Integrand size = 31, antiderivative size = 379 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\frac {a^3 A c^3 (e x)^{1+m}}{e (1+m)}+\frac {a^2 c^2 (a B c+3 A (b c+a d)) (e x)^{3+m}}{e^3 (3+m)}+\frac {3 a c \left (a B c (b c+a d)+A \left (b^2 c^2+3 a b c d+a^2 d^2\right )\right ) (e x)^{5+m}}{e^5 (5+m)}+\frac {\left (3 a B c \left (b^2 c^2+3 a b c d+a^2 d^2\right )+A \left (b^3 c^3+9 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right )\right ) (e x)^{7+m}}{e^7 (7+m)}+\frac {\left (a^3 B d^3+9 a b^2 c d (B c+A d)+3 a^2 b d^2 (3 B c+A d)+b^3 c^2 (B c+3 A d)\right ) (e x)^{9+m}}{e^9 (9+m)}+\frac {3 b d \left (a^2 B d^2+b^2 c (B c+A d)+a b d (3 B c+A d)\right ) (e x)^{11+m}}{e^{11} (11+m)}+\frac {b^2 d^2 (3 b B c+A b d+3 a B d) (e x)^{13+m}}{e^{13} (13+m)}+\frac {b^3 B d^3 (e x)^{15+m}}{e^{15} (15+m)} \]
a^3*A*c^3*(e*x)^(1+m)/e/(1+m)+a^2*c^2*(B*a*c+3*A*(a*d+b*c))*(e*x)^(3+m)/e^ 3/(3+m)+3*a*c*(a*B*c*(a*d+b*c)+A*(a^2*d^2+3*a*b*c*d+b^2*c^2))*(e*x)^(5+m)/ e^5/(5+m)+(3*a*B*c*(a^2*d^2+3*a*b*c*d+b^2*c^2)+A*(a^3*d^3+9*a^2*b*c*d^2+9* a*b^2*c^2*d+b^3*c^3))*(e*x)^(7+m)/e^7/(7+m)+(a^3*B*d^3+9*a*b^2*c*d*(A*d+B* c)+3*a^2*b*d^2*(A*d+3*B*c)+b^3*c^2*(3*A*d+B*c))*(e*x)^(9+m)/e^9/(9+m)+3*b* d*(a^2*B*d^2+b^2*c*(A*d+B*c)+a*b*d*(A*d+3*B*c))*(e*x)^(11+m)/e^11/(11+m)+b ^2*d^2*(A*b*d+3*B*a*d+3*B*b*c)*(e*x)^(13+m)/e^13/(13+m)+b^3*B*d^3*(e*x)^(1 5+m)/e^15/(15+m)
Time = 0.96 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.86 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=x (e x)^m \left (\frac {a^3 A c^3}{1+m}+\frac {a^2 c^2 (a B c+3 A (b c+a d)) x^2}{3+m}+\frac {3 a c \left (a B c (b c+a d)+A \left (b^2 c^2+3 a b c d+a^2 d^2\right )\right ) x^4}{5+m}+\frac {\left (3 a B c \left (b^2 c^2+3 a b c d+a^2 d^2\right )+A \left (b^3 c^3+9 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right )\right ) x^6}{7+m}+\frac {\left (a^3 B d^3+9 a b^2 c d (B c+A d)+3 a^2 b d^2 (3 B c+A d)+b^3 c^2 (B c+3 A d)\right ) x^8}{9+m}+\frac {3 b d \left (a^2 B d^2+b^2 c (B c+A d)+a b d (3 B c+A d)\right ) x^{10}}{11+m}+\frac {b^2 d^2 (3 b B c+A b d+3 a B d) x^{12}}{13+m}+\frac {b^3 B d^3 x^{14}}{15+m}\right ) \]
x*(e*x)^m*((a^3*A*c^3)/(1 + m) + (a^2*c^2*(a*B*c + 3*A*(b*c + a*d))*x^2)/( 3 + m) + (3*a*c*(a*B*c*(b*c + a*d) + A*(b^2*c^2 + 3*a*b*c*d + a^2*d^2))*x^ 4)/(5 + m) + ((3*a*B*c*(b^2*c^2 + 3*a*b*c*d + a^2*d^2) + A*(b^3*c^3 + 9*a* b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3))*x^6)/(7 + m) + ((a^3*B*d^3 + 9*a*b^2 *c*d*(B*c + A*d) + 3*a^2*b*d^2*(3*B*c + A*d) + b^3*c^2*(B*c + 3*A*d))*x^8) /(9 + m) + (3*b*d*(a^2*B*d^2 + b^2*c*(B*c + A*d) + a*b*d*(3*B*c + A*d))*x^ 10)/(11 + m) + (b^2*d^2*(3*b*B*c + A*b*d + 3*a*B*d)*x^12)/(13 + m) + (b^3* B*d^3*x^14)/(15 + m))
Time = 0.65 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {437, 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.
