Integrand size = 22, antiderivative size = 151 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {a^2 c^3 x^{1+m}}{1+m}+\frac {a c^2 (2 b c+3 a d) x^{3+m}}{3+m}+\frac {c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{5+m}}{5+m}+\frac {d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{7+m}}{7+m}+\frac {b d^2 (3 b c+2 a d) x^{9+m}}{9+m}+\frac {b^2 d^3 x^{11+m}}{11+m} \]
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)
Time = 0.17 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.93 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=x^m \left (\frac {a^2 c^3 x}{1+m}+\frac {a c^2 (2 b c+3 a d) x^3}{3+m}+\frac {c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5}{5+m}+\frac {d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7}{7+m}+\frac {b d^2 (3 b c+2 a d) x^9}{9+m}+\frac {b^2 d^3 x^{11}}{11+m}\right ) \]
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))
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 |
| \(\displaystyle \frac {c x^{m+5} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )}{m+5}+\frac {d x^{m+7} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )}{m+7}+\frac {a^2 c^3 x^{m+1}}{m+1}+\frac {a c^2 x^{m+3} (3 a d+2 b c)}{m+3}+\frac {b d^2 x^{m+9} (2 a d+3 b c)}{m+9}+\frac {b^2 d^3 x^{m+11}}{m+11}\) |
(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)
3.4.25.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(974\) vs. \(2(151)=302\).
Time = 2.82 (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 x^{6} d^{2} a b c 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 x^{6} d^{2} a b c +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 x^{6} d^{2} a b c 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 x^{6} d^{2} a b c +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 | \(\text {Expression too large to display}\) | \(1193\) |
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)
Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (151) = 302\).
Time = 0.26 (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=\frac {{\left ({\left (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}\right )} x^{11} + {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m^{5} + 3465 \, b^{2} c d^{2} + 2310 \, a b d^{3} + 27 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m^{4} + 262 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m^{3} + 1122 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m^{2} + 2041 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} m\right )} x^{9} + {\left ({\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m^{5} + 4455 \, b^{2} c^{2} d + 8910 \, a b c d^{2} + 1485 \, a^{2} d^{3} + 29 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m^{4} + 302 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m^{3} + 1366 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m^{2} + 2577 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} m\right )} x^{7} + {\left ({\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m^{5} + 2079 \, b^{2} c^{3} + 12474 \, a b c^{2} d + 6237 \, a^{2} c d^{2} + 31 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m^{4} + 350 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m^{3} + 1730 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m^{2} + 3489 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} m\right )} x^{5} + {\left ({\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m^{5} + 6930 \, a b c^{3} + 10395 \, a^{2} c^{2} d + 33 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m^{4} + 406 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m^{3} + 2262 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m^{2} + 5353 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} m\right )} x^{3} + {\left (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}\right )} x\right )} x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \]
((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)
Leaf count of result is larger than twice the leaf count of optimal. 4345 vs. \(2 (144) = 288\).
Time = 0.90 (sec) , antiderivative size = 4345, normalized size of antiderivative = 28.77 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\text {Too large to display} \]
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*...
Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.42 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {b^{2} d^{3} x^{m + 11}}{m + 11} + \frac {3 \, b^{2} c d^{2} x^{m + 9}}{m + 9} + \frac {2 \, a b d^{3} x^{m + 9}}{m + 9} + \frac {3 \, b^{2} c^{2} d x^{m + 7}}{m + 7} + \frac {6 \, a b c d^{2} x^{m + 7}}{m + 7} + \frac {a^{2} d^{3} x^{m + 7}}{m + 7} + \frac {b^{2} c^{3} x^{m + 5}}{m + 5} + \frac {6 \, a b c^{2} d x^{m + 5}}{m + 5} + \frac {3 \, a^{2} c d^{2} x^{m + 5}}{m + 5} + \frac {2 \, a b c^{3} x^{m + 3}}{m + 3} + \frac {3 \, a^{2} c^{2} d x^{m + 3}}{m + 3} + \frac {a^{2} c^{3} x^{m + 1}}{m + 1} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (151) = 302\).
Time = 0.31 (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=\frac {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^{2} d m x^{5} x^{m} + 10467 \, a^{2} c d^{2} m x^{5} x^{m} + 35 \, a^{2} c^{3} m^{4} x x^{m} + 4524 \, a b c^{3} m^{2} x^{3} x^{m} + 6786 \, a^{2} c^{2} d m^{2} x^{3} x^{m} + 2079 \, b^{2} c^{3} x^{5} x^{m} + 12474 \, a b c^{2} d x^{5} x^{m} + 6237 \, a^{2} c d^{2} x^{5} x^{m} + 470 \, a^{2} c^{3} m^{3} x x^{m} + 10706 \, a b c^{3} m x^{3} x^{m} + 16059 \, a^{2} c^{2} d m x^{3} x^{m} + 3010 \, a^{2} c^{3} m^{2} x x^{m} + 6930 \, a b c^{3} x^{3} x^{m} + 10395 \, a^{2} c^{2} d x^{3} x^{m} + 9129 \, a^{2} c^{3} m x x^{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} \]
(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...
Time = 5.33 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.93 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {a^2\,c^3\,x\,x^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {c\,x^m\,x^5\,\left (3\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {d\,x^m\,x^7\,\left (a^2\,d^2+6\,a\,b\,c\,d+3\,b^2\,c^2\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b^2\,d^3\,x^m\,x^{11}\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a\,c^2\,x^m\,x^3\,\left (3\,a\,d+2\,b\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b\,d^2\,x^m\,x^9\,\left (2\,a\,d+3\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \]
(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)