Integrand size = 31, antiderivative size = 216 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {a^2 A c^2 (e x)^{1+m}}{e (1+m)}+\frac {a c (a B c+2 A (b c+a d)) (e x)^{3+m}}{e^3 (3+m)}+\frac {\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) (e x)^{5+m}}{e^5 (5+m)}+\frac {\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{7+m}}{e^7 (7+m)}+\frac {b d (2 b B c+A b d+2 a B d) (e x)^{9+m}}{e^9 (9+m)}+\frac {b^2 B d^2 (e x)^{11+m}}{e^{11} (11+m)} \]
a^2*A*c^2*(e*x)^(1+m)/e/(1+m)+a*c*(B*a*c+2*A*(a*d+b*c))*(e*x)^(3+m)/e^3/(3 +m)+(2*a*B*c*(a*d+b*c)+A*(a^2*d^2+4*a*b*c*d+b^2*c^2))*(e*x)^(5+m)/e^5/(5+m )+(a^2*B*d^2+2*a*b*d*(A*d+2*B*c)+b^2*c*(2*A*d+B*c))*(e*x)^(7+m)/e^7/(7+m)+ b*d*(A*b*d+2*B*a*d+2*B*b*c)*(e*x)^(9+m)/e^9/(9+m)+b^2*B*d^2*(e*x)^(11+m)/e ^11/(11+m)
Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.82 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=x (e x)^m \left (\frac {a^2 A c^2}{1+m}+\frac {a c (a B c+2 A (b c+a d)) x^2}{3+m}+\frac {\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^4}{5+m}+\frac {\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^6}{7+m}+\frac {b d (2 b B c+A b d+2 a B d) x^8}{9+m}+\frac {b^2 B d^2 x^{10}}{11+m}\right ) \]
x*(e*x)^m*((a^2*A*c^2)/(1 + m) + (a*c*(a*B*c + 2*A*(b*c + a*d))*x^2)/(3 + m) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^4)/(5 + m) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^6)/(7 + m) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^8)/(9 + m) + (b^2*B*d^2*x^10)/(11 + m))
Time = 0.42 (sec) , antiderivative size = 216, 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.
\(\displaystyle \int \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 (e x)^m \, dx\) |
\(\Big \downarrow \) 437 |
\(\displaystyle \int \left (\frac {(e x)^{m+4} \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{e^4}+\frac {(e x)^{m+6} \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{e^6}+a^2 A c^2 (e x)^m+\frac {b d (e x)^{m+8} (2 a B d+A b d+2 b B c)}{e^8}+\frac {a c (e x)^{m+2} (2 A (a d+b c)+a B c)}{e^2}+\frac {b^2 B d^2 (e x)^{m+10}}{e^{10}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e x)^{m+5} \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{e^5 (m+5)}+\frac {(e x)^{m+7} \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{e^7 (m+7)}+\frac {a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac {b d (e x)^{m+9} (2 a B d+A b d+2 b B c)}{e^9 (m+9)}+\frac {a c (e x)^{m+3} (2 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac {b^2 B d^2 (e x)^{m+11}}{e^{11} (m+11)}\) |
(a^2*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (a*c*(a*B*c + 2*A*(b*c + a*d))*(e* x)^(3 + m))/(e^3*(3 + m)) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*(e*x)^(5 + m))/(e^5*(5 + m)) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*(e*x)^(7 + m))/(e^7*(7 + m)) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*(e*x)^(9 + m))/(e^9*(9 + m)) + (b^2*B*d^2*(e*x)^(11 + m ))/(e^11*(11 + m))
3.1.9.3.1 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. \(1470\) vs. \(2(216)=432\).