| \(\displaystyle \int \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 (e x)^m \, dx\) |
| \(\Big \downarrow \) 437 |
| \(\displaystyle \int \left (a^3 A c^3 (e x)^m+\frac {3 a c (e x)^{m+4} \left (A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a B c (a d+b c)\right )}{e^4}+\frac {3 b d (e x)^{m+10} \left (a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right )}{e^{10}}+\frac {a^2 c^2 (e x)^{m+2} (3 A (a d+b c)+a B c)}{e^2}+\frac {(e x)^{m+8} \left (a^3 B d^3+3 a^2 b d^2 (A d+3 B c)+9 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)\right )}{e^8}+\frac {(e x)^{m+6} \left (3 a B c \left (a^2 d^2+3 a b c d+b^2 c^2\right )+A \left (a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )\right )}{e^6}+\frac {b^2 d^2 (e x)^{m+12} (3 a B d+A b d+3 b B c)}{e^{12}}+\frac {b^3 B d^3 (e x)^{m+14}}{e^{14}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 A c^3 (e x)^{m+1}}{e (m+1)}+\frac {3 a c (e x)^{m+5} \left (A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a B c (a d+b c)\right )}{e^5 (m+5)}+\frac {3 b d (e x)^{m+11} \left (a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right )}{e^{11} (m+11)}+\frac {a^2 c^2 (e x)^{m+3} (3 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac {(e x)^{m+9} \left (a^3 B d^3+3 a^2 b d^2 (A d+3 B c)+9 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)\right )}{e^9 (m+9)}+\frac {(e x)^{m+7} \left (3 a B c \left (a^2 d^2+3 a b c d+b^2 c^2\right )+A \left (a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )\right )}{e^7 (m+7)}+\frac {b^2 d^2 (e x)^{m+13} (3 a B d+A b d+3 b B c)}{e^{13} (m+13)}+\frac {b^3 B d^3 (e x)^{m+15}}{e^{15} (m+15)}\) |
(a^3*A*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*c^2*(a*B*c + 3*A*(b*c + a*d)) *(e*x)^(3 + m))/(e^3*(3 + m)) + (3*a*c*(a*B*c*(b*c + a*d) + A*(b^2*c^2 + 3 *a*b*c*d + a^2*d^2))*(e*x)^(5 + m))/(e^5*(5 + m)) + ((3*a*B*c*(b^2*c^2 + 3 *a*b*c*d + a^2*d^2) + A*(b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3 ))*(e*x)^(7 + m))/(e^7*(7 + m)) + ((a^3*B*d^3 + 9*a*b^2*c*d*(B*c + A*d) + 3*a^2*b*d^2*(3*B*c + A*d) + b^3*c^2*(B*c + 3*A*d))*(e*x)^(9 + m))/(e^9*(9 + m)) + (3*b*d*(a^2*B*d^2 + b^2*c*(B*c + A*d) + a*b*d*(3*B*c + A*d))*(e*x) ^(11 + m))/(e^11*(11 + m)) + (b^2*d^2*(3*b*B*c + A*b*d + 3*a*B*d)*(e*x)^(1 3 + m))/(e^13*(13 + m)) + (b^3*B*d^3*(e*x)^(15 + m))/(e^15*(15 + m))
3.1.15.3.1 Defintions of rubi rules used
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*( a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f , g, m}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3952\) vs. \(2(379)=758\).
