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

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

[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 = 0.8616, 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 \[ \frac{5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac{120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac{210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac{126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac{70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac{30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac{9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac{5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac{(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac{d^{10} (a+b x)^3}{3 b^{11}}+\frac{45 d^8 x (b c-a d)^2}{b^{10}} \]

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

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Rubi in Sympy [A]  time = 122.667, size = 240, normalized size = 0.93 \[ \frac{45 d^{8} x \left (a d - b c\right )^{2}}{b^{10}} + \frac{d^{10} \left (a + b x\right )^{3}}{3 b^{11}} - \frac{5 d^{9} \left (a + b x\right )^{2} \left (a d - b c\right )}{b^{11}} - \frac{120 d^{7} \left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{11}} - \frac{210 d^{6} \left (a d - b c\right )^{4}}{b^{11} \left (a + b x\right )} + \frac{126 d^{5} \left (a d - b c\right )^{5}}{b^{11} \left (a + b x\right )^{2}} - \frac{70 d^{4} \left (a d - b c\right )^{6}}{b^{11} \left (a + b x\right )^{3}} + \frac{30 d^{3} \left (a d - b c\right )^{7}}{b^{11} \left (a + b x\right )^{4}} - \frac{9 d^{2} \left (a d - b c\right )^{8}}{b^{11} \left (a + b x\right )^{5}} + \frac{5 d \left (a d - b c\right )^{9}}{3 b^{11} \left (a + b x\right )^{6}} - \frac{\left (a d - b c\right )^{10}}{7 b^{11} \left (a + b x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**8,x)

[Out]

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

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Mathematica [A]  time = 0.439378, size = 239, normalized size = 0.93 \[ \frac{21 b d^8 x \left (36 a^2 d^2-80 a b c d+45 b^2 c^2\right )+21 b^2 d^9 x^2 (5 b c-4 a d)+2520 d^7 (b c-a d)^3 \log (a+b x)-\frac{4410 d^6 (b c-a d)^4}{a+b x}+\frac{2646 d^5 (a d-b c)^5}{(a+b x)^2}-\frac{1470 d^4 (b c-a d)^6}{(a+b x)^3}+\frac{630 d^3 (a d-b c)^7}{(a+b x)^4}-\frac{189 d^2 (b c-a d)^8}{(a+b x)^5}+\frac{35 d (a d-b c)^9}{(a+b x)^6}-\frac{3 (b c-a d)^{10}}{(a+b x)^7}+7 b^3 d^{10} x^3}{21 b^{11}} \]

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 = 0.029, size = 1241, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^8,x)

[Out]

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

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Maxima [A]  time = 1.48837, size = 1261, normalized size = 4.89 \[ \text{result too large to display} \]

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 + 12
6*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 [A]  time = 0.224649, size = 1839, normalized size = 7.13 \[ \text{result too large to display} \]

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 + 121*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^1
0)*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 + 10647*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 + 240
1*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 + 2
1*(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 + 3
5*(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 + 2
1*(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^1
4*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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.223096, size = 1177, normalized size = 4.56 \[ \text{result too large to display} \]

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)*ln(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 + 2
1*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 - 12*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