Time = 3.48 (sec) , antiderivative size = 1471, normalized size of antiderivative = 6.81
method | result | size |
gosper | \(\text {Expression too large to display}\) | \(1471\) |
risch | \(\text {Expression too large to display}\) | \(1471\) |
parallelrisch | \(\text {Expression too large to display}\) | \(2011\) |
x*(B*b^2*d^2*m^5*x^10+25*B*b^2*d^2*m^4*x^10+A*b^2*d^2*m^5*x^8+2*B*a*b*d^2* m^5*x^8+2*B*b^2*c*d*m^5*x^8+230*B*b^2*d^2*m^3*x^10+27*A*b^2*d^2*m^4*x^8+54 *B*a*b*d^2*m^4*x^8+54*B*b^2*c*d*m^4*x^8+950*B*b^2*d^2*m^2*x^10+2*A*a*b*d^2 *m^5*x^6+2*A*b^2*c*d*m^5*x^6+262*A*b^2*d^2*m^3*x^8+B*a^2*d^2*m^5*x^6+4*B*a *b*c*d*m^5*x^6+524*B*a*b*d^2*m^3*x^8+B*b^2*c^2*m^5*x^6+524*B*b^2*c*d*m^3*x ^8+1689*B*b^2*d^2*m*x^10+58*A*a*b*d^2*m^4*x^6+58*A*b^2*c*d*m^4*x^6+1122*A* b^2*d^2*m^2*x^8+29*B*a^2*d^2*m^4*x^6+116*B*a*b*c*d*m^4*x^6+2244*B*a*b*d^2* m^2*x^8+29*B*b^2*c^2*m^4*x^6+2244*B*b^2*c*d*m^2*x^8+945*B*b^2*d^2*x^10+A*a ^2*d^2*m^5*x^4+4*A*a*b*c*d*m^5*x^4+604*A*a*b*d^2*m^3*x^6+A*b^2*c^2*m^5*x^4 +604*A*b^2*c*d*m^3*x^6+2041*A*b^2*d^2*m*x^8+2*B*a^2*c*d*m^5*x^4+302*B*a^2* d^2*m^3*x^6+2*B*a*b*c^2*m^5*x^4+1208*B*a*b*c*d*m^3*x^6+4082*B*a*b*d^2*m*x^ 8+302*B*b^2*c^2*m^3*x^6+4082*B*b^2*c*d*m*x^8+31*A*a^2*d^2*m^4*x^4+124*A*a* b*c*d*m^4*x^4+2732*A*a*b*d^2*m^2*x^6+31*A*b^2*c^2*m^4*x^4+2732*A*b^2*c*d*m ^2*x^6+1155*A*b^2*d^2*x^8+62*B*a^2*c*d*m^4*x^4+1366*B*a^2*d^2*m^2*x^6+62*B *a*b*c^2*m^4*x^4+5464*B*a*b*c*d*m^2*x^6+2310*B*a*b*d^2*x^8+1366*B*b^2*c^2* m^2*x^6+2310*B*b^2*c*d*x^8+2*A*a^2*c*d*m^5*x^2+350*A*a^2*d^2*m^3*x^4+2*A*a *b*c^2*m^5*x^2+1400*A*a*b*c*d*m^3*x^4+5154*A*a*b*d^2*m*x^6+350*A*b^2*c^2*m ^3*x^4+5154*A*b^2*c*d*m*x^6+B*a^2*c^2*m^5*x^2+700*B*a^2*c*d*m^3*x^4+2577*B *a^2*d^2*m*x^6+700*B*a*b*c^2*m^3*x^4+10308*B*a*b*c*d*m*x^6+2577*B*b^2*c^2* m*x^6+66*A*a^2*c*d*m^4*x^2+1730*A*a^2*d^2*m^2*x^4+66*A*a*b*c^2*m^4*x^2+...
Leaf count of result is larger than twice the leaf count of optimal. 1043 vs. \(2 (216) = 432\).