Time = 3.96 (sec) , antiderivative size = 3953, normalized size of antiderivative = 10.43
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(3953\) |
| risch | \(\text {Expression too large to display}\) | \(3953\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(5235\) |
x*(B*b^3*d^3*m^7*x^14+49*B*b^3*d^3*m^6*x^14+A*b^3*d^3*m^7*x^12+3*B*a*b^2*d ^3*m^7*x^12+3*B*b^3*c*d^2*m^7*x^12+973*B*b^3*d^3*m^5*x^14+51*A*b^3*d^3*m^6 *x^12+153*B*a*b^2*d^3*m^6*x^12+153*B*b^3*c*d^2*m^6*x^12+10045*B*b^3*d^3*m^ 4*x^14+3*A*a*b^2*d^3*m^7*x^10+3*A*b^3*c*d^2*m^7*x^10+1045*A*b^3*d^3*m^5*x^ 12+3*B*a^2*b*d^3*m^7*x^10+9*B*a*b^2*c*d^2*m^7*x^10+3135*B*a*b^2*d^3*m^5*x^ 12+3*B*b^3*c^2*d*m^7*x^10+3135*B*b^3*c*d^2*m^5*x^12+57379*B*b^3*d^3*m^3*x^ 14+159*A*a*b^2*d^3*m^6*x^10+159*A*b^3*c*d^2*m^6*x^10+11055*A*b^3*d^3*m^4*x ^12+159*B*a^2*b*d^3*m^6*x^10+477*B*a*b^2*c*d^2*m^6*x^10+33165*B*a*b^2*d^3* m^4*x^12+159*B*b^3*c^2*d*m^6*x^10+33165*B*b^3*c*d^2*m^4*x^12+177331*B*b^3* d^3*m^2*x^14+3*A*a^2*b*d^3*m^7*x^8+9*A*a*b^2*c*d^2*m^7*x^8+3375*A*a*b^2*d^ 3*m^5*x^10+3*A*b^3*c^2*d*m^7*x^8+3375*A*b^3*c*d^2*m^5*x^10+64339*A*b^3*d^3 *m^3*x^12+B*a^3*d^3*m^7*x^8+9*B*a^2*b*c*d^2*m^7*x^8+3375*B*a^2*b*d^3*m^5*x ^10+9*B*a*b^2*c^2*d*m^7*x^8+10125*B*a*b^2*c*d^2*m^5*x^10+193017*B*a*b^2*d^ 3*m^3*x^12+B*b^3*c^3*m^7*x^8+3375*B*b^3*c^2*d*m^5*x^10+193017*B*b^3*c*d^2* m^3*x^12+264207*B*b^3*d^3*m*x^14+165*A*a^2*b*d^3*m^6*x^8+495*A*a*b^2*c*d^2 *m^6*x^8+36795*A*a*b^2*d^3*m^4*x^10+165*A*b^3*c^2*d*m^6*x^8+36795*A*b^3*c* d^2*m^4*x^10+201609*A*b^3*d^3*m^2*x^12+55*B*a^3*d^3*m^6*x^8+495*B*a^2*b*c* d^2*m^6*x^8+36795*B*a^2*b*d^3*m^4*x^10+495*B*a*b^2*c^2*d*m^6*x^8+110385*B* a*b^2*c*d^2*m^4*x^10+604827*B*a*b^2*d^3*m^2*x^12+55*B*b^3*c^3*m^6*x^8+3679 5*B*b^3*c^2*d*m^4*x^10+604827*B*b^3*c*d^2*m^2*x^12+135135*B*b^3*d^3*x^1...
Leaf count of result is larger than twice the leaf count of optimal. 2657 vs. \(2 (379) = 758\).
