Integrand size = 30, antiderivative size = 399 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^3 A (d x)^{1+m}}{d (1+m)}+\frac {a^3 B (d x)^{2+m}}{d^2 (2+m)}+\frac {a^2 (3 A b+a C) (d x)^{3+m}}{d^3 (3+m)}+\frac {3 a^2 b B (d x)^{4+m}}{d^4 (4+m)}+\frac {3 a \left (A \left (b^2+a c\right )+a b C\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {3 a B \left (b^2+a c\right ) (d x)^{6+m}}{d^6 (6+m)}+\frac {\left (A \left (b^3+6 a b c\right )+3 a \left (b^2+a c\right ) C\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {b B \left (b^2+6 a c\right ) (d x)^{8+m}}{d^8 (8+m)}+\frac {\left (3 A c \left (b^2+a c\right )+b \left (b^2+6 a c\right ) C\right ) (d x)^{9+m}}{d^9 (9+m)}+\frac {3 B c \left (b^2+a c\right ) (d x)^{10+m}}{d^{10} (10+m)}+\frac {3 c \left (A b c+\left (b^2+a c\right ) C\right ) (d x)^{11+m}}{d^{11} (11+m)}+\frac {3 b B c^2 (d x)^{12+m}}{d^{12} (12+m)}+\frac {c^2 (A c+3 b C) (d x)^{13+m}}{d^{13} (13+m)}+\frac {B c^3 (d x)^{14+m}}{d^{14} (14+m)}+\frac {c^3 C (d x)^{15+m}}{d^{15} (15+m)} \]
a^3*A*(d*x)^(1+m)/d/(1+m)+a^3*B*(d*x)^(2+m)/d^2/(2+m)+a^2*(3*A*b+C*a)*(d*x )^(3+m)/d^3/(3+m)+3*a^2*b*B*(d*x)^(4+m)/d^4/(4+m)+3*a*(A*(a*c+b^2)+a*b*C)* (d*x)^(5+m)/d^5/(5+m)+3*a*B*(a*c+b^2)*(d*x)^(6+m)/d^6/(6+m)+(A*(6*a*b*c+b^ 3)+3*a*(a*c+b^2)*C)*(d*x)^(7+m)/d^7/(7+m)+b*B*(6*a*c+b^2)*(d*x)^(8+m)/d^8/ (8+m)+(3*A*c*(a*c+b^2)+b*(6*a*c+b^2)*C)*(d*x)^(9+m)/d^9/(9+m)+3*B*c*(a*c+b ^2)*(d*x)^(10+m)/d^10/(10+m)+3*c*(A*b*c+(a*c+b^2)*C)*(d*x)^(11+m)/d^11/(11 +m)+3*b*B*c^2*(d*x)^(12+m)/d^12/(12+m)+c^2*(A*c+3*C*b)*(d*x)^(13+m)/d^13/( 13+m)+B*c^3*(d*x)^(14+m)/d^14/(14+m)+c^3*C*(d*x)^(15+m)/d^15/(15+m)
Time = 2.10 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.74 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=x (d x)^m \left (\frac {a^3 A}{1+m}+\frac {a^3 B x}{2+m}+\frac {a^2 (3 A b+a C) x^2}{3+m}+\frac {3 a^2 b B x^3}{4+m}+\frac {3 a \left (A \left (b^2+a c\right )+a b C\right ) x^4}{5+m}+\frac {3 a B \left (b^2+a c\right ) x^5}{6+m}+\frac {\left (A \left (b^3+6 a b c\right )+3 a \left (b^2+a c\right ) C\right ) x^6}{7+m}+\frac {b B \left (b^2+6 a c\right ) x^7}{8+m}+\frac {\left (3 A c \left (b^2+a c\right )+b \left (b^2+6 a c\right ) C\right ) x^8}{9+m}+\frac {3 B c \left (b^2+a c\right ) x^9}{10+m}+\frac {3 c \left (A b c+\left (b^2+a c\right ) C\right ) x^{10}}{11+m}+\frac {3 b B c^2 x^{11}}{12+m}+\frac {c^2 (A c+3 b C) x^{12}}{13+m}+\frac {B c^3 x^{13}}{14+m}+\frac {c^3 C x^{14}}{15+m}\right ) \]
x*(d*x)^m*((a^3*A)/(1 + m) + (a^3*B*x)/(2 + m) + (a^2*(3*A*b + a*C)*x^2)/( 3 + m) + (3*a^2*b*B*x^3)/(4 + m) + (3*a*(A*(b^2 + a*c) + a*b*C)*x^4)/(5 + m) + (3*a*B*(b^2 + a*c)*x^5)/(6 + m) + ((A*(b^3 + 6*a*b*c) + 3*a*(b^2 + a* c)*C)*x^6)/(7 + m) + (b*B*(b^2 + 6*a*c)*x^7)/(8 + m) + ((3*A*c*(b^2 + a*c) + b*(b^2 + 6*a*c)*C)*x^8)/(9 + m) + (3*B*c*(b^2 + a*c)*x^9)/(10 + m) + (3 *c*(A*b*c + (b^2 + a*c)*C)*x^10)/(11 + m) + (3*b*B*c^2*x^11)/(12 + m) + (c ^2*(A*c + 3*b*C)*x^12)/(13 + m) + (B*c^3*x^13)/(14 + m) + (c^3*C*x^14)/(15 + m))
Time = 0.76 (sec) , antiderivative size = 399, 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.
