Integrand size = 22, antiderivative size = 109 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {a^2 c^2 x^{1+m}}{1+m}+\frac {2 a c (b c+a d) x^{3+m}}{3+m}+\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5+m}}{5+m}+\frac {2 b d (b c+a d) x^{7+m}}{7+m}+\frac {b^2 d^2 x^{9+m}}{9+m} \] Output:
a^2*c^2*x^(1+m)/(1+m)+2*a*c*(a*d+b*c)*x^(3+m)/(3+m)+(a^2*d^2+4*a*b*c*d+b^2 *c^2)*x^(5+m)/(5+m)+2*b*d*(a*d+b*c)*x^(7+m)/(7+m)+b^2*d^2*x^(9+m)/(9+m)
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^m \left (\frac {a^2 c^2 x}{1+m}+\frac {2 a c (b c+a d) x^3}{3+m}+\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5}{5+m}+\frac {2 b d (b c+a d) x^7}{7+m}+\frac {b^2 d^2 x^9}{9+m}\right ) \] Input:
Integrate[x^m*(a + b*x^2)^2*(c + d*x^2)^2,x]
Output:
x^m*((a^2*c^2*x)/(1 + m) + (2*a*c*(b*c + a*d)*x^3)/(3 + m) + ((b^2*c^2 + 4 *a*b*c*d + a^2*d^2)*x^5)/(5 + m) + (2*b*d*(b*c + a*d)*x^7)/(7 + m) + (b^2* d^2*x^9)/(9 + m))
Time = 0.25 (sec) , antiderivative size = 109, 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.
| \(\displaystyle \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 355 |
| \(\displaystyle \int \left (x^{m+4} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x^m+2 a c x^{m+2} (a d+b c)+2 b d x^{m+6} (a d+b c)+b^2 d^2 x^{m+8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{m+5} \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{m+5}+\frac {a^2 c^2 x^{m+1}}{m+1}+\frac {2 a c x^{m+3} (a d+b c)}{m+3}+\frac {2 b d x^{m+7} (a d+b c)}{m+7}+\frac {b^2 d^2 x^{m+9}}{m+9}\) |
Input:
Int[x^m*(a + b*x^2)^2*(c + d*x^2)^2,x]
Output:
(a^2*c^2*x^(1 + m))/(1 + m) + (2*a*c*(b*c + a*d)*x^(3 + m))/(3 + m) + ((b^ 2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5 + m))/(5 + m) + (2*b*d*(b*c + a*d)*x^(7 + m))/(7 + m) + (b^2*d^2*x^(9 + m))/(9 + m)
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. \(567\) vs. \(2(109)=218\).
Time = 0.55 (sec) , antiderivative size = 568, normalized size of antiderivative = 5.21
| method | result | size |
| risch | \(\frac {x \left (b^{2} d^{2} m^{4} x^{8}+16 b^{2} d^{2} m^{3} x^{8}+2 a b \,d^{2} m^{4} x^{6}+2 b^{2} c d \,m^{4} x^{6}+86 b^{2} d^{2} m^{2} x^{8}+36 a b \,d^{2} m^{3} x^{6}+36 b^{2} c d \,m^{3} x^{6}+176 m \,x^{8} b^{2} d^{2}+a^{2} d^{2} m^{4} x^{4}+4 a b c d \,m^{4} x^{4}+208 a b \,d^{2} m^{2} x^{6}+b^{2} c^{2} m^{4} x^{4}+208 b^{2} c d \,m^{2} x^{6}+105 b^{2} d^{2} x^{8}+20 a^{2} d^{2} m^{3} x^{4}+80 