Time = 0.32 (sec) , antiderivative size = 2657, normalized size of antiderivative = 7.01 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\text {Too large to display} \]
((B*b^3*d^3*m^7 + 49*B*b^3*d^3*m^6 + 973*B*b^3*d^3*m^5 + 10045*B*b^3*d^3*m ^4 + 57379*B*b^3*d^3*m^3 + 177331*B*b^3*d^3*m^2 + 264207*B*b^3*d^3*m + 135 135*B*b^3*d^3)*x^15 + ((3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m^7 + 467 775*B*b^3*c*d^2 + 51*(3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m^6 + 1045* (3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m^5 + 11055*(3*B*b^3*c*d^2 + (3* B*a*b^2 + A*b^3)*d^3)*m^4 + 155925*(3*B*a*b^2 + A*b^3)*d^3 + 64339*(3*B*b^ 3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m^3 + 201609*(3*B*b^3*c*d^2 + (3*B*a*b^ 2 + A*b^3)*d^3)*m^2 + 303255*(3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m)* x^13 + 3*((B*b^3*c^2*d + (3*B*a*b^2 + A*b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d ^3)*m^7 + 184275*B*b^3*c^2*d + 53*(B*b^3*c^2*d + (3*B*a*b^2 + A*b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^6 + 1125*(B*b^3*c^2*d + (3*B*a*b^2 + A*b^3)* c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^5 + 12265*(B*b^3*c^2*d + (3*B*a*b^2 + A *b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^4 + 184275*(3*B*a*b^2 + A*b^3)*c* d^2 + 184275*(B*a^2*b + A*a*b^2)*d^3 + 73139*(B*b^3*c^2*d + (3*B*a*b^2 + A *b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^3 + 233487*(B*b^3*c^2*d + (3*B*a* b^2 + A*b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^2 + 355815*(B*b^3*c^2*d + (3*B*a*b^2 + A*b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m)*x^11 + ((B*b^3*c^3 + 3*(3*B*a*b^2 + A*b^3)*c^2*d + 9*(B*a^2*b + A*a*b^2)*c*d^2 + (B*a^3 + 3* A*a^2*b)*d^3)*m^7 + 225225*B*b^3*c^3 + 55*(B*b^3*c^3 + 3*(3*B*a*b^2 + A*b^ 3)*c^2*d + 9*(B*a^2*b + A*a*b^2)*c*d^2 + (B*a^3 + 3*A*a^2*b)*d^3)*m^6 +...
Leaf count of result is larger than twice the leaf count of optimal. 20086 vs. \(2 (377) = 754\).
Time = 1.93 (sec) , antiderivative size = 20086, normalized size of antiderivative = 53.00 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise(((-A*a**3*c**3/(14*x**14) - A*a**3*c**2*d/(4*x**12) - 3*A*a**3*c *d**2/(10*x**10) - A*a**3*d**3/(8*x**8) - A*a**2*b*c**3/(4*x**12) - 9*A*a* *2*b*c**2*d/(10*x**10) - 9*A*a**2*b*c*d**2/(8*x**8) - A*a**2*b*d**3/(2*x** 6) - 3*A*a*b**2*c**3/(10*x**10) - 9*A*a*b**2*c**2*d/(8*x**8) - 3*A*a*b**2* c*d**2/(2*x**6) - 3*A*a*b**2*d**3/(4*x**4) - A*b**3*c**3/(8*x**8) - A*b**3 *c**2*d/(2*x**6) - 3*A*b**3*c*d**2/(4*x**4) - A*b**3*d**3/(2*x**2) - B*a** 3*c**3/(12*x**12) - 3*B*a**3*c**2*d/(10*x**10) - 3*B*a**3*c*d**2/(8*x**8) - B*a**3*d**3/(6*x**6) - 3*B*a**2*b*c**3/(10*x**10) - 9*B*a**2*b*c**2*d/(8 *x**8) - 3*B*a**2*b*c*d**2/(2*x**6) - 3*B*a**2*b*d**3/(4*x**4) - 3*B*a*b** 2*c**3/(8*x**8) - 3*B*a*b**2*c**2*d/(2*x**6) - 9*B*a*b**2*c*d**2/(4*x**4) - 3*B*a*b**2*d**3/(2*x**2) - B*b**3*c**3/(6*x**6) - 3*B*b**3*c**2*d/(4*x** 4) - 3*B*b**3*c*d**2/(2*x**2) + B*b**3*d**3*log(x))/e**15, Eq(m, -15)), (( -A*a**3*c**3/(12*x**12) - 3*A*a**3*c**2*d/(10*x**10) - 3*A*a**3*c*d**2/(8* x**8) - A*a**3*d**3/(6*x**6) - 3*A*a**2*b*c**3/(10*x**10) - 9*A*a**2*b*c** 2*d/(8*x**8) - 3*A*a**2*b*c*d**2/(2*x**6) - 3*A*a**2*b*d**3/(4*x**4) - 3*A *a*b**2*c**3/(8*x**8) - 3*A*a*b**2*c**2*d/(2*x**6) - 9*A*a*b**2*c*d**2/(4* x**4) - 3*A*a*b**2*d**3/(2*x**2) - A*b**3*c**3/(6*x**6) - 3*A*b**3*c**2*d/ (4*x**4) - 3*A*b**3*c*d**2/(2*x**2) + A*b**3*d**3*log(x) - B*a**3*c**3/(10 *x**10) - 3*B*a**3*c**2*d/(8*x**8) - B*a**3*c*d**2/(2*x**6) - B*a**3*d**3/ (4*x**4) - 3*B*a**2*b*c**3/(8*x**8) - 3*B*a**2*b*c**2*d/(2*x**6) - 9*B*...
Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (379) = 758\).
Time = 0.29 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.01 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\frac {B b^{3} d^{3} e^{m} x^{15} x^{m}}{m + 15} + \frac {3 \, B b^{3} c d^{2} e^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, B a b^{2} d^{3} e^{m} x^{13} x^{m}}{m + 13} + \frac {A b^{3} d^{3} e^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, B b^{3} c^{2} d e^{m} x^{11} x^{m}}{m + 11} + \frac {9 \, B a b^{2} c d^{2} e^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, A b^{3} c d^{2} e^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, B a^{2} b d^{3} e^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, A a b^{2} d^{3} e^{m} x^{11} x^{m}}{m + 11} + \frac {B b^{3} c^{3} e^{m} x^{9} x^{m}}{m + 9} + \frac {9 \, B a b^{2} c^{2} d e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, A b^{3} c^{2} d e^{m} x^{9} x^{m}}{m + 9} + \frac {9 \, B a^{2} b c d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {9 \, A a b^{2} c d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {B a^{3} d^{3} e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, A a^{2} b d^{3} e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B a b^{2} c^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{3} c^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {9 \, B a^{2} b c^{2} d e^{m} x^{7} x^{m}}{m + 7} + \frac {9 \, A a b^{2} c^{2} d e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a^{3} c d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {9 \, A a^{2} b c d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {A a^{3} d^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a^{2} b c^{3} e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a b^{2} c^{3} e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, B a^{3} c^{2} d e^{m} x^{5} x^{m}}{m + 5} + \frac {9 \, A a^{2} b c^{2} d e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a^{3} c d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{3} c^{3} e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{2} b c^{3} e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{3} c^{2} d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a^{3} c^{3}}{e {\left (m + 1\right )}} \]
B*b^3*d^3*e^m*x^15*x^m/(m + 15) + 3*B*b^3*c*d^2*e^m*x^13*x^m/(m + 13) + 3* B*a*b^2*d^3*e^m*x^13*x^m/(m + 13) + A*b^3*d^3*e^m*x^13*x^m/(m + 13) + 3*B* b^3*c^2*d*e^m*x^11*x^m/(m + 11) + 9*B*a*b^2*c*d^2*e^m*x^11*x^m/(m + 11) + 3*A*b^3*c*d^2*e^m*x^11*x^m/(m + 11) + 3*B*a^2*b*d^3*e^m*x^11*x^m/(m + 11) + 3*A*a*b^2*d^3*e^m*x^11*x^m/(m + 11) + B*b^3*c^3*e^m*x^9*x^m/(m + 9) + 9* B*a*b^2*c^2*d*e^m*x^9*x^m/(m + 9) + 3*A*b^3*c^2*d*e^m*x^9*x^m/(m + 9) + 9* B*a^2*b*c*d^2*e^m*x^9*x^m/(m + 9) + 9*A*a*b^2*c*d^2*e^m*x^9*x^m/(m + 9) + B*a^3*d^3*e^m*x^9*x^m/(m + 9) + 3*A*a^2*b*d^3*e^m*x^9*x^m/(m + 9) + 3*B*a* b^2*c^3*e^m*x^7*x^m/(m + 7) + A*b^3*c^3*e^m*x^7*x^m/(m + 7) + 9*B*a^2*b*c^ 2*d*e^m*x^7*x^m/(m + 7) + 9*A*a*b^2*c^2*d*e^m*x^7*x^m/(m + 7) + 3*B*a^3*c* d^2*e^m*x^7*x^m/(m + 7) + 9*A*a^2*b*c*d^2*e^m*x^7*x^m/(m + 7) + A*a^3*d^3* e^m*x^7*x^m/(m + 7) + 3*B*a^2*b*c^3*e^m*x^5*x^m/(m + 5) + 3*A*a*b^2*c^3*e^ m*x^5*x^m/(m + 5) + 3*B*a^3*c^2*d*e^m*x^5*x^m/(m + 5) + 9*A*a^2*b*c^2*d*e^ m*x^5*x^m/(m + 5) + 3*A*a^3*c*d^2*e^m*x^5*x^m/(m + 5) + B*a^3*c^3*e^m*x^3* x^m/(m + 3) + 3*A*a^2*b*c^3*e^m*x^3*x^m/(m + 3) + 3*A*a^3*c^2*d*e^m*x^3*x^ m/(m + 3) + (e*x)^(m + 1)*A*a^3*c^3/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 5234 vs. \(2 (379) = 758\).