| \(\displaystyle \int (d x)^m \left (a+b x^2+c x^4\right )^3 \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (a^3 A (d x)^m+\frac {a^3 B (d x)^{m+1}}{d}+\frac {a^2 (d x)^{m+2} (a C+3 A b)}{d^2}+\frac {3 a^2 b B (d x)^{m+3}}{d^3}+\frac {3 c (d x)^{m+10} \left (C \left (a c+b^2\right )+A b c\right )}{d^{10}}+\frac {(d x)^{m+8} \left (3 A c \left (a c+b^2\right )+b C \left (6 a c+b^2\right )\right )}{d^8}+\frac {3 a (d x)^{m+4} \left (A \left (a c+b^2\right )+a b C\right )}{d^4}+\frac {(d x)^{m+6} \left (A \left (6 a b c+b^3\right )+3 a C \left (a c+b^2\right )\right )}{d^6}+\frac {3 B c \left (a c+b^2\right ) (d x)^{m+9}}{d^9}+\frac {b B \left (6 a c+b^2\right ) (d x)^{m+7}}{d^7}+\frac {3 a B \left (a c+b^2\right ) (d x)^{m+5}}{d^5}+\frac {c^2 (d x)^{m+12} (A c+3 b C)}{d^{12}}+\frac {3 b B c^2 (d x)^{m+11}}{d^{11}}+\frac {B c^3 (d x)^{m+13}}{d^{13}}+\frac {c^3 C (d x)^{m+14}}{d^{14}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 A (d x)^{m+1}}{d (m+1)}+\frac {a^3 B (d x)^{m+2}}{d^2 (m+2)}+\frac {a^2 (d x)^{m+3} (a C+3 A b)}{d^3 (m+3)}+\frac {3 a^2 b B (d x)^{m+4}}{d^4 (m+4)}+\frac {3 c (d x)^{m+11} \left (C \left (a c+b^2\right )+A b c\right )}{d^{11} (m+11)}+\frac {(d x)^{m+9} \left (3 A c \left (a c+b^2\right )+b C \left (6 a c+b^2\right )\right )}{d^9 (m+9)}+\frac {3 a (d x)^{m+5} \left (A \left (a c+b^2\right )+a b C\right )}{d^5 (m+5)}+\frac {(d x)^{m+7} \left (A \left (6 a b c+b^3\right )+3 a C \left (a c+b^2\right )\right )}{d^7 (m+7)}+\frac {3 B c \left (a c+b^2\right ) (d x)^{m+10}}{d^{10} (m+10)}+\frac {b B \left (6 a c+b^2\right ) (d x)^{m+8}}{d^8 (m+8)}+\frac {3 a B \left (a c+b^2\right ) (d x)^{m+6}}{d^6 (m+6)}+\frac {c^2 (d x)^{m+13} (A c+3 b C)}{d^{13} (m+13)}+\frac {3 b B c^2 (d x)^{m+12}}{d^{12} (m+12)}+\frac {B c^3 (d x)^{m+14}}{d^{14} (m+14)}+\frac {c^3 C (d x)^{m+15}}{d^{15} (m+15)}\) |
(a^3*A*(d*x)^(1 + m))/(d*(1 + m)) + (a^3*B*(d*x)^(2 + m))/(d^2*(2 + m)) + (a^2*(3*A*b + a*C)*(d*x)^(3 + m))/(d^3*(3 + m)) + (3*a^2*b*B*(d*x)^(4 + m) )/(d^4*(4 + m)) + (3*a*(A*(b^2 + a*c) + a*b*C)*(d*x)^(5 + m))/(d^5*(5 + m) ) + (3*a*B*(b^2 + a*c)*(d*x)^(6 + m))/(d^6*(6 + m)) + ((A*(b^3 + 6*a*b*c) + 3*a*(b^2 + a*c)*C)*(d*x)^(7 + m))/(d^7*(7 + m)) + (b*B*(b^2 + 6*a*c)*(d* x)^(8 + m))/(d^8*(8 + m)) + ((3*A*c*(b^2 + a*c) + b*(b^2 + 6*a*c)*C)*(d*x) ^(9 + m))/(d^9*(9 + m)) + (3*B*c*(b^2 + a*c)*(d*x)^(10 + m))/(d^10*(10 + m )) + (3*c*(A*b*c + (b^2 + a*c)*C)*(d*x)^(11 + m))/(d^11*(11 + m)) + (3*b*B *c^2*(d*x)^(12 + m))/(d^12*(12 + m)) + (c^2*(A*c + 3*b*C)*(d*x)^(13 + m))/ (d^13*(13 + m)) + (B*c^3*(d*x)^(14 + m))/(d^14*(14 + m)) + (c^3*C*(d*x)^(1 5 + m))/(d^15*(15 + m))
