Integrand size = 29, antiderivative size = 189 \[ \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\frac {a A c^3 (e x)^{1+m}}{e (1+m)}+\frac {c^2 (A b c+a B c+3 a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {c (3 a d (B c+A d)+b c (B c+3 A d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {d (3 b c (B c+A d)+a d (3 B c+A d)) (e x)^{7+m}}{e^7 (7+m)}+\frac {d^2 (3 b B c+A b d+a B d) (e x)^{9+m}}{e^9 (9+m)}+\frac {b B d^3 (e x)^{11+m}}{e^{11} (11+m)} \]
a*A*c^3*(e*x)^(1+m)/e/(1+m)+c^2*(3*A*a*d+A*b*c+B*a*c)*(e*x)^(3+m)/e^3/(3+m )+c*(3*a*d*(A*d+B*c)+b*c*(3*A*d+B*c))*(e*x)^(5+m)/e^5/(5+m)+d*(3*b*c*(A*d+ B*c)+a*d*(A*d+3*B*c))*(e*x)^(7+m)/e^7/(7+m)+d^2*(A*b*d+B*a*d+3*B*b*c)*(e*x )^(9+m)/e^9/(9+m)+b*B*d^3*(e*x)^(11+m)/e^11/(11+m)
Time = 0.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=x (e x)^m \left (\frac {a A c^3}{1+m}+\frac {c^2 (A b c+a B c+3 a A d) x^2}{3+m}+\frac {c (3 a d (B c+A d)+b c (B c+3 A d)) x^4}{5+m}+\frac {d (3 b c (B c+A d)+a d (3 B c+A d)) x^6}{7+m}+\frac {d^2 (3 b B c+A b d+a B d) x^8}{9+m}+\frac {b B d^3 x^{10}}{11+m}\right ) \]
x*(e*x)^m*((a*A*c^3)/(1 + m) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*x^2)/(3 + m) + (c*(3*a*d*(B*c + A*d) + b*c*(B*c + 3*A*d))*x^4)/(5 + m) + (d*(3*b*c*(B* c + A*d) + a*d*(3*B*c + A*d))*x^6)/(7 + m) + (d^2*(3*b*B*c + A*b*d + a*B*d )*x^8)/(9 + m) + (b*B*d^3*x^10)/(11 + m))
Time = 0.38 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
| \(\displaystyle \int \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 (e x)^m \, dx\) |
| \(\Big \downarrow \) 437 |
| \(\displaystyle \int \left (\frac {c^2 (e x)^{m+2} (3 a A d+a B c+A b c)}{e^2}+\frac {d^2 (e x)^{m+8} (a B d+A b d+3 b B c)}{e^8}+\frac {d (e x)^{m+6} (a d (A d+3 B c)+3 b c (A d+B c))}{e^6}+\frac {c (e x)^{m+4} (3 a d (A d+B c)+b c (3 A d+B c))}{e^4}+a A c^3 (e x)^m+\frac {b B d^3 (e x)^{m+10}}{e^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^2 (e x)^{m+3} (3 a A d+a B c+A b c)}{e^3 (m+3)}+\frac {d^2 (e x)^{m+9} (a B d+A b d+3 b B c)}{e^9 (m+9)}+\frac {d (e x)^{m+7} (a d (A d+3 B c)+3 b c (A d+B c))}{e^7 (m+7)}+\frac {c (e x)^{m+5} (3 a d (A d+B c)+b c (3 A d+B c))}{e^5 (m+5)}+\frac {a A c^3 (e x)^{m+1}}{e (m+1)}+\frac {b B d^3 (e x)^{m+11}}{e^{11} (m+11)}\) |
(a*A*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*(e*x) ^(3 + m))/(e^3*(3 + m)) + (c*(3*a*d*(B*c + A*d) + b*c*(B*c + 3*A*d))*(e*x) ^(5 + m))/(e^5*(5 + m)) + (d*(3*b*c*(B*c + A*d) + a*d*(3*B*c + A*d))*(e*x) ^(7 + m))/(e^7*(7 + m)) + (d^2*(3*b*B*c + A*b*d + a*B*d)*(e*x)^(9 + m))/(e ^9*(9 + m)) + (b*B*d^3*(e*x)^(11 + m))/(e^11*(11 + m))
3.1.17.3.1 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. \(1228\) vs. \(2(189)=378\).
