\(\int (A+B x) (d+e x)^m (a+c x^2)^2 \, dx\) [296]

Optimal result

Integrand size = 22, antiderivative size = 234 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {c^2 (5 B d-A e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {B c^2 (d+e x)^{6+m}}{e^6 (6+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.52 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (-\left ((B d-A e) (6+m) \left (e^4 (1+m) (2+m) (3+m) (4+m) \left (a+c x^2\right )^2+4 \left (c d^2+a e^2\right ) (4+m) \left (a e^2 \left (6+5 m+m^2\right )+c \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )\right )-4 c d (1+m) (d+e x) \left (a e^2 \left (12+7 m+m^2\right )+c \left (2 d^2-2 d e (2+m) x+e^2 \left (6+5 m+m^2\right ) x^2\right )\right )\right )\right )+B (1+m) (d+e x) \left (e^4 (2+m) (3+m) (4+m) (5+m) \left (a+c x^2\right )^2+4 \left (c d^2+a e^2\right ) (5+m) \left (a e^2 \left (12+7 m+m^2\right )+c \left (2 d^2-2 d e (2+m) x+e^2 \left (6+5 m+m^2\right ) x^2\right )\right )-4 c d (2+m) (d+e x) \left (a e^2 \left (20+9 m+m^2\right )+c \left (2 d^2-2 d e (3+m) x+e^2 \left (12+7 m+m^2\right ) x^2\right )\right )\right )\right )}{e^6 (1+m) (2+m) (3+m) (4+m) (5+m) (6+m)} \] Input:

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

Output:

((d + e*x)^(1 + m)*(-((B*d - A*e)*(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)))) + B*(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*(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)))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 234, 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.

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

\(\Big \downarrow \) 652

\(\displaystyle \int \left (\frac {\left (a e^2+c d^2\right )^2 (A e-B d) (d+e x)^m}{e^5}+\frac {\left (a e^2+c d^2\right ) (d+e x)^{m+1} \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5}-\frac {2 c (d+e x)^{m+3} \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5}+\frac {2 c (d+e x)^{m+2} \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5}+\frac {c^2 (A e-5 B d) (d+e x)^{m+4}}{e^5}+\frac {B c^2 (d+e x)^{m+5}}{e^5}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(((B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + ((c*d 
^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2)*(d + e*x)^(2 + m))/(e^6*(2 + 
 m)) - (2*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3 
 + m))/(e^6*(3 + m)) + (2*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(4 
 + m))/(e^6*(4 + m)) - (c^2*(5*B*d - A*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) 
 + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + 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. \(1083\) vs. \(2(234)=468\).

Time = 0.96 (sec) , antiderivative size = 1084, normalized size of antiderivative = 4.63

method	result	size
norman	\(\text {Expression too large to display}\)	\(1084\)
gosper	\(\text {Expression too large to display}\)	\(1271\)
orering	\(\text {Expression too large to display}\)	\(1274\)
risch	\(\text {Expression too large to display}\)	\(1639\)
parallelrisch	\(\text {Expression too large to display}\)	\(2517\)

Input:

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

Output:

B*c^2/(6+m)*x^6*exp(m*ln(e*x+d))+d*(A*a^2*e^5*m^5+20*A*a^2*e^5*m^4-B*a^2*d 
*e^4*m^4+155*A*a^2*e^5*m^3+4*A*a*c*d^2*e^3*m^3-18*B*a^2*d*e^4*m^3+580*A*a^ 
2*e^5*m^2+60*A*a*c*d^2*e^3*m^2-119*B*a^2*d*e^4*m^2-12*B*a*c*d^3*e^2*m^2+10 
44*A*a^2*e^5*m+296*A*a*c*d^2*e^3*m+24*A*c^2*d^4*e*m-342*B*a^2*d*e^4*m-132* 
B*a*c*d^3*e^2*m+720*A*a^2*e^5+480*A*a*c*d^2*e^3+144*A*c^2*d^4*e-360*B*a^2* 
d*e^4-360*B*a*c*d^3*e^2-120*B*c^2*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+162 
4*m^2+1764*m+720)*exp(m*ln(e*x+d))+(2*A*a*c*d*e^3*m^4+B*a^2*e^4*m^4+30*A*a 
*c*d*e^3*m^3+18*B*a^2*e^4*m^3-6*B*a*c*d^2*e^2*m^3+148*A*a*c*d*e^3*m^2+12*A 
*c^2*d^3*e*m^2+119*B*a^2*e^4*m^2-66*B*a*c*d^2*e^2*m^2+240*A*a*c*d*e^3*m+72 
*A*c^2*d^3*e*m+342*B*a^2*e^4*m-180*B*a*c*d^2*e^2*m-60*B*c^2*d^4*m+360*B*a^ 
2*e^4)/e^4/(m^5+20*m^4+155*m^3+580*m^2+1044*m+720)*x^2*exp(m*ln(e*x+d))+(A 
*a^2*e^5*m^5+B*a^2*d*e^4*m^5+20*A*a^2*e^5*m^4-4*A*a*c*d^2*e^3*m^4+18*B*a^2 
*d*e^4*m^4+155*A*a^2*e^5*m^3-60*A*a*c*d^2*e^3*m^3+119*B*a^2*d*e^4*m^3+12*B 
*a*c*d^3*e^2*m^3+580*A*a^2*e^5*m^2-296*A*a*c*d^2*e^3*m^2-24*A*c^2*d^4*e*m^ 
2+342*B*a^2*d*e^4*m^2+132*B*a*c*d^3*e^2*m^2+1044*A*a^2*e^5*m-480*A*a*c*d^2 
*e^3*m-144*A*c^2*d^4*e*m+360*B*a^2*d*e^4*m+360*B*a*c*d^3*e^2*m+120*B*c^2*d 
^5*m+720*A*a^2*e^5)/e^5/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)*x 
*exp(m*ln(e*x+d))+c*(A*c*d*e*m^2+2*B*a*e^2*m^2+6*A*c*d*e*m+22*B*a*e^2*m-5* 
B*c*d^2*m+60*B*a*e^2)/e^2/(m^3+15*m^2+74*m+120)*x^4*exp(m*ln(e*x+d))+(A*e* 
m+B*d*m+6*A*e)*c^2/e/(m^2+11*m+30)*x^5*exp(m*ln(e*x+d))+2*(A*a*e^3*m^3+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1373 vs. \(2 (234) = 468\).

Time = 0.11 (sec) , antiderivative size = 1373, normalized size of antiderivative = 5.87 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

(A*a^2*d*e^5*m^5 - 120*B*c^2*d^6 + 144*A*c^2*d^5*e - 360*B*a*c*d^4*e^2 + 4 
80*A*a*c*d^3*e^3 - 360*B*a^2*d^2*e^4 + 720*A*a^2*d*e^5 + (B*c^2*e^6*m^5 + 
15*B*c^2*e^6*m^4 + 85*B*c^2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*m 
+ 120*B*c^2*e^6)*x^6 + (144*A*c^2*e^6 + (B*c^2*d*e^5 + A*c^2*e^6)*m^5 + 2* 
(5*B*c^2*d*e^5 + 8*A*c^2*e^6)*m^4 + 5*(7*B*c^2*d*e^5 + 19*A*c^2*e^6)*m^3 + 
 10*(5*B*c^2*d*e^5 + 26*A*c^2*e^6)*m^2 + 12*(2*B*c^2*d*e^5 + 27*A*c^2*e^6) 
*m)*x^5 - (B*a^2*d^2*e^4 - 20*A*a^2*d*e^5)*m^4 + (360*B*a*c*e^6 + (A*c^2*d 
*e^5 + 2*B*a*c*e^6)*m^5 - (5*B*c^2*d^2*e^4 - 12*A*c^2*d*e^5 - 34*B*a*c*e^6 
)*m^4 - (30*B*c^2*d^2*e^4 - 47*A*c^2*d*e^5 - 214*B*a*c*e^6)*m^3 - (55*B*c^ 
2*d^2*e^4 - 72*A*c^2*d*e^5 - 614*B*a*c*e^6)*m^2 - 6*(5*B*c^2*d^2*e^4 - 6*A 
*c^2*d*e^5 - 132*B*a*c*e^6)*m)*x^4 + (4*A*a*c*d^3*e^3 - 18*B*a^2*d^2*e^4 + 
 155*A*a^2*d*e^5)*m^3 + 2*(240*A*a*c*e^6 + (B*a*c*d*e^5 + A*a*c*e^6)*m^5 - 
 2*(A*c^2*d^2*e^4 - 7*B*a*c*d*e^5 - 9*A*a*c*e^6)*m^4 + (10*B*c^2*d^3*e^3 - 
 18*A*c^2*d^2*e^4 + 65*B*a*c*d*e^5 + 121*A*a*c*e^6)*m^3 + 2*(15*B*c^2*d^3* 
e^3 - 20*A*c^2*d^2*e^4 + 56*B*a*c*d*e^5 + 186*A*a*c*e^6)*m^2 + 4*(5*B*c^2* 
d^3*e^3 - 6*A*c^2*d^2*e^4 + 15*B*a*c*d*e^5 + 127*A*a*c*e^6)*m)*x^3 - (12*B 
*a*c*d^4*e^2 - 60*A*a*c*d^3*e^3 + 119*B*a^2*d^2*e^4 - 580*A*a^2*d*e^5)*m^2 
 + (360*B*a^2*e^6 + (2*A*a*c*d*e^5 + B*a^2*e^6)*m^5 - (6*B*a*c*d^2*e^4 - 3 
2*A*a*c*d*e^5 - 19*B*a^2*e^6)*m^4 + (12*A*c^2*d^3*e^3 - 72*B*a*c*d^2*e^4 + 
 178*A*a*c*d*e^5 + 137*B*a^2*e^6)*m^3 - (60*B*c^2*d^4*e^2 - 84*A*c^2*d^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16458 vs. \(2 (226) = 452\).

