Integrand size = 30, antiderivative size = 260 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {a^2 A (d x)^{1+m}}{d (1+m)}+\frac {a^2 B (d x)^{2+m}}{d^2 (2+m)}+\frac {a (2 A b+a C) (d x)^{3+m}}{d^3 (3+m)}+\frac {2 a b B (d x)^{4+m}}{d^4 (4+m)}+\frac {\left (A \left (b^2+2 a c\right )+2 a b C\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {B \left (b^2+2 a c\right ) (d x)^{6+m}}{d^6 (6+m)}+\frac {\left (2 A b c+\left (b^2+2 a c\right ) C\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {2 b B c (d x)^{8+m}}{d^8 (8+m)}+\frac {c (A c+2 b C) (d x)^{9+m}}{d^9 (9+m)}+\frac {B c^2 (d x)^{10+m}}{d^{10} (10+m)}+\frac {c^2 C (d x)^{11+m}}{d^{11} (11+m)} \]
a^2*A*(d*x)^(1+m)/d/(1+m)+a^2*B*(d*x)^(2+m)/d^2/(2+m)+a*(2*A*b+C*a)*(d*x)^ (3+m)/d^3/(3+m)+2*a*b*B*(d*x)^(4+m)/d^4/(4+m)+(A*(2*a*c+b^2)+2*a*b*C)*(d*x )^(5+m)/d^5/(5+m)+B*(2*a*c+b^2)*(d*x)^(6+m)/d^6/(6+m)+(2*A*b*c+(2*a*c+b^2) *C)*(d*x)^(7+m)/d^7/(7+m)+2*b*B*c*(d*x)^(8+m)/d^8/(8+m)+c*(A*c+2*C*b)*(d*x )^(9+m)/d^9/(9+m)+B*c^2*(d*x)^(10+m)/d^10/(10+m)+c^2*C*(d*x)^(11+m)/d^11/( 11+m)
Time = 0.98 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.71 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=x (d x)^m \left (\frac {a^2 A}{1+m}+\frac {a^2 B x}{2+m}+\frac {a (2 A b+a C) x^2}{3+m}+\frac {2 a b B x^3}{4+m}+\frac {\left (A \left (b^2+2 a c\right )+2 a b C\right ) x^4}{5+m}+\frac {B \left (b^2+2 a c\right ) x^5}{6+m}+\frac {\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^6}{7+m}+\frac {2 b B c x^7}{8+m}+\frac {c (A c+2 b C) x^8}{9+m}+\frac {B c^2 x^9}{10+m}+\frac {c^2 C x^{10}}{11+m}\right ) \]
x*(d*x)^m*((a^2*A)/(1 + m) + (a^2*B*x)/(2 + m) + (a*(2*A*b + a*C)*x^2)/(3 + m) + (2*a*b*B*x^3)/(4 + m) + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^4)/(5 + m) + (B*(b^2 + 2*a*c)*x^5)/(6 + m) + ((2*A*b*c + (b^2 + 2*a*c)*C)*x^6)/(7 + m) + (2*b*B*c*x^7)/(8 + m) + (c*(A*c + 2*b*C)*x^8)/(9 + m) + (B*c^2*x^9)/(10 + m) + (c^2*C*x^10)/(11 + m))
Time = 0.51 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2159, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d x)^m \left (a+b x^2+c x^4\right )^2 \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (a^2 A (d x)^m+\frac {a^2 B (d x)^{m+1}}{d}+\frac {(d x)^{m+6} \left (C \left (2 a c+b^2\right )+2 A b c\right )}{d^6}+\frac {(d x)^{m+4} \left (A \left (2 a c+b^2\right )+2 a b C\right )}{d^4}+\frac {a (d x)^{m+2} (a C+2 A b)}{d^2}+\frac {B \left (2 a c+b^2\right ) (d x)^{m+5}}{d^5}+\frac {2 a b B (d x)^{m+3}}{d^3}+\frac {c (d x)^{m+8} (A c+2 b C)}{d^8}+\frac {2 b B c (d x)^{m+7}}{d^7}+\frac {B c^2 (d x)^{m+9}}{d^9}+\frac {c^2 C (d x)^{m+10}}{d^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 A (d x)^{m+1}}{d (m+1)}+\frac {a^2 B (d x)^{m+2}}{d^2 (m+2)}+\frac {(d x)^{m+7} \left (C \left (2 a c+b^2\right )+2 A b c\right )}{d^7 (m+7)}+\frac {(d x)^{m+5} \left (A \left (2 a c+b^2\right )+2 a b C\right )}{d^5 (m+5)}+\frac {a (d x)^{m+3} (a C+2 A b)}{d^3 (m+3)}+\frac {B \left (2 a c+b^2\right ) (d x)^{m+6}}{d^6 (m+6)}+\frac {2 a b B (d x)^{m+4}}{d^4 (m+4)}+\frac {c (d x)^{m+9} (A c+2 b C)}{d^9 (m+9)}+\frac {2 b B c (d x)^{m+8}}{d^8 (m+8)}+\frac {B c^2 (d x)^{m+10}}{d^{10} (m+10)}+\frac {c^2 C (d x)^{m+11}}{d^{11} (m+11)}\) |
(a^2*A*(d*x)^(1 + m))/(d*(1 + m)) + (a^2*B*(d*x)^(2 + m))/(d^2*(2 + m)) + (a*(2*A*b + a*C)*(d*x)^(3 + m))/(d^3*(3 + m)) + (2*a*b*B*(d*x)^(4 + m))/(d ^4*(4 + m)) + ((A*(b^2 + 2*a*c) + 2*a*b*C)*(d*x)^(5 + m))/(d^5*(5 + m)) + (B*(b^2 + 2*a*c)*(d*x)^(6 + m))/(d^6*(6 + m)) + ((2*A*b*c + (b^2 + 2*a*c)* C)*(d*x)^(7 + m))/(d^7*(7 + m)) + (2*b*B*c*(d*x)^(8 + m))/(d^8*(8 + m)) + (c*(A*c + 2*b*C)*(d*x)^(9 + m))/(d^9*(9 + m)) + (B*c^2*(d*x)^(10 + m))/(d^ 10*(10 + m)) + (c^2*C*(d*x)^(11 + m))/(d^11*(11 + m))
3.1.38.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(2186\) vs. \(2(260)=520\).
Time = 0.18 (sec) , antiderivative size = 2187, normalized size of antiderivative = 8.41
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(2187\) |
| risch | \(\text {Expression too large to display}\) | \(2187\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(3204\) |
x*(C*c^2*m^10*x^10+B*c^2*m^10*x^9+55*C*c^2*m^9*x^10+A*c^2*m^10*x^8+56*B*c^ 2*m^9*x^9+2*C*b*c*m^10*x^8+1320*C*c^2*m^8*x^10+57*A*c^2*m^9*x^8+2*B*b*c*m^ 10*x^7+1365*B*c^2*m^8*x^9+114*C*b*c*m^9*x^8+18150*C*c^2*m^7*x^10+2*A*b*c*m ^10*x^6+1412*A*c^2*m^8*x^8+116*B*b*c*m^9*x^7+19020*B*c^2*m^7*x^9+2*C*a*c*m ^10*x^6+C*b^2*m^10*x^6+2824*C*b*c*m^8*x^8+157773*C*c^2*m^6*x^10+118*A*b*c* m^9*x^6+19962*A*c^2*m^7*x^8+2*B*a*c*m^10*x^5+B*b^2*m^10*x^5+2922*B*b*c*m^8 *x^7+167223*B*c^2*m^6*x^9+118*C*a*c*m^9*x^6+59*C*b^2*m^9*x^6+39924*C*b*c*m ^7*x^8+902055*C*c^2*m^5*x^10+2*A*a*c*m^10*x^4+A*b^2*m^10*x^4+3024*A*b*c*m^ 8*x^6+177765*A*c^2*m^6*x^8+120*B*a*c*m^9*x^5+60*B*b^2*m^9*x^5+41964*B*b*c* m^7*x^7+965328*B*c^2*m^5*x^9+2*C*a*b*m^10*x^4+3024*C*a*c*m^8*x^6+1512*C*b^ 2*m^8*x^6+355530*C*b*c*m^6*x^8+3416930*C*c^2*m^4*x^10+122*A*a*c*m^9*x^4+61 *A*b^2*m^9*x^4+44172*A*b*c*m^7*x^6+1037673*A*c^2*m^5*x^8+2*B*a*b*m^10*x^3+ 3130*B*a*c*m^8*x^5+1565*B*b^2*m^8*x^5+379134*B*b*c*m^6*x^7+3686255*B*c^2*m ^4*x^9+122*C*a*b*m^9*x^4+44172*C*a*c*m^7*x^6+22086*C*b^2*m^7*x^6+2075346*C *b*c*m^5*x^8+8409500*C*c^2*m^3*x^10+2*A*a*b*m^10*x^2+3240*A*a*c*m^8*x^4+16 20*A*b^2*m^8*x^4+405642*A*b*c*m^6*x^6+4000478*A*c^2*m^4*x^8+124*B*a*b*m^9* x^3+46560*B*a*c*m^7*x^5+23280*B*b^2*m^7*x^5+2242044*B*b*c*m^5*x^7+9133180* B*c^2*m^3*x^9+C*a^2*m^10*x^2+3240*C*a*b*m^8*x^4+405642*C*a*c*m^6*x^6+20282 1*C*b^2*m^6*x^6+8000956*C*b*c*m^4*x^8+12753576*C*c^2*m^2*x^10+126*A*a*b*m^ 9*x^2+49140*A*a*c*m^7*x^4+24570*A*b^2*m^7*x^4+2435622*A*b*c*m^5*x^6+999...
Leaf count of result is larger than twice the leaf count of optimal. 1603 vs. \(2 (260) = 520\).
Time = 0.33 (sec) , antiderivative size = 1603, normalized size of antiderivative = 6.17 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\text {Too large to display} \]
((C*c^2*m^10 + 55*C*c^2*m^9 + 1320*C*c^2*m^8 + 18150*C*c^2*m^7 + 157773*C* c^2*m^6 + 902055*C*c^2*m^5 + 3416930*C*c^2*m^4 + 8409500*C*c^2*m^3 + 12753 576*C*c^2*m^2 + 10628640*C*c^2*m + 3628800*C*c^2)*x^11 + (B*c^2*m^10 + 56* B*c^2*m^9 + 1365*B*c^2*m^8 + 19020*B*c^2*m^7 + 167223*B*c^2*m^6 + 965328*B *c^2*m^5 + 3686255*B*c^2*m^4 + 9133180*B*c^2*m^3 + 13926276*B*c^2*m^2 + 11 655216*B*c^2*m + 3991680*B*c^2)*x^10 + ((2*C*b*c + A*c^2)*m^10 + 57*(2*C*b *c + A*c^2)*m^9 + 1412*(2*C*b*c + A*c^2)*m^8 + 19962*(2*C*b*c + A*c^2)*m^7 + 177765*(2*C*b*c + A*c^2)*m^6 + 1037673*(2*C*b*c + A*c^2)*m^5 + 4000478* (2*C*b*c + A*c^2)*m^4 + 9991428*(2*C*b*c + A*c^2)*m^3 + 8870400*C*b*c + 44 35200*A*c^2 + 15335224*(2*C*b*c + A*c^2)*m^2 + 12900960*(2*C*b*c + A*c^2)* m)*x^9 + 2*(B*b*c*m^10 + 58*B*b*c*m^9 + 1461*B*b*c*m^8 + 20982*B*b*c*m^7 + 189567*B*b*c*m^6 + 1121022*B*b*c*m^5 + 4371359*B*b*c*m^4 + 11024858*B*b*c *m^3 + 17059212*B*b*c*m^2 + 14444280*B*b*c*m + 4989600*B*b*c)*x^8 + ((C*b^ 2 + 2*(C*a + A*b)*c)*m^10 + 59*(C*b^2 + 2*(C*a + A*b)*c)*m^9 + 1512*(C*b^2 + 2*(C*a + A*b)*c)*m^8 + 22086*(C*b^2 + 2*(C*a + A*b)*c)*m^7 + 202821*(C* b^2 + 2*(C*a + A*b)*c)*m^6 + 1217811*(C*b^2 + 2*(C*a + A*b)*c)*m^5 + 48148 58*(C*b^2 + 2*(C*a + A*b)*c)*m^4 + 12291724*(C*b^2 + 2*(C*a + A*b)*c)*m^3 + 5702400*C*b^2 + 19216008*(C*b^2 + 2*(C*a + A*b)*c)*m^2 + 11404800*(C*a + A*b)*c + 16405920*(C*b^2 + 2*(C*a + A*b)*c)*m)*x^7 + ((B*b^2 + 2*B*a*c)*m ^10 + 60*(B*b^2 + 2*B*a*c)*m^9 + 1565*(B*b^2 + 2*B*a*c)*m^8 + 23280*(B*...
