\(\int (A+B x) (d+e x)^m (a+c x^2)^3 \, dx\) [295]

Optimal result

Integrand size = 22, antiderivative size = 372 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^{2+m}}{e^8 (2+m)}-\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{3+m}}{e^8 (3+m)}-\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}-\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{5+m}}{e^8 (5+m)}+\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac {c^3 (7 B d-A e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac {B c^3 (d+e x)^{8+m}}{e^8 (8+m)} \] Output:

-(-A*e+B*d)*(a*e^2+c*d^2)^3*(e*x+d)^(1+m)/e^8/(1+m)+(a*e^2+c*d^2)^2*(-6*A* 
c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(2+m)/e^8/(2+m)-3*c*(a*e^2+c*d^2)*(-A*a*e 
^3-5*A*c*d^2*e+3*B*a*d*e^2+7*B*c*d^3)*(e*x+d)^(3+m)/e^8/(3+m)-c*(4*A*c*d*e 
*(3*a*e^2+5*c*d^2)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))*(e*x+d)^(4+m)/ 
e^8/(4+m)-c^2*(-3*A*a*e^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)*(e*x+d)^(5 
+m)/e^8/(5+m)+3*c^2*(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(6+m)/e^8/(6+m) 
-c^3*(-A*e+7*B*d)*(e*x+d)^(7+m)/e^8/(7+m)+B*c^3*(e*x+d)^(8+m)/e^8/(8+m)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(773\) vs. \(2(372)=744\).

Time = 1.75 (sec) , antiderivative size = 773, normalized size of antiderivative = 2.08 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left (-\left ((B d-A e) (8+m) \left (e^6 (1+m) (2+m) (3+m) (4+m) (5+m) (6+m) \left (a+c x^2\right )^3+6 \left (c d^2+a e^2\right ) (6+m) \left (e^4 (1+m) (2+m) (3+m) (4+m) \left (a+c x^2\right )^2+4 \left (c d^2+a e^2\right ) (4+m) \left (a e^2 \left (6+5 m+m^2\right )+c \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )\right )-4 c d (1+m) (d+e x) \left (a e^2 \left (12+7 m+m^2\right )+c \left (2 d^2-2 d e (2+m) x+e^2 \left (6+5 m+m^2\right ) x^2\right )\right )\right )-6 c d (1+m) (d+e x) \left (e^4 (2+m) (3+m) (4+m) (5+m) \left (a+c x^2\right )^2+4 \left (c d^2+a e^2\right ) (5+m) \left (a e^2 \left (12+7 m+m^2\right )+c \left (2 d^2-2 d e (2+m) x+e^2 \left (6+5 m+m^2\right ) x^2\right )\right )-4 c d (2+m) (d+e x) \left (a e^2 \left (20+9 m+m^2\right )+c \left (2 d^2-2 d e (3+m) x+e^2 \left (12+7 m+m^2\right ) x^2\right )\right )\right )\right )\right )+B (1+m) (d+e x) \left (e^6 (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \left (a+c x^2\right )^3+6 \left (c d^2+a e^2\right ) (7+m) \left (e^4 (2+m) (3+m) (4+m) (5+m) \left (a+c x^2\right )^2+4 \left (c d^2+a e^2\right ) (5+m) \left (a e^2 \left (12+7 m+m^2\right )+c \left (2 d^2-2 d e (2+m) x+e^2 \left (6+5 m+m^2\right ) x^2\right )\right )-4 c d (2+m) (d+e x) \left (a e^2 \left (20+9 m+m^2\right )+c \left (2 d^2-2 d e (3+m) x+e^2 \left (12+7 m+m^2\right ) x^2\right )\right )\right )-6 c d (2+m) (d+e x) \left (e^4 (3+m) (4+m) (5+m) (6+m) \left (a+c x^2\right )^2+4 \left (c d^2+a e^2\right ) (6+m) \left (a e^2 \left (20+9 m+m^2\right )+c \left (2 d^2-2 d e (3+m) x+e^2 \left (12+7 m+m^2\right ) x^2\right )\right )-4 c d (3+m) (d+e x) \left (a e^2 \left (30+11 m+m^2\right )+c \left (2 d^2-2 d e (4+m) x+e^2 \left (20+9 m+m^2\right ) x^2\right )\right )\right )\right )\right )}{e^8 (1+m) (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) (8+m)} \] Input:

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

Output:

((d + e*x)^(1 + m)*(-((B*d - A*e)*(8 + m)*(e^6*(1 + m)*(2 + m)*(3 + m)*(4 
+ m)*(5 + m)*(6 + m)*(a + c*x^2)^3 + 6*(c*d^2 + a*e^2)*(6 + m)*(e^4*(1 + m 
)*(2 + m)*(3 + m)*(4 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(4 + m)*(a*e^2 
*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)) 
- 4*c*d*(1 + m)*(d + e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + 
m)*x + e^2*(6 + 5*m + m^2)*x^2))) - 6*c*d*(1 + m)*(d + e*x)*(e^4*(2 + m)*( 
3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(5 + m)*(a*e^2*(1 
2 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)) - 
4*c*d*(2 + m)*(d + e*x)*(a*e^2*(20 + 9*m + m^2) + c*(2*d^2 - 2*d*e*(3 + m) 
*x + e^2*(12 + 7*m + m^2)*x^2))))) + B*(1 + m)*(d + e*x)*(e^6*(2 + m)*(3 + 
 m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(a + c*x^2)^3 + 6*(c*d^2 + a*e^2)*(7 + 
 m)*(e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2) 
*(5 + m)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5 
*m + m^2)*x^2)) - 4*c*d*(2 + m)*(d + e*x)*(a*e^2*(20 + 9*m + m^2) + c*(2*d 
^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2))) - 6*c*d*(2 + m)*(d + e* 
x)*(e^4*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)* 
(6 + m)*(a*e^2*(20 + 9*m + m^2) + c*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7 
*m + m^2)*x^2)) - 4*c*d*(3 + m)*(d + e*x)*(a*e^2*(30 + 11*m + m^2) + c*(2* 
d^2 - 2*d*e*(4 + m)*x + e^2*(20 + 9*m + m^2)*x^2))))))/(e^8*(1 + m)*(2 + m 
)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 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.

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

\(\Big \downarrow \) 652

\(\displaystyle \int \left (-\frac {c (d+e x)^{m+3} \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7}-\frac {3 c^2 (d+e x)^{m+5} \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7}+\frac {c^2 (d+e x)^{m+4} \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d) (d+e x)^m}{e^7}+\frac {\left (a e^2+c d^2\right )^2 (d+e x)^{m+1} \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7}+\frac {3 c \left (a e^2+c d^2\right ) (d+e x)^{m+2} \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7}+\frac {c^3 (A e-7 B d) (d+e x)^{m+6}}{e^7}+\frac {B c^3 (d+e x)^{m+7}}{e^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c (d+e x)^{m+4} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 (m+4)}+\frac {3 c^2 (d+e x)^{m+6} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (m+6)}-\frac {c^2 (d+e x)^{m+5} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (m+5)}-\frac {\left (a e^2+c d^2\right )^3 (B d-A e) (d+e x)^{m+1}}{e^8 (m+1)}+\frac {\left (a e^2+c d^2\right )^2 (d+e x)^{m+2} \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 (m+2)}-\frac {3 c \left (a e^2+c d^2\right ) (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (m+3)}-\frac {c^3 (7 B d-A e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac {B c^3 (d+e x)^{m+8}}{e^8 (m+8)}\)

Input:

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

Output:

-(((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + ((c*d 
^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d + e*x)^(2 + m))/(e^8*(2 
 + m)) - (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A 
*e^3)*(d + e*x)^(3 + m))/(e^8*(3 + m)) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) 
 - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^(4 + m))/(e^8*(4 
 + m)) - (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + 
e*x)^(5 + m))/(e^8*(5 + m)) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d 
+ e*x)^(6 + m))/(e^8*(6 + m)) - (c^3*(7*B*d - A*e)*(d + e*x)^(7 + m))/(e^8 
*(7 + m)) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(8 + m))
 

Defintions of rubi rules used

rule 652

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ 
)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c 
*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 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. \(2311\) vs. \(2(372)=744\).