Time = 4.08 (sec) , antiderivative size = 16458, normalized size of antiderivative = 70.33 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((d**m*(A*a**2*x + 2*A*a*c*x**3/3 + A*c**2*x**5/5 + B*a**2*x**2/2 
 + B*a*c*x**4/2 + B*c**2*x**6/6), Eq(e, 0)), (-12*A*a**2*e**5/(60*d**5*e** 
6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**1 
0*x**4 + 60*e**11*x**5) - 4*A*a*c*d**2*e**3/(60*d**5*e**6 + 300*d**4*e**7* 
x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11* 
x**5) - 20*A*a*c*d*e**4*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8* 
x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 40*A*a*c*e 
**5*x**2/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e 
**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 12*A*c**2*d**4*e/(60*d**5*e 
**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e* 
*10*x**4 + 60*e**11*x**5) - 60*A*c**2*d**3*e**2*x/(60*d**5*e**6 + 300*d**4 
*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60* 
e**11*x**5) - 120*A*c**2*d**2*e**3*x**2/(60*d**5*e**6 + 300*d**4*e**7*x + 
600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5 
) - 120*A*c**2*d*e**4*x**3/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8 
*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 60*A*c**2 
*e**5*x**4/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2 
*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 3*B*a**2*d*e**4/(60*d**5* 
e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e 
**10*x**4 + 60*e**11*x**5) - 15*B*a**2*e**5*x/(60*d**5*e**6 + 300*d**4*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (234) = 468\).

Time = 0.08 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.46 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a^{2}}{e {\left (m + 1\right )}} + \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A a c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B a c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} A c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} B c^{2}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} \] Input:

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

Output:

(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a^2/((m^2 + 3*m + 2)*e^2) 
+ (e*x + d)^(m + 1)*A*a^2/(e*(m + 1)) + 2*((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*c/((m^3 + 6*m^2 + 11 
*m + 6)*e^3) + 2*((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* 
c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((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*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + (( 
m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 3 
5*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*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 
+ 1624*m^2 + 1764*m + 720)*e^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2435 vs. \(2 (234) = 468\).

Time = 0.13 (sec) , antiderivative size = 2435, normalized size of antiderivative = 10.41 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

