3.1319 \(\int \frac{(c+d x)^{10}}{(a+b x)^8} \, dx\)

Optimal. Leaf size=258 \[ \frac{5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac{45 d^8 x (b c-a d)^2}{b^{10}}-\frac{210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac{126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac{70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac{30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac{9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}+\frac{120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac{5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac{(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac{d^{10} (a+b x)^3}{3 b^{11}} \]

[Out]

(45*d^8*(b*c - a*d)^2*x)/b^10 - (b*c - a*d)^10/(7*b^11*(a + b*x)^7) - (5*d*(b*c - a*d)^9)/(3*b^11*(a + b*x)^6)
- (9*d^2*(b*c - a*d)^8)/(b^11*(a + b*x)^5) - (30*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^4) - (70*d^4*(b*c - a*d)^
6)/(b^11*(a + b*x)^3) - (126*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^2) - (210*d^6*(b*c - a*d)^4)/(b^11*(a + b*x))
+ (5*d^9*(b*c - a*d)*(a + b*x)^2)/b^11 + (d^10*(a + b*x)^3)/(3*b^11) + (120*d^7*(b*c - a*d)^3*Log[a + b*x])/b^
11

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Rubi [A]  time = 0.3649, antiderivative size = 258, 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} \[ \frac{5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac{45 d^8 x (b c-a d)^2}{b^{10}}-\frac{210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac{126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac{70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac{30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac{9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}+\frac{120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac{5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac{(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac{d^{10} (a+b x)^3}{3 b^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^8,x]

[Out]

(45*d^8*(b*c - a*d)^2*x)/b^10 - (b*c - a*d)^10/(7*b^11*(a + b*x)^7) - (5*d*(b*c - a*d)^9)/(3*b^11*(a + b*x)^6)
- (9*d^2*(b*c - a*d)^8)/(b^11*(a + b*x)^5) - (30*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^4) - (70*d^4*(b*c - a*d)^
6)/(b^11*(a + b*x)^3) - (126*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^2) - (210*d^6*(b*c - a*d)^4)/(b^11*(a + b*x))
+ (5*d^9*(b*c - a*d)*(a + b*x)^2)/b^11 + (d^10*(a + b*x)^3)/(3*b^11) + (120*d^7*(b*c - a*d)^3*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)^8} \, dx &=\int \left (\frac{45 d^8 (b c-a d)^2}{b^{10}}+\frac{(b c-a d)^{10}}{b^{10} (a+b x)^8}+\frac{10 d (b c-a d)^9}{b^{10} (a+b x)^7}+\frac{45 d^2 (b c-a d)^8}{b^{10} (a+b x)^6}+\frac{120 d^3 (b c-a d)^7}{b^{10} (a+b x)^5}+\frac{210 d^4 (b c-a d)^6}{b^{10} (a+b x)^4}+\frac{252 d^5 (b c-a d)^5}{b^{10} (a+b x)^3}+\frac{210 d^6 (b c-a d)^4}{b^{10} (a+b x)^2}+\frac{120 d^7 (b c-a d)^3}{b^{10} (a+b x)}+\frac{10 d^9 (b c-a d) (a+b x)}{b^{10}}+\frac{d^{10} (a+b x)^2}{b^{10}}\right ) \, dx\\ &=\frac{45 d^8 (b c-a d)^2 x}{b^{10}}-\frac{(b c-a d)^{10}}{7 b^{11} (a+b x)^7}-\frac{5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac{9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac{30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac{70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac{126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac{210 d^6 (b c-a d)^4}{b^{11} (a+b x)}+\frac{5 d^9 (b c-a d) (a+b x)^2}{b^{11}}+\frac{d^{10} (a+b x)^3}{3 b^{11}}+\frac{120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}\\ \end{align*}

Mathematica [A]  time = 0.265018, size = 239, normalized size = 0.93 \[ \frac{21 b d^8 x \left (36 a^2 d^2-80 a b c d+45 b^2 c^2\right )+21 b^2 d^9 x^2 (5 b c-4 a d)-\frac{4410 d^6 (b c-a d)^4}{a+b x}+\frac{2646 d^5 (a d-b c)^5}{(a+b x)^2}-\frac{1470 d^4 (b c-a d)^6}{(a+b x)^3}+\frac{630 d^3 (a d-b c)^7}{(a+b x)^4}-\frac{189 d^2 (b c-a d)^8}{(a+b x)^5}+2520 d^7 (b c-a d)^3 \log (a+b x)+\frac{35 d (a d-b c)^9}{(a+b x)^6}-\frac{3 (b c-a d)^{10}}{(a+b x)^7}+7 b^3 d^{10} x^3}{21 b^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^8,x]

