^4 - 6*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 + 15*a^4*b^6*c^2*d^
8 - 6*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 600*(b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 - 63*a^2*b^8*c^5*d^5 + 175*a^3*b
^7*c^4*d^6 - 245*a^4*b^6*c^3*d^7 + 189*a^5*b^5*c^2*d^8 - 77*a^6*b^4*c*d^9 + 13*a^7*b^3*d^10)*x^3 + 150*(b^10*c
^8*d^2 + 4*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 - 308*a^3*b^7*c^5*d^5 + 910*a^4*b^6*c^4*d^6 - 1316*a^5*b^5*c^3*d
^7 + 1036*a^6*b^4*c^2*d^8 - 428*a^7*b^3*c*d^9 + 73*a^8*b^2*d^10)*x^2 + 25*(b^10*c^9*d + 3*a*b^9*c^8*d^2 + 12*a
^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 - 1050*a^4*b^6*c^5*d^5 + 3234*a^5*b^5*c^4*d^6 - 4788*a^6*b^4*c^3*d^7 + 382
8*a^7*b^3*c^2*d^8 - 1599*a^8*b^2*c*d^9 + 275*a^9*b*d^10)*x)/(b^16*x^5 + 5*a*b^15*x^4 + 10*a^2*b^14*x^3 + 10*a^
3*b^13*x^2 + 5*a^4*b^12*x + a^5*b^11) + 1/10*(2*b^4*d^10*x^5 + 5*(5*b^4*c*d^9 - 3*a*b^3*d^10)*x^4 + 10*(15*b^4
*c^2*d^8 - 20*a*b^3*c*d^9 + 7*a^2*b^2*d^10)*x^3 + 10*(60*b^4*c^3*d^7 - 135*a*b^3*c^2*d^8 + 105*a^2*b^2*c*d^9 -
28*a^3*b*d^10)*x^2 + 10*(210*b^4*c^4*d^6 - 720*a*b^3*c^3*d^7 + 945*a^2*b^2*c^2*d^8 - 560*a^3*b*c*d^9 + 126*a^
4*d^10)*x)/b^10 + 252*(b^5*c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2*c^2*d^8 + 5*a^4*b*c*d^9
- a^5*d^10)*log(b*x + a)/b^11

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Fricas [B]  time = 1.84126, size = 2981, normalized size = 11.47 \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)^6,x, algorithm="fricas")

[Out]

1/10*(2*b^10*d^10*x^10 - 2*b^10*c^10 - 5*a*b^9*c^9*d - 15*a^2*b^8*c^8*d^2 - 60*a^3*b^7*c^7*d^3 - 420*a^4*b^6*c
^6*d^4 + 5754*a^5*b^5*c^5*d^5 - 18270*a^6*b^4*c^4*d^6 + 27540*a^7*b^3*c^3*d^7 - 22290*a^8*b^2*c^2*d^8 + 9395*a
^9*b*c*d^9 - 1627*a^10*d^10 + 5*(5*b^10*c*d^9 - a*b^9*d^10)*x^9 + 15*(10*b^10*c^2*d^8 - 5*a*b^9*c*d^9 + a^2*b^
8*d^10)*x^8 + 60*(10*b^10*c^3*d^7 - 10*a*b^9*c^2*d^8 + 5*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 420*(5*b^10*c^4*d
^6 - 10*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*d^8 - 5*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + (10500*a*b^9*c^4*d^6 - 3000
0*a^2*b^8*c^3*d^7 + 35250*a^3*b^7*c^2*d^8 - 19375*a^4*b^6*c*d^9 + 4127*a^5*b^5*d^10)*x^5 - 5*(420*b^10*c^6*d^4
- 2520*a*b^9*c^5*d^5 + 2100*a^2*b^8*c^4*d^6 + 4800*a^3*b^7*c^3*d^7 - 10050*a^4*b^6*c^2*d^8 + 6775*a^5*b^5*c*d
^9 - 1607*a^6*b^4*d^10)*x^4 - 10*(60*b^10*c^7*d^3 + 420*a*b^9*c^6*d^4 - 3780*a^2*b^8*c^5*d^5 + 8400*a^3*b^7*c^
4*d^6 - 7800*a^4*b^6*c^3*d^7 + 2550*a^5*b^5*c^2*d^8 + 475*a^6*b^4*c*d^9 - 347*a^7*b^3*d^10)*x^3 - 10*(15*b^10*
c^8*d^2 + 60*a*b^9*c^7*d^3 + 420*a^2*b^8*c^6*d^4 - 4620*a^3*b^7*c^5*d^5 + 12600*a^4*b^6*c^4*d^6 - 16200*a^5*b^
5*c^3*d^7 + 10950*a^6*b^4*c^2*d^8 - 3725*a^7*b^3*c*d^9 + 493*a^8*b^2*d^10)*x^2 - 5*(5*b^10*c^9*d + 15*a*b^9*c^
8*d^2 + 60*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 - 5250*a^4*b^6*c^5*d^5 + 15750*a^5*b^5*c^4*d^6 - 22500*a^6*b^
4*c^3*d^7 + 17250*a^7*b^3*c^2*d^8 - 6875*a^8*b^2*c*d^9 + 1123*a^9*b*d^10)*x + 2520*(a^5*b^5*c^5*d^5 - 5*a^6*b^
4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 - 10*a^8*b^2*c^2*d^8 + 5*a^9*b*c*d^9 - a^10*d^10 + (b^10*c^5*d^5 - 5*a*b^9*c^4*
d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 5*(a*b^9*c^5*d^5 - 5*a^2
*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 - 10*a^4*b^6*c^2*d^8 + 5*a^5*b^5*c*d^9 - a^6*b^4*d^10)*x^4 + 10*(a^2*b^8*c^5
*d^5 - 5*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7 - 10*a^5*b^5*c^2*d^8 + 5*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 10*
(a^3*b^7*c^5*d^5 - 5*a^4*b^6*c^4*d^6 + 10*a^5*b^5*c^3*d^7 - 10*a^6*b^4*c^2*d^8 + 5*a^7*b^3*c*d^9 - a^8*b^2*d^1
0)*x^2 + 5*(a^4*b^6*c^5*d^5 - 5*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c^3*d^7 - 10*a^7*b^3*c^2*d^8 + 5*a^8*b^2*c*d^9 -
a^9*b*d^10)*x)*log(b*x + a))/(b^16*x^5 + 5*a*b^15*x^4 + 10*a^2*b^14*x^3 + 10*a^3*b^13*x^2 + 5*a^4*b^12*x + a^5
*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)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.06812, size = 1192, normalized size = 4.58 \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)^6,x, algorithm="giac")

