\(\int (e x)^m (a+b x^2)^3 (A+B x^2) (c+d x^2) \, dx\) [1]

Optimal result

Integrand size = 29, antiderivative size = 189 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {a^3 A c (e x)^{1+m}}{e (1+m)}+\frac {a^2 (3 A b c+a B c+a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) (e x)^{7+m}}{e^7 (7+m)}+\frac {b^2 (b B c+A b d+3 a B d) (e x)^{9+m}}{e^9 (9+m)}+\frac {b^3 B d (e x)^{11+m}}{e^{11} (11+m)} \] Output:

a^3*A*c*(e*x)^(1+m)/e/(1+m)+a^2*(A*a*d+3*A*b*c+B*a*c)*(e*x)^(3+m)/e^3/(3+m 
)+a*(3*A*b*(a*d+b*c)+a*B*(a*d+3*b*c))*(e*x)^(5+m)/e^5/(5+m)+b*(3*a*B*(a*d+ 
b*c)+A*b*(3*a*d+b*c))*(e*x)^(7+m)/e^7/(7+m)+b^2*(A*b*d+3*B*a*d+B*b*c)*(e*x 
)^(9+m)/e^9/(9+m)+b^3*B*d*(e*x)^(11+m)/e^11/(11+m)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=x (e x)^m \left (\frac {a^3 A c}{1+m}+\frac {a^2 (3 A b c+a B c+a A d) x^2}{3+m}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) x^4}{5+m}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) x^6}{7+m}+\frac {b^2 (b B c+A b d+3 a B d) x^8}{9+m}+\frac {b^3 B d x^{10}}{11+m}\right ) \] Input:

Integrate[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2),x]
 

Output:

x*(e*x)^m*((a^3*A*c)/(1 + m) + (a^2*(3*A*b*c + a*B*c + a*A*d)*x^2)/(3 + m) 
 + (a*(3*A*b*(b*c + a*d) + a*B*(3*b*c + a*d))*x^4)/(5 + m) + (b*(3*a*B*(b* 
c + a*d) + A*b*(b*c + 3*a*d))*x^6)/(7 + m) + (b^2*(b*B*c + A*b*d + 3*a*B*d 
)*x^8)/(9 + m) + (b^3*B*d*x^10)/(11 + m))
 

Rubi [A] (verified)

Time = 0.55 (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.

\(\displaystyle \int \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) (e x)^m \, dx\)

\(\Big \downarrow \) 437

\(\displaystyle \int \left (a^3 A c (e x)^m+\frac {a^2 (e x)^{m+2} (a A d+a B c+3 A b c)}{e^2}+\frac {b^2 (e x)^{m+8} (3 a B d+A b d+b B c)}{e^8}+\frac {b (e x)^{m+6} (A b (3 a d+b c)+3 a B (a d+b c))}{e^6}+\frac {a (e x)^{m+4} (3 A b (a d+b c)+a B (a d+3 b c))}{e^4}+\frac {b^3 B d (e x)^{m+10}}{e^{10}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 A c (e x)^{m+1}}{e (m+1)}+\frac {a^2 (e x)^{m+3} (a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac {b^2 (e x)^{m+9} (3 a B d+A b d+b B c)}{e^9 (m+9)}+\frac {b (e x)^{m+7} (A b (3 a d+b c)+3 a B (a d+b c))}{e^7 (m+7)}+\frac {a (e x)^{m+5} (3 A b (a d+b c)+a B (a d+3 b c))}{e^5 (m+5)}+\frac {b^3 B d (e x)^{m+11}}{e^{11} (m+11)}\)

Input:

Int[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2),x]
 

Output:

(a^3*A*c*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*(3*A*b*c + a*B*c + a*A*d)*(e*x) 
^(3 + m))/(e^3*(3 + m)) + (a*(3*A*b*(b*c + a*d) + a*B*(3*b*c + a*d))*(e*x) 
^(5 + m))/(e^5*(5 + m)) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*(e*x) 
^(7 + m))/(e^7*(7 + m)) + (b^2*(b*B*c + A*b*d + 3*a*B*d)*(e*x)^(9 + m))/(e 
^9*(9 + m)) + (b^3*B*d*(e*x)^(11 + m))/(e^11*(11 + m))
 

Defintions of rubi rules used

rule 437

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1228\) vs. \(2(189)=378\).