[Out]

(21*b*d^8*(45*b^2*c^2 - 80*a*b*c*d + 36*a^2*d^2)*x + 21*b^2*d^9*(5*b*c - 4*a*d)*x^2 + 7*b^3*d^10*x^3 - (3*(b*c
- a*d)^10)/(a + b*x)^7 + (35*d*(-(b*c) + a*d)^9)/(a + b*x)^6 - (189*d^2*(b*c - a*d)^8)/(a + b*x)^5 + (630*d^3
*(-(b*c) + a*d)^7)/(a + b*x)^4 - (1470*d^4*(b*c - a*d)^6)/(a + b*x)^3 + (2646*d^5*(-(b*c) + a*d)^5)/(a + b*x)^
2 - (4410*d^6*(b*c - a*d)^4)/(a + b*x) + 2520*d^7*(b*c - a*d)^3*Log[a + b*x])/(21*b^11)

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Maple [B]  time = 0.02, size = 1241, 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)^8,x)

[Out]

15/b^3*d^2/(b*x+a)^6*a*c^8-80*d^9/b^9*a*c*x+1400/b^8*d^7/(b*x+a)^3*a^3*c^3-1050/b^7*d^6/(b*x+a)^3*a^2*c^4+420/
b^6*d^5/(b*x+a)^3*a*c^5+72/b^10*d^9/(b*x+a)^5*a^7*c-252/b^9*d^8/(b*x+a)^5*a^6*c^2+504/b^8*d^7/(b*x+a)^5*a^5*c^
3-630/b^7*d^6/(b*x+a)^5*a^4*c^4+1260/b^9*d^8/(b*x+a)^2*a^3*c^2-1260/b^8*d^7/(b*x+a)^2*a^2*c^3+630/b^7*d^6/(b*x
+a)^2*a*c^4+420/b^10*d^9/(b*x+a)^3*a^5*c-1050/b^9*d^8/(b*x+a)^3*a^4*c^2+630/b^9*d^8/(b*x+a)^4*a^5*c^2-1050/b^8
*d^7/(b*x+a)^4*a^4*c^3+1050/b^7*d^6/(b*x+a)^4*a^3*c^4-45/7/b^3/(b*x+a)^7*a^2*c^8*d^2+10/7/b^2/(b*x+a)^7*a*c^9*
d+840/b^10*d^9/(b*x+a)*a^3*c-1260/b^9*d^8/(b*x+a)*a^2*c^2+840/b^8*d^7/(b*x+a)*a*c^3-15/b^10*d^9/(b*x+a)^6*a^8*
c+60/b^9*d^8/(b*x+a)^6*a^7*c^2-140/b^8*d^7/(b*x+a)^6*a^6*c^3+210/b^7*d^6/(b*x+a)^6*a^5*c^4-210/b^6*d^5/(b*x+a)
^6*a^4*c^5+140/b^5*d^4/(b*x+a)^6*a^3*c^6-60/b^4*d^3/(b*x+a)^6*a^2*c^7+30/b^11*d^10/(b*x+a)^4*a^7-30/b^4*d^3/(b
*x+a)^4*c^7-30/b^7/(b*x+a)^7*a^6*c^4*d^6+36/b^6/(b*x+a)^7*a^5*c^5*d^5-30/b^5/(b*x+a)^7*a^4*c^6*d^4+120/7/b^4/(
b*x+a)^7*a^3*c^7*d^3-210/b^10*d^9/(b*x+a)^4*a^6*c+120/7/b^8/(b*x+a)^7*a^7*c^3*d^7+1/3*d^10/b^8*x^3-1/7/b/(b*x+
a)^7*c^10-1/7/b^11/(b*x+a)^7*a^10*d^10-120/b^11*d^10*ln(b*x+a)*a^3+120/b^8*d^7*ln(b*x+a)*c^3-210/b^11*d^10/(b*
x+a)*a^4-210/b^7*d^6/(b*x+a)*c^4+5/3/b^11*d^10/(b*x+a)^6*a^9-5/3/b^2*d/(b*x+a)^6*c^9-4*d^10/b^9*x^2*a+5*d^9/b^
8*x^2*c+36*d^10/b^10*a^2*x+45*d^8/b^8*c^2*x+126/b^11*d^10/(b*x+a)^2*a^5-126/b^6*d^5/(b*x+a)^2*c^5-9/b^11*d^10/
(b*x+a)^5*a^8-9/b^3*d^2/(b*x+a)^5*c^8-70/b^11*d^10/(b*x+a)^3*a^6-70/b^5*d^4/(b*x+a)^3*c^6-630/b^6*d^5/(b*x+a)^
4*a^2*c^5+210/b^5*d^4/(b*x+a)^4*a*c^6+10/7/b^10/(b*x+a)^7*a^9*c*d^9-630/b^10*d^9/(b*x+a)^2*a^4*c-45/7/b^9/(b*x
+a)^7*a^8*c^2*d^8+504/b^6*d^5/(b*x+a)^5*a^3*c^5-252/b^5*d^4/(b*x+a)^5*a^2*c^6+72/b^4*d^3/(b*x+a)^5*a*c^7+360/b
^10*d^9*ln(b*x+a)*a^2*c-360/b^9*d^8*ln(b*x+a)*a*c^2