[Out]

252*(b^5*c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2*c^2*d^8 + 5*a^4*b*c*d^9 - a^5*d^10)*log(a
bs(b*x + a))/b^11 - 1/10*(2*b^10*c^10 + 5*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 420*a^4*b^6*
c^6*d^4 - 5754*a^5*b^5*c^5*d^5 + 18270*a^6*b^4*c^4*d^6 - 27540*a^7*b^3*c^3*d^7 + 22290*a^8*b^2*c^2*d^8 - 9395*
a^9*b*c*d^9 + 1627*a^10*d^10 + 2100*(b^10*c^6*d^4 - 6*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
+ 15*a^4*b^6*c^2*d^8 - 6*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 600*(b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 - 63*a^2*b^8*
c^5*d^5 + 175*a^3*b^7*c^4*d^6 - 245*a^4*b^6*c^3*d^7 + 189*a^5*b^5*c^2*d^8 - 77*a^6*b^4*c*d^9 + 13*a^7*b^3*d^10
)*x^3 + 150*(b^10*c^8*d^2 + 4*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 - 308*a^3*b^7*c^5*d^5 + 910*a^4*b^6*c^4*d^6 -
1316*a^5*b^5*c^3*d^7 + 1036*a^6*b^4*c^2*d^8 - 428*a^7*b^3*c*d^9 + 73*a^8*b^2*d^10)*x^2 + 25*(b^10*c^9*d + 3*a
*b^9*c^8*d^2 + 12*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 - 1050*a^4*b^6*c^5*d^5 + 3234*a^5*b^5*c^4*d^6 - 4788*a^
6*b^4*c^3*d^7 + 3828*a^7*b^3*c^2*d^8 - 1599*a^8*b^2*c*d^9 + 275*a^9*b*d^10)*x)/((b*x + a)^5*b^11) + 1/10*(2*b^
24*d^10*x^5 + 25*b^24*c*d^9*x^4 - 15*a*b^23*d^10*x^4 + 150*b^24*c^2*d^8*x^3 - 200*a*b^23*c*d^9*x^3 + 70*a^2*b^
22*d^10*x^3 + 600*b^24*c^3*d^7*x^2 - 1350*a*b^23*c^2*d^8*x^2 + 1050*a^2*b^22*c*d^9*x^2 - 280*a^3*b^21*d^10*x^2
+ 2100*b^24*c^4*d^6*x - 7200*a*b^23*c^3*d^7*x + 9450*a^2*b^22*c^2*d^8*x - 5600*a^3*b^21*c*d^9*x + 1260*a^4*b^
20*d^10*x)/b^30