3.1.37.3.1 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. \(5519\) vs. \(2(399)=798\).
Time = 0.41 (sec) , antiderivative size = 5520, normalized size of antiderivative = 13.83
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(5520\) |
| risch | \(\text {Expression too large to display}\) | \(5520\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(7809\) |
Leaf count of result is larger than twice the leaf count of optimal. 3898 vs. \(2 (399) = 798\).
Time = 0.38 (sec) , antiderivative size = 3898, normalized size of antiderivative = 9.77 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\text {Too large to display} \]
((C*c^3*m^14 + 105*C*c^3*m^13 + 5005*C*c^3*m^12 + 143325*C*c^3*m^11 + 2749 747*C*c^3*m^10 + 37312275*C*c^3*m^9 + 368411615*C*c^3*m^8 + 2681453775*C*c ^3*m^7 + 14409322928*C*c^3*m^6 + 56663366760*C*c^3*m^5 + 159721605680*C*c^ 3*m^4 + 310989260400*C*c^3*m^3 + 392156797824*C*c^3*m^2 + 283465647360*C*c ^3*m + 87178291200*C*c^3)*x^15 + (B*c^3*m^14 + 106*B*c^3*m^13 + 5096*B*c^3 *m^12 + 147056*B*c^3*m^11 + 2840838*B*c^3*m^10 + 38786748*B*c^3*m^9 + 3850 81268*B*c^3*m^8 + 2816490248*B*c^3*m^7 + 15200266081*B*c^3*m^6 + 599994855 46*B*c^3*m^5 + 169679309436*B*c^3*m^4 + 331303013496*B*c^3*m^3 + 418753514 880*B*c^3*m^2 + 303268406400*B*c^3*m + 93405312000*B*c^3)*x^14 + ((3*C*b*c ^2 + A*c^3)*m^14 + 107*(3*C*b*c^2 + A*c^3)*m^13 + 5189*(3*C*b*c^2 + A*c^3) *m^12 + 150943*(3*C*b*c^2 + A*c^3)*m^11 + 2937363*(3*C*b*c^2 + A*c^3)*m^10 + 40372761*(3*C*b*c^2 + A*c^3)*m^9 + 403249847*(3*C*b*c^2 + A*c^3)*m^8 + 2965379989*(3*C*b*c^2 + A*c^3)*m^7 + 16081189696*(3*C*b*c^2 + A*c^3)*m^6 + 63747744632*(3*C*b*c^2 + A*c^3)*m^5 + 180951426864*(3*C*b*c^2 + A*c^3)*m^ 4 + 301771008000*C*b*c^2 + 100590336000*A*c^3 + 354444796368*(3*C*b*c^2 + A*c^3)*m^3 + 449213351040*(3*C*b*c^2 + A*c^3)*m^2 + 326044051200*(3*C*b*c^ 2 + A*c^3)*m)*x^13 + 3*(B*b*c^2*m^14 + 108*B*b*c^2*m^13 + 5284*B*b*c^2*m^1 2 + 154992*B*b*c^2*m^11 + 3039718*B*b*c^2*m^10 + 42081864*B*b*c^2*m^9 + 42 3113372*B*b*c^2*m^8 + 3130267536*B*b*c^2*m^7 + 17067919121*B*b*c^2*m^6 + 6 7988181228*B*b*c^2*m^5 + 193813932344*B*b*c^2*m^4 + 381046157472*B*b*c^...