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Maxima [B]  time = 1.34837, size = 1261, normalized size = 4.89 \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)^8,x, algorithm="maxima")

[Out]

-1/21*(3*b^10*c^10 + 5*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 126*a^5*b^5
*c^5*d^5 + 630*a^6*b^4*c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 - 10047*a^9*b*c*d^9 + 2761*a^10*
d^10 + 4410*(b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2646*(
b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 - 30*a^2*b^8*c^3*d^7 + 50*a^3*b^7*c^2*d^8 - 35*a^4*b^6*c*d^9 + 9*a^5*b^5*d^10)*
x^5 + 1470*(b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 110*a^3*b^7*c^3*d^7 + 195*a^4*b^6*c^2*d^8 -
141*a^5*b^5*c*d^9 + 37*a^6*b^4*d^10)*x^4 + 210*(3*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 105*a^
3*b^7*c^4*d^6 - 875*a^4*b^6*c^3*d^7 + 1617*a^5*b^5*c^2*d^8 - 1197*a^6*b^4*c*d^9 + 319*a^7*b^3*d^10)*x^3 + 63*(
3*b^10*c^8*d^2 + 6*a*b^9*c^7*d^3 + 14*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 1918*a^5*b^
5*c^3*d^7 + 3654*a^6*b^4*c^2*d^8 - 2754*a^7*b^3*c*d^9 + 743*a^8*b^2*d^10)*x^2 + 7*(5*b^10*c^9*d + 9*a*b^9*c^8*
d^2 + 18*a^2*b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d
^7 + 12042*a^7*b^3*c^2*d^8 - 9207*a^8*b^2*c*d^9 + 2509*a^9*b*d^10)*x)/(b^18*x^7 + 7*a*b^17*x^6 + 21*a^2*b^16*x
^5 + 35*a^3*b^15*x^4 + 35*a^4*b^14*x^3 + 21*a^5*b^13*x^2 + 7*a^6*b^12*x + a^7*b^11) + 1/3*(b^2*d^10*x^3 + 3*(5
*b^2*c*d^9 - 4*a*b*d^10)*x^2 + 3*(45*b^2*c^2*d^8 - 80*a*b*c*d^9 + 36*a^2*d^10)*x)/b^10 + 120*(b^3*c^3*d^7 - 3*
a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10)*log(b*x + a)/b^11

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Fricas [B]  time = 1.95742, size = 2878, normalized size = 11.16 \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)^8,x, algorithm="fricas")

[Out]

