\(\int x^m (a+b x^2)^2 (c+d x^2)^3 \, dx\) [1589]

Optimal result

Integrand size = 22, antiderivative size = 151 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {a^2 c^3 x^{1+m}}{1+m}+\frac {a c^2 (2 b c+3 a d) x^{3+m}}{3+m}+\frac {c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{5+m}}{5+m}+\frac {d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{7+m}}{7+m}+\frac {b d^2 (3 b c+2 a d) x^{9+m}}{9+m}+\frac {b^2 d^3 x^{11+m}}{11+m} \] Output:

a^2*c^3*x^(1+m)/(1+m)+a*c^2*(3*a*d+2*b*c)*x^(3+m)/(3+m)+c*(3*a^2*d^2+6*a*b 
*c*d+b^2*c^2)*x^(5+m)/(5+m)+d*(a^2*d^2+6*a*b*c*d+3*b^2*c^2)*x^(7+m)/(7+m)+ 
b*d^2*(2*a*d+3*b*c)*x^(9+m)/(9+m)+b^2*d^3*x^(11+m)/(11+m)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.93 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=x^m \left (\frac {a^2 c^3 x}{1+m}+\frac {a c^2 (2 b c+3 a d) x^3}{3+m}+\frac {c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5}{5+m}+\frac {d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7}{7+m}+\frac {b d^2 (3 b c+2 a d) x^9}{9+m}+\frac {b^2 d^3 x^{11}}{11+m}\right ) \] Input:

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

Output:

x^m*((a^2*c^3*x)/(1 + m) + (a*c^2*(2*b*c + 3*a*d)*x^3)/(3 + m) + (c*(b^2*c 
^2 + 6*a*b*c*d + 3*a^2*d^2)*x^5)/(5 + m) + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2 
*d^2)*x^7)/(7 + m) + (b*d^2*(3*b*c + 2*a*d)*x^9)/(9 + m) + (b^2*d^3*x^11)/ 
(11 + m))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 151, 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 = {355, 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.

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

\(\Big \downarrow \) 355

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c x^{m+5} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )}{m+5}+\frac {d x^{m+7} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )}{m+7}+\frac {a^2 c^3 x^{m+1}}{m+1}+\frac {a c^2 x^{m+3} (3 a d+2 b c)}{m+3}+\frac {b d^2 x^{m+9} (2 a d+3 b c)}{m+9}+\frac {b^2 d^3 x^{m+11}}{m+11}\)

Input:

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

Output:

(a^2*c^3*x^(1 + m))/(1 + m) + (a*c^2*(2*b*c + 3*a*d)*x^(3 + m))/(3 + m) + 
(c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(5 + m))/(5 + m) + (d*(3*b^2*c^2 + 
6*a*b*c*d + a^2*d^2)*x^(7 + m))/(7 + m) + (b*d^2*(3*b*c + 2*a*d)*x^(9 + m) 
)/(9 + m) + (b^2*d^3*x^(11 + m))/(11 + m)
 

Defintions of rubi rules used

rule 355

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, 
x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & 
& IGtQ[q, 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. \(974\) vs. \(2(151)=302\).

Time = 0.65 (sec) , antiderivative size = 975, normalized size of antiderivative = 6.46