Time = 0.41 (sec) , antiderivative size = 5234, normalized size of antiderivative = 13.81 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\text {Too large to display} \]
((e*x)^m*B*b^3*d^3*m^7*x^15 + 49*(e*x)^m*B*b^3*d^3*m^6*x^15 + 3*(e*x)^m*B* b^3*c*d^2*m^7*x^13 + 3*(e*x)^m*B*a*b^2*d^3*m^7*x^13 + (e*x)^m*A*b^3*d^3*m^ 7*x^13 + 973*(e*x)^m*B*b^3*d^3*m^5*x^15 + 153*(e*x)^m*B*b^3*c*d^2*m^6*x^13 + 153*(e*x)^m*B*a*b^2*d^3*m^6*x^13 + 51*(e*x)^m*A*b^3*d^3*m^6*x^13 + 1004 5*(e*x)^m*B*b^3*d^3*m^4*x^15 + 3*(e*x)^m*B*b^3*c^2*d*m^7*x^11 + 9*(e*x)^m* B*a*b^2*c*d^2*m^7*x^11 + 3*(e*x)^m*A*b^3*c*d^2*m^7*x^11 + 3*(e*x)^m*B*a^2* b*d^3*m^7*x^11 + 3*(e*x)^m*A*a*b^2*d^3*m^7*x^11 + 3135*(e*x)^m*B*b^3*c*d^2 *m^5*x^13 + 3135*(e*x)^m*B*a*b^2*d^3*m^5*x^13 + 1045*(e*x)^m*A*b^3*d^3*m^5 *x^13 + 57379*(e*x)^m*B*b^3*d^3*m^3*x^15 + 159*(e*x)^m*B*b^3*c^2*d*m^6*x^1 1 + 477*(e*x)^m*B*a*b^2*c*d^2*m^6*x^11 + 159*(e*x)^m*A*b^3*c*d^2*m^6*x^11 + 159*(e*x)^m*B*a^2*b*d^3*m^6*x^11 + 159*(e*x)^m*A*a*b^2*d^3*m^6*x^11 + 33 165*(e*x)^m*B*b^3*c*d^2*m^4*x^13 + 33165*(e*x)^m*B*a*b^2*d^3*m^4*x^13 + 11 055*(e*x)^m*A*b^3*d^3*m^4*x^13 + 177331*(e*x)^m*B*b^3*d^3*m^2*x^15 + (e*x) ^m*B*b^3*c^3*m^7*x^9 + 9*(e*x)^m*B*a*b^2*c^2*d*m^7*x^9 + 3*(e*x)^m*A*b^3*c ^2*d*m^7*x^9 + 9*(e*x)^m*B*a^2*b*c*d^2*m^7*x^9 + 9*(e*x)^m*A*a*b^2*c*d^2*m ^7*x^9 + (e*x)^m*B*a^3*d^3*m^7*x^9 + 3*(e*x)^m*A*a^2*b*d^3*m^7*x^9 + 3375* (e*x)^m*B*b^3*c^2*d*m^5*x^11 + 10125*(e*x)^m*B*a*b^2*c*d^2*m^5*x^11 + 3375 *(e*x)^m*A*b^3*c*d^2*m^5*x^11 + 3375*(e*x)^m*B*a^2*b*d^3*m^5*x^11 + 3375*( e*x)^m*A*a*b^2*d^3*m^5*x^11 + 193017*(e*x)^m*B*b^3*c*d^2*m^3*x^13 + 193017 *(e*x)^m*B*a*b^2*d^3*m^3*x^13 + 64339*(e*x)^m*A*b^3*d^3*m^3*x^13 + 2642...