Time = 0.51 (sec) , antiderivative size = 1229, normalized size of antiderivative = 6.50

method	result	size
gosper	\(\text {Expression too large to display}\)	\(1229\)
risch	\(\text {Expression too large to display}\)	\(1229\)
orering	\(\text {Expression too large to display}\)	\(1229\)
parallelrisch	\(\text {Expression too large to display}\)	\(1709\)

Input:

int((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

x*(B*b^3*d*m^5*x^10+25*B*b^3*d*m^4*x^10+A*b^3*d*m^5*x^8+3*B*a*b^2*d*m^5*x^ 
8+B*b^3*c*m^5*x^8+230*B*b^3*d*m^3*x^10+27*A*b^3*d*m^4*x^8+81*B*a*b^2*d*m^4 
*x^8+27*B*b^3*c*m^4*x^8+950*B*b^3*d*m^2*x^10+3*A*a*b^2*d*m^5*x^6+A*b^3*c*m 
^5*x^6+262*A*b^3*d*m^3*x^8+3*B*a^2*b*d*m^5*x^6+3*B*a*b^2*c*m^5*x^6+786*B*a 
*b^2*d*m^3*x^8+262*B*b^3*c*m^3*x^8+1689*B*b^3*d*m*x^10+87*A*a*b^2*d*m^4*x^ 
6+29*A*b^3*c*m^4*x^6+1122*A*b^3*d*m^2*x^8+87*B*a^2*b*d*m^4*x^6+87*B*a*b^2* 
c*m^4*x^6+3366*B*a*b^2*d*m^2*x^8+1122*B*b^3*c*m^2*x^8+945*B*b^3*d*x^10+3*A 
*a^2*b*d*m^5*x^4+3*A*a*b^2*c*m^5*x^4+906*A*a*b^2*d*m^3*x^6+302*A*b^3*c*m^3 
*x^6+2041*A*b^3*d*m*x^8+B*a^3*d*m^5*x^4+3*B*a^2*b*c*m^5*x^4+906*B*a^2*b*d* 
m^3*x^6+906*B*a*b^2*c*m^3*x^6+6123*B*a*b^2*d*m*x^8+2041*B*b^3*c*m*x^8+93*A 
*a^2*b*d*m^4*x^4+93*A*a*b^2*c*m^4*x^4+4098*A*a*b^2*d*m^2*x^6+1366*A*b^3*c* 
m^2*x^6+1155*A*b^3*d*x^8+31*B*a^3*d*m^4*x^4+93*B*a^2*b*c*m^4*x^4+4098*B*a^ 
2*b*d*m^2*x^6+4098*B*a*b^2*c*m^2*x^6+3465*B*a*b^2*d*x^8+1155*B*b^3*c*x^8+A 
*a^3*d*m^5*x^2+3*A*a^2*b*c*m^5*x^2+1050*A*a^2*b*d*m^3*x^4+1050*A*a*b^2*c*m 
^3*x^4+7731*A*a*b^2*d*m*x^6+2577*A*b^3*c*m*x^6+B*a^3*c*m^5*x^2+350*B*a^3*d 
*m^3*x^4+1050*B*a^2*b*c*m^3*x^4+7731*B*a^2*b*d*m*x^6+7731*B*a*b^2*c*m*x^6+ 
33*A*a^3*d*m^4*x^2+99*A*a^2*b*c*m^4*x^2+5190*A*a^2*b*d*m^2*x^4+5190*A*a*b^ 
2*c*m^2*x^4+4455*A*a*b^2*d*x^6+1485*A*b^3*c*x^6+33*B*a^3*c*m^4*x^2+1730*B* 
a^3*d*m^2*x^4+5190*B*a^2*b*c*m^2*x^4+4455*B*a^2*b*d*x^6+4455*B*a*b^2*c*x^6 
+A*a^3*c*m^5+406*A*a^3*d*m^3*x^2+1218*A*a^2*b*c*m^3*x^2+10467*A*a^2*b*d...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 911 vs. \(2 (189) = 378\).