Time = 3.47 (sec) , antiderivative size = 1229, normalized size of antiderivative = 6.50
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(1229\) |
| risch | \(\text {Expression too large to display}\) | \(1229\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1709\) |
x*(B*b*d^3*m^5*x^10+25*B*b*d^3*m^4*x^10+A*b*d^3*m^5*x^8+B*a*d^3*m^5*x^8+3* B*b*c*d^2*m^5*x^8+230*B*b*d^3*m^3*x^10+27*A*b*d^3*m^4*x^8+27*B*a*d^3*m^4*x ^8+81*B*b*c*d^2*m^4*x^8+950*B*b*d^3*m^2*x^10+A*a*d^3*m^5*x^6+3*A*b*c*d^2*m ^5*x^6+262*A*b*d^3*m^3*x^8+3*B*a*c*d^2*m^5*x^6+262*B*a*d^3*m^3*x^8+3*B*b*c ^2*d*m^5*x^6+786*B*b*c*d^2*m^3*x^8+1689*B*b*d^3*m*x^10+29*A*a*d^3*m^4*x^6+ 87*A*b*c*d^2*m^4*x^6+1122*A*b*d^3*m^2*x^8+87*B*a*c*d^2*m^4*x^6+1122*B*a*d^ 3*m^2*x^8+87*B*b*c^2*d*m^4*x^6+3366*B*b*c*d^2*m^2*x^8+945*B*b*d^3*x^10+3*A *a*c*d^2*m^5*x^4+302*A*a*d^3*m^3*x^6+3*A*b*c^2*d*m^5*x^4+906*A*b*c*d^2*m^3 *x^6+2041*A*b*d^3*m*x^8+3*B*a*c^2*d*m^5*x^4+906*B*a*c*d^2*m^3*x^6+2041*B*a *d^3*m*x^8+B*b*c^3*m^5*x^4+906*B*b*c^2*d*m^3*x^6+6123*B*b*c*d^2*m*x^8+93*A *a*c*d^2*m^4*x^4+1366*A*a*d^3*m^2*x^6+93*A*b*c^2*d*m^4*x^4+4098*A*b*c*d^2* m^2*x^6+1155*A*b*d^3*x^8+93*B*a*c^2*d*m^4*x^4+4098*B*a*c*d^2*m^2*x^6+1155* B*a*d^3*x^8+31*B*b*c^3*m^4*x^4+4098*B*b*c^2*d*m^2*x^6+3465*B*b*c*d^2*x^8+3 *A*a*c^2*d*m^5*x^2+1050*A*a*c*d^2*m^3*x^4+2577*A*a*d^3*m*x^6+A*b*c^3*m^5*x ^2+1050*A*b*c^2*d*m^3*x^4+7731*A*b*c*d^2*m*x^6+B*a*c^3*m^5*x^2+1050*B*a*c^ 2*d*m^3*x^4+7731*B*a*c*d^2*m*x^6+350*B*b*c^3*m^3*x^4+7731*B*b*c^2*d*m*x^6+ 99*A*a*c^2*d*m^4*x^2+5190*A*a*c*d^2*m^2*x^4+1485*A*a*d^3*x^6+33*A*b*c^3*m^ 4*x^2+5190*A*b*c^2*d*m^2*x^4+4455*A*b*c*d^2*x^6+33*B*a*c^3*m^4*x^2+5190*B* a*c^2*d*m^2*x^4+4455*B*a*c*d^2*x^6+1730*B*b*c^3*m^2*x^4+4455*B*b*c^2*d*x^6 +A*a*c^3*m^5+1218*A*a*c^2*d*m^3*x^2+10467*A*a*c*d^2*m*x^4+406*A*b*c^3*m...
Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (189) = 378\).
Time = 0.28 (sec) , antiderivative size = 837, normalized size of antiderivative = 4.43 \[ \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\frac {{\left ({\left (B b d^{3} m^{5} + 25 \, B b d^{3} m^{4} + 230 \, B b d^{3} m^{3} + 950 \, B b d^{3} m^{2} + 1689 \, B b d^{3} m + 945 \, B b d^{3}\right )} x^{11} + {\left ({\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m^{5} + 3465 \, B b c d^{2} + 27 \, {\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m^{4} + 1155 \, {\left (B a + A b\right )} d^{3} + 262 \, {\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m^{3} + 1122 \, {\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m^{2} + 2041 \, {\left (3 \, B b c d^{2} + {\left (B a + A b\right )} d^{3}\right )} m\right )} x^{9} + {\left ({\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m^{5} + 4455 \, B b c^{2} d + 1485 \, A a d^{3} + 29 \, {\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m^{4} + 4455 \, {\left (B a + A b\right )} c d^{2} + 302 \, {\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m^{3} + 1366 \, {\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m^{2} + 2577 \, {\left (3 \, B b c^{2} d + A a d^{3} + 3 \, {\left (B a + A b\right )} c d^{2}\right )} m\right )} x^{7} + {\left ({\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m^{5} + 2079 \, B b c^{3} + 6237 \, A a c d^{2} + 31 \, {\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m^{4} + 6237 \, {\left (B a + A b\right )} c^{2} d + 350 \, {\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m^{3} + 1730 \, {\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m^{2} + 3489 \, {\left (B b c^{3} + 3 \, A a c d^{2} + 3 \, {\left (B a + A b\right )} c^{2} d\right )} m\right )} x^{5} + {\left ({\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m^{5} + 10395 \, A a c^{2} d + 33 \, {\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m^{4} + 3465 \, {\left (B a + A b\right )} c^{3} + 406 \, {\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m^{3} + 2262 \, {\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m^{2} + 5353 \, {\left (3 \, A a c^{2} d + {\left (B a + A b\right )} c^{3}\right )} m\right )} x^{3} + {\left (A a c^{3} m^{5} + 35 \, A a c^{3} m^{4} + 470 \, A a c^{3} m^{3} + 3010 \, A a c^{3} m^{2} + 9129 \, A a c^{3} m + 10395 \, A a c^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \]
((B*b*d^3*m^5 + 25*B*b*d^3*m^4 + 230*B*b*d^3*m^3 + 950*B*b*d^3*m^2 + 1689* B*b*d^3*m + 945*B*b*d^3)*x^11 + ((3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^5 + 346 5*B*b*c*d^2 + 27*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^4 + 1155*(B*a + A*b)*d^ 3 + 262*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^3 + 1122*(3*B*b*c*d^2 + (B*a + A *b)*d^3)*m^2 + 2041*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m)*x^9 + ((3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^5 + 4455*B*b*c^2*d + 1485*A*a*d^3 + 29 *(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^4 + 4455*(B*a + A*b)*c*d^ 2 + 302*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^3 + 1366*(3*B*b*c^ 2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 2577*(3*B*b*c^2*d + A*a*d^3 + 3 *(B*a + A*b)*c*d^2)*m)*x^7 + ((B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d )*m^5 + 2079*B*b*c^3 + 6237*A*a*c*d^2 + 31*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^4 + 6237*(B*a + A*b)*c^2*d + 350*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^3 + 1730*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^ 2*d)*m^2 + 3489*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m)*x^5 + ((3 *A*a*c^2*d + (B*a + A*b)*c^3)*m^5 + 10395*A*a*c^2*d + 33*(3*A*a*c^2*d + (B *a + A*b)*c^3)*m^4 + 3465*(B*a + A*b)*c^3 + 406*(3*A*a*c^2*d + (B*a + A*b) *c^3)*m^3 + 2262*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^2 + 5353*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*x^3 + (A*a*c^3*m^5 + 35*A*a*c^3*m^4 + 470*A*a*c^3*m^3 + 3010*A*a*c^3*m^2 + 9129*A*a*c^3*m + 10395*A*a*c^3)*x)*(e*x)^m/(m^6 + 36 *m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)
Leaf count of result is larger than twice the leaf count of optimal. 5992 vs. \(2 (184) = 368\).