method	result	size
risch	\(\frac {x \left (b^{2} d^{3} m^{5} x^{10}+25 b^{2} d^{3} m^{4} x^{10}+2 a b \,d^{3} m^{5} x^{8}+3 b^{2} c \,d^{2} m^{5} x^{8}+230 b^{2} d^{3} m^{3} x^{10}+54 a b \,d^{3} m^{4} x^{8}+81 b^{2} c \,d^{2} m^{4} x^{8}+950 b^{2} d^{3} m^{2} x^{10}+a^{2} d^{3} m^{5} x^{6}+6 a b c \,d^{2} m^{5} x^{6}+524 a b \,d^{3} m^{3} x^{8}+3 b^{2} c^{2} d \,m^{5} x^{6}+786 b^{2} c \,d^{2} m^{3} x^{8}+1689 m \,x^{10} b^{2} d^{3}+29 a^{2} d^{3} m^{4} x^{6}+174 a b c \,d^{2} m^{4} x^{6}+2244 a b \,d^{3} m^{2} x^{8}+87 b^{2} c^{2} d \,m^{4} x^{6}+3366 b^{2} c \,d^{2} m^{2} x^{8}+945 b^{2} d^{3} x^{10}+3 a^{2} c \,d^{2} m^{5} x^{4}+302 a^{2} d^{3} m^{3} x^{6}+6 a b \,c^{2} d \,m^{5} x^{4}+1812 a b c \,d^{2} m^{3} x^{6}+4082 a b \,d^{3} x^{8} m +b^{2} c^{3} m^{5} x^{4}+906 b^{2} c^{2} d \,m^{3} x^{6}+6123 b^{2} c \,d^{2} x^{8} m +93 a^{2} c \,d^{2} m^{4} x^{4}+1366 a^{2} d^{3} m^{2} x^{6}+186 a b \,c^{2} d \,m^{4} x^{4}+8196 a b c \,d^{2} m^{2} x^{6}+2310 a b \,d^{3} x^{8}+31 b^{2} c^{3} m^{4} x^{4}+4098 b^{2} c^{2} d \,m^{2} x^{6}+3465 b^{2} c \,d^{2} x^{8}+3 a^{2} c^{2} d \,m^{5} x^{2}+1050 a^{2} c \,d^{2} m^{3} x^{4}+2577 a^{2} d^{3} x^{6} m +2 a b \,c^{3} m^{5} x^{2}+2100 a b \,c^{2} d \,m^{3} x^{4}+15462 a b c \,d^{2} x^{6} m +350 b^{2} c^{3} m^{3} x^{4}+7731 b^{2} c^{2} d \,x^{6} m +99 a^{2} c^{2} d \,m^{4} x^{2}+5190 a^{2} c \,d^{2} m^{2} x^{4}+1485 a^{2} d^{3} x^{6}+66 a b \,c^{3} m^{4} x^{2}+10380 a b \,c^{2} d \,m^{2} x^{4}+8910 a b c \,d^{2} x^{6}+1730 b^{2} c^{3} m^{2} x^{4}+4455 b^{2} c^{2} d \,x^{6}+a^{2} c^{3} m^{5}+1218 a^{2} c^{2} d \,m^{3} x^{2}+10467 a^{2} c \,d^{2} x^{4} m +812 a b \,c^{3} m^{3} x^{2}+20934 a b \,c^{2} d \,x^{4} m +3489 b^{2} c^{3} x^{4} m +35 a^{2} c^{3} m^{4}+6786 a^{2} c^{2} d \,m^{2} x^{2}+6237 a^{2} c \,d^{2} x^{4}+4524 a b \,c^{3} m^{2} x^{2}+12474 a b \,c^{2} d \,x^{4}+2079 b^{2} c^{3} x^{4}+470 a^{2} c^{3} m^{3}+16059 a^{2} c^{2} d \,x^{2} m +10706 a b \,c^{3} x^{2} m +3010 a^{2} c^{3} m^{2}+10395 a^{2} c^{2} d \,x^{2}+6930 a b \,c^{3} x^{2}+9129 a^{2} c^{3} m +10395 a^{2} c^{3}\right ) x^{m}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(975\)
