3.1318 \(\int \frac{(c+d x)^{10}}{(a+b x)^7} \, dx\)

Optimal. Leaf size=262 \[ \frac{10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac{45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac{120 d^7 x (b c-a d)^3}{b^{10}}-\frac{252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac{105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac{40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac{45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}+\frac{210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac{2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac{(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac{d^{10} (a+b x)^4}{4 b^{11}} \]

[Out]

(120*d^7*(b*c - a*d)^3*x)/b^10 - (b*c - a*d)^10/(6*b^11*(a + b*x)^6) - (2*d*(b*c - a*d)^9)/(b^11*(a + b*x)^5)
- (45*d^2*(b*c - a*d)^8)/(4*b^11*(a + b*x)^4) - (40*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^3) - (105*d^4*(b*c - a*
d)^6)/(b^11*(a + b*x)^2) - (252*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)) + (45*d^8*(b*c - a*d)^2*(a + b*x)^2)/(2*b^
11) + (10*d^9*(b*c - a*d)*(a + b*x)^3)/(3*b^11) + (d^10*(a + b*x)^4)/(4*b^11) + (210*d^6*(b*c - a*d)^4*Log[a +
b*x])/b^11

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Rubi [A]  time = 0.386594, antiderivative size = 262, 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} \[ \frac{10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac{45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac{120 d^7 x (b c-a d)^3}{b^{10}}-\frac{252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac{105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac{40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac{45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}+\frac{210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac{2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac{(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac{d^{10} (a+b x)^4}{4 b^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^7,x]

[Out]

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

Mathematica [A]  time = 0.236312, size = 265, normalized size = 1.01 \[ \frac{6 b^2 d^8 x^2 \left (28 a^2 d^2-70 a b c d+45 b^2 c^2\right )+12 b d^7 x \left (280 a^2 b c d^2-84 a^3 d^3-315 a b^2 c^2 d+120 b^3 c^3\right )+4 b^3 d^9 x^3 (10 b c-7 a d)+\frac{3024 d^5 (a d-b c)^5}{a+b x}-\frac{1260 d^4 (b c-a d)^6}{(a+b x)^2}+\frac{480 d^3 (a d-b c)^7}{(a+b x)^3}-\frac{135 d^2 (b c-a d)^8}{(a+b x)^4}+2520 d^6 (b c-a d)^4 \log (a+b x)+\frac{24 d (a d-b c)^9}{(a+b x)^5}-\frac{2 (b c-a d)^{10}}{(a+b x)^6}+3 b^4 d^{10} x^4}{12 b^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^7,x]

[Out]

(12*b*d^7*(120*b^3*c^3 - 315*a*b^2*c^2*d + 280*a^2*b*c*d^2 - 84*a^3*d^3)*x + 6*b^2*d^8*(45*b^2*c^2 - 70*a*b*c*
d + 28*a^2*d^2)*x^2 + 4*b^3*d^9*(10*b*c - 7*a*d)*x^3 + 3*b^4*d^10*x^4 - (2*(b*c - a*d)^10)/(a + b*x)^6 + (24*d
*(-(b*c) + a*d)^9)/(a + b*x)^5 - (135*d^2*(b*c - a*d)^8)/(a + b*x)^4 + (480*d^3*(-(b*c) + a*d)^7)/(a + b*x)^3
- (1260*d^4*(b*c - a*d)^6)/(a + b*x)^2 + (3024*d^5*(-(b*c) + a*d)^5)/(a + b*x) + 2520*d^6*(b*c - a*d)^4*Log[a
+ b*x])/(12*b^11)

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Maple [B]  time = 0.02, size = 1222, 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)^7,x)

[Out]