a b c d \,m^{3} x^{4}+444 a b \,d^{2} x^{6} m +20 b^{2} c^{2} m^{3} x^{4}+444 b^{2} c d \,x^{6} m +2 a^{2} c d \,m^{4} x^{2}+130 a^{2} d^{2} m^{2} x^{4}+2 a b \,c^{2} m^{4} x^{2}+520 a b c d \,m^{2} x^{4}+270 a b \,d^{2} x^{6}+130 b^{2} c^{2} m^{2} x^{4}+270 b^{2} c d \,x^{6}+44 a^{2} c d \,m^{3} x^{2}+300 a^{2} d^{2} x^{4} m +44 a b \,c^{2} m^{3} x^{2}+1200 a b c d \,x^{4} m +300 b^{2} c^{2} x^{4} m +a^{2} c^{2} m^{4}+328 a^{2} c d \,m^{2} x^{2}+189 a^{2} d^{2} x^{4}+328 a b \,c^{2} m^{2} x^{2}+756 a b c d \,x^{4}+189 b^{2} c^{2} x^{4}+24 a^{2} c^{2} m^{3}+916 a^{2} c d \,x^{2} m +916 a b \,c^{2} x^{2} m +206 a^{2} c^{2} m^{2}+630 a^{2} c d \,x^{2}+630 a b \,c^{2} x^{2}+744 a^{2} c^{2} m +945 a^{2} c^{2}\right ) x^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(568\) |
| orering | \(\frac {x \left (b^{2} d^{2} m^{4} x^{8}+16 b^{2} d^{2} m^{3} x^{8}+2 a b \,d^{2} m^{4} x^{6}+2 b^{2} c d \,m^{4} x^{6}+86 b^{2} d^{2} m^{2} x^{8}+36 a b \,d^{2} m^{3} x^{6}+36 b^{2} c d \,m^{3} x^{6}+176 m \,x^{8} b^{2} d^{2}+a^{2} d^{2} m^{4} x^{4}+4 a b c d \,m^{4} x^{4}+208 a b \,d^{2} m^{2} x^{6}+b^{2} c^{2} m^{4} x^{4}+208 b^{2} c d \,m^{2} x^{6}+105 b^{2} d^{2} x^{8}+20 a^{2} d^{2} m^{3} x^{4}+80 a b c d \,m^{3} x^{4}+444 a b \,d^{2} x^{6} m +20 b^{2} c^{2} m^{3} x^{4}+444 b^{2} c d \,x^{6} m +2 a^{2} c d \,m^{4} x^{2}+130 a^{2} d^{2} m^{2} x^{4}+2 a b \,c^{2} m^{4} x^{2}+520 a b c d \,m^{2} x^{4}+270 a b \,d^{2} x^{6}+130 b^{2} c^{2} m^{2} x^{4}+270 b^{2} c d \,x^{6}+44 a^{2} c d \,m^{3} x^{2}+300 a^{2} d^{2} x^{4} m +44 a b \,c^{2} m^{3} x^{2}+1200 a b c d \,x^{4} m +300 b^{2} c^{2} x^{4} m +a^{2} c^{2} m^{4}+328 a^{2} c d \,m^{2} x^{2}+189 a^{2} d^{2} x^{4}+328 a b \,c^{2} m^{2} x^{2}+756 a b c d \,x^{4}+189 b^{2} c^{2} x^{4}+24 a^{2} c^{2} m^{3}+916 a^{2} c d \,x^{2} m +916 a b \,c^{2} x^{2} m +206 a^{2} c^{2} m^{2}+630 a^{2} c d \,x^{2}+630 a b \,c^{2} x^{2}+744 a^{2} c^{2} m +945 a^{2} c^{2}\right ) x^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(568\) |
| gosper | \(\frac {x^{1+m} \left (b^{2} d^{2} m^{4} x^{8}+16 b^{2} d^{2} m^{3} x^{8}+2 a b \,d^{2} m^{4} x^{6}+2 b^{2} c d \,m^{4} x^{6}+86 b^{2} d^{2} m^{2} x^{8}+36 a b \,d^{2} m^{3} x^{6}+36 b^{2} c d \,m^{3} x^{6}+176 m \,x^{8} b^{2} d^{2}+a^{2} d^{2} m^{4} x^{4}+4 a b c d \,m^{4} x^{4}+208 a b \,d^{2} m^{2} x^{6}+b^{2} c^{2} m^{4} x^{4}+208 b^{2} c d \,m^{2} x^{6}+105 b^{2} d^{2} x^{8}+20 