Leaf count of result is larger than twice the leaf count of optimal. 47658 vs. \(2 (379) = 758\).
Time = 3.03 (sec) , antiderivative size = 47658, normalized size of antiderivative = 119.44 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\text {Too large to display} \]
Piecewise(((-A*a**3/(14*x**14) - A*a**2*b/(4*x**12) - 3*A*a**2*c/(10*x**10 ) - 3*A*a*b**2/(10*x**10) - 3*A*a*b*c/(4*x**8) - A*a*c**2/(2*x**6) - A*b** 3/(8*x**8) - A*b**2*c/(2*x**6) - 3*A*b*c**2/(4*x**4) - A*c**3/(2*x**2) - B *a**3/(13*x**13) - 3*B*a**2*b/(11*x**11) - B*a**2*c/(3*x**9) - B*a*b**2/(3 *x**9) - 6*B*a*b*c/(7*x**7) - 3*B*a*c**2/(5*x**5) - B*b**3/(7*x**7) - 3*B* b**2*c/(5*x**5) - B*b*c**2/x**3 - B*c**3/x - C*a**3/(12*x**12) - 3*C*a**2* b/(10*x**10) - 3*C*a**2*c/(8*x**8) - 3*C*a*b**2/(8*x**8) - C*a*b*c/x**6 - 3*C*a*c**2/(4*x**4) - C*b**3/(6*x**6) - 3*C*b**2*c/(4*x**4) - 3*C*b*c**2/( 2*x**2) + C*c**3*log(x))/d**15, Eq(m, -15)), ((-A*a**3/(13*x**13) - 3*A*a* *2*b/(11*x**11) - A*a**2*c/(3*x**9) - A*a*b**2/(3*x**9) - 6*A*a*b*c/(7*x** 7) - 3*A*a*c**2/(5*x**5) - A*b**3/(7*x**7) - 3*A*b**2*c/(5*x**5) - A*b*c** 2/x**3 - A*c**3/x - B*a**3/(12*x**12) - 3*B*a**2*b/(10*x**10) - 3*B*a**2*c /(8*x**8) - 3*B*a*b**2/(8*x**8) - B*a*b*c/x**6 - 3*B*a*c**2/(4*x**4) - B*b **3/(6*x**6) - 3*B*b**2*c/(4*x**4) - 3*B*b*c**2/(2*x**2) + B*c**3*log(x) - C*a**3/(11*x**11) - C*a**2*b/(3*x**9) - 3*C*a**2*c/(7*x**7) - 3*C*a*b**2/ (7*x**7) - 6*C*a*b*c/(5*x**5) - C*a*c**2/x**3 - C*b**3/(5*x**5) - C*b**2*c /x**3 - 3*C*b*c**2/x + C*c**3*x)/d**14, Eq(m, -14)), ((-A*a**3/(12*x**12) - 3*A*a**2*b/(10*x**10) - 3*A*a**2*c/(8*x**8) - 3*A*a*b**2/(8*x**8) - A*a* b*c/x**6 - 3*A*a*c**2/(4*x**4) - A*b**3/(6*x**6) - 3*A*b**2*c/(4*x**4) - 3 *A*b*c**2/(2*x**2) + A*c**3*log(x) - B*a**3/(11*x**11) - B*a**2*b/(3*x*...
