Integrand size = 27, antiderivative size = 243 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\frac {a^3 d (f x)^{1+m}}{f (1+m)}+\frac {a^2 (3 b d+a e) (f x)^{3+m}}{f^3 (3+m)}+\frac {3 a \left (b^2 d+a c d+a b e\right ) (f x)^{5+m}}{f^5 (5+m)}+\frac {\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) (f x)^{7+m}}{f^7 (7+m)}+\frac {\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) (f x)^{9+m}}{f^9 (9+m)}+\frac {3 c \left (b c d+b^2 e+a c e\right ) (f x)^{11+m}}{f^{11} (11+m)}+\frac {c^2 (c d+3 b e) (f x)^{13+m}}{f^{13} (13+m)}+\frac {c^3 e (f x)^{15+m}}{f^{15} (15+m)} \]
a^3*d*(f*x)^(1+m)/f/(1+m)+a^2*(a*e+3*b*d)*(f*x)^(3+m)/f^3/(3+m)+3*a*(a*b*e +a*c*d+b^2*d)*(f*x)^(5+m)/f^5/(5+m)+(3*a^2*c*e+3*a*b^2*e+6*a*b*c*d+b^3*d)* (f*x)^(7+m)/f^7/(7+m)+(6*a*b*c*e+3*a*c^2*d+b^3*e+3*b^2*c*d)*(f*x)^(9+m)/f^ 9/(9+m)+3*c*(a*c*e+b^2*e+b*c*d)*(f*x)^(11+m)/f^11/(11+m)+c^2*(3*b*e+c*d)*( f*x)^(13+m)/f^13/(13+m)+c^3*e*(f*x)^(15+m)/f^15/(15+m)
Time = 0.81 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.79 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=x (f x)^m \left (\frac {a^3 d}{1+m}+\frac {a^2 (3 b d+a e) x^2}{3+m}+\frac {3 a \left (b^2 d+a c d+a b e\right ) x^4}{5+m}+\frac {\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^6}{7+m}+\frac {\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^8}{9+m}+\frac {3 c \left (b c d+b^2 e+a c e\right ) x^{10}}{11+m}+\frac {c^2 (c d+3 b e) x^{12}}{13+m}+\frac {c^3 e x^{14}}{15+m}\right ) \]
x*(f*x)^m*((a^3*d)/(1 + m) + (a^2*(3*b*d + a*e)*x^2)/(3 + m) + (3*a*(b^2*d + a*c*d + a*b*e)*x^4)/(5 + m) + ((b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c *e)*x^6)/(7 + m) + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^8)/(9 + m) + (3*c*(b*c*d + b^2*e + a*c*e)*x^10)/(11 + m) + (c^2*(c*d + 3*b*e)*x^12 )/(13 + m) + (c^3*e*x^14)/(15 + m))
Time = 0.43 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1584, 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.
| \(\displaystyle \int \left (d+e x^2\right ) (f x)^m \left (a+b x^2+c x^4\right )^3 \, dx\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \int \left (a^3 d (f x)^m+\frac {(f x)^{m+6} \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )}{f^6}+\frac {a^2 (f x)^{m+2} (a e+3 b d)}{f^2}+\frac {3 c (f x)^{m+10} \left (a c e+b^2 e+b c d\right )}{f^{10}}+\frac {3 a (f x)^{m+4} \left (a b e+a c d+b^2 d\right )}{f^4}+\frac {(f x)^{m+8} \left (6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right )}{f^8}+\frac {c^2 (f x)^{m+12} (3 b e+c d)}{f^{12}}+\frac {c^3 e (f x)^{m+14}}{f^{14}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 d (f x)^{m+1}}{f (m+1)}+\frac {(f x)^{m+7} \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )}{f^7 (m+7)}+\frac {a^2 (f x)^{m+3} (a e+3 b d)}{f^3 (m+3)}+\frac {3 c (f x)^{m+11} \left (a c e+b^2 e+b c d\right )}{f^{11} (m+11)}+\frac {3 a (f x)^{m+5} \left (a b e+a c d+b^2 d\right )}{f^5 (m+5)}+\frac {(f x)^{m+9} \left (6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right )}{f^9 (m+9)}+\frac {c^2 (f x)^{m+13} (3 b e+c d)}{f^{13} (m+13)}+\frac {c^3 e (f x)^{m+15}}{f^{15} (m+15)}\) |
(a^3*d*(f*x)^(1 + m))/(f*(1 + m)) + (a^2*(3*b*d + a*e)*(f*x)^(3 + m))/(f^3 *(3 + m)) + (3*a*(b^2*d + a*c*d + a*b*e)*(f*x)^(5 + m))/(f^5*(5 + m)) + (( b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*(f*x)^(7 + m))/(f^7*(7 + m)) + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*(f*x)^(9 + m))/(f^9*(9 + m)) + (3*c*(b*c*d + b^2*e + a*c*e)*(f*x)^(11 + m))/(f^11*(11 + m)) + (c^2*(c*d + 3*b*e)*(f*x)^(13 + m))/(f^13*(13 + m)) + (c^3*e*(f*x)^(15 + m))/(f^15*( 15 + m))
3.3.20.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(1934\) vs. \(2(243)=486\).