a^{2} d^{2} m^{3} x^{4}+80 a b c d \,m^{3} x^{4}+444 a b \,d^{2} x^{6} m +20 b^{2} c^{2} m^{3} x^{4}+444 b^{2} c d \,x^{6} m +2 a^{2} c d \,m^{4} x^{2}+130 a^{2} d^{2} m^{2} x^{4}+2 a b \,c^{2} m^{4} x^{2}+520 a b c d \,m^{2} x^{4}+270 a b \,d^{2} x^{6}+130 b^{2} c^{2} m^{2} x^{4}+270 b^{2} c d \,x^{6}+44 a^{2} c d \,m^{3} x^{2}+300 a^{2} d^{2} x^{4} m +44 a b \,c^{2} m^{3} x^{2}+1200 a b c d \,x^{4} m +300 b^{2} c^{2} x^{4} m +a^{2} c^{2} m^{4}+328 a^{2} c d \,m^{2} x^{2}+189 a^{2} d^{2} x^{4}+328 a b \,c^{2} m^{2} x^{2}+756 a b c d \,x^{4}+189 b^{2} c^{2} x^{4}+24 a^{2} c^{2} m^{3}+916 a^{2} c d \,x^{2} m +916 a b \,c^{2} x^{2} m +206 a^{2} c^{2} m^{2}+630 a^{2} c d \,x^{2}+630 a b \,c^{2} x^{2}+744 a^{2} c^{2} m +945 a^{2} c^{2}\right )}{\left (1+m \right ) \left (3+m \right ) \left (5+m \right ) \left (7+m \right ) \left (9+m \right )}\) | \(569\) |
| parallelrisch | \(\frac {328 x^{3} x^{m} a b \,c^{2} m^{2}+916 x^{3} x^{m} a^{2} c d m +916 x^{3} x^{m} a b \,c^{2} m +4 x^{5} x^{m} a b c d \,m^{4}+80 x^{5} x^{m} a b c d \,m^{3}+520 x^{5} x^{m} a b c d \,m^{2}+1200 x^{5} x^{m} a b c d m +x^{9} x^{m} b^{2} d^{2} m^{4}+16 x^{9} x^{m} b^{2} d^{2} m^{3}+86 x^{9} x^{m} b^{2} d^{2} m^{2}+176 x^{9} x^{m} b^{2} d^{2} m +x^{5} x^{m} a^{2} d^{2} m^{4}+x^{5} x^{m} b^{2} c^{2} m^{4}+20 x^{5} x^{m} a^{2} d^{2} m^{3}+20 x^{5} x^{m} b^{2} c^{2} m^{3}+270 x^{7} x^{m} a b \,d^{2}+270 x^{7} x^{m} b^{2} c d +130 x^{5} x^{m} a^{2} d^{2} m^{2}+2 x^{7} x^{m} b^{2} c d \,m^{4}+130 x^{5} x^{m} b^{2} c^{2} m^{2}+300 x^{5} x^{m} a^{2} d^{2} m +300 x^{5} x^{m} b^{2} c^{2} m +x \,x^{m} a^{2} c^{2} m^{4}+24 x \,x^{m} a^{2} c^{2} m^{3}+630 x^{3} x^{m} a^{2} c d +2 x^{7} x^{m} a b \,d^{2} m^{4}+630 x^{3} x^{m} a b \,c^{2}+206 x \,x^{m} a^{2} c^{2} m^{2}+744 x \,x^{m} a^{2} c^{2} m +2 x^{3} x^{m} a^{2} c d \,m^{4}+2 x^{3} x^{m} a b \,c^{2} m^{4}+44 x^{3} x^{m} a^{2} c d \,m^{3}+44 x^{3} x^{m} a b \,c^{2} m^{3}+756 x^{5} x^{m} a b c d +328 x^{3} x^{m} a^{2} c d \,m^{2}+444 x^{7} x^{m} b^{2} c d m +208 x^{7} x^{m} a b \,d^{2} m^{2}+208 x^{7} x^{m} b^{2} c d \,m^{2}+105 x^{9} x^{m} b^{2} d^{2}+189 x^{5} x^{m} a^{2} d^{2}+189 x^{5} x^{m} b^{2} c^{2}+945 x \,x^{m} a^{2} c^{2}+36 x^{7} x^{m} a b \,d^{2} m^{3}+36 x^{7} x^{m} b^{2} c d \,m^{3}+444 x^{7} x^{m} a b \,d^{2} m}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(704\) |
Input:
int(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