orering	\(\frac {x \left (b^{2} d^{3} m^{5} x^{10}+25 b^{2} d^{3} m^{4} x^{10}+2 a b \,d^{3} m^{5} x^{8}+3 b^{2} c \,d^{2} m^{5} x^{8}+230 b^{2} d^{3} m^{3} x^{10}+54 a b \,d^{3} m^{4} x^{8}+81 b^{2} c \,d^{2} m^{4} x^{8}+950 b^{2} d^{3} m^{2} x^{10}+a^{2} d^{3} m^{5} x^{6}+6 a b c \,d^{2} m^{5} x^{6}+524 a b \,d^{3} m^{3} x^{8}+3 b^{2} c^{2} d \,m^{5} x^{6}+786 b^{2} c \,d^{2} m^{3} x^{8}+1689 m \,x^{10} b^{2} d^{3}+29 a^{2} d^{3} m^{4} x^{6}+174 a b c \,d^{2} m^{4} x^{6}+2244 a b \,d^{3} m^{2} x^{8}+87 b^{2} c^{2} d \,m^{4} x^{6}+3366 b^{2} c \,d^{2} m^{2} x^{8}+945 b^{2} d^{3} x^{10}+3 a^{2} c \,d^{2} m^{5} x^{4}+302 a^{2} d^{3} m^{3} x^{6}+6 a b \,c^{2} d \,m^{5} x^{4}+1812 a b c \,d^{2} m^{3} x^{6}+4082 a b \,d^{3} x^{8} m +b^{2} c^{3} m^{5} x^{4}+906 b^{2} c^{2} d \,m^{3} x^{6}+6123 b^{2} c \,d^{2} x^{8} m +93 a^{2} c \,d^{2} m^{4} x^{4}+1366 a^{2} d^{3} m^{2} x^{6}+186 a b \,c^{2} d \,m^{4} x^{4}+8196 a b c \,d^{2} m^{2} x^{6}+2310 a b \,d^{3} x^{8}+31 b^{2} c^{3} m^{4} x^{4}+4098 b^{2} c^{2} d \,m^{2} x^{6}+3465 b^{2} c \,d^{2} x^{8}+3 a^{2} c^{2} d \,m^{5} x^{2}+1050 a^{2} c \,d^{2} m^{3} x^{4}+2577 a^{2} d^{3} x^{6} m +2 a b \,c^{3} m^{5} x^{2}+2100 a b \,c^{2} d \,m^{3} x^{4}+15462 a b c \,d^{2} x^{6} m +350 b^{2} c^{3} m^{3} x^{4}+7731 b^{2} c^{2} d \,x^{6} m +99 a^{2} c^{2} d \,m^{4} x^{2}+5190 a^{2} c \,d^{2} m^{2} x^{4}+1485 a^{2} d^{3} x^{6}+66 a b \,c^{3} m^{4} x^{2}+10380 a b \,c^{2} d \,m^{2} x^{4}+8910 a b c \,d^{2} x^{6}+1730 b^{2} c^{3} m^{2} x^{4}+4455 b^{2} c^{2} d \,x^{6}+a^{2} c^{3} m^{5}+1218 a^{2} c^{2} d \,m^{3} x^{2}+10467 a^{2} c \,d^{2} x^{4} m +812 a b \,c^{3} m^{3} x^{2}+20934 a b \,c^{2} d \,x^{4} m +3489 b^{2} c^{3} x^{4} m +35 a^{2} c^{3} m^{4}+6786 a^{2} c^{2} d \,m^{2} x^{2}+6237 a^{2} c \,d^{2} x^{4}+4524 a b \,c^{3} m^{2} x^{2}+12474 a b \,c^{2} d \,x^{4}+2079 b^{2} c^{3} x^{4}+470 a^{2} c^{3} m^{3}+16059 a^{2} c^{2} d \,x^{2} m +10706 a b \,c^{3} x^{2} m +3010 a^{2} c^{3} m^{2}+10395 a^{2} c^{2} d \,x^{2}+6930 a b \,c^{3} x^{2}+9129 a^{2} c^{3} m +10395 a^{2} c^{3}\right ) x^{m}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(975\)