Time = 0.35 (sec) , antiderivative size = 1935, normalized size of antiderivative = 7.96
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(1935\) |
| risch | \(\text {Expression too large to display}\) | \(1935\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2737\) |
x*(c^3*e*m^7*x^14+49*c^3*e*m^6*x^14+3*b*c^2*e*m^7*x^12+c^3*d*m^7*x^12+973* c^3*e*m^5*x^14+153*b*c^2*e*m^6*x^12+51*c^3*d*m^6*x^12+10045*c^3*e*m^4*x^14 +3*a*c^2*e*m^7*x^10+3*b^2*c*e*m^7*x^10+3*b*c^2*d*m^7*x^10+3135*b*c^2*e*m^5 *x^12+1045*c^3*d*m^5*x^12+57379*c^3*e*m^3*x^14+159*a*c^2*e*m^6*x^10+159*b^ 2*c*e*m^6*x^10+159*b*c^2*d*m^6*x^10+33165*b*c^2*e*m^4*x^12+11055*c^3*d*m^4 *x^12+177331*c^3*e*m^2*x^14+6*a*b*c*e*m^7*x^8+3*a*c^2*d*m^7*x^8+3375*a*c^2 *e*m^5*x^10+b^3*e*m^7*x^8+3*b^2*c*d*m^7*x^8+3375*b^2*c*e*m^5*x^10+3375*b*c ^2*d*m^5*x^10+193017*b*c^2*e*m^3*x^12+64339*c^3*d*m^3*x^12+264207*c^3*e*m* x^14+330*a*b*c*e*m^6*x^8+165*a*c^2*d*m^6*x^8+36795*a*c^2*e*m^4*x^10+55*b^3 *e*m^6*x^8+165*b^2*c*d*m^6*x^8+36795*b^2*c*e*m^4*x^10+36795*b*c^2*d*m^4*x^ 10+604827*b*c^2*e*m^2*x^12+201609*c^3*d*m^2*x^12+135135*c^3*e*x^14+3*a^2*c *e*m^7*x^6+3*a*b^2*e*m^7*x^6+6*a*b*c*d*m^7*x^6+7278*a*b*c*e*m^5*x^8+3639*a *c^2*d*m^5*x^8+219417*a*c^2*e*m^3*x^10+b^3*d*m^7*x^6+1213*b^3*e*m^5*x^8+36 39*b^2*c*d*m^5*x^8+219417*b^2*c*e*m^3*x^10+219417*b*c^2*d*m^3*x^10+909765* b*c^2*e*m*x^12+303255*c^3*d*m*x^12+171*a^2*c*e*m^6*x^6+171*a*b^2*e*m^6*x^6 +342*a*b*c*d*m^6*x^6+82338*a*b*c*e*m^4*x^8+41169*a*c^2*d*m^4*x^8+700461*a* c^2*e*m^2*x^10+57*b^3*d*m^6*x^6+13723*b^3*e*m^4*x^8+41169*b^2*c*d*m^4*x^8+ 700461*b^2*c*e*m^2*x^10+700461*b*c^2*d*m^2*x^10+467775*b*c^2*e*x^12+155925 *c^3*d*x^12+3*a^2*b*e*m^7*x^4+3*a^2*c*d*m^7*x^4+3927*a^2*c*e*m^5*x^6+3*a*b ^2*d*m^7*x^4+3927*a*b^2*e*m^5*x^6+7854*a*b*c*d*m^5*x^6+507282*a*b*c*e*m...