Time = 0.12 (sec) , antiderivative size = 911, normalized size of antiderivative = 4.82 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x, algorithm="fricas")
 

Output:

((B*b^3*d*m^5 + 25*B*b^3*d*m^4 + 230*B*b^3*d*m^3 + 950*B*b^3*d*m^2 + 1689* 
B*b^3*d*m + 945*B*b^3*d)*x^11 + ((B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^5 + 1 
155*B*b^3*c + 27*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^4 + 262*(B*b^3*c + (3 
*B*a*b^2 + A*b^3)*d)*m^3 + 1122*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + 11 
55*(3*B*a*b^2 + A*b^3)*d + 2041*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*x^9 + 
 (((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^5 + 29*((3*B*a*b^2 + 
 A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^4 + 302*((3*B*a*b^2 + A*b^3)*c + 3* 
(B*a^2*b + A*a*b^2)*d)*m^3 + 1366*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A* 
a*b^2)*d)*m^2 + 1485*(3*B*a*b^2 + A*b^3)*c + 4455*(B*a^2*b + A*a*b^2)*d + 
2577*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m)*x^7 + ((3*(B*a^2 
*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^5 + 31*(3*(B*a^2*b + A*a*b^2)*c 
 + (B*a^3 + 3*A*a^2*b)*d)*m^4 + 350*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3* 
A*a^2*b)*d)*m^3 + 1730*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m 
^2 + 6237*(B*a^2*b + A*a*b^2)*c + 2079*(B*a^3 + 3*A*a^2*b)*d + 3489*(3*(B* 
a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m)*x^5 + ((A*a^3*d + (B*a^3 + 
3*A*a^2*b)*c)*m^5 + 3465*A*a^3*d + 33*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^ 
4 + 406*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^3 + 2262*(A*a^3*d + (B*a^3 + 3 
*A*a^2*b)*c)*m^2 + 3465*(B*a^3 + 3*A*a^2*b)*c + 5353*(A*a^3*d + (B*a^3 + 3 
*A*a^2*b)*c)*m)*x^3 + (A*a^3*c*m^5 + 35*A*a^3*c*m^4 + 470*A*a^3*c*m^3 + 30 
10*A*a^3*c*m^2 + 9129*A*a^3*c*m + 10395*A*a^3*c)*x)*(e*x)^m/(m^6 + 36*m...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5992 vs. \(2 (184) = 368\).

Time = 0.92 (sec) , antiderivative size = 5992, normalized size of antiderivative = 31.70 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(b*x**2+a)**3*(B*x**2+A)*(d*x**2+c),x)
 

Output:

Piecewise(((-A*a**3*c/(10*x**10) - A*a**3*d/(8*x**8) - 3*A*a**2*b*c/(8*x** 
8) - A*a**2*b*d/(2*x**6) - A*a*b**2*c/(2*x**6) - 3*A*a*b**2*d/(4*x**4) - A 
*b**3*c/(4*x**4) - A*b**3*d/(2*x**2) - B*a**3*c/(8*x**8) - B*a**3*d/(6*x** 
6) - B*a**2*b*c/(2*x**6) - 3*B*a**2*b*d/(4*x**4) - 3*B*a*b**2*c/(4*x**4) - 
 3*B*a*b**2*d/(2*x**2) - B*b**3*c/(2*x**2) + B*b**3*d*log(x))/e**11, Eq(m, 
 -11)), ((-A*a**3*c/(8*x**8) - A*a**3*d/(6*x**6) - A*a**2*b*c/(2*x**6) - 3 
*A*a**2*b*d/(4*x**4) - 3*A*a*b**2*c/(4*x**4) - 3*A*a*b**2*d/(2*x**2) - A*b 
**3*c/(2*x**2) + A*b**3*d*log(x) - B*a**3*c/(6*x**6) - B*a**3*d/(4*x**4) - 
 3*B*a**2*b*c/(4*x**4) - 3*B*a**2*b*d/(2*x**2) - 3*B*a*b**2*c/(2*x**2) + 3 
*B*a*b**2*d*log(x) + B*b**3*c*log(x) + B*b**3*d*x**2/2)/e**9, Eq(m, -9)), 
((-A*a**3*c/(6*x**6) - A*a**3*d/(4*x**4) - 3*A*a**2*b*c/(4*x**4) - 3*A*a** 
2*b*d/(2*x**2) - 3*A*a*b**2*c/(2*x**2) + 3*A*a*b**2*d*log(x) + A*b**3*c*lo 
g(x) + A*b**3*d*x**2/2 - B*a**3*c/(4*x**4) - B*a**3*d/(2*x**2) - 3*B*a**2* 
b*c/(2*x**2) + 3*B*a**2*b*d*log(x) + 3*B*a*b**2*c*log(x) + 3*B*a*b**2*d*x* 
*2/2 + B*b**3*c*x**2/2 + B*b**3*d*x**4/4)/e**7, Eq(m, -7)), ((-A*a**3*c/(4 
*x**4) - A*a**3*d/(2*x**2) - 3*A*a**2*b*c/(2*x**2) + 3*A*a**2*b*d*log(x) + 
 3*A*a*b**2*c*log(x) + 3*A*a*b**2*d*x**2/2 + A*b**3*c*x**2/2 + A*b**3*d*x* 
*4/4 - B*a**3*c/(2*x**2) + B*a**3*d*log(x) + 3*B*a**2*b*c*log(x) + 3*B*a** 
2*b*d*x**2/2 + 3*B*a*b**2*c*x**2/2 + 3*B*a*b**2*d*x**4/4 + B*b**3*c*x**4/4 
 + B*b**3*d*x**6/6)/e**5, Eq(m, -5)), ((-A*a**3*c/(2*x**2) + A*a**3*d*l...
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.79 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {B b^{3} d e^{m} x^{11} x^{m}}{m + 11} + \frac {B b^{3} c e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B a b^{2} d e^{m} x^{9} x^{m}}{m + 9} + \frac {A b^{3} d e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B a b^{2} c e^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{3} c e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a^{2} b d e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, A a b^{2} d e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a^{2} b c e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a b^{2} c e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{3} d e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a^{2} b d e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{3} c e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{2} b c e^{m} x^{3} x^{m}}{m + 3} + \frac {A a^{3} d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a^{3} c}{e {\left (m + 1\right )}} \] Input:

integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x, algorithm="maxima")
 

Output:

B*b^3*d*e^m*x^11*x^m/(m + 11) + B*b^3*c*e^m*x^9*x^m/(m + 9) + 3*B*a*b^2*d* 
e^m*x^9*x^m/(m + 9) + A*b^3*d*e^m*x^9*x^m/(m + 9) + 3*B*a*b^2*c*e^m*x^7*x^ 
m/(m + 7) + A*b^3*c*e^m*x^7*x^m/(m + 7) + 3*B*a^2*b*d*e^m*x^7*x^m/(m + 7) 
+ 3*A*a*b^2*d*e^m*x^7*x^m/(m + 7) + 3*B*a^2*b*c*e^m*x^5*x^m/(m + 5) + 3*A* 
a*b^2*c*e^m*x^5*x^m/(m + 5) + B*a^3*d*e^m*x^5*x^m/(m + 5) + 3*A*a^2*b*d*e^ 
m*x^5*x^m/(m + 5) + B*a^3*c*e^m*x^3*x^m/(m + 3) + 3*A*a^2*b*c*e^m*x^3*x^m/ 
(m + 3) + A*a^3*d*e^m*x^3*x^m/(m + 3) + (e*x)^(m + 1)*A*a^3*c/(e*(m + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1708 vs. \(2 (189) = 378\).