((e*x + d)^m*B*c^2*e^6*m^5*x^6 + (e*x + d)^m*B*c^2*d*e^5*m^5*x^5 + (e*x + 
d)^m*A*c^2*e^6*m^5*x^5 + 15*(e*x + d)^m*B*c^2*e^6*m^4*x^6 + (e*x + d)^m*A* 
c^2*d*e^5*m^5*x^4 + 2*(e*x + d)^m*B*a*c*e^6*m^5*x^4 + 10*(e*x + d)^m*B*c^2 
*d*e^5*m^4*x^5 + 16*(e*x + d)^m*A*c^2*e^6*m^4*x^5 + 85*(e*x + d)^m*B*c^2*e 
^6*m^3*x^6 + 2*(e*x + d)^m*B*a*c*d*e^5*m^5*x^3 + 2*(e*x + d)^m*A*a*c*e^6*m 
^5*x^3 - 5*(e*x + d)^m*B*c^2*d^2*e^4*m^4*x^4 + 12*(e*x + d)^m*A*c^2*d*e^5* 
m^4*x^4 + 34*(e*x + d)^m*B*a*c*e^6*m^4*x^4 + 35*(e*x + d)^m*B*c^2*d*e^5*m^ 
3*x^5 + 95*(e*x + d)^m*A*c^2*e^6*m^3*x^5 + 225*(e*x + d)^m*B*c^2*e^6*m^2*x 
^6 + 2*(e*x + d)^m*A*a*c*d*e^5*m^5*x^2 + (e*x + d)^m*B*a^2*e^6*m^5*x^2 - 4 
*(e*x + d)^m*A*c^2*d^2*e^4*m^4*x^3 + 28*(e*x + d)^m*B*a*c*d*e^5*m^4*x^3 + 
36*(e*x + d)^m*A*a*c*e^6*m^4*x^3 - 30*(e*x + d)^m*B*c^2*d^2*e^4*m^3*x^4 + 
47*(e*x + d)^m*A*c^2*d*e^5*m^3*x^4 + 214*(e*x + d)^m*B*a*c*e^6*m^3*x^4 + 5 
0*(e*x + d)^m*B*c^2*d*e^5*m^2*x^5 + 260*(e*x + d)^m*A*c^2*e^6*m^2*x^5 + 27 
4*(e*x + d)^m*B*c^2*e^6*m*x^6 + (e*x + d)^m*B*a^2*d*e^5*m^5*x + (e*x + d)^ 
m*A*a^2*e^6*m^5*x - 6*(e*x + d)^m*B*a*c*d^2*e^4*m^4*x^2 + 32*(e*x + d)^m*A 
*a*c*d*e^5*m^4*x^2 + 19*(e*x + d)^m*B*a^2*e^6*m^4*x^2 + 20*(e*x + d)^m*B*c 
^2*d^3*e^3*m^3*x^3 - 36*(e*x + d)^m*A*c^2*d^2*e^4*m^3*x^3 + 130*(e*x + d)^ 
m*B*a*c*d*e^5*m^3*x^3 + 242*(e*x + d)^m*A*a*c*e^6*m^3*x^3 - 55*(e*x + d)^m 
*B*c^2*d^2*e^4*m^2*x^4 + 72*(e*x + d)^m*A*c^2*d*e^5*m^2*x^4 + 614*(e*x + d 
)^m*B*a*c*e^6*m^2*x^4 + 24*(e*x + d)^m*B*c^2*d*e^5*m*x^5 + 324*(e*x + d...
 

Mupad [B] (verification not implemented)

Time = 6.34 (sec) , antiderivative size = 1229, normalized size of antiderivative = 5.25 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

((d + e*x)^m*(720*A*a^2*d*e^5 - 120*B*c^2*d^6 + 144*A*c^2*d^5*e - 360*B*a^ 
2*d^2*e^4 + 580*A*a^2*d*e^5*m^2 + 155*A*a^2*d*e^5*m^3 + 20*A*a^2*d*e^5*m^4 
 + A*a^2*d*e^5*m^5 - 342*B*a^2*d^2*e^4*m - 119*B*a^2*d^2*e^4*m^2 - 18*B*a^ 
2*d^2*e^4*m^3 - B*a^2*d^2*e^4*m^4 + 480*A*a*c*d^3*e^3 - 360*B*a*c*d^4*e^2 
+ 1044*A*a^2*d*e^5*m + 24*A*c^2*d^5*e*m + 296*A*a*c*d^3*e^3*m - 132*B*a*c* 
d^4*e^2*m + 60*A*a*c*d^3*e^3*m^2 + 4*A*a*c*d^3*e^3*m^3 - 12*B*a*c*d^4*e^2* 
m^2))/(e^6*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + 
 (x*(d + e*x)^m*(720*A*a^2*e^6 + 1044*A*a^2*e^6*m + 580*A*a^2*e^6*m^2 + 15 
5*A*a^2*e^6*m^3 + 20*A*a^2*e^6*m^4 + A*a^2*e^6*m^5 + 342*B*a^2*d*e^5*m^2 + 
 119*B*a^2*d*e^5*m^3 + 18*B*a^2*d*e^5*m^4 + B*a^2*d*e^5*m^5 - 144*A*c^2*d^ 
4*e^2*m - 24*A*c^2*d^4*e^2*m^2 + 360*B*a^2*d*e^5*m + 120*B*c^2*d^5*e*m - 4 
80*A*a*c*d^2*e^4*m + 360*B*a*c*d^3*e^3*m - 296*A*a*c*d^2*e^4*m^2 - 60*A*a* 
c*d^2*e^4*m^3 - 4*A*a*c*d^2*e^4*m^4 + 132*B*a*c*d^3*e^3*m^2 + 12*B*a*c*d^3 
*e^3*m^3))/(e^6*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 72 
0)) + (x^2*(m + 1)*(d + e*x)^m*(360*B*a^2*e^4 + 342*B*a^2*e^4*m - 60*B*c^2 
*d^4*m + 119*B*a^2*e^4*m^2 + 18*B*a^2*e^4*m^3 + B*a^2*e^4*m^4 + 12*A*c^2*d 
^3*e*m^2 + 72*A*c^2*d^3*e*m + 148*A*a*c*d*e^3*m^2 + 30*A*a*c*d*e^3*m^3 + 2 
*A*a*c*d*e^3*m^4 - 180*B*a*c*d^2*e^2*m - 66*B*a*c*d^2*e^2*m^2 - 6*B*a*c*d^ 
2*e^2*m^3 + 240*A*a*c*d*e^3*m))/(e^4*(1764*m + 1624*m^2 + 735*m^3 + 175*m^ 
4 + 21*m^5 + m^6 + 720)) + (B*c^2*x^6*(d + e*x)^m*(274*m + 225*m^2 + 85...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1645, normalized size of antiderivative = 7.03 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