Time = 0.26 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.53 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\frac {C c^{3} d^{m} x^{15} x^{m}}{m + 15} + \frac {B c^{3} d^{m} x^{14} x^{m}}{m + 14} + \frac {3 \, C b c^{2} d^{m} x^{13} x^{m}}{m + 13} + \frac {A c^{3} d^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, B b c^{2} d^{m} x^{12} x^{m}}{m + 12} + \frac {3 \, C b^{2} c d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, C a c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, A b c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, B b^{2} c d^{m} x^{10} x^{m}}{m + 10} + \frac {3 \, B a c^{2} d^{m} x^{10} x^{m}}{m + 10} + \frac {C b^{3} d^{m} x^{9} x^{m}}{m + 9} + \frac {6 \, C a b c d^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, A b^{2} c d^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, A a c^{2} d^{m} x^{9} x^{m}}{m + 9} + \frac {B b^{3} d^{m} x^{8} x^{m}}{m + 8} + \frac {6 \, B a b c d^{m} x^{8} x^{m}}{m + 8} + \frac {3 \, C a b^{2} d^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, C a^{2} c d^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, A a b c d^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a b^{2} d^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, B a^{2} c d^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, C a^{2} b d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a^{2} c d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, B a^{2} b d^{m} x^{4} x^{m}}{m + 4} + \frac {C a^{3} d^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{2} b d^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{3} d^{m} x^{2} x^{m}}{m + 2} + \frac {\left (d x\right )^{m + 1} A a^{3}}{d {\left (m + 1\right )}} \]
C*c^3*d^m*x^15*x^m/(m + 15) + B*c^3*d^m*x^14*x^m/(m + 14) + 3*C*b*c^2*d^m* x^13*x^m/(m + 13) + A*c^3*d^m*x^13*x^m/(m + 13) + 3*B*b*c^2*d^m*x^12*x^m/( m + 12) + 3*C*b^2*c*d^m*x^11*x^m/(m + 11) + 3*C*a*c^2*d^m*x^11*x^m/(m + 11 ) + 3*A*b*c^2*d^m*x^11*x^m/(m + 11) + 3*B*b^2*c*d^m*x^10*x^m/(m + 10) + 3* B*a*c^2*d^m*x^10*x^m/(m + 10) + C*b^3*d^m*x^9*x^m/(m + 9) + 6*C*a*b*c*d^m* x^9*x^m/(m + 9) + 3*A*b^2*c*d^m*x^9*x^m/(m + 9) + 3*A*a*c^2*d^m*x^9*x^m/(m + 9) + B*b^3*d^m*x^8*x^m/(m + 8) + 6*B*a*b*c*d^m*x^8*x^m/(m + 8) + 3*C*a* b^2*d^m*x^7*x^m/(m + 7) + A*b^3*d^m*x^7*x^m/(m + 7) + 3*C*a^2*c*d^m*x^7*x^ m/(m + 7) + 6*A*a*b*c*d^m*x^7*x^m/(m + 7) + 3*B*a*b^2*d^m*x^6*x^m/(m + 6) + 3*B*a^2*c*d^m*x^6*x^m/(m + 6) + 3*C*a^2*b*d^m*x^5*x^m/(m + 5) + 3*A*a*b^ 2*d^m*x^5*x^m/(m + 5) + 3*A*a^2*c*d^m*x^5*x^m/(m + 5) + 3*B*a^2*b*d^m*x^4* x^m/(m + 4) + C*a^3*d^m*x^3*x^m/(m + 3) + 3*A*a^2*b*d^m*x^3*x^m/(m + 3) + B*a^3*d^m*x^2*x^m/(m + 2) + (d*x)^(m + 1)*A*a^3/(d*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 7808 vs. \(2 (399) = 798\).