Leaf count of result is larger than twice the leaf count of optimal. 1357 vs. \(2 (243) = 486\).
Time = 0.26 (sec) , antiderivative size = 1357, normalized size of antiderivative = 5.58 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\text {Too large to display} \]
((c^3*e*m^7 + 49*c^3*e*m^6 + 973*c^3*e*m^5 + 10045*c^3*e*m^4 + 57379*c^3*e *m^3 + 177331*c^3*e*m^2 + 264207*c^3*e*m + 135135*c^3*e)*x^15 + ((c^3*d + 3*b*c^2*e)*m^7 + 51*(c^3*d + 3*b*c^2*e)*m^6 + 1045*(c^3*d + 3*b*c^2*e)*m^5 + 11055*(c^3*d + 3*b*c^2*e)*m^4 + 155925*c^3*d + 467775*b*c^2*e + 64339*( c^3*d + 3*b*c^2*e)*m^3 + 201609*(c^3*d + 3*b*c^2*e)*m^2 + 303255*(c^3*d + 3*b*c^2*e)*m)*x^13 + 3*((b*c^2*d + (b^2*c + a*c^2)*e)*m^7 + 53*(b*c^2*d + (b^2*c + a*c^2)*e)*m^6 + 1125*(b*c^2*d + (b^2*c + a*c^2)*e)*m^5 + 12265*(b *c^2*d + (b^2*c + a*c^2)*e)*m^4 + 184275*b*c^2*d + 73139*(b*c^2*d + (b^2*c + a*c^2)*e)*m^3 + 233487*(b*c^2*d + (b^2*c + a*c^2)*e)*m^2 + 184275*(b^2* c + a*c^2)*e + 355815*(b*c^2*d + (b^2*c + a*c^2)*e)*m)*x^11 + ((3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*m^7 + 55*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a* b*c)*e)*m^6 + 1213*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*m^5 + 13723*( 3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*m^4 + 84547*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*m^3 + 277093*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e )*m^2 + 675675*(b^2*c + a*c^2)*d + 225225*(b^3 + 6*a*b*c)*e + 430335*(3*(b ^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*m)*x^9 + (((b^3 + 6*a*b*c)*d + 3*(a*b ^2 + a^2*c)*e)*m^7 + 57*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*m^6 + 13 09*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*m^5 + 15477*((b^3 + 6*a*b*c)* d + 3*(a*b^2 + a^2*c)*e)*m^4 + 99715*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c )*e)*m^3 + 340011*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*m^2 + 28957...
Leaf count of result is larger than twice the leaf count of optimal. 11266 vs. \(2 (238) = 476\).