Time = 0.94 (sec) , antiderivative size = 5992, normalized size of antiderivative = 31.70 \[ \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise(((-A*a*c**3/(10*x**10) - 3*A*a*c**2*d/(8*x**8) - A*a*c*d**2/(2*x **6) - A*a*d**3/(4*x**4) - A*b*c**3/(8*x**8) - A*b*c**2*d/(2*x**6) - 3*A*b *c*d**2/(4*x**4) - A*b*d**3/(2*x**2) - B*a*c**3/(8*x**8) - B*a*c**2*d/(2*x **6) - 3*B*a*c*d**2/(4*x**4) - B*a*d**3/(2*x**2) - B*b*c**3/(6*x**6) - 3*B *b*c**2*d/(4*x**4) - 3*B*b*c*d**2/(2*x**2) + B*b*d**3*log(x))/e**11, Eq(m, -11)), ((-A*a*c**3/(8*x**8) - A*a*c**2*d/(2*x**6) - 3*A*a*c*d**2/(4*x**4) - A*a*d**3/(2*x**2) - A*b*c**3/(6*x**6) - 3*A*b*c**2*d/(4*x**4) - 3*A*b*c *d**2/(2*x**2) + A*b*d**3*log(x) - B*a*c**3/(6*x**6) - 3*B*a*c**2*d/(4*x** 4) - 3*B*a*c*d**2/(2*x**2) + B*a*d**3*log(x) - B*b*c**3/(4*x**4) - 3*B*b*c **2*d/(2*x**2) + 3*B*b*c*d**2*log(x) + B*b*d**3*x**2/2)/e**9, Eq(m, -9)), ((-A*a*c**3/(6*x**6) - 3*A*a*c**2*d/(4*x**4) - 3*A*a*c*d**2/(2*x**2) + A*a *d**3*log(x) - A*b*c**3/(4*x**4) - 3*A*b*c**2*d/(2*x**2) + 3*A*b*c*d**2*lo g(x) + A*b*d**3*x**2/2 - B*a*c**3/(4*x**4) - 3*B*a*c**2*d/(2*x**2) + 3*B*a *c*d**2*log(x) + B*a*d**3*x**2/2 - B*b*c**3/(2*x**2) + 3*B*b*c**2*d*log(x) + 3*B*b*c*d**2*x**2/2 + B*b*d**3*x**4/4)/e**7, Eq(m, -7)), ((-A*a*c**3/(4 *x**4) - 3*A*a*c**2*d/(2*x**2) + 3*A*a*c*d**2*log(x) + A*a*d**3*x**2/2 - A *b*c**3/(2*x**2) + 3*A*b*c**2*d*log(x) + 3*A*b*c*d**2*x**2/2 + A*b*d**3*x* *4/4 - B*a*c**3/(2*x**2) + 3*B*a*c**2*d*log(x) + 3*B*a*c*d**2*x**2/2 + B*a *d**3*x**4/4 + B*b*c**3*log(x) + 3*B*b*c**2*d*x**2/2 + 3*B*b*c*d**2*x**4/4 + B*b*d**3*x**6/6)/e**5, Eq(m, -5)), ((-A*a*c**3/(2*x**2) + 3*A*a*c**2...