gosper	\(\frac {x^{1+m} \left (b^{2} d^{3} m^{5} x^{10}+25 b^{2} d^{3} m^{4} x^{10}+2 a b \,d^{3} m^{5} x^{8}+3 b^{2} c \,d^{2} m^{5} x^{8}+230 b^{2} d^{3} m^{3} x^{10}+54 a b \,d^{3} m^{4} x^{8}+81 b^{2} c \,d^{2} m^{4} x^{8}+950 b^{2} d^{3} m^{2} x^{10}+a^{2} d^{3} m^{5} x^{6}+6 a b c \,d^{2} m^{5} x^{6}+524 a b \,d^{3} m^{3} x^{8}+3 b^{2} c^{2} d \,m^{5} x^{6}+786 b^{2} c \,d^{2} m^{3} x^{8}+1689 m \,x^{10} b^{2} d^{3}+29 a^{2} d^{3} m^{4} x^{6}+174 a b c \,d^{2} m^{4} x^{6}+2244 a b \,d^{3} m^{2} x^{8}+87 b^{2} c^{2} d \,m^{4} x^{6}+3366 b^{2} c \,d^{2} m^{2} x^{8}+945 b^{2} d^{3} x^{10}+3 a^{2} c \,d^{2} m^{5} x^{4}+302 a^{2} d^{3} m^{3} x^{6}+6 a b \,c^{2} d \,m^{5} x^{4}+1812 a b c \,d^{2} m^{3} x^{6}+4082 a b \,d^{3} x^{8} m +b^{2} c^{3} m^{5} x^{4}+906 b^{2} c^{2} d \,m^{3} x^{6}+6123 b^{2} c \,d^{2} x^{8} m +93 a^{2} c \,d^{2} m^{4} x^{4}+1366 a^{2} d^{3} m^{2} x^{6}+186 a b \,c^{2} d \,m^{4} x^{4}+8196 a b c \,d^{2} m^{2} x^{6}+2310 a b \,d^{3} x^{8}+31 b^{2} c^{3} m^{4} x^{4}+4098 b^{2} c^{2} d \,m^{2} x^{6}+3465 b^{2} c \,d^{2} x^{8}+3 a^{2} c^{2} d \,m^{5} x^{2}+1050 a^{2} c \,d^{2} m^{3} x^{4}+2577 a^{2} d^{3} x^{6} m +2 a b \,c^{3} m^{5} x^{2}+2100 a b \,c^{2} d \,m^{3} x^{4}+15462 a b c \,d^{2} x^{6} m +350 b^{2} c^{3} m^{3} x^{4}+7731 b^{2} c^{2} d \,x^{6} m +99 a^{2} c^{2} d \,m^{4} x^{2}+5190 a^{2} c \,d^{2} m^{2} x^{4}+1485 a^{2} d^{3} x^{6}+66 a b \,c^{3} m^{4} x^{2}+10380 a b \,c^{2} d \,m^{2} x^{4}+8910 a b c \,d^{2} x^{6}+1730 b^{2} c^{3} m^{2} x^{4}+4455 b^{2} c^{2} d \,x^{6}+a^{2} c^{3} m^{5}+1218 a^{2} c^{2} d \,m^{3} x^{2}+10467 a^{2} c \,d^{2} x^{4} m +812 a b \,c^{3} m^{3} x^{2}+20934 a b \,c^{2} d \,x^{4} m +3489 b^{2} c^{3} x^{4} m +35 a^{2} c^{3} m^{4}+6786 a^{2} c^{2} d \,m^{2} x^{2}+6237 a^{2} c \,d^{2} x^{4}+4524 a b \,c^{3} m^{2} x^{2}+12474 a b \,c^{2} d \,x^{4}+2079 b^{2} c^{3} x^{4}+470 a^{2} c^{3} m^{3}+16059 a^{2} c^{2} d \,x^{2} m +10706 a b \,c^{3} x^{2} m +3010 a^{2} c^{3} m^{2}+10395 a^{2} c^{2} d \,x^{2}+6930 a b \,c^{3} x^{2}+9129 a^{2} c^{3} m +10395 a^{2} c^{3}\right )}{\left (1+m \right ) \left (3+m \right ) \left (5+m \right ) \left (7+m \right ) \left (9+m \right ) \left (11+m \right )}\)	\(976\)
parallelrisch	\(\text {Expression too large to display}\)	\(1193\)