Time = 1.53 (sec) , antiderivative size = 11266, normalized size of antiderivative = 46.36 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\text {Too large to display} \]
Piecewise(((-a**3*d/(14*x**14) - a**3*e/(12*x**12) - a**2*b*d/(4*x**12) - 3*a**2*b*e/(10*x**10) - 3*a**2*c*d/(10*x**10) - 3*a**2*c*e/(8*x**8) - 3*a* b**2*d/(10*x**10) - 3*a*b**2*e/(8*x**8) - 3*a*b*c*d/(4*x**8) - a*b*c*e/x** 6 - a*c**2*d/(2*x**6) - 3*a*c**2*e/(4*x**4) - b**3*d/(8*x**8) - b**3*e/(6* x**6) - b**2*c*d/(2*x**6) - 3*b**2*c*e/(4*x**4) - 3*b*c**2*d/(4*x**4) - 3* b*c**2*e/(2*x**2) - c**3*d/(2*x**2) + c**3*e*log(x))/f**15, Eq(m, -15)), ( (-a**3*d/(12*x**12) - a**3*e/(10*x**10) - 3*a**2*b*d/(10*x**10) - 3*a**2*b *e/(8*x**8) - 3*a**2*c*d/(8*x**8) - a**2*c*e/(2*x**6) - 3*a*b**2*d/(8*x**8 ) - a*b**2*e/(2*x**6) - a*b*c*d/x**6 - 3*a*b*c*e/(2*x**4) - 3*a*c**2*d/(4* x**4) - 3*a*c**2*e/(2*x**2) - b**3*d/(6*x**6) - b**3*e/(4*x**4) - 3*b**2*c *d/(4*x**4) - 3*b**2*c*e/(2*x**2) - 3*b*c**2*d/(2*x**2) + 3*b*c**2*e*log(x ) + c**3*d*log(x) + c**3*e*x**2/2)/f**13, Eq(m, -13)), ((-a**3*d/(10*x**10 ) - a**3*e/(8*x**8) - 3*a**2*b*d/(8*x**8) - a**2*b*e/(2*x**6) - a**2*c*d/( 2*x**6) - 3*a**2*c*e/(4*x**4) - a*b**2*d/(2*x**6) - 3*a*b**2*e/(4*x**4) - 3*a*b*c*d/(2*x**4) - 3*a*b*c*e/x**2 - 3*a*c**2*d/(2*x**2) + 3*a*c**2*e*log (x) - b**3*d/(4*x**4) - b**3*e/(2*x**2) - 3*b**2*c*d/(2*x**2) + 3*b**2*c*e *log(x) + 3*b*c**2*d*log(x) + 3*b*c**2*e*x**2/2 + c**3*d*x**2/2 + c**3*e*x **4/4)/f**11, Eq(m, -11)), ((-a**3*d/(8*x**8) - a**3*e/(6*x**6) - a**2*b*d /(2*x**6) - 3*a**2*b*e/(4*x**4) - 3*a**2*c*d/(4*x**4) - 3*a**2*c*e/(2*x**2 ) - 3*a*b**2*d/(4*x**4) - 3*a*b**2*e/(2*x**2) - 3*a*b*c*d/x**2 + 6*a*b*...
Time = 0.25 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.68 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\frac {c^{3} e f^{m} x^{15} x^{m}}{m + 15} + \frac {c^{3} d f^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, b c^{2} e f^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, b c^{2} d f^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, b^{2} c e f^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, a c^{2} e f^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, b^{2} c d f^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, a c^{2} d f^{m} x^{9} x^{m}}{m + 9} + \frac {b^{3} e f^{m} x^{9} x^{m}}{m + 9} + \frac {6 \, a b c e f^{m} x^{9} x^{m}}{m + 9} + \frac {b^{3} d f^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, a b c d f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a b^{2} e f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a^{2} c e f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a b^{2} d f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} c d f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} b e f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} b d f^{m} x^{3} x^{m}}{m + 3} + \frac {a^{3} e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a^{3} d}{f {\left (m + 1\right )}} \]
c^3*e*f^m*x^15*x^m/(m + 15) + c^3*d*f^m*x^13*x^m/(m + 13) + 3*b*c^2*e*f^m* x^13*x^m/(m + 13) + 3*b*c^2*d*f^m*x^11*x^m/(m + 11) + 3*b^2*c*e*f^m*x^11*x ^m/(m + 11) + 3*a*c^2*e*f^m*x^11*x^m/(m + 11) + 3*b^2*c*d*f^m*x^9*x^m/(m + 9) + 3*a*c^2*d*f^m*x^9*x^m/(m + 9) + b^3*e*f^m*x^9*x^m/(m + 9) + 6*a*b*c* e*f^m*x^9*x^m/(m + 9) + b^3*d*f^m*x^7*x^m/(m + 7) + 6*a*b*c*d*f^m*x^7*x^m/ (m + 7) + 3*a*b^2*e*f^m*x^7*x^m/(m + 7) + 3*a^2*c*e*f^m*x^7*x^m/(m + 7) + 3*a*b^2*d*f^m*x^5*x^m/(m + 5) + 3*a^2*c*d*f^m*x^5*x^m/(m + 5) + 3*a^2*b*e* f^m*x^5*x^m/(m + 5) + 3*a^2*b*d*f^m*x^3*x^m/(m + 3) + a^3*e*f^m*x^3*x^m/(m + 3) + (f*x)^(m + 1)*a^3*d/(f*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 2736 vs. \(2 (243) = 486\).