Time = 0.40 (sec) , antiderivative size = 7808, normalized size of antiderivative = 19.57 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\text {Too large to display} \]
((d*x)^m*C*c^3*m^14*x^15 + (d*x)^m*B*c^3*m^14*x^14 + 105*(d*x)^m*C*c^3*m^1 3*x^15 + 3*(d*x)^m*C*b*c^2*m^14*x^13 + (d*x)^m*A*c^3*m^14*x^13 + 106*(d*x) ^m*B*c^3*m^13*x^14 + 5005*(d*x)^m*C*c^3*m^12*x^15 + 3*(d*x)^m*B*b*c^2*m^14 *x^12 + 321*(d*x)^m*C*b*c^2*m^13*x^13 + 107*(d*x)^m*A*c^3*m^13*x^13 + 5096 *(d*x)^m*B*c^3*m^12*x^14 + 143325*(d*x)^m*C*c^3*m^11*x^15 + 3*(d*x)^m*C*b^ 2*c*m^14*x^11 + 3*(d*x)^m*C*a*c^2*m^14*x^11 + 3*(d*x)^m*A*b*c^2*m^14*x^11 + 324*(d*x)^m*B*b*c^2*m^13*x^12 + 15567*(d*x)^m*C*b*c^2*m^12*x^13 + 5189*( d*x)^m*A*c^3*m^12*x^13 + 147056*(d*x)^m*B*c^3*m^11*x^14 + 2749747*(d*x)^m* C*c^3*m^10*x^15 + 3*(d*x)^m*B*b^2*c*m^14*x^10 + 3*(d*x)^m*B*a*c^2*m^14*x^1 0 + 327*(d*x)^m*C*b^2*c*m^13*x^11 + 327*(d*x)^m*C*a*c^2*m^13*x^11 + 327*(d *x)^m*A*b*c^2*m^13*x^11 + 15852*(d*x)^m*B*b*c^2*m^12*x^12 + 452829*(d*x)^m *C*b*c^2*m^11*x^13 + 150943*(d*x)^m*A*c^3*m^11*x^13 + 2840838*(d*x)^m*B*c^ 3*m^10*x^14 + 37312275*(d*x)^m*C*c^3*m^9*x^15 + (d*x)^m*C*b^3*m^14*x^9 + 6 *(d*x)^m*C*a*b*c*m^14*x^9 + 3*(d*x)^m*A*b^2*c*m^14*x^9 + 3*(d*x)^m*A*a*c^2 *m^14*x^9 + 330*(d*x)^m*B*b^2*c*m^13*x^10 + 330*(d*x)^m*B*a*c^2*m^13*x^10 + 16143*(d*x)^m*C*b^2*c*m^12*x^11 + 16143*(d*x)^m*C*a*c^2*m^12*x^11 + 1614 3*(d*x)^m*A*b*c^2*m^12*x^11 + 464976*(d*x)^m*B*b*c^2*m^11*x^12 + 8812089*( d*x)^m*C*b*c^2*m^10*x^13 + 2937363*(d*x)^m*A*c^3*m^10*x^13 + 38786748*(d*x )^m*B*c^3*m^9*x^14 + 368411615*(d*x)^m*C*c^3*m^8*x^15 + (d*x)^m*B*b^3*m^14 *x^8 + 6*(d*x)^m*B*a*b*c*m^14*x^8 + 111*(d*x)^m*C*b^3*m^13*x^9 + 666*(d...
Time = 9.25 (sec) , antiderivative size = 2443, normalized size of antiderivative = 6.12 \[ \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\text {Too large to display} \]
(x^7*(d*x)^m*(A*b^3 + 3*C*a*b^2 + 3*C*a^2*c + 6*A*a*b*c)*(593193196800*m + 796089202560*m^2 + 608700928752*m^3 + 299730345264*m^4 + 101420251688*m^5 + 24483279856*m^6 + 4306835671*m^7 + 557256047*m^8 + 52977099*m^9 + 36544 83*m^10 + 177877*m^11 + 5789*m^12 + 113*m^13 + m^14 + 186810624000))/(4339 163001600*m + 6165817614720*m^2 + 5056995703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 54631129553*m^7 + 8207628000*m^8 + 928095740*m^9 + 78558480*m^10 + 4899622*m^11 + 218400*m^12 + 6580*m^13 + 1 20*m^14 + m^15 + 1307674368000) + (x^9*(d*x)^m*(C*b^3 + 3*A*a*c^2 + 3*A*b^ 2*c + 6*C*a*b*c)*(465985094400*m + 633314724480*m^2 + 491520108816*m^3 + 2 46143692976*m^4 + 84836490456*m^5 + 20885191136*m^6 + 3749548713*m^7 + 495 342143*m^8 + 48083733*m^9 + 3386083*m^10 + 168171*m^11 + 5581*m^12 + 111*m ^13 + m^14 + 145297152000))/(4339163001600*m + 6165817614720*m^2 + 5056995 703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 54 631129553*m^7 + 8207628000*m^8 + 928095740*m^9 + 78558480*m^10 + 4899622*m ^11 + 218400*m^12 + 6580*m^13 + 120*m^14 + m^15 + 1307674368000) + (B*c^3* x^14*(d*x)^m*(303268406400*m + 418753514880*m^2 + 331303013496*m^3 + 16967 9309436*m^4 + 59999485546*m^5 + 15200266081*m^6 + 2816490248*m^7 + 3850812 68*m^8 + 38786748*m^9 + 2840838*m^10 + 147056*m^11 + 5096*m^12 + 106*m^13 + m^14 + 93405312000))/(4339163001600*m + 6165817614720*m^2 + 505699570382 4*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 5463...