((d + e*x)**m*(a**3*d*e**5*m**5 + 20*a**3*d*e**5*m**4 + 155*a**3*d*e**5*m* 
*3 + 580*a**3*d*e**5*m**2 + 1044*a**3*d*e**5*m + 720*a**3*d*e**5 + a**3*e* 
*6*m**5*x + 20*a**3*e**6*m**4*x + 155*a**3*e**6*m**3*x + 580*a**3*e**6*m** 
2*x + 1044*a**3*e**6*m*x + 720*a**3*e**6*x - a**2*b*d**2*e**4*m**4 - 18*a* 
*2*b*d**2*e**4*m**3 - 119*a**2*b*d**2*e**4*m**2 - 342*a**2*b*d**2*e**4*m - 
 360*a**2*b*d**2*e**4 + a**2*b*d*e**5*m**5*x + 18*a**2*b*d*e**5*m**4*x + 1 
19*a**2*b*d*e**5*m**3*x + 342*a**2*b*d*e**5*m**2*x + 360*a**2*b*d*e**5*m*x 
 + a**2*b*e**6*m**5*x**2 + 19*a**2*b*e**6*m**4*x**2 + 137*a**2*b*e**6*m**3 
*x**2 + 461*a**2*b*e**6*m**2*x**2 + 702*a**2*b*e**6*m*x**2 + 360*a**2*b*e* 
*6*x**2 + 4*a**2*c*d**3*e**3*m**3 + 60*a**2*c*d**3*e**3*m**2 + 296*a**2*c* 
d**3*e**3*m + 480*a**2*c*d**3*e**3 - 4*a**2*c*d**2*e**4*m**4*x - 60*a**2*c 
*d**2*e**4*m**3*x - 296*a**2*c*d**2*e**4*m**2*x - 480*a**2*c*d**2*e**4*m*x 
 + 2*a**2*c*d*e**5*m**5*x**2 + 32*a**2*c*d*e**5*m**4*x**2 + 178*a**2*c*d*e 
**5*m**3*x**2 + 388*a**2*c*d*e**5*m**2*x**2 + 240*a**2*c*d*e**5*m*x**2 + 2 
*a**2*c*e**6*m**5*x**3 + 36*a**2*c*e**6*m**4*x**3 + 242*a**2*c*e**6*m**3*x 
**3 + 744*a**2*c*e**6*m**2*x**3 + 1016*a**2*c*e**6*m*x**3 + 480*a**2*c*e** 
6*x**3 - 12*a*b*c*d**4*e**2*m**2 - 132*a*b*c*d**4*e**2*m - 360*a*b*c*d**4* 
e**2 + 12*a*b*c*d**3*e**3*m**3*x + 132*a*b*c*d**3*e**3*m**2*x + 360*a*b*c* 
d**3*e**3*m*x - 6*a*b*c*d**2*e**4*m**4*x**2 - 72*a*b*c*d**2*e**4*m**3*x**2 
 - 246*a*b*c*d**2*e**4*m**2*x**2 - 180*a*b*c*d**2*e**4*m*x**2 + 2*a*b*c...