Leaf count of result is larger than twice the leaf count of optimal. 16323 vs. \(2 (245) = 490\).
Time = 1.42 (sec) , antiderivative size = 16323, normalized size of antiderivative = 62.78 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\text {Too large to display} \]
Piecewise(((-A*a**2/(10*x**10) - A*a*b/(4*x**8) - A*a*c/(3*x**6) - A*b**2/ (6*x**6) - A*b*c/(2*x**4) - A*c**2/(2*x**2) - B*a**2/(9*x**9) - 2*B*a*b/(7 *x**7) - 2*B*a*c/(5*x**5) - B*b**2/(5*x**5) - 2*B*b*c/(3*x**3) - B*c**2/x - C*a**2/(8*x**8) - C*a*b/(3*x**6) - C*a*c/(2*x**4) - C*b**2/(4*x**4) - C* b*c/x**2 + C*c**2*log(x))/d**11, Eq(m, -11)), ((-A*a**2/(9*x**9) - 2*A*a*b /(7*x**7) - 2*A*a*c/(5*x**5) - A*b**2/(5*x**5) - 2*A*b*c/(3*x**3) - A*c**2 /x - B*a**2/(8*x**8) - B*a*b/(3*x**6) - B*a*c/(2*x**4) - B*b**2/(4*x**4) - B*b*c/x**2 + B*c**2*log(x) - C*a**2/(7*x**7) - 2*C*a*b/(5*x**5) - 2*C*a*c /(3*x**3) - C*b**2/(3*x**3) - 2*C*b*c/x + C*c**2*x)/d**10, Eq(m, -10)), (( -A*a**2/(8*x**8) - A*a*b/(3*x**6) - A*a*c/(2*x**4) - A*b**2/(4*x**4) - A*b *c/x**2 + A*c**2*log(x) - B*a**2/(7*x**7) - 2*B*a*b/(5*x**5) - 2*B*a*c/(3* x**3) - B*b**2/(3*x**3) - 2*B*b*c/x + B*c**2*x - C*a**2/(6*x**6) - C*a*b/( 2*x**4) - C*a*c/x**2 - C*b**2/(2*x**2) + 2*C*b*c*log(x) + C*c**2*x**2/2)/d **9, Eq(m, -9)), ((-A*a**2/(7*x**7) - 2*A*a*b/(5*x**5) - 2*A*a*c/(3*x**3) - A*b**2/(3*x**3) - 2*A*b*c/x + A*c**2*x - B*a**2/(6*x**6) - B*a*b/(2*x**4 ) - B*a*c/x**2 - B*b**2/(2*x**2) + 2*B*b*c*log(x) + B*c**2*x**2/2 - C*a**2 /(5*x**5) - 2*C*a*b/(3*x**3) - 2*C*a*c/x - C*b**2/x + 2*C*b*c*x + C*c**2*x **3/3)/d**8, Eq(m, -8)), ((-A*a**2/(6*x**6) - A*a*b/(2*x**4) - A*a*c/x**2 - A*b**2/(2*x**2) + 2*A*b*c*log(x) + A*c**2*x**2/2 - B*a**2/(5*x**5) - 2*B *a*b/(3*x**3) - 2*B*a*c/x - B*b**2/x + 2*B*b*c*x + B*c**2*x**3/3 - C*a*...