Input:

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

Output:

x*(b^2*d^3*m^5*x^10+25*b^2*d^3*m^4*x^10+2*a*b*d^3*m^5*x^8+3*b^2*c*d^2*m^5* 
x^8+230*b^2*d^3*m^3*x^10+54*a*b*d^3*m^4*x^8+81*b^2*c*d^2*m^4*x^8+950*b^2*d 
^3*m^2*x^10+a^2*d^3*m^5*x^6+6*a*b*c*d^2*m^5*x^6+524*a*b*d^3*m^3*x^8+3*b^2* 
c^2*d*m^5*x^6+786*b^2*c*d^2*m^3*x^8+1689*b^2*d^3*m*x^10+29*a^2*d^3*m^4*x^6 
+174*a*b*c*d^2*m^4*x^6+2244*a*b*d^3*m^2*x^8+87*b^2*c^2*d*m^4*x^6+3366*b^2* 
c*d^2*m^2*x^8+945*b^2*d^3*x^10+3*a^2*c*d^2*m^5*x^4+302*a^2*d^3*m^3*x^6+6*a 
*b*c^2*d*m^5*x^4+1812*a*b*c*d^2*m^3*x^6+4082*a*b*d^3*m*x^8+b^2*c^3*m^5*x^4 
+906*b^2*c^2*d*m^3*x^6+6123*b^2*c*d^2*m*x^8+93*a^2*c*d^2*m^4*x^4+1366*a^2* 
d^3*m^2*x^6+186*a*b*c^2*d*m^4*x^4+8196*a*b*c*d^2*m^2*x^6+2310*a*b*d^3*x^8+ 
31*b^2*c^3*m^4*x^4+4098*b^2*c^2*d*m^2*x^6+3465*b^2*c*d^2*x^8+3*a^2*c^2*d*m 
^5*x^2+1050*a^2*c*d^2*m^3*x^4+2577*a^2*d^3*m*x^6+2*a*b*c^3*m^5*x^2+2100*a* 
b*c^2*d*m^3*x^4+15462*a*b*c*d^2*m*x^6+350*b^2*c^3*m^3*x^4+7731*b^2*c^2*d*m 
*x^6+99*a^2*c^2*d*m^4*x^2+5190*a^2*c*d^2*m^2*x^4+1485*a^2*d^3*x^6+66*a*b*c 
^3*m^4*x^2+10380*a*b*c^2*d*m^2*x^4+8910*a*b*c*d^2*x^6+1730*b^2*c^3*m^2*x^4 
+4455*b^2*c^2*d*x^6+a^2*c^3*m^5+1218*a^2*c^2*d*m^3*x^2+10467*a^2*c*d^2*m*x 
^4+812*a*b*c^3*m^3*x^2+20934*a*b*c^2*d*m*x^4+3489*b^2*c^3*m*x^4+35*a^2*c^3 
*m^4+6786*a^2*c^2*d*m^2*x^2+6237*a^2*c*d^2*x^4+4524*a*b*c^3*m^2*x^2+12474* 
a*b*c^2*d*x^4+2079*b^2*c^3*x^4+470*a^2*c^3*m^3+16059*a^2*c^2*d*m*x^2+10706 
*a*b*c^3*m*x^2+3010*a^2*c^3*m^2+10395*a^2*c^2*d*x^2+6930*a*b*c^3*x^2+9129* 
a^2*c^3*m+10395*a^2*c^3)*x^m/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (151) = 302\).