Time = 0.36 (sec) , antiderivative size = 2736, normalized size of antiderivative = 11.26 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\text {Too large to display} \]
((f*x)^m*c^3*e*m^7*x^15 + 49*(f*x)^m*c^3*e*m^6*x^15 + (f*x)^m*c^3*d*m^7*x^ 13 + 3*(f*x)^m*b*c^2*e*m^7*x^13 + 973*(f*x)^m*c^3*e*m^5*x^15 + 51*(f*x)^m* c^3*d*m^6*x^13 + 153*(f*x)^m*b*c^2*e*m^6*x^13 + 10045*(f*x)^m*c^3*e*m^4*x^ 15 + 3*(f*x)^m*b*c^2*d*m^7*x^11 + 3*(f*x)^m*b^2*c*e*m^7*x^11 + 3*(f*x)^m*a *c^2*e*m^7*x^11 + 1045*(f*x)^m*c^3*d*m^5*x^13 + 3135*(f*x)^m*b*c^2*e*m^5*x ^13 + 57379*(f*x)^m*c^3*e*m^3*x^15 + 159*(f*x)^m*b*c^2*d*m^6*x^11 + 159*(f *x)^m*b^2*c*e*m^6*x^11 + 159*(f*x)^m*a*c^2*e*m^6*x^11 + 11055*(f*x)^m*c^3* d*m^4*x^13 + 33165*(f*x)^m*b*c^2*e*m^4*x^13 + 177331*(f*x)^m*c^3*e*m^2*x^1 5 + 3*(f*x)^m*b^2*c*d*m^7*x^9 + 3*(f*x)^m*a*c^2*d*m^7*x^9 + (f*x)^m*b^3*e* m^7*x^9 + 6*(f*x)^m*a*b*c*e*m^7*x^9 + 3375*(f*x)^m*b*c^2*d*m^5*x^11 + 3375 *(f*x)^m*b^2*c*e*m^5*x^11 + 3375*(f*x)^m*a*c^2*e*m^5*x^11 + 64339*(f*x)^m* c^3*d*m^3*x^13 + 193017*(f*x)^m*b*c^2*e*m^3*x^13 + 264207*(f*x)^m*c^3*e*m* x^15 + 165*(f*x)^m*b^2*c*d*m^6*x^9 + 165*(f*x)^m*a*c^2*d*m^6*x^9 + 55*(f*x )^m*b^3*e*m^6*x^9 + 330*(f*x)^m*a*b*c*e*m^6*x^9 + 36795*(f*x)^m*b*c^2*d*m^ 4*x^11 + 36795*(f*x)^m*b^2*c*e*m^4*x^11 + 36795*(f*x)^m*a*c^2*e*m^4*x^11 + 201609*(f*x)^m*c^3*d*m^2*x^13 + 604827*(f*x)^m*b*c^2*e*m^2*x^13 + 135135* (f*x)^m*c^3*e*x^15 + (f*x)^m*b^3*d*m^7*x^7 + 6*(f*x)^m*a*b*c*d*m^7*x^7 + 3 *(f*x)^m*a*b^2*e*m^7*x^7 + 3*(f*x)^m*a^2*c*e*m^7*x^7 + 3639*(f*x)^m*b^2*c* d*m^5*x^9 + 3639*(f*x)^m*a*c^2*d*m^5*x^9 + 1213*(f*x)^m*b^3*e*m^5*x^9 + 72 78*(f*x)^m*a*b*c*e*m^5*x^9 + 219417*(f*x)^m*b*c^2*d*m^3*x^11 + 219417*(...