x*(b^2*d^2*m^4*x^8+16*b^2*d^2*m^3*x^8+2*a*b*d^2*m^4*x^6+2*b^2*c*d*m^4*x^6+ 86*b^2*d^2*m^2*x^8+36*a*b*d^2*m^3*x^6+36*b^2*c*d*m^3*x^6+176*b^2*d^2*m*x^8 +a^2*d^2*m^4*x^4+4*a*b*c*d*m^4*x^4+208*a*b*d^2*m^2*x^6+b^2*c^2*m^4*x^4+208 *b^2*c*d*m^2*x^6+105*b^2*d^2*x^8+20*a^2*d^2*m^3*x^4+80*a*b*c*d*m^3*x^4+444 *a*b*d^2*m*x^6+20*b^2*c^2*m^3*x^4+444*b^2*c*d*m*x^6+2*a^2*c*d*m^4*x^2+130* a^2*d^2*m^2*x^4+2*a*b*c^2*m^4*x^2+520*a*b*c*d*m^2*x^4+270*a*b*d^2*x^6+130* b^2*c^2*m^2*x^4+270*b^2*c*d*x^6+44*a^2*c*d*m^3*x^2+300*a^2*d^2*m*x^4+44*a* b*c^2*m^3*x^2+1200*a*b*c*d*m*x^4+300*b^2*c^2*m*x^4+a^2*c^2*m^4+328*a^2*c*d *m^2*x^2+189*a^2*d^2*x^4+328*a*b*c^2*m^2*x^2+756*a*b*c*d*x^4+189*b^2*c^2*x ^4+24*a^2*c^2*m^3+916*a^2*c*d*m*x^2+916*a*b*c^2*m*x^2+206*a^2*c^2*m^2+630* a^2*c*d*x^2+630*a*b*c^2*x^2+744*a^2*c^2*m+945*a^2*c^2)*x^m/(9+m)/(7+m)/(5+ m)/(3+m)/(1+m)
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (109) = 218\).
Time = 0.11 (sec) , antiderivative size = 442, normalized size of antiderivative = 4.06 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {{\left ({\left (b^{2} d^{2} m^{4} + 16 \, b^{2} d^{2} m^{3} + 86 \, b^{2} d^{2} m^{2} + 176 \, b^{2} d^{2} m + 105 \, b^{2} d^{2}\right )} x^{9} + 2 \, {\left ({\left (b^{2} c d + a b d^{2}\right )} m^{4} + 135 \, b^{2} c d + 135 \, a b d^{2} + 18 \, {\left (b^{2} c d + a b d^{2}\right )} m^{3} + 104 \, {\left (b^{2} c d + a b d^{2}\right )} m^{2} + 222 \, {\left (b^{2} c d + a b d^{2}\right )} m\right )} x^{7} + {\left ({\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{4} + 189 \, b^{2} c^{2} + 756 \, a b c d + 189 \, a^{2} d^{2} + 20 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{3} + 130 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{2} + 300 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m\right )} x^{5} + 2 \, {\left ({\left (a b c^{2} + a^{2} c d\right )} m^{4} + 315 \, a b c^{2} + 315 \, a^{2} c d + 22 \, {\left (a b c^{2} + a^{2} c d\right )} m^{3} + 164 \, {\left (a b c^{2} + a^{2} c d\right )} m^{2} + 458 \, {\left (a b c^{2} + a^{2} c d\right )} m\right )} x^{3} + {\left (a^{2} c^{2} m^{4} + 24 \, a^{2} c^{2} m^{3} + 206 \, a^{2} c^{2} m^{2} + 744 \, a^{2} c^{2} m + 945 \, a^{2} c^{2}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="fricas")
Output:
((b^2*d^2*m^4 + 16*b^2*d^2*m^3 + 86*b^2*d^2*m^2 + 176*b^2*d^2*m + 105*b^2* d^2)*x^9 + 2*((b^2*c*d + a*b*d^2)*m^4 + 135*b^2*c*d + 135*a*b*d^2 + 18*(b^ 2*c*d + a*b*d^2)*m^3 + 104*(b^2*c*d + a*b*d^2)*m^2 + 222*(b^2*c*d + a*b*d^ 2)*m)*x^7 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m^4 + 189*b^2*c^2 + 756*a*b*c *d + 189*a^2*d^2 + 20*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m^3 + 130*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m^2 + 300*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m)*x^5 + 2 *((a*b*c^2 + a^2*c*d)*m^4 + 315*a*b*c^2 + 315*a^2*c*d + 22*(a*b*c^2 + a^2* c*d)*m^3 + 164*(a*b*c^2 + a^2*c*d)*m^2 + 458*(a*b*c^2 + a^2*c*d)*m)*x^3 + (a^2*c^2*m^4 + 24*a^2*c^2*m^3 + 206*a^2*c^2*m^2 + 744*a^2*c^2*m + 945*a^2* c^2)*x)*x^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)
Leaf count of result is larger than twice the leaf count of optimal. 2363 vs. \(2 (100) = 200\).
Time = 0.61 (sec) , antiderivative size = 2363, normalized size of antiderivative = 21.68 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x**m*(b*x**2+a)**2*(d*x**2+c)**2,x)
Output:
Piecewise((-a**2*c**2/(8*x**8) - a**2*c*d/(3*x**6) - a**2*d**2/(4*x**4) - a*b*c**2/(3*x**6) - a*b*c*d/x**4 - a*b*d**2/x**2 - b**2*c**2/(4*x**4) - b* *2*c*d/x**2 + b**2*d**2*log(x), Eq(m, -9)), (-a**2*c**2/(6*x**6) - a**2*c* d/(2*x**4) - a**2*d**2/(2*x**2) - a*b*c**2/(2*x**4) - 2*a*b*c*d/x**2 + 2*a *b*d**2*log(x) - b**2*c**2/(2*x**2) + 2*b**2*c*d*log(x) + b**2*d**2*x**2/2 , Eq(m, -7)), (-a**2*c**2/(4*x**4) - a**2*c*d/x**2 + a**2*d**2*log(x) - a* b*c**2/x**2 + 4*a*b*c*d*log(x) + a*b*d**2*x**2 + b**2*c**2*log(x) + b**2*c *d*x**2 + b**2*d**2*x**4/4, Eq(m, -5)), (-a**2*c**2/(2*x**2) + 2*a**2*c*d* log(x) + a**2*d**2*x**2/2 + 2*a*b*c**2*log(x) + 2*a*b*c*d*x**2 + a*b*d**2* x**4/2 + b**2*c**2*x**2/2 + b**2*c*d*x**4/2 + b**2*d**2*x**6/6, Eq(m, -3)) , (a**2*c**2*log(x) + a**2*c*d*x**2 + a**2*d**2*x**4/4 + a*b*c**2*x**2 + a *b*c*d*x**4 + a*b*d**2*x**6/3 + b**2*c**2*x**4/4 + b**2*c*d*x**6/3 + b**2* d**2*x**8/8, Eq(m, -1)), (a**2*c**2*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*a**2*c**2*m**3*x*x**m/(m**5 + 25*m**4 + 2 30*m**3 + 950*m**2 + 1689*m + 945) + 206*a**2*c**2*m**2*x*x**m/(m**5 + 25* m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 744*a**2*c**2*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*a**2*c**2*x*x**m/(m* *5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a**2*c*d*m**4*x**3* x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 44*a**2*c*d*m **3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3...