Time = 0.29 (sec) , antiderivative size = 1043, normalized size of antiderivative = 4.83 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \]
((B*b^2*d^2*m^5 + 25*B*b^2*d^2*m^4 + 230*B*b^2*d^2*m^3 + 950*B*b^2*d^2*m^2 + 1689*B*b^2*d^2*m + 945*B*b^2*d^2)*x^11 + ((2*B*b^2*c*d + (2*B*a*b + A*b ^2)*d^2)*m^5 + 2310*B*b^2*c*d + 27*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m ^4 + 262*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + 1155*(2*B*a*b + A*b^2 )*d^2 + 1122*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2 + 2041*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*x^9 + ((B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^5 + 1485*B*b^2*c^2 + 29*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^4 + 302*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^3 + 2970*(2*B*a*b + A*b^2)*c*d + 148 5*(B*a^2 + 2*A*a*b)*d^2 + 1366*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a ^2 + 2*A*a*b)*d^2)*m^2 + 2577*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^ 2 + 2*A*a*b)*d^2)*m)*x^7 + ((A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^5 + 2079*A*a^2*d^2 + 31*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c ^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^4 + 350*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + 2079*(2*B*a*b + A*b^2)*c^2 + 4158*(B*a^2 + 2*A*a*b)*c*d + 1730*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A *a*b)*c*d)*m^2 + 3489*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A* a*b)*c*d)*m)*x^5 + ((2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^5 + 6930*A*a^2 *c*d + 33*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^4 + 406*(2*A*a^2*c*d + ( B*a^2 + 2*A*a*b)*c^2)*m^3 + 3465*(B*a^2 + 2*A*a*b)*c^2 + 2262*(2*A*a^2*...
Leaf count of result is larger than twice the leaf count of optimal. 6836 vs. \(2 (211) = 422\).
Time = 0.96 (sec) , antiderivative size = 6836, normalized size of antiderivative = 31.65 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \]
Piecewise(((-A*a**2*c**2/(10*x**10) - A*a**2*c*d/(4*x**8) - A*a**2*d**2/(6 *x**6) - A*a*b*c**2/(4*x**8) - 2*A*a*b*c*d/(3*x**6) - A*a*b*d**2/(2*x**4) - A*b**2*c**2/(6*x**6) - A*b**2*c*d/(2*x**4) - A*b**2*d**2/(2*x**2) - B*a* *2*c**2/(8*x**8) - B*a**2*c*d/(3*x**6) - B*a**2*d**2/(4*x**4) - B*a*b*c**2 /(3*x**6) - B*a*b*c*d/x**4 - B*a*b*d**2/x**2 - B*b**2*c**2/(4*x**4) - B*b* *2*c*d/x**2 + B*b**2*d**2*log(x))/e**11, Eq(m, -11)), ((-A*a**2*c**2/(8*x* *8) - A*a**2*c*d/(3*x**6) - A*a**2*d**2/(4*x**4) - A*a*b*c**2/(3*x**6) - A *a*b*c*d/x**4 - A*a*b*d**2/x**2 - A*b**2*c**2/(4*x**4) - A*b**2*c*d/x**2 + A*b**2*d**2*log(x) - B*a**2*c**2/(6*x**6) - B*a**2*c*d/(2*x**4) - B*a**2* d**2/(2*x**2) - B*a*b*c**2/(2*x**4) - 2*B*a*b*c*d/x**2 + 2*B*a*b*d**2*log( x) - B*b**2*c**2/(2*x**2) + 2*B*b**2*c*d*log(x) + B*b**2*d**2*x**2/2)/e**9 , Eq(m, -9)), ((-A*a**2*c**2/(6*x**6) - A*a**2*c*d/(2*x**4) - A*a**2*d**2/ (2*x**2) - A*a*b*c**2/(2*x**4) - 2*A*a*b*c*d/x**2 + 2*A*a*b*d**2*log(x) - A*b**2*c**2/(2*x**2) + 2*A*b**2*c*d*log(x) + A*b**2*d**2*x**2/2 - B*a**2*c **2/(4*x**4) - B*a**2*c*d/x**2 + B*a**2*d**2*log(x) - B*a*b*c**2/x**2 + 4* B*a*b*c*d*log(x) + B*a*b*d**2*x**2 + B*b**2*c**2*log(x) + B*b**2*c*d*x**2 + B*b**2*d**2*x**4/4)/e**7, Eq(m, -7)), ((-A*a**2*c**2/(4*x**4) - A*a**2*c *d/x**2 + A*a**2*d**2*log(x) - A*a*b*c**2/x**2 + 4*A*a*b*c*d*log(x) + A*a* b*d**2*x**2 + A*b**2*c**2*log(x) + A*b**2*c*d*x**2 + A*b**2*d**2*x**4/4 - B*a**2*c**2/(2*x**2) + 2*B*a**2*c*d*log(x) + B*a**2*d**2*x**2/2 + 2*B*a...