Time = 0.24 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.79 \[ \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\frac {B b d^{3} e^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, B b c d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {B a d^{3} e^{m} x^{9} x^{m}}{m + 9} + \frac {A b d^{3} e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B b c^{2} d e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a c d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, A b c d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {A a d^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {B b c^{3} e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, B a c^{2} d e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A b c^{2} d e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a c d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B a c^{3} e^{m} x^{3} x^{m}}{m + 3} + \frac {A b c^{3} e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a c^{2} d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a c^{3}}{e {\left (m + 1\right )}} \]
B*b*d^3*e^m*x^11*x^m/(m + 11) + 3*B*b*c*d^2*e^m*x^9*x^m/(m + 9) + B*a*d^3* e^m*x^9*x^m/(m + 9) + A*b*d^3*e^m*x^9*x^m/(m + 9) + 3*B*b*c^2*d*e^m*x^7*x^ m/(m + 7) + 3*B*a*c*d^2*e^m*x^7*x^m/(m + 7) + 3*A*b*c*d^2*e^m*x^7*x^m/(m + 7) + A*a*d^3*e^m*x^7*x^m/(m + 7) + B*b*c^3*e^m*x^5*x^m/(m + 5) + 3*B*a*c^ 2*d*e^m*x^5*x^m/(m + 5) + 3*A*b*c^2*d*e^m*x^5*x^m/(m + 5) + 3*A*a*c*d^2*e^ m*x^5*x^m/(m + 5) + B*a*c^3*e^m*x^3*x^m/(m + 3) + A*b*c^3*e^m*x^3*x^m/(m + 3) + 3*A*a*c^2*d*e^m*x^3*x^m/(m + 3) + (e*x)^(m + 1)*A*a*c^3/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 1708 vs. \(2 (189) = 378\).
Time = 0.35 (sec) , antiderivative size = 1708, normalized size of antiderivative = 9.04 \[ \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\text {Too large to display} \]
((e*x)^m*B*b*d^3*m^5*x^11 + 25*(e*x)^m*B*b*d^3*m^4*x^11 + 3*(e*x)^m*B*b*c* d^2*m^5*x^9 + (e*x)^m*B*a*d^3*m^5*x^9 + (e*x)^m*A*b*d^3*m^5*x^9 + 230*(e*x )^m*B*b*d^3*m^3*x^11 + 81*(e*x)^m*B*b*c*d^2*m^4*x^9 + 27*(e*x)^m*B*a*d^3*m ^4*x^9 + 27*(e*x)^m*A*b*d^3*m^4*x^9 + 950*(e*x)^m*B*b*d^3*m^2*x^11 + 3*(e* x)^m*B*b*c^2*d*m^5*x^7 + 3*(e*x)^m*B*a*c*d^2*m^5*x^7 + 3*(e*x)^m*A*b*c*d^2 *m^5*x^7 + (e*x)^m*A*a*d^3*m^5*x^7 + 786*(e*x)^m*B*b*c*d^2*m^3*x^9 + 262*( e*x)^m*B*a*d^3*m^3*x^9 + 262*(e*x)^m*A*b*d^3*m^3*x^9 + 1689*(e*x)^m*B*b*d^ 3*m*x^11 + 87*(e*x)^m*B*b*c^2*d*m^4*x^7 + 87*(e*x)^m*B*a*c*d^2*m^4*x^7 + 8 7*(e*x)^m*A*b*c*d^2*m^4*x^7 + 29*(e*x)^m*A*a*d^3*m^4*x^7 + 3366*(e*x)^m*B* b*c*d^2*m^2*x^9 + 1122*(e*x)^m*B*a*d^3*m^2*x^9 + 1122*(e*x)^m*A*b*d^3*m^2* x^9 + 945*(e*x)^m*B*b*d^3*x^11 + (e*x)^m*B*b*c^3*m^5*x^5 + 3*(e*x)^m*B*a*c ^2*d*m^5*x^5 + 3*(e*x)^m*A*b*c^2*d*m^5*x^5 + 3*(e*x)^m*A*a*c*d^2*m^5*x^5 + 906*(e*x)^m*B*b*c^2*d*m^3*x^7 + 906*(e*x)^m*B*a*c*d^2*m^3*x^7 + 906*(e*x) ^m*A*b*c*d^2*m^3*x^7 + 302*(e*x)^m*A*a*d^3*m^3*x^7 + 6123*(e*x)^m*B*b*c*d^ 2*m*x^9 + 2041*(e*x)^m*B*a*d^3*m*x^9 + 2041*(e*x)^m*A*b*d^3*m*x^9 + 31*(e* x)^m*B*b*c^3*m^4*x^5 + 93*(e*x)^m*B*a*c^2*d*m^4*x^5 + 93*(e*x)^m*A*b*c^2*d *m^4*x^5 + 93*(e*x)^m*A*a*c*d^2*m^4*x^5 + 4098*(e*x)^m*B*b*c^2*d*m^2*x^7 + 4098*(e*x)^m*B*a*c*d^2*m^2*x^7 + 4098*(e*x)^m*A*b*c*d^2*m^2*x^7 + 1366*(e *x)^m*A*a*d^3*m^2*x^7 + 3465*(e*x)^m*B*b*c*d^2*x^9 + 1155*(e*x)^m*B*a*d^3* x^9 + 1155*(e*x)^m*A*b*d^3*x^9 + (e*x)^m*B*a*c^3*m^5*x^3 + (e*x)^m*A*b*...
Time = 5.91 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.48 \[ \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\frac {c^2\,x^3\,{\left (e\,x\right )}^m\,\left (3\,A\,a\,d+A\,b\,c+B\,a\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {d^2\,x^9\,{\left (e\,x\right )}^m\,\left (A\,b\,d+B\,a\,d+3\,B\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {c\,x^5\,{\left (e\,x\right )}^m\,\left (3\,A\,a\,d^2+B\,b\,c^2+3\,A\,b\,c\,d+3\,B\,a\,c\,d\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {d\,x^7\,{\left (e\,x\right )}^m\,\left (A\,a\,d^2+3\,B\,b\,c^2+3\,A\,b\,c\,d+3\,B\,a\,c\,d\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {B\,b\,d^3\,x^{11}\,{\left (e\,x\right )}^m\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {A\,a\,c^3\,x\,{\left (e\,x\right )}^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \]
(c^2*x^3*(e*x)^m*(3*A*a*d + A*b*c + B*a*c)*(5353*m + 2262*m^2 + 406*m^3 + 33*m^4 + m^5 + 3465))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (d^2*x^9*(e*x)^m*(A*b*d + B*a*d + 3*B*b*c)*(2041*m + 1122* m^2 + 262*m^3 + 27*m^4 + m^5 + 1155))/(19524*m + 12139*m^2 + 3480*m^3 + 50 5*m^4 + 36*m^5 + m^6 + 10395) + (c*x^5*(e*x)^m*(3*A*a*d^2 + B*b*c^2 + 3*A* b*c*d + 3*B*a*c*d)*(3489*m + 1730*m^2 + 350*m^3 + 31*m^4 + m^5 + 2079))/(1 9524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (d*x^7*( e*x)^m*(A*a*d^2 + 3*B*b*c^2 + 3*A*b*c*d + 3*B*a*c*d)*(2577*m + 1366*m^2 + 302*m^3 + 29*m^4 + m^5 + 1485))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (B*b*d^3*x^11*(e*x)^m*(1689*m + 950*m^2 + 230*m^ 3 + 25*m^4 + m^5 + 945))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^ 5 + m^6 + 10395) + (A*a*c^3*x*(e*x)^m*(9129*m + 3010*m^2 + 470*m^3 + 35*m^ 4 + m^5 + 10395))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395)