Time = 0.86 (sec) , antiderivative size = 2312, normalized size of antiderivative = 6.22

method	result	size
norman	\(\text {Expression too large to display}\)	\(2312\)
gosper	\(\text {Expression too large to display}\)	\(3176\)
orering	\(\text {Expression too large to display}\)	\(3179\)
risch	\(\text {Expression too large to display}\)	\(3864\)
parallelrisch	\(\text {Expression too large to display}\)	\(5537\)

Input:

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

Output:

B*c^3/(8+m)*x^8*exp(m*ln(e*x+d))+d*(A*a^3*e^7*m^7+35*A*a^3*e^7*m^6-B*a^3*d 
*e^6*m^6+511*A*a^3*e^7*m^5+6*A*a^2*c*d^2*e^5*m^5-33*B*a^3*d*e^6*m^5+4025*A 
*a^3*e^7*m^4+180*A*a^2*c*d^2*e^5*m^4-445*B*a^3*d*e^6*m^4-18*B*a^2*c*d^3*e^ 
4*m^4+18424*A*a^3*e^7*m^3+2130*A*a^2*c*d^2*e^5*m^3+72*A*a*c^2*d^4*e^3*m^3- 
3135*B*a^3*d*e^6*m^3-468*B*a^2*c*d^3*e^4*m^3+48860*A*a^3*e^7*m^2+12420*A*a 
^2*c*d^2*e^5*m^2+1512*A*a*c^2*d^4*e^3*m^2-12154*B*a^3*d*e^6*m^2-4518*B*a^2 
*c*d^3*e^4*m^2-360*B*a*c^2*d^5*e^2*m^2+69264*A*a^3*e^7*m+35664*A*a^2*c*d^2 
*e^5*m+10512*A*a*c^2*d^4*e^3*m+720*A*c^3*d^6*e*m-24552*B*a^3*d*e^6*m-19188 
*B*a^2*c*d^3*e^4*m-5400*B*a*c^2*d^5*e^2*m+40320*A*a^3*e^7+40320*A*a^2*c*d^ 
2*e^5+24192*A*a*c^2*d^4*e^3+5760*A*c^3*d^6*e-20160*B*a^3*d*e^6-30240*B*a^2 
*c*d^3*e^4-20160*B*a*c^2*d^5*e^2-5040*B*c^3*d^7)/e^8/(m^8+36*m^7+546*m^6+4 
536*m^5+22449*m^4+67284*m^3+118124*m^2+109584*m+40320)*exp(m*ln(e*x+d))+(3 
*A*a^2*c*d*e^5*m^6+B*a^3*e^6*m^6+90*A*a^2*c*d*e^5*m^5+33*B*a^3*e^6*m^5-9*B 
*a^2*c*d^2*e^4*m^5+1065*A*a^2*c*d*e^5*m^4+36*A*a*c^2*d^3*e^3*m^4+445*B*a^3 
*e^6*m^4-234*B*a^2*c*d^2*e^4*m^4+6210*A*a^2*c*d*e^5*m^3+756*A*a*c^2*d^3*e^ 
3*m^3+3135*B*a^3*e^6*m^3-2259*B*a^2*c*d^2*e^4*m^3-180*B*a*c^2*d^4*e^2*m^3+ 
17832*A*a^2*c*d*e^5*m^2+5256*A*a*c^2*d^3*e^3*m^2+360*A*c^3*d^5*e*m^2+12154 
*B*a^3*e^6*m^2-9594*B*a^2*c*d^2*e^4*m^2-2700*B*a*c^2*d^4*e^2*m^2+20160*A*a 
^2*c*d*e^5*m+12096*A*a*c^2*d^3*e^3*m+2880*A*c^3*d^5*e*m+24552*B*a^3*e^6*m- 
15120*B*a^2*c*d^2*e^4*m-10080*B*a*c^2*d^4*e^2*m-2520*B*c^3*d^6*m+20160*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3116 vs. \(2 (372) = 744\).

Time = 0.18 (sec) , antiderivative size = 3116, normalized size of antiderivative = 8.38 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46038 vs. \(2 (366) = 732\).