Time = 0.23 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.32 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {C c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {B c^{2} d^{m} x^{10} x^{m}}{m + 10} + \frac {2 \, C b c d^{m} x^{9} x^{m}}{m + 9} + \frac {A c^{2} d^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, B b c d^{m} x^{8} x^{m}}{m + 8} + \frac {C b^{2} d^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, C a c d^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, A b c d^{m} x^{7} x^{m}}{m + 7} + \frac {B b^{2} d^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, B a c d^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, C a b d^{m} x^{5} x^{m}}{m + 5} + \frac {A b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, A a c d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, B a b d^{m} x^{4} x^{m}}{m + 4} + \frac {C a^{2} d^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a b d^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{2} d^{m} x^{2} x^{m}}{m + 2} + \frac {\left (d x\right )^{m + 1} A a^{2}}{d {\left (m + 1\right )}} \]
C*c^2*d^m*x^11*x^m/(m + 11) + B*c^2*d^m*x^10*x^m/(m + 10) + 2*C*b*c*d^m*x^ 9*x^m/(m + 9) + A*c^2*d^m*x^9*x^m/(m + 9) + 2*B*b*c*d^m*x^8*x^m/(m + 8) + C*b^2*d^m*x^7*x^m/(m + 7) + 2*C*a*c*d^m*x^7*x^m/(m + 7) + 2*A*b*c*d^m*x^7* x^m/(m + 7) + B*b^2*d^m*x^6*x^m/(m + 6) + 2*B*a*c*d^m*x^6*x^m/(m + 6) + 2* C*a*b*d^m*x^5*x^m/(m + 5) + A*b^2*d^m*x^5*x^m/(m + 5) + 2*A*a*c*d^m*x^5*x^ m/(m + 5) + 2*B*a*b*d^m*x^4*x^m/(m + 4) + C*a^2*d^m*x^3*x^m/(m + 3) + 2*A* a*b*d^m*x^3*x^m/(m + 3) + B*a^2*d^m*x^2*x^m/(m + 2) + (d*x)^(m + 1)*A*a^2/ (d*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 3203 vs. \(2 (260) = 520\).
Time = 0.37 (sec) , antiderivative size = 3203, normalized size of antiderivative = 12.32 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\text {Too large to display} \]
((d*x)^m*C*c^2*m^10*x^11 + (d*x)^m*B*c^2*m^10*x^10 + 55*(d*x)^m*C*c^2*m^9* x^11 + 2*(d*x)^m*C*b*c*m^10*x^9 + (d*x)^m*A*c^2*m^10*x^9 + 56*(d*x)^m*B*c^ 2*m^9*x^10 + 1320*(d*x)^m*C*c^2*m^8*x^11 + 2*(d*x)^m*B*b*c*m^10*x^8 + 114* (d*x)^m*C*b*c*m^9*x^9 + 57*(d*x)^m*A*c^2*m^9*x^9 + 1365*(d*x)^m*B*c^2*m^8* x^10 + 18150*(d*x)^m*C*c^2*m^7*x^11 + (d*x)^m*C*b^2*m^10*x^7 + 2*(d*x)^m*C *a*c*m^10*x^7 + 2*(d*x)^m*A*b*c*m^10*x^7 + 116*(d*x)^m*B*b*c*m^9*x^8 + 282 4*(d*x)^m*C*b*c*m^8*x^9 + 1412*(d*x)^m*A*c^2*m^8*x^9 + 19020*(d*x)^m*B*c^2 *m^7*x^10 + 