Time = 0.12 (sec) , antiderivative size = 773, normalized size of antiderivative = 5.12 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((b^2*d^3*m^5 + 25*b^2*d^3*m^4 + 230*b^2*d^3*m^3 + 950*b^2*d^3*m^2 + 1689* 
b^2*d^3*m + 945*b^2*d^3)*x^11 + ((3*b^2*c*d^2 + 2*a*b*d^3)*m^5 + 3465*b^2* 
c*d^2 + 2310*a*b*d^3 + 27*(3*b^2*c*d^2 + 2*a*b*d^3)*m^4 + 262*(3*b^2*c*d^2 
 + 2*a*b*d^3)*m^3 + 1122*(3*b^2*c*d^2 + 2*a*b*d^3)*m^2 + 2041*(3*b^2*c*d^2 
 + 2*a*b*d^3)*m)*x^9 + ((3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*m^5 + 4455*b 
^2*c^2*d + 8910*a*b*c*d^2 + 1485*a^2*d^3 + 29*(3*b^2*c^2*d + 6*a*b*c*d^2 + 
 a^2*d^3)*m^4 + 302*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*m^3 + 1366*(3*b^ 
2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*m^2 + 2577*(3*b^2*c^2*d + 6*a*b*c*d^2 + a 
^2*d^3)*m)*x^7 + ((b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*m^5 + 2079*b^2*c^3 
 + 12474*a*b*c^2*d + 6237*a^2*c*d^2 + 31*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c* 
d^2)*m^4 + 350*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*m^3 + 1730*(b^2*c^3 + 
 6*a*b*c^2*d + 3*a^2*c*d^2)*m^2 + 3489*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^ 
2)*m)*x^5 + ((2*a*b*c^3 + 3*a^2*c^2*d)*m^5 + 6930*a*b*c^3 + 10395*a^2*c^2* 
d + 33*(2*a*b*c^3 + 3*a^2*c^2*d)*m^4 + 406*(2*a*b*c^3 + 3*a^2*c^2*d)*m^3 + 
 2262*(2*a*b*c^3 + 3*a^2*c^2*d)*m^2 + 5353*(2*a*b*c^3 + 3*a^2*c^2*d)*m)*x^ 
3 + (a^2*c^3*m^5 + 35*a^2*c^3*m^4 + 470*a^2*c^3*m^3 + 3010*a^2*c^3*m^2 + 9 
129*a^2*c^3*m + 10395*a^2*c^3)*x)*x^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 
 12139*m^2 + 19524*m + 10395)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4345 vs. \(2 (144) = 288\).

Time = 0.81 (sec) , antiderivative size = 4345, normalized size of antiderivative = 28.77 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.42 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {b^{2} d^{3} x^{m + 11}}{m + 11} + \frac {3 \, b^{2} c d^{2} x^{m + 9}}{m + 9} + \frac {2 \, a b d^{3} x^{m + 9}}{m + 9} + \frac {3 \, b^{2} c^{2} d x^{m + 7}}{m + 7} + \frac {6 \, a b c d^{2} x^{m + 7}}{m + 7} + \frac {a^{2} d^{3} x^{m + 7}}{m + 7} + \frac {b^{2} c^{3} x^{m + 5}}{m + 5} + \frac {6 \, a b c^{2} d x^{m + 5}}{m + 5} + \frac {3 \, a^{2} c d^{2} x^{m + 5}}{m + 5} + \frac {2 \, a b c^{3} x^{m + 3}}{m + 3} + \frac {3 \, a^{2} c^{2} d x^{m + 3}}{m + 3} + \frac {a^{2} c^{3} x^{m + 1}}{m + 1} \] Input:

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

Output:

b^2*d^3*x^(m + 11)/(m + 11) + 3*b^2*c*d^2*x^(m + 9)/(m + 9) + 2*a*b*d^3*x^ 
(m + 9)/(m + 9) + 3*b^2*c^2*d*x^(m + 7)/(m + 7) + 6*a*b*c*d^2*x^(m + 7)/(m 
 + 7) + a^2*d^3*x^(m + 7)/(m + 7) + b^2*c^3*x^(m + 5)/(m + 5) + 6*a*b*c^2* 
d*x^(m + 5)/(m + 5) + 3*a^2*c*d^2*x^(m + 5)/(m + 5) + 2*a*b*c^3*x^(m + 3)/ 
(m + 3) + 3*a^2*c^2*d*x^(m + 3)/(m + 3) + a^2*c^3*x^(m + 1)/(m + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (151) = 302\).