-1400/b^8*d^7/(b*x+a)^3*a^4*c^3+1400/b^7*d^6/(b*x+a)^3*a^3*c^4-1575/b^9*d^8/(b*x+a)^2*a^4*c^2+2100/b^8*d^7/(b*
x+a)^2*a^3*c^3-1575/b^7*d^6/(b*x+a)^2*a^2*c^4+42/b^6/(b*x+a)^6*a^5*c^5*d^5-35/b^5/(b*x+a)^6*a^4*c^6*d^4+18/b^3
*d^2/(b*x+a)^5*a*c^8-315*d^8/b^8*a*c^2*x-1260/b^10*d^9/(b*x+a)*a^4*c+2520/b^9*d^8/(b*x+a)*a^3*c^2-2520/b^8*d^7
/(b*x+a)*a^2*c^3+1260/b^7*d^6/(b*x+a)*a*c^4+5/3/b^10/(b*x+a)^6*a^9*c*d^9-15/2/b^9/(b*x+a)^6*a^8*c^2*d^8+20/b^8
/(b*x+a)^6*a^7*c^3*d^7-35/b^7/(b*x+a)^6*a^6*c^4*d^6+20/b^4/(b*x+a)^6*a^3*c^7*d^3-15/2/b^3/(b*x+a)^6*a^2*c^8*d^
2+5/3/b^2/(b*x+a)^6*a*c^9*d+630/b^10*d^9/(b*x+a)^2*a^5*c-840/b^6*d^5/(b*x+a)^3*a^2*c^5+280/b^5*d^4/(b*x+a)^3*a
*c^6+630/b^6*d^5/(b*x+a)^2*a*c^5-840/b^10*d^9*ln(b*x+a)*a^3*c+1260/b^9*d^8*ln(b*x+a)*a^2*c^2-840/b^8*d^7*ln(b*
x+a)*a*c^3-280/b^10*d^9/(b*x+a)^3*a^6*c+840/b^9*d^8/(b*x+a)^3*a^5*c^2+1/4*d^10/b^7*x^4-1/6/b/(b*x+a)^6*c^10-31
5/b^9*d^8/(b*x+a)^4*a^6*c^2+252/b^11*d^10/(b*x+a)*a^5-252/b^6*d^5/(b*x+a)*c^5-1/6/b^11/(b*x+a)^6*a^10*d^10+90/
b^10*d^9/(b*x+a)^4*a^7*c-105/b^11*d^10/(b*x+a)^2*a^6-105/b^5*d^4/(b*x+a)^2*c^6+40/b^11*d^10/(b*x+a)^3*a^7-40/b
^4*d^3/(b*x+a)^3*c^7-7/3*d^10/b^8*x^3*a+10/3*d^9/b^7*x^3*c+14*d^10/b^9*x^2*a^2+45/2*d^8/b^7*x^2*c^2-84*d^10/b^
10*a^3*x+120*d^7/b^7*c^3*x+2/b^11*d^10/(b*x+a)^5*a^9-2/b^2*d/(b*x+a)^5*c^9-45/4/b^11*d^10/(b*x+a)^4*a^8-45/4/b
^3*d^2/(b*x+a)^4*c^8+210/b^11*d^10*ln(b*x+a)*a^4+210/b^7*d^6*ln(b*x+a)*c^4+630/b^8*d^7/(b*x+a)^4*a^5*c^3-1575/
2/b^7*d^6/(b*x+a)^4*a^4*c^4+630/b^6*d^5/(b*x+a)^4*a^3*c^5-315/b^5*d^4/(b*x+a)^4*a^2*c^6+90/b^4*d^3/(b*x+a)^4*a
*c^7-35*d^9/b^8*x^2*a*c+280*d^9/b^9*a^2*c*x-18/b^10*d^9/(b*x+a)^5*a^8*c+72/b^9*d^8/(b*x+a)^5*a^7*c^2-168/b^8*d
^7/(b*x+a)^5*a^6*c^3+252/b^7*d^6/(b*x+a)^5*a^5*c^4-252/b^6*d^5/(b*x+a)^5*a^4*c^5+168/b^5*d^4/(b*x+a)^5*a^3*c^6
-72/b^4*d^3/(b*x+a)^5*a^2*c^7

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Maxima [B]  time = 1.33374, size = 1249, normalized size = 4.77 \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)^7,x, algorithm="maxima")

[Out]

-1/12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5
*c^5*d^5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 2131*a^1
0*d^10 + 3024*(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 + 1260*(b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100*a^3*b^7*c^3*d^7 - 105*a^4*b^
6*c^2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^4*d^10)*x^4 + 240*(2*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d
^5 - 385*a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3*d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4*c*d^9 - 107*a^7*b^3*d^10)*x
^3 + 45*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 - 1750*a^4*b^6*c^4*d^6 +
4312*a^5*b^5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)*x^2 + 6*(4*b^10*c^9*d + 9
*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14616*
a^6*b^4*c^3*d^7 - 16524*a^7*b^3*c^2*d^8 + 8916*a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/(b^17*x^6 + 6*a*b^16*x^5 +
15*a^2*b^15*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*b^11) + 1/12*(3*b^3*d^10*x^4 + 4*(10*
b^3*c*d^9 - 7*a*b^2*d^10)*x^3 + 6*(45*b^3*c^2*d^8 - 70*a*b^2*c*d^9 + 28*a^2*b*d^10)*x^2 + 12*(120*b^3*c^3*d^7
- 315*a*b^2*c^2*d^8 + 280*a^2*b*c*d^9 - 84*a^3*d^10)*x)/b^10 + 210*(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*
c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*log(b*x + a)/b^11

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Fricas [B]  time = 1.94954, size = 2954, normalized size = 11.27 \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)^7,x, algorithm="fricas")

[Out]