Time = 0.25 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.83 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {B b^{2} d^{2} e^{m} x^{11} x^{m}}{m + 11} + \frac {2 \, B b^{2} c d e^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, B a b d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {A b^{2} d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {B b^{2} c^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {4 \, B a b c d e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, A b^{2} c d e^{m} x^{7} x^{m}}{m + 7} + \frac {B a^{2} d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, A a b d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, B a b c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {A b^{2} c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, B a^{2} c d e^{m} x^{5} x^{m}}{m + 5} + \frac {4 \, A a b c d e^{m} x^{5} x^{m}}{m + 5} + \frac {A a^{2} d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{2} c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a b c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a^{2} c d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a^{2} c^{2}}{e {\left (m + 1\right )}} \]
B*b^2*d^2*e^m*x^11*x^m/(m + 11) + 2*B*b^2*c*d*e^m*x^9*x^m/(m + 9) + 2*B*a* b*d^2*e^m*x^9*x^m/(m + 9) + A*b^2*d^2*e^m*x^9*x^m/(m + 9) + B*b^2*c^2*e^m* x^7*x^m/(m + 7) + 4*B*a*b*c*d*e^m*x^7*x^m/(m + 7) + 2*A*b^2*c*d*e^m*x^7*x^ m/(m + 7) + B*a^2*d^2*e^m*x^7*x^m/(m + 7) + 2*A*a*b*d^2*e^m*x^7*x^m/(m + 7 ) + 2*B*a*b*c^2*e^m*x^5*x^m/(m + 5) + A*b^2*c^2*e^m*x^5*x^m/(m + 5) + 2*B* a^2*c*d*e^m*x^5*x^m/(m + 5) + 4*A*a*b*c*d*e^m*x^5*x^m/(m + 5) + A*a^2*d^2* e^m*x^5*x^m/(m + 5) + B*a^2*c^2*e^m*x^3*x^m/(m + 3) + 2*A*a*b*c^2*e^m*x^3* x^m/(m + 3) + 2*A*a^2*c*d*e^m*x^3*x^m/(m + 3) + (e*x)^(m + 1)*A*a^2*c^2/(e *(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 2010 vs. \(2 (216) = 432\).
Time = 0.36 (sec) , antiderivative size = 2010, normalized size of antiderivative = 9.31 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \]
((e*x)^m*B*b^2*d^2*m^5*x^11 + 25*(e*x)^m*B*b^2*d^2*m^4*x^11 + 2*(e*x)^m*B* b^2*c*d*m^5*x^9 + 2*(e*x)^m*B*a*b*d^2*m^5*x^9 + (e*x)^m*A*b^2*d^2*m^5*x^9 + 230*(e*x)^m*B*b^2*d^2*m^3*x^11 + 54*(e*x)^m*B*b^2*c*d*m^4*x^9 + 54*(e*x) ^m*B*a*b*d^2*m^4*x^9 + 27*(e*x)^m*A*b^2*d^2*m^4*x^9 + 950*(e*x)^m*B*b^2*d^ 2*m^2*x^11 + (e*x)^m*B*b^2*c^2*m^5*x^7 + 4*(e*x)^m*B*a*b*c*d*m^5*x^7 + 2*( e*x)^m*A*b^2*c*d*m^5*x^7 + (e*x)^m*B*a^2*d^2*m^5*x^7 + 2*(e*x)^m*A*a*b*d^2 *m^5*x^7 + 524*(e*x)^m*B*b^2*c*d*m^3*x^9 + 524*(e*x)^m*B*a*b*d^2*m^3*x^9 + 262*(e*x)^m*A*b^2*d^2*m^3*x^9 + 1689*(e*x)^m*B*b^2*d^2*m*x^11 + 