Time = 0.15 (sec) , antiderivative size = 1192, normalized size of antiderivative = 7.89 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(b^2*d^3*m^5*x^11*x^m + 25*b^2*d^3*m^4*x^11*x^m + 3*b^2*c*d^2*m^5*x^9*x^m 
+ 2*a*b*d^3*m^5*x^9*x^m + 230*b^2*d^3*m^3*x^11*x^m + 81*b^2*c*d^2*m^4*x^9* 
x^m + 54*a*b*d^3*m^4*x^9*x^m + 950*b^2*d^3*m^2*x^11*x^m + 3*b^2*c^2*d*m^5* 
x^7*x^m + 6*a*b*c*d^2*m^5*x^7*x^m + a^2*d^3*m^5*x^7*x^m + 786*b^2*c*d^2*m^ 
3*x^9*x^m + 524*a*b*d^3*m^3*x^9*x^m + 1689*b^2*d^3*m*x^11*x^m + 87*b^2*c^2 
*d*m^4*x^7*x^m + 174*a*b*c*d^2*m^4*x^7*x^m + 29*a^2*d^3*m^4*x^7*x^m + 3366 
*b^2*c*d^2*m^2*x^9*x^m + 2244*a*b*d^3*m^2*x^9*x^m + 945*b^2*d^3*x^11*x^m + 
 b^2*c^3*m^5*x^5*x^m + 6*a*b*c^2*d*m^5*x^5*x^m + 3*a^2*c*d^2*m^5*x^5*x^m + 
 906*b^2*c^2*d*m^3*x^7*x^m + 1812*a*b*c*d^2*m^3*x^7*x^m + 302*a^2*d^3*m^3* 
x^7*x^m + 6123*b^2*c*d^2*m*x^9*x^m + 4082*a*b*d^3*m*x^9*x^m + 31*b^2*c^3*m 
^4*x^5*x^m + 186*a*b*c^2*d*m^4*x^5*x^m + 93*a^2*c*d^2*m^4*x^5*x^m + 4098*b 
^2*c^2*d*m^2*x^7*x^m + 8196*a*b*c*d^2*m^2*x^7*x^m + 1366*a^2*d^3*m^2*x^7*x 
^m + 3465*b^2*c*d^2*x^9*x^m + 2310*a*b*d^3*x^9*x^m + 2*a*b*c^3*m^5*x^3*x^m 
 + 3*a^2*c^2*d*m^5*x^3*x^m + 350*b^2*c^3*m^3*x^5*x^m + 2100*a*b*c^2*d*m^3* 
x^5*x^m + 1050*a^2*c*d^2*m^3*x^5*x^m + 7731*b^2*c^2*d*m*x^7*x^m + 15462*a* 
b*c*d^2*m*x^7*x^m + 2577*a^2*d^3*m*x^7*x^m + 66*a*b*c^3*m^4*x^3*x^m + 99*a 
^2*c^2*d*m^4*x^3*x^m + 1730*b^2*c^3*m^2*x^5*x^m + 10380*a*b*c^2*d*m^2*x^5* 
x^m + 5190*a^2*c*d^2*m^2*x^5*x^m + 4455*b^2*c^2*d*x^7*x^m + 8910*a*b*c*d^2 
*x^7*x^m + 1485*a^2*d^3*x^7*x^m + a^2*c^3*m^5*x*x^m + 812*a*b*c^3*m^3*x^3* 
x^m + 1218*a^2*c^2*d*m^3*x^3*x^m + 3489*b^2*c^3*m*x^5*x^m + 20934*a*b*c...
 