Time = 6.61 (sec) , antiderivative size = 933, normalized size of antiderivative = 2.46 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\frac {x^7\,{\left (e\,x\right )}^m\,\left (3\,B\,a^3\,c\,d^2+A\,a^3\,d^3+9\,B\,a^2\,b\,c^2\,d+9\,A\,a^2\,b\,c\,d^2+3\,B\,a\,b^2\,c^3+9\,A\,a\,b^2\,c^2\,d+A\,b^3\,c^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 (e\,x\right )}^m\,\left (B\,a^3\,d^3+9\,B\,a^2\,b\,c\,d^2+3\,A\,a^2\,b\,d^3+9\,B\,a\,b^2\,c^2\,d+9\,A\,a\,b^2\,c\,d^2+B\,b^3\,c^3+3\,A\,b^3\,c^2\,d\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 {B\,b^3\,d^3\,x^{15}\,{\left (e\,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\,c\,x^5\,{\left (e\,x\right )}^m\,\left (B\,a^2\,c\,d+A\,a^2\,d^2+B\,a\,b\,c^2+3\,A\,a\,b\,c\,d+A\,b^2\,c^2\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\,b\,d\,x^{11}\,{\left (e\,x\right )}^m\,\left (B\,a^2\,d^2+3\,B\,a\,b\,c\,d+A\,a\,b\,d^2+B\,b^2\,c^2+A\,b^2\,c\,d\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\,c^2\,x^3\,{\left (e\,x\right )}^m\,\left (3\,A\,a\,d+3\,A\,b\,c+B\,a\,c\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 {b^2\,d^2\,x^{13}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+3\,B\,a\,d+3\,B\,b\,c\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}+\frac {A\,a^3\,c^3\,x\,{\left (e\,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} \]
(x^7*(e*x)^m*(A*a^3*d^3 + A*b^3*c^3 + 3*B*a*b^2*c^3 + 3*B*a^3*c*d^2 + 9*A* a*b^2*c^2*d + 9*A*a^2*b*c*d^2 + 9*B*a^2*b*c^2*d)*(544095*m + 340011*m^2 + 99715*m^3 + 15477*m^4 + 1309*m^5 + 57*m^6 + m^7 + 289575))/(4098240*m + 29 24172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64*m^7 + m^8 + 2027025) + (x^9*(e*x)^m*(B*a^3*d^3 + B*b^3*c^3 + 3*A*a^2*b*d^3 + 3*A*b^ 3*c^2*d + 9*A*a*b^2*c*d^2 + 9*B*a*b^2*c^2*d + 9*B*a^2*b*c*d^2)*(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) + (B*b^3*d^3*x^15*(e*x)^m*(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64*m^7 + m ^8 + 2027025) + (3*a*c*x^5*(e*x)^m*(A*a^2*d^2 + A*b^2*c^2 + B*a*b*c^2 + B* a^2*c*d + 3*A*a*b*c*d)*(738567*m + 437121*m^2 + 120179*m^3 + 17575*m^4 + 1 413*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*b*d*x^11 *(e*x)^m*(B*a^2*d^2 + B*b^2*c^2 + A*a*b*d^2 + A*b^2*c*d + 3*B*a*b*c*d)*(35 5815*m + 233487*m^2 + 73139*m^3 + 12265*m^4 + 1125*m^5 + 53*m^6 + m^7 + 18 4275))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1 708*m^6 + 64*m^7 + m^8 + 2027025) + (a^2*c^2*x^3*(e*x)^m*(3*A*a*d + 3*A*b* c + B*a*c)*(1140855*m + 594439*m^2 + 147859*m^3 + 20065*m^4 + 1525*m^5 ...