29*(e*x)^ m*B*b^2*c^2*m^4*x^7 + 116*(e*x)^m*B*a*b*c*d*m^4*x^7 + 58*(e*x)^m*A*b^2*c*d *m^4*x^7 + 29*(e*x)^m*B*a^2*d^2*m^4*x^7 + 58*(e*x)^m*A*a*b*d^2*m^4*x^7 + 2 244*(e*x)^m*B*b^2*c*d*m^2*x^9 + 2244*(e*x)^m*B*a*b*d^2*m^2*x^9 + 1122*(e*x )^m*A*b^2*d^2*m^2*x^9 + 945*(e*x)^m*B*b^2*d^2*x^11 + 2*(e*x)^m*B*a*b*c^2*m ^5*x^5 + (e*x)^m*A*b^2*c^2*m^5*x^5 + 2*(e*x)^m*B*a^2*c*d*m^5*x^5 + 4*(e*x) ^m*A*a*b*c*d*m^5*x^5 + (e*x)^m*A*a^2*d^2*m^5*x^5 + 302*(e*x)^m*B*b^2*c^2*m ^3*x^7 + 1208*(e*x)^m*B*a*b*c*d*m^3*x^7 + 604*(e*x)^m*A*b^2*c*d*m^3*x^7 + 302*(e*x)^m*B*a^2*d^2*m^3*x^7 + 604*(e*x)^m*A*a*b*d^2*m^3*x^7 + 4082*(e*x) ^m*B*b^2*c*d*m*x^9 + 4082*(e*x)^m*B*a*b*d^2*m*x^9 + 2041*(e*x)^m*A*b^2*d^2 *m*x^9 + 62*(e*x)^m*B*a*b*c^2*m^4*x^5 + 31*(e*x)^m*A*b^2*c^2*m^4*x^5 + 62* (e*x)^m*B*a^2*c*d*m^4*x^5 + 124*(e*x)^m*A*a*b*c*d*m^4*x^5 + 31*(e*x)^m*A*a ^2*d^2*m^4*x^5 + 1366*(e*x)^m*B*b^2*c^2*m^2*x^7 + 5464*(e*x)^m*B*a*b*c*...
Time = 5.88 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.31 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {x^5\,{\left (e\,x\right )}^m\,\left (2\,B\,a^2\,c\,d+A\,a^2\,d^2+2\,B\,a\,b\,c^2+4\,A\,a\,b\,c\,d+A\,b^2\,c^2\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {x^7\,{\left (e\,x\right )}^m\,\left (B\,a^2\,d^2+4\,B\,a\,b\,c\,d+2\,A\,a\,b\,d^2+B\,b^2\,c^2+2\,A\,b^2\,c\,d\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a\,c\,x^3\,{\left (e\,x\right )}^m\,\left (2\,A\,a\,d+2\,A\,b\,c+B\,a\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b\,d\,x^9\,{\left (e\,x\right )}^m\,\left (A\,b\,d+2\,B\,a\,d+2\,B\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {A\,a^2\,c^2\,x\,{\left (e\,x\right )}^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {B\,b^2\,d^2\,x^{11}\,{\left (e\,x\right )}^m\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \]
(x^5*(e*x)^m*(A*a^2*d^2 + A*b^2*c^2 + 2*B*a*b*c^2 + 2*B*a^2*c*d + 4*A*a*b* c*d)*(3489*m + 1730*m^2 + 350*m^3 + 31*m^4 + m^5 + 2079))/(19524*m + 12139 *m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (x^7*(e*x)^m*(B*a^2*d^ 2 + B*b^2*c^2 + 2*A*a*b*d^2 + 2*A*b^2*c*d + 4*B*a*b*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) + (a*c*x^3*(e*x)^m*(2*A*a*d + 2*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) + (b*d*x^9*(e*x)^m*(A*b*d + 2*B*a*d + 2*B*b*c)*(2041*m + 1122*m^2 + 262*m^3 + 27*m^4 + m^5 + 1155))/(1 9524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (A*a^2*c ^2*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) + (B*b^2*d^2*x^ 11*(e*x)^m*(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945))/(19524*m + 1 2139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395)