Time = 0.16 (sec) , antiderivative size = 1708, normalized size of antiderivative = 9.04 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\text {Too large to display} \] Input:

integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x, algorithm="giac")
 

Output:

((e*x)^m*B*b^3*d*m^5*x^11 + 25*(e*x)^m*B*b^3*d*m^4*x^11 + (e*x)^m*B*b^3*c* 
m^5*x^9 + 3*(e*x)^m*B*a*b^2*d*m^5*x^9 + (e*x)^m*A*b^3*d*m^5*x^9 + 230*(e*x 
)^m*B*b^3*d*m^3*x^11 + 27*(e*x)^m*B*b^3*c*m^4*x^9 + 81*(e*x)^m*B*a*b^2*d*m 
^4*x^9 + 27*(e*x)^m*A*b^3*d*m^4*x^9 + 950*(e*x)^m*B*b^3*d*m^2*x^11 + 3*(e* 
x)^m*B*a*b^2*c*m^5*x^7 + (e*x)^m*A*b^3*c*m^5*x^7 + 3*(e*x)^m*B*a^2*b*d*m^5 
*x^7 + 3*(e*x)^m*A*a*b^2*d*m^5*x^7 + 262*(e*x)^m*B*b^3*c*m^3*x^9 + 786*(e* 
x)^m*B*a*b^2*d*m^3*x^9 + 262*(e*x)^m*A*b^3*d*m^3*x^9 + 1689*(e*x)^m*B*b^3* 
d*m*x^11 + 87*(e*x)^m*B*a*b^2*c*m^4*x^7 + 29*(e*x)^m*A*b^3*c*m^4*x^7 + 87* 
(e*x)^m*B*a^2*b*d*m^4*x^7 + 87*(e*x)^m*A*a*b^2*d*m^4*x^7 + 1122*(e*x)^m*B* 
b^3*c*m^2*x^9 + 3366*(e*x)^m*B*a*b^2*d*m^2*x^9 + 1122*(e*x)^m*A*b^3*d*m^2* 
x^9 + 945*(e*x)^m*B*b^3*d*x^11 + 3*(e*x)^m*B*a^2*b*c*m^5*x^5 + 3*(e*x)^m*A 
*a*b^2*c*m^5*x^5 + (e*x)^m*B*a^3*d*m^5*x^5 + 3*(e*x)^m*A*a^2*b*d*m^5*x^5 + 
 906*(e*x)^m*B*a*b^2*c*m^3*x^7 + 302*(e*x)^m*A*b^3*c*m^3*x^7 + 906*(e*x)^m 
*B*a^2*b*d*m^3*x^7 + 906*(e*x)^m*A*a*b^2*d*m^3*x^7 + 2041*(e*x)^m*B*b^3*c* 
m*x^9 + 6123*(e*x)^m*B*a*b^2*d*m*x^9 + 2041*(e*x)^m*A*b^3*d*m*x^9 + 93*(e* 
x)^m*B*a^2*b*c*m^4*x^5 + 93*(e*x)^m*A*a*b^2*c*m^4*x^5 + 31*(e*x)^m*B*a^3*d 
*m^4*x^5 + 93*(e*x)^m*A*a^2*b*d*m^4*x^5 + 4098*(e*x)^m*B*a*b^2*c*m^2*x^7 + 
 1366*(e*x)^m*A*b^3*c*m^2*x^7 + 4098*(e*x)^m*B*a^2*b*d*m^2*x^7 + 4098*(e*x 
)^m*A*a*b^2*d*m^2*x^7 + 1155*(e*x)^m*B*b^3*c*x^9 + 3465*(e*x)^m*B*a*b^2*d* 
x^9 + 1155*(e*x)^m*A*b^3*d*x^9 + (e*x)^m*B*a^3*c*m^5*x^3 + 3*(e*x)^m*A*...
 