Time = 8.07 (sec) , antiderivative size = 769, normalized size of antiderivative = 3.16 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\frac {x^7\,{\left (f\,x\right )}^m\,\left (3\,c\,e\,a^2+3\,e\,a\,b^2+6\,c\,d\,a\,b+d\,b^3\right )\,\left (m^7+57\,m^6+1309\,m^5+15477\,m^4+99715\,m^3+340011\,m^2+544095\,m+289575\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {x^9\,{\left (f\,x\right )}^m\,\left (e\,b^3+3\,d\,b^2\,c+6\,a\,e\,b\,c+3\,a\,d\,c^2\right )\,\left (m^7+55\,m^6+1213\,m^5+13723\,m^4+84547\,m^3+277093\,m^2+430335\,m+225225\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {a^3\,d\,x\,{\left (f\,x\right )}^m\,\left (m^7+63\,m^6+1645\,m^5+22995\,m^4+185059\,m^3+852957\,m^2+2071215\,m+2027025\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {c^3\,e\,x^{15}\,{\left (f\,x\right )}^m\,\left (m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {3\,a\,x^5\,{\left (f\,x\right )}^m\,\left (d\,b^2+a\,e\,b+a\,c\,d\right )\,\left (m^7+59\,m^6+1413\,m^5+17575\,m^4+120179\,m^3+437121\,m^2+738567\,m+405405\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {3\,c\,x^{11}\,{\left (f\,x\right )}^m\,\left (e\,b^2+c\,d\,b+a\,c\,e\right )\,\left (m^7+53\,m^6+1125\,m^5+12265\,m^4+73139\,m^3+233487\,m^2+355815\,m+184275\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {a^2\,x^3\,{\left (f\,x\right )}^m\,\left (a\,e+3\,b\,d\right )\,\left (m^7+61\,m^6+1525\,m^5+20065\,m^4+147859\,m^3+594439\,m^2+1140855\,m+675675\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {c^2\,x^{13}\,{\left (f\,x\right )}^m\,\left (3\,b\,e+c\,d\right )\,\left (m^7+51\,m^6+1045\,m^5+11055\,m^4+64339\,m^3+201609\,m^2+303255\,m+155925\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025} \]
(x^7*(f*x)^m*(b^3*d + 3*a*b^2*e + 3*a^2*c*e + 6*a*b*c*d)*(544095*m + 34001 1*m^2 + 99715*m^3 + 15477*m^4 + 1309*m^5 + 57*m^6 + m^7 + 289575))/(409824 0*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64*m ^7 + m^8 + 2027025) + (x^9*(f*x)^m*(b^3*e + 3*a*c^2*d + 3*b^2*c*d + 6*a*b* c*e)*(430335*m + 277093*m^2 + 84547*m^3 + 13723*m^4 + 1213*m^5 + 55*m^6 + m^7 + 225225))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640 *m^5 + 1708*m^6 + 64*m^7 + m^8 + 2027025) + (a^3*d*x*(f*x)^m*(2071215*m + 852957*m^2 + 185059*m^3 + 22995*m^4 + 1645*m^5 + 63*m^6 + m^7 + 2027025))/ (4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64*m^7 + m^8 + 2027025) + (c^3*e*x^15*(f*x)^m*(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135))/(4098240*m + 292 4172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64*m^7 + m^8 + 2027025) + (3*a*x^5*(f*x)^m*(b^2*d + a*b*e + a*c*d)*(738567*m + 437121*m ^2 + 120179*m^3 + 17575*m^4 + 1413*m^5 + 59*m^6 + m^7 + 405405))/(4098240* m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64*m^7 + m^8 + 2027025) + (3*c*x^11*(f*x)^m*(b^2*e + a*c*e + b*c*d)*(355815*m + 233487*m^2 + 73139*m^3 + 12265*m^4 + 1125*m^5 + 53*m^6 + m^7 + 184275))/(4 098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64*m^7 + m^8 + 2027025) + (a^2*x^3*(f*x)^m*(a*e + 3*b*d)*(1140855*m + 594 439*m^2 + 147859*m^3 + 20065*m^4 + 1525*m^5 + 61*m^6 + m^7 + 675675))/(...