Time = 11.93 (sec) , antiderivative size = 46038, normalized size of antiderivative = 123.76 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((d**m*(A*a**3*x + A*a**2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c**3*x** 
7/7 + B*a**3*x**2/2 + 3*B*a**2*c*x**4/4 + B*a*c**2*x**6/2 + B*c**3*x**8/8) 
, Eq(e, 0)), (-60*A*a**3*e**7/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d** 
5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e 
**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 12*A*a**2*c*d**2*e**5/(4 
20*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11* 
x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 
420*e**15*x**7) - 84*A*a**2*c*d*e**6*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 
 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 88 
20*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 252*A*a**2*c*e* 
*7*x**2/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d 
**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e** 
14*x**6 + 420*e**15*x**7) - 12*A*a*c**2*d**4*e**3/(420*d**7*e**8 + 2940*d* 
*6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**1 
2*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 84*A 
*a*c**2*d**3*e**4*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x* 
*2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 
+ 2940*d*e**14*x**6 + 420*e**15*x**7) - 252*A*a*c**2*d**2*e**5*x**2/(420*d 
**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 
 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 4...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1104 vs. \(2 (372) = 744\).

Time = 0.09 (sec) , antiderivative size = 1104, normalized size of antiderivative = 2.97 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a^3/((m^2 + 3*m + 2)*e^2) 
+ (e*x + d)^(m + 1)*A*a^3/(e*(m + 1)) + 3*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 
+ m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*a^2*c/((m^3 + 6*m^2 + 
11*m + 6)*e^3) + 3*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m) 
*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*B* 
a^2*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 3*((m^4 + 10*m^3 + 35*m^ 
2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 
 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d 
^5)*(e*x + d)^m*A*a*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e 
^5) + 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 
10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m 
)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2* 
x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*B*a*c^2/((m^6 + 21*m^5 + 175*m^ 
4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6) + ((m^6 + 21*m^5 + 175*m^4 + 7 
35*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m 
^3 + 274*m^2 + 120*m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m 
)*d^2*e^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3 
*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720* 
d^7)*(e*x + d)^m*A*c^3/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13 
132*m^2 + 13068*m + 5040)*e^7) + ((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5562 vs. \(2 (372) = 744\).

Time = 0.15 (sec) , antiderivative size = 5562, normalized size of antiderivative = 14.95 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