1/12*(3*b^10*d^10*x^10 - 2*b^10*c^10 - 4*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 24*a^3*b^7*c^7*d^3 - 84*a^4*b^6*c^6
*d^4 - 504*a^5*b^5*c^5*d^5 + 6174*a^6*b^4*c^4*d^6 - 16056*a^7*b^3*c^3*d^7 + 18414*a^8*b^2*c^2*d^8 - 10036*a^9*
b*c*d^9 + 2131*a^10*d^10 + 10*(4*b^10*c*d^9 - a*b^9*d^10)*x^9 + 45*(6*b^10*c^2*d^8 - 4*a*b^9*c*d^9 + a^2*b^8*d
^10)*x^8 + 360*(4*b^10*c^3*d^7 - 6*a*b^9*c^2*d^8 + 4*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + (8640*a*b^9*c^3*d^7 -
18630*a^2*b^8*c^2*d^8 + 14660*a^3*b^7*c*d^9 - 4043*a^4*b^6*d^10)*x^6 - 6*(504*b^10*c^5*d^5 - 2520*a*b^9*c^4*d
^6 + 1440*a^2*b^8*c^3*d^7 + 3510*a^3*b^7*c^2*d^8 - 4580*a^4*b^6*c*d^9 + 1523*a^5*b^5*d^10)*x^5 - 15*(84*b^10*c
^6*d^4 + 504*a*b^9*c^5*d^5 - 3780*a^2*b^8*c^4*d^6 + 6480*a^3*b^7*c^3*d^7 - 4050*a^4*b^6*c^2*d^8 + 460*a^5*b^5*
c*d^9 + 263*a^6*b^4*d^10)*x^4 - 20*(24*b^10*c^7*d^3 + 84*a*b^9*c^6*d^4 + 504*a^2*b^8*c^5*d^5 - 4620*a^3*b^7*c^
4*d^6 + 9840*a^4*b^6*c^3*d^7 - 9090*a^5*b^5*c^2*d^8 + 3820*a^6*b^4*c*d^9 - 577*a^7*b^3*d^10)*x^3 - 15*(9*b^10*
c^8*d^2 + 24*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 + 504*a^3*b^7*c^5*d^5 - 5250*a^4*b^6*c^4*d^6 + 12360*a^5*b^5*c
^3*d^7 - 12870*a^6*b^4*c^2*d^8 + 6340*a^7*b^3*c*d^9 - 1207*a^8*b^2*d^10)*x^2 - 6*(4*b^10*c^9*d + 9*a*b^9*c^8*d
^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14376*a^6*b^4*c^3*
d^7 - 15894*a^7*b^3*c^2*d^8 + 8356*a^8*b^2*c*d^9 - 1711*a^9*b*d^10)*x + 2520*(a^6*b^4*c^4*d^6 - 4*a^7*b^3*c^3*
d^7 + 6*a^8*b^2*c^2*d^8 - 4*a^9*b*c*d^9 + a^10*d^10 + (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 + 6*(a*b^9*c^4*d^6 - 4*a^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d^8 - 4*a^4*b^6*c*d^9
+ a^5*b^5*d^10)*x^5 + 15*(a^2*b^8*c^4*d^6 - 4*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 - 4*a^5*b^5*c*d^9 + a^6*b^4
*d^10)*x^4 + 20*(a^3*b^7*c^4*d^6 - 4*a^4*b^6*c^3*d^7 + 6*a^5*b^5*c^2*d^8 - 4*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3
+ 15*(a^4*b^6*c^4*d^6 - 4*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 - 4*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 6*(a^5*
b^5*c^4*d^6 - 4*a^6*b^4*c^3*d^7 + 6*a^7*b^3*c^2*d^8 - 4*a^8*b^2*c*d^9 + a^9*b*d^10)*x)*log(b*x + a))/(b^17*x^6
+ 6*a*b^16*x^5 + 15*a^2*b^15*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*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)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.05499, size = 1185, normalized size = 4.52 \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)^7,x, algorithm="giac")

[Out]

210*(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*log(abs(b*x + a))/b^11 - 1/
12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^
5*d^5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 2131*a^10*d
^10 + 3024*(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 + 1260*(b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100*a^3*b^7*c^3*d^7 - 105*a^4*b^6*c
^2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^4*d^10)*x^4 + 240*(2*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5
- 385*a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3*d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4*c*d^9 - 107*a^7*b^3*d^10)*x^3
+ 45*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 - 1750*a^4*b^6*c^4*d^6 + 431
2*a^5*b^5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)*x^2 + 6*(4*b^10*c^9*d + 9*a*
b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14616*a^6
*b^4*c^3*d^7 - 16524*a^7*b^3*c^2*d^8 + 8916*a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/((b*x + a)^6*b^11) + 1/12*(3*b
^21*d^10*x^4 + 40*b^21*c*d^9*x^3 - 28*a*b^20*d^10*x^3 + 270*b^21*c^2*d^8*x^2 - 420*a*b^20*c*d^9*x^2 + 168*a^2*
b^19*d^10*x^2 + 1440*b^21*c^3*d^7*x - 3780*a*b^20*c^2*d^8*x + 3360*a^2*b^19*c*d^9*x - 1008*a^3*b^18*d^10*x)/b^
28