157773*(d*x)^m*C*c^2*m^6*x^11 + (d*x)^m*B*b^2*m^10*x^6 + 2*(d* x)^m*B*a*c*m^10*x^6 + 59*(d*x)^m*C*b^2*m^9*x^7 + 118*(d*x)^m*C*a*c*m^9*x^7 + 118*(d*x)^m*A*b*c*m^9*x^7 + 2922*(d*x)^m*B*b*c*m^8*x^8 + 39924*(d*x)^m* C*b*c*m^7*x^9 + 19962*(d*x)^m*A*c^2*m^7*x^9 + 167223*(d*x)^m*B*c^2*m^6*x^1 0 + 902055*(d*x)^m*C*c^2*m^5*x^11 + 2*(d*x)^m*C*a*b*m^10*x^5 + (d*x)^m*A*b ^2*m^10*x^5 + 2*(d*x)^m*A*a*c*m^10*x^5 + 60*(d*x)^m*B*b^2*m^9*x^6 + 120*(d *x)^m*B*a*c*m^9*x^6 + 1512*(d*x)^m*C*b^2*m^8*x^7 + 3024*(d*x)^m*C*a*c*m^8* x^7 + 3024*(d*x)^m*A*b*c*m^8*x^7 + 41964*(d*x)^m*B*b*c*m^7*x^8 + 355530*(d *x)^m*C*b*c*m^6*x^9 + 177765*(d*x)^m*A*c^2*m^6*x^9 + 965328*(d*x)^m*B*c^2* m^5*x^10 + 3416930*(d*x)^m*C*c^2*m^4*x^11 + 2*(d*x)^m*B*a*b*m^10*x^4 + 122 *(d*x)^m*C*a*b*m^9*x^5 + 61*(d*x)^m*A*b^2*m^9*x^5 + 122*(d*x)^m*A*a*c*m^9* x^5 + 1565*(d*x)^m*B*b^2*m^8*x^6 + 3130*(d*x)^m*B*a*c*m^8*x^6 + 22086*(d*x )^m*C*b^2*m^7*x^7 + 44172*(d*x)^m*C*a*c*m^7*x^7 + 44172*(d*x)^m*A*b*c*m...
Time = 8.39 (sec) , antiderivative size = 1314, normalized size of antiderivative = 5.05 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\text {Too large to display} \]
(x^5*(d*x)^m*(A*b^2 + 2*A*a*c + 2*C*a*b)*(22512096*m + 25681176*m^2 + 1591 5380*m^3 + 6016070*m^4 + 1464693*m^5 + 234573*m^6 + 24570*m^7 + 1620*m^8 + 61*m^9 + m^10 + 7983360))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925* m^9 + 66*m^10 + m^11 + 39916800) + (x^7*(d*x)^m*(C*b^2 + 2*A*b*c + 2*C*a*c )*(16405920*m + 19216008*m^2 + 12291724*m^3 + 4814858*m^4 + 1217811*m^5 + 202821*m^6 + 22086*m^7 + 1512*m^8 + 59*m^9 + m^10 + 5702400))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 13339535*m^5 + 2637558*m ^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (B*x ^6*(d*x)^m*(2*a*c + b^2)*(18981840*m + 21989356*m^2 + 13878120*m^3 + 53529 35*m^4 + 1331100*m^5 + 217743*m^6 + 23280*m^7 + 1565*m^8 + 60*m^9 + m^10 + 6652800))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 1 3339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (A*a^2*x*(d*x)^m*(80627040*m + 70290936*m^2 + 34967140* m^3 + 11028590*m^4 + 2310945*m^5 + 326613*m^6 + 30810*m^7 + 1860*m^8 + 65* m^9 + m^10 + 39916800))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 459 95730*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (c*x^9*(d*x)^m*(A*c + 2*C*b)*(12900960*m + 15335224*m^2 + 9991428*m^3 + 4000478*m^4 + 1037673*m^5 + 177765*m^6 + 199 62*m^7 + 1412*m^8 + 57*m^9 + m^10 + 4435200))/(120543840*m + 150917976*...