((e*x + d)^m*B*c^3*e^8*m^7*x^8 + (e*x + d)^m*B*c^3*d*e^7*m^7*x^7 + (e*x + 
d)^m*A*c^3*e^8*m^7*x^7 + 28*(e*x + d)^m*B*c^3*e^8*m^6*x^8 + (e*x + d)^m*A* 
c^3*d*e^7*m^7*x^6 + 3*(e*x + d)^m*B*a*c^2*e^8*m^7*x^6 + 21*(e*x + d)^m*B*c 
^3*d*e^7*m^6*x^7 + 29*(e*x + d)^m*A*c^3*e^8*m^6*x^7 + 322*(e*x + d)^m*B*c^ 
3*e^8*m^5*x^8 + 3*(e*x + d)^m*B*a*c^2*d*e^7*m^7*x^5 + 3*(e*x + d)^m*A*a*c^ 
2*e^8*m^7*x^5 - 7*(e*x + d)^m*B*c^3*d^2*e^6*m^6*x^6 + 23*(e*x + d)^m*A*c^3 
*d*e^7*m^6*x^6 + 90*(e*x + d)^m*B*a*c^2*e^8*m^6*x^6 + 175*(e*x + d)^m*B*c^ 
3*d*e^7*m^5*x^7 + 343*(e*x + d)^m*A*c^3*e^8*m^5*x^7 + 1960*(e*x + d)^m*B*c 
^3*e^8*m^4*x^8 + 3*(e*x + d)^m*A*a*c^2*d*e^7*m^7*x^4 + 3*(e*x + d)^m*B*a^2 
*c*e^8*m^7*x^4 - 6*(e*x + d)^m*A*c^3*d^2*e^6*m^6*x^5 + 75*(e*x + d)^m*B*a* 
c^2*d*e^7*m^6*x^5 + 93*(e*x + d)^m*A*a*c^2*e^8*m^6*x^5 - 105*(e*x + d)^m*B 
*c^3*d^2*e^6*m^5*x^6 + 205*(e*x + d)^m*A*c^3*d*e^7*m^5*x^6 + 1098*(e*x + d 
)^m*B*a*c^2*e^8*m^5*x^6 + 735*(e*x + d)^m*B*c^3*d*e^7*m^4*x^7 + 2135*(e*x 
+ d)^m*A*c^3*e^8*m^4*x^7 + 6769*(e*x + d)^m*B*c^3*e^8*m^3*x^8 + 3*(e*x + d 
)^m*B*a^2*c*d*e^7*m^7*x^3 + 3*(e*x + d)^m*A*a^2*c*e^8*m^7*x^3 - 15*(e*x + 
d)^m*B*a*c^2*d^2*e^6*m^6*x^4 + 81*(e*x + d)^m*A*a*c^2*d*e^7*m^6*x^4 + 96*( 
e*x + d)^m*B*a^2*c*e^8*m^6*x^4 + 42*(e*x + d)^m*B*c^3*d^3*e^5*m^5*x^5 - 10 
8*(e*x + d)^m*A*c^3*d^2*e^6*m^5*x^5 + 723*(e*x + d)^m*B*a*c^2*d*e^7*m^5*x^ 
5 + 1173*(e*x + d)^m*A*a*c^2*e^8*m^5*x^5 - 595*(e*x + d)^m*B*c^3*d^2*e^6*m 
^4*x^6 + 905*(e*x + d)^m*A*c^3*d*e^7*m^4*x^6 + 7020*(e*x + d)^m*B*a*c^2...
 

Mupad [B] (verification not implemented)

Time = 6.93 (sec) , antiderivative size = 2585, normalized size of antiderivative = 6.95 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

((d + e*x)^m*(40320*A*a^3*d*e^7 - 5040*B*c^3*d^8 + 5760*A*c^3*d^7*e - 2016 
0*B*a^3*d^2*e^6 + 24192*A*a*c^2*d^5*e^3 + 40320*A*a^2*c*d^3*e^5 - 20160*B* 
a*c^2*d^6*e^2 - 30240*B*a^2*c*d^4*e^4 + 48860*A*a^3*d*e^7*m^2 + 18424*A*a^ 
3*d*e^7*m^3 + 4025*A*a^3*d*e^7*m^4 + 511*A*a^3*d*e^7*m^5 + 35*A*a^3*d*e^7* 
m^6 + A*a^3*d*e^7*m^7 - 24552*B*a^3*d^2*e^6*m - 12154*B*a^3*d^2*e^6*m^2 - 
3135*B*a^3*d^2*e^6*m^3 - 445*B*a^3*d^2*e^6*m^4 - 33*B*a^3*d^2*e^6*m^5 - B* 
a^3*d^2*e^6*m^6 + 69264*A*a^3*d*e^7*m + 720*A*c^3*d^7*e*m + 1512*A*a*c^2*d 
^5*e^3*m^2 + 12420*A*a^2*c*d^3*e^5*m^2 + 72*A*a*c^2*d^5*e^3*m^3 + 2130*A*a 
^2*c*d^3*e^5*m^3 + 180*A*a^2*c*d^3*e^5*m^4 + 6*A*a^2*c*d^3*e^5*m^5 - 360*B 
*a*c^2*d^6*e^2*m^2 - 4518*B*a^2*c*d^4*e^4*m^2 - 468*B*a^2*c*d^4*e^4*m^3 - 
18*B*a^2*c*d^4*e^4*m^4 + 10512*A*a*c^2*d^5*e^3*m + 35664*A*a^2*c*d^3*e^5*m 
 - 5400*B*a*c^2*d^6*e^2*m - 19188*B*a^2*c*d^4*e^4*m))/(e^8*(109584*m + 118 
124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 4032 
0)) + (x*(d + e*x)^m*(40320*A*a^3*e^8 + 69264*A*a^3*e^8*m + 48860*A*a^3*e^ 
8*m^2 + 18424*A*a^3*e^8*m^3 + 4025*A*a^3*e^8*m^4 + 511*A*a^3*e^8*m^5 + 35* 
A*a^3*e^8*m^6 + A*a^3*e^8*m^7 + 24552*B*a^3*d*e^7*m^2 + 12154*B*a^3*d*e^7* 
m^3 + 3135*B*a^3*d*e^7*m^4 + 445*B*a^3*d*e^7*m^5 + 33*B*a^3*d*e^7*m^6 + B* 
a^3*d*e^7*m^7 - 5760*A*c^3*d^6*e^2*m - 720*A*c^3*d^6*e^2*m^2 + 20160*B*a^3 
*d*e^7*m + 5040*B*c^3*d^7*e*m - 10512*A*a*c^2*d^4*e^4*m^2 - 35664*A*a^2*c* 
d^2*e^6*m^2 - 1512*A*a*c^2*d^4*e^4*m^3 - 12420*A*a^2*c*d^2*e^6*m^3 - 72...
 