Mupad [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.48 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {a^2\,x^3\,{\left (e\,x\right )}^m\,\left (A\,a\,d+3\,A\,b\,c+B\,a\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b^2\,x^9\,{\left (e\,x\right )}^m\,\left (A\,b\,d+3\,B\,a\,d+B\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a\,x^5\,{\left (e\,x\right )}^m\,\left (3\,A\,b^2\,c+B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b\,x^7\,{\left (e\,x\right )}^m\,\left (A\,b^2\,c+3\,B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {B\,b^3\,d\,x^{11}\,{\left (e\,x\right )}^m\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {A\,a^3\,c\,x\,{\left (e\,x\right )}^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \] Input:

int((A + B*x^2)*(e*x)^m*(a + b*x^2)^3*(c + d*x^2),x)
 

Output:

(a^2*x^3*(e*x)^m*(A*a*d + 3*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^2*x^9*(e*x)^m*(A*b*d + 3*B*a*d + 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) + (a*x^5*(e*x)^m*(3*A*b^2*c + B*a^2*d + 3*A* 
a*b*d + 3*B*a*b*c)*(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) + (b*x^7*( 
e*x)^m*(A*b^2*c + 3*B*a^2*d + 3*A*a*b*d + 3*B*a*b*c)*(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^3*d*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^3*c*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)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 747, normalized size of antiderivative = 3.95 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {x^{m} e^{m} x \left (b^{4} d \,m^{5} x^{10}+25 b^{4} d \,m^{4} x^{10}+4 a \,b^{3} d \,m^{5} x^{8}+b^{4} c \,m^{5} x^{8}+230 b^{4} d \,m^{3} x^{10}+108 a \,b^{3} d \,m^{4} x^{8}+27 b^{4} c \,m^{4} x^{8}+950 b^{4} d \,m^{2} x^{10}+6 a^{2} b^{2} d \,m^{5} x^{6}+4 a \,b^{3} c \,m^{5} x^{6}+1048 a \,b^{3} d \,m^{3} x^{8}+262 b^{4} c \,m^{3} x^{8}+1689 b^{4} d m \,x^{10}+174 a^{2} b^{2} d \,m^{4} x^{6}+116 a \,b^{3} c \,m^{4} x^{6}+4488 a \,b^{3} d \,m^{2} x^{8}+1122 b^{4} c \,m^{2} x^{8}+945 b^{4} d \,x^{10}+4 a^{3} b d \,m^{5} x^{4}+6 a^{2} b^{2} c \,m^{5} x^{4}+1812 a^{2} b^{2} d \,m^{3} x^{6}+1208 a \,b^{3} c \,m^{3} x^{6}+8164 a \,b^{3} d m \,x^{8}+2041 b^{4} c m \,x^{8}+124 a^{3} b d \,m^{4} x^{4}+186 a^{2} b^{2} c \,m^{4} x^{4}+8196 a^{2} b^{2} d \,m^{2} x^{6}+5464 a \,b^{3} c \,m^{2} x^{6}+4620 a \,b^{3} d \,x^{8}+1155 b^{4} c \,x^{8}+a^{4} d \,m^{5} x^{2}+4 a^{3} b c \,m^{5} x^{2}+1400 a^{3} b d \,m^{3} x^{4}+2100 a^{2} b^{2} c \,m^{3} x^{4}+15462 a^{2} b^{2} d m \,x^{6}+10308 a \,b^{3} c m \,x^{6}+33 a^{4} d \,m^{4} x^{2}+132 a^{3} b c \,m^{4} x^{2}+6920 a^{3} b d \,m^{2} x^{4}+10380 a^{2} b^{2} c \,m^{2} x^{4}+8910 a^{2} b^{2} d \,x^{6}+5940 a \,b^{3} c \,x^{6}+a^{4} c \,m^{5}+406 a^{4} d \,m^{3} x^{2}+1624 a^{3} b c \,m^{3} x^{2}+13956 a^{3} b d m \,x^{4}+20934 a^{2} b^{2} c m \,x^{4}+35 a^{4} c \,m^{4}+2262 a^{4} d \,m^{2} x^{2}+9048 a^{3} b c \,m^{2} x^{2}+8316 a^{3} b d \,x^{4}+12474 a^{2} b^{2} c \,x^{4}+470 a^{4} c \,m^{3}+5353 a^{4} d m \,x^{2}+21412 a^{3} b c m \,x^{2}+3010 a^{4} c \,m^{2}+3465 a^{4} d \,x^{2}+13860 a^{3} b c \,x^{2}+9129 a^{4} c m +10395 a^{4} c \right )}{m^{6}+36 m^{5}+505 m^{4}+3480 m^{3}+12139 m^{2}+19524 m +10395} \] Input:

int((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x)
 

Output:

(x**m*e**m*x*(a**4*c*m**5 + 35*a**4*c*m**4 + 470*a**4*c*m**3 + 3010*a**4*c 
*m**2 + 9129*a**4*c*m + 10395*a**4*c + a**4*d*m**5*x**2 + 33*a**4*d*m**4*x 
**2 + 406*a**4*d*m**3*x**2 + 2262*a**4*d*m**2*x**2 + 5353*a**4*d*m*x**2 + 
3465*a**4*d*x**2 + 4*a**3*b*c*m**5*x**2 + 132*a**3*b*c*m**4*x**2 + 1624*a* 
*3*b*c*m**3*x**2 + 9048*a**3*b*c*m**2*x**2 + 21412*a**3*b*c*m*x**2 + 13860 
*a**3*b*c*x**2 + 4*a**3*b*d*m**5*x**4 + 124*a**3*b*d*m**4*x**4 + 1400*a**3 
*b*d*m**3*x**4 + 6920*a**3*b*d*m**2*x**4 + 13956*a**3*b*d*m*x**4 + 8316*a* 
*3*b*d*x**4 + 6*a**2*b**2*c*m**5*x**4 + 186*a**2*b**2*c*m**4*x**4 + 2100*a 
**2*b**2*c*m**3*x**4 + 10380*a**2*b**2*c*m**2*x**4 + 20934*a**2*b**2*c*m*x 
**4 + 12474*a**2*b**2*c*x**4 + 6*a**2*b**2*d*m**5*x**6 + 174*a**2*b**2*d*m 
**4*x**6 + 1812*a**2*b**2*d*m**3*x**6 + 8196*a**2*b**2*d*m**2*x**6 + 15462 
*a**2*b**2*d*m*x**6 + 8910*a**2*b**2*d*x**6 + 4*a*b**3*c*m**5*x**6 + 116*a 
*b**3*c*m**4*x**6 + 1208*a*b**3*c*m**3*x**6 + 5464*a*b**3*c*m**2*x**6 + 10 
308*a*b**3*c*m*x**6 + 5940*a*b**3*c*x**6 + 4*a*b**3*d*m**5*x**8 + 108*a*b* 
*3*d*m**4*x**8 + 1048*a*b**3*d*m**3*x**8 + 4488*a*b**3*d*m**2*x**8 + 8164* 
a*b**3*d*m*x**8 + 4620*a*b**3*d*x**8 + b**4*c*m**5*x**8 + 27*b**4*c*m**4*x 
**8 + 262*b**4*c*m**3*x**8 + 1122*b**4*c*m**2*x**8 + 2041*b**4*c*m*x**8 + 
1155*b**4*c*x**8 + b**4*d*m**5*x**10 + 25*b**4*d*m**4*x**10 + 230*b**4*d*m 
**3*x**10 + 950*b**4*d*m**2*x**10 + 1689*b**4*d*m*x**10 + 945*b**4*d*x**10 
))/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039...