Mupad [B] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.93 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {a^2\,c^3\,x\,x^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {c\,x^m\,x^5\,\left (3\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {d\,x^m\,x^7\,\left (a^2\,d^2+6\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b^2\,d^3\,x^m\,x^{11}\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a\,c^2\,x^m\,x^3\,\left (3\,a\,d+2\,b\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b\,d^2\,x^m\,x^9\,\left (2\,a\,d+3\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \] Input:

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

Output:

(a^2*c^3*x*x^m*(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5 + 10395))/(1952 
4*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (c*x^m*x^5* 
(3*a^2*d^2 + b^2*c^2 + 6*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 + 10 
395) + (d*x^m*x^7*(a^2*d^2 + 3*b^2*c^2 + 6*a*b*c*d)*(2577*m + 1366*m^2 + 3 
02*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^2*d^3*x^m*x^11*(1689*m + 950*m^2 + 230*m^3 + 2 
5*m^4 + m^5 + 945))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m 
^6 + 10395) + (a*c^2*x^m*x^3*(3*a*d + 2*b*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^2*x^m*x^9*(2*a*d + 3*b*c)*(2041*m + 1122*m^2 + 262* 
m^3 + 27*m^4 + m^5 + 1155))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36 
*m^5 + m^6 + 10395)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 974, normalized size of antiderivative = 6.45 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(x**m*x*(a**2*c**3*m**5 + 35*a**2*c**3*m**4 + 470*a**2*c**3*m**3 + 3010*a* 
*2*c**3*m**2 + 9129*a**2*c**3*m + 10395*a**2*c**3 + 3*a**2*c**2*d*m**5*x** 
2 + 99*a**2*c**2*d*m**4*x**2 + 1218*a**2*c**2*d*m**3*x**2 + 6786*a**2*c**2 
*d*m**2*x**2 + 16059*a**2*c**2*d*m*x**2 + 10395*a**2*c**2*d*x**2 + 3*a**2* 
c*d**2*m**5*x**4 + 93*a**2*c*d**2*m**4*x**4 + 1050*a**2*c*d**2*m**3*x**4 + 
 5190*a**2*c*d**2*m**2*x**4 + 10467*a**2*c*d**2*m*x**4 + 6237*a**2*c*d**2* 
x**4 + a**2*d**3*m**5*x**6 + 29*a**2*d**3*m**4*x**6 + 302*a**2*d**3*m**3*x 
**6 + 1366*a**2*d**3*m**2*x**6 + 2577*a**2*d**3*m*x**6 + 1485*a**2*d**3*x* 
*6 + 2*a*b*c**3*m**5*x**2 + 66*a*b*c**3*m**4*x**2 + 812*a*b*c**3*m**3*x**2 
 + 4524*a*b*c**3*m**2*x**2 + 10706*a*b*c**3*m*x**2 + 6930*a*b*c**3*x**2 + 
6*a*b*c**2*d*m**5*x**4 + 186*a*b*c**2*d*m**4*x**4 + 2100*a*b*c**2*d*m**3*x 
**4 + 10380*a*b*c**2*d*m**2*x**4 + 20934*a*b*c**2*d*m*x**4 + 12474*a*b*c** 
2*d*x**4 + 6*a*b*c*d**2*m**5*x**6 + 174*a*b*c*d**2*m**4*x**6 + 1812*a*b*c* 
d**2*m**3*x**6 + 8196*a*b*c*d**2*m**2*x**6 + 15462*a*b*c*d**2*m*x**6 + 891 
0*a*b*c*d**2*x**6 + 2*a*b*d**3*m**5*x**8 + 54*a*b*d**3*m**4*x**8 + 524*a*b 
*d**3*m**3*x**8 + 2244*a*b*d**3*m**2*x**8 + 4082*a*b*d**3*m*x**8 + 2310*a* 
b*d**3*x**8 + b**2*c**3*m**5*x**4 + 31*b**2*c**3*m**4*x**4 + 350*b**2*c**3 
*m**3*x**4 + 1730*b**2*c**3*m**2*x**4 + 3489*b**2*c**3*m*x**4 + 2079*b**2* 
c**3*x**4 + 3*b**2*c**2*d*m**5*x**6 + 87*b**2*c**2*d*m**4*x**6 + 906*b**2* 
c**2*d*m**3*x**6 + 4098*b**2*c**2*d*m**2*x**6 + 7731*b**2*c**2*d*m*x**6...