Reduce [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 3854, normalized size of antiderivative = 10.36 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((d + e*x)**m*(a**4*d*e**7*m**7 + 35*a**4*d*e**7*m**6 + 511*a**4*d*e**7*m* 
*5 + 4025*a**4*d*e**7*m**4 + 18424*a**4*d*e**7*m**3 + 48860*a**4*d*e**7*m* 
*2 + 69264*a**4*d*e**7*m + 40320*a**4*d*e**7 + a**4*e**8*m**7*x + 35*a**4* 
e**8*m**6*x + 511*a**4*e**8*m**5*x + 4025*a**4*e**8*m**4*x + 18424*a**4*e* 
*8*m**3*x + 48860*a**4*e**8*m**2*x + 69264*a**4*e**8*m*x + 40320*a**4*e**8 
*x - a**3*b*d**2*e**6*m**6 - 33*a**3*b*d**2*e**6*m**5 - 445*a**3*b*d**2*e* 
*6*m**4 - 3135*a**3*b*d**2*e**6*m**3 - 12154*a**3*b*d**2*e**6*m**2 - 24552 
*a**3*b*d**2*e**6*m - 20160*a**3*b*d**2*e**6 + a**3*b*d*e**7*m**7*x + 33*a 
**3*b*d*e**7*m**6*x + 445*a**3*b*d*e**7*m**5*x + 3135*a**3*b*d*e**7*m**4*x 
 + 12154*a**3*b*d*e**7*m**3*x + 24552*a**3*b*d*e**7*m**2*x + 20160*a**3*b* 
d*e**7*m*x + a**3*b*e**8*m**7*x**2 + 34*a**3*b*e**8*m**6*x**2 + 478*a**3*b 
*e**8*m**5*x**2 + 3580*a**3*b*e**8*m**4*x**2 + 15289*a**3*b*e**8*m**3*x**2 
 + 36706*a**3*b*e**8*m**2*x**2 + 44712*a**3*b*e**8*m*x**2 + 20160*a**3*b*e 
**8*x**2 + 6*a**3*c*d**3*e**5*m**5 + 180*a**3*c*d**3*e**5*m**4 + 2130*a**3 
*c*d**3*e**5*m**3 + 12420*a**3*c*d**3*e**5*m**2 + 35664*a**3*c*d**3*e**5*m 
 + 40320*a**3*c*d**3*e**5 - 6*a**3*c*d**2*e**6*m**6*x - 180*a**3*c*d**2*e* 
*6*m**5*x - 2130*a**3*c*d**2*e**6*m**4*x - 12420*a**3*c*d**2*e**6*m**3*x - 
 35664*a**3*c*d**2*e**6*m**2*x - 40320*a**3*c*d**2*e**6*m*x + 3*a**3*c*d*e 
**7*m**7*x**2 + 93*a**3*c*d*e**7*m**6*x**2 + 1155*a**3*c*d*e**7*m**5*x**2 
+ 7275*a**3*c*d*e**7*m**4*x**2 + 24042*a**3*c*d*e**7*m**3*x**2 + 37992*...