1/21*(7*b^10*d^10*x^10 - 3*b^10*c^10 - 5*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 18*a^3*b^7*c^7*d^3 - 42*a^4*b^6*c^6
*d^4 - 126*a^5*b^5*c^5*d^5 - 630*a^6*b^4*c^4*d^6 + 6534*a^7*b^3*c^3*d^7 - 12987*a^8*b^2*c^2*d^8 + 10047*a^9*b*
c*d^9 - 2761*a^10*d^10 + 35*(3*b^10*c*d^9 - a*b^9*d^10)*x^9 + 315*(3*b^10*c^2*d^8 - 3*a*b^9*c*d^9 + a^2*b^8*d^
10)*x^8 + 49*(135*a*b^9*c^2*d^8 - 195*a^2*b^8*c*d^9 + 77*a^3*b^7*d^10)*x^7 - 49*(90*b^10*c^4*d^6 - 360*a*b^9*c
^3*d^7 + 135*a^2*b^8*c^2*d^8 + 285*a^3*b^7*c*d^9 - 179*a^4*b^6*d^10)*x^6 - 147*(18*b^10*c^5*d^5 + 90*a*b^9*c^4
*d^6 - 540*a^2*b^8*c^3*d^7 + 675*a^3*b^7*c^2*d^8 - 255*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 - 245*(6*b^10*c^6*d^4
+ 18*a*b^9*c^5*d^5 + 90*a^2*b^8*c^4*d^6 - 660*a^3*b^7*c^3*d^7 + 1035*a^4*b^6*c^2*d^8 - 615*a^5*b^5*c*d^9 + 12
1*a^6*b^4*d^10)*x^4 - 35*(18*b^10*c^7*d^3 + 42*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 630*a^3*b^7*c^4*d^6 - 525
0*a^4*b^6*c^3*d^7 + 9135*a^5*b^5*c^2*d^8 - 6195*a^6*b^4*c*d^9 + 1477*a^7*b^3*d^10)*x^3 - 21*(9*b^10*c^8*d^2 +
18*a*b^9*c^7*d^3 + 42*a^2*b^8*c^6*d^4 + 126*a^3*b^7*c^5*d^5 + 630*a^4*b^6*c^4*d^6 - 5754*a^5*b^5*c^3*d^7 + 106
47*a^6*b^4*c^2*d^8 - 7707*a^7*b^3*c*d^9 + 1981*a^8*b^2*d^10)*x^2 - 7*(5*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 18*a^2*
b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d^7 + 11907*a^
7*b^3*c^2*d^8 - 8967*a^8*b^2*c*d^9 + 2401*a^9*b*d^10)*x + 2520*(a^7*b^3*c^3*d^7 - 3*a^8*b^2*c^2*d^8 + 3*a^9*b*
c*d^9 - a^10*d^10 + (b^10*c^3*d^7 - 3*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 7*(a*b^9*c^3*d^7 -
3*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 - a^4*b^6*d^10)*x^6 + 21*(a^2*b^8*c^3*d^7 - 3*a^3*b^7*c^2*d^8 + 3*a^4*b^6
*c*d^9 - a^5*b^5*d^10)*x^5 + 35*(a^3*b^7*c^3*d^7 - 3*a^4*b^6*c^2*d^8 + 3*a^5*b^5*c*d^9 - a^6*b^4*d^10)*x^4 + 3
5*(a^4*b^6*c^3*d^7 - 3*a^5*b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 21*(a^5*b^5*c^3*d^7 - 3*a^6*b^4
*c^2*d^8 + 3*a^7*b^3*c*d^9 - a^8*b^2*d^10)*x^2 + 7*(a^6*b^4*c^3*d^7 - 3*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 - a^
9*b*d^10)*x)*log(b*x + a))/(b^18*x^7 + 7*a*b^17*x^6 + 21*a^2*b^16*x^5 + 35*a^3*b^15*x^4 + 35*a^4*b^14*x^3 + 21
*a^5*b^13*x^2 + 7*a^6*b^12*x + a^7*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)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.06426, size = 1177, normalized size = 4.56 \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)^8,x, algorithm="giac")

[Out]

120*(b^3*c^3*d^7 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10)*log(abs(b*x + a))/b^11 - 1/21*(3*b^10*c^10 + 5*
a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 630*a^6*b^4*
c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 - 10047*a^9*b*c*d^9 + 2761*a^10*d^10 + 4410*(b^10*c^4*d
^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2646*(b^10*c^5*d^5 + 5*a*b^9*
c^4*d^6 - 30*a^2*b^8*c^3*d^7 + 50*a^3*b^7*c^2*d^8 - 35*a^4*b^6*c*d^9 + 9*a^5*b^5*d^10)*x^5 + 1470*(b^10*c^6*d^
4 + 3*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 110*a^3*b^7*c^3*d^7 + 195*a^4*b^6*c^2*d^8 - 141*a^5*b^5*c*d^9 + 37*
a^6*b^4*d^10)*x^4 + 210*(3*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6 - 875*a^4
*b^6*c^3*d^7 + 1617*a^5*b^5*c^2*d^8 - 1197*a^6*b^4*c*d^9 + 319*a^7*b^3*d^10)*x^3 + 63*(3*b^10*c^8*d^2 + 6*a*b^
9*c^7*d^3 + 14*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 1918*a^5*b^5*c^3*d^7 + 3654*a^6*b^
4*c^2*d^8 - 2754*a^7*b^3*c*d^9 + 743*a^8*b^2*d^10)*x^2 + 7*(5*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 18*a^2*b^8*c^7*d^
3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d^7 + 12042*a^7*b^3*c^2*
d^8 - 9207*a^8*b^2*c*d^9 + 2509*a^9*b*d^10)*x)/((b*x + a)^7*b^11) + 1/3*(b^16*d^10*x^3 + 15*b^16*c*d^9*x^2 - 1
2*a*b^15*d^10*x^2 + 135*b^16*c^2*d^8*x - 240*a*b^15*c*d^9*x + 108*a^2*b^14*d^10*x)/b^24