Time = 0.04 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {b^{2} d^{2} x^{m + 9}}{m + 9} + \frac {2 \, b^{2} c d x^{m + 7}}{m + 7} + \frac {2 \, a b d^{2} x^{m + 7}}{m + 7} + \frac {b^{2} c^{2} x^{m + 5}}{m + 5} + \frac {4 \, a b c d x^{m + 5}}{m + 5} + \frac {a^{2} d^{2} x^{m + 5}}{m + 5} + \frac {2 \, a b c^{2} x^{m + 3}}{m + 3} + \frac {2 \, a^{2} c d x^{m + 3}}{m + 3} + \frac {a^{2} c^{2} x^{m + 1}}{m + 1} \] Input:
integrate(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="maxima")
Output:
b^2*d^2*x^(m + 9)/(m + 9) + 2*b^2*c*d*x^(m + 7)/(m + 7) + 2*a*b*d^2*x^(m + 7)/(m + 7) + b^2*c^2*x^(m + 5)/(m + 5) + 4*a*b*c*d*x^(m + 5)/(m + 5) + a^ 2*d^2*x^(m + 5)/(m + 5) + 2*a*b*c^2*x^(m + 3)/(m + 3) + 2*a^2*c*d*x^(m + 3 )/(m + 3) + a^2*c^2*x^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (109) = 218\).
Time = 0.14 (sec) , antiderivative size = 703, normalized size of antiderivative = 6.45 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {b^{2} d^{2} m^{4} x^{9} x^{m} + 16 \, b^{2} d^{2} m^{3} x^{9} x^{m} + 2 \, b^{2} c d m^{4} x^{7} x^{m} + 2 \, a b d^{2} m^{4} x^{7} x^{m} + 86 \, b^{2} d^{2} m^{2} x^{9} x^{m} + 36 \, b^{2} c d m^{3} x^{7} x^{m} + 36 \, a b d^{2} m^{3} x^{7} x^{m} + 176 \, b^{2} d^{2} m x^{9} x^{m} + b^{2} c^{2} m^{4} x^{5} x^{m} + 4 \, a b c d m^{4} x^{5} x^{m} + a^{2} d^{2} m^{4} x^{5} x^{m} + 208 \, b^{2} c d m^{2} x^{7} x^{m} + 208 \, a b d^{2} m^{2} x^{7} x^{m} + 105 \, b^{2} d^{2} x^{9} x^{m} + 20 \, b^{2} c^{2} m^{3} x^{5} x^{m} + 80 \, a b c d m^{3} x^{5} x^{m} + 20 \, a^{2} d^{2} m^{3} x^{5} x^{m} + 444 \, b^{2} c d m x^{7} x^{m} + 444 \, a b d^{2} m x^{7} x^{m} + 2 \, a b c^{2} m^{4} x^{3} x^{m} + 2 \, a^{2} c d m^{4} x^{3} x^{m} + 130 \, b^{2} c^{2} m^{2} x^{5} x^{m} + 520 \, a b c d m^{2} x^{5} x^{m} + 130 \, a^{2} d^{2} m^{2} x^{5} x^{m} + 270 \, b^{2} c d x^{7} x^{m} + 270 \, a b d^{2} x^{7} x^{m} + 44 \, a b c^{2} m^{3} x^{3} x^{m} + 44 \, a^{2} c d m^{3} x^{3} x^{m} + 300 \, b^{2} c^{2} m x^{5} x^{m} + 1200 \, a b c d m x^{5} x^{m} + 300 \, a^{2} d^{2} m x^{5} x^{m} + a^{2} c^{2} m^{4} x x^{m} + 328 \, a b c^{2} m^{2} x^{3} x^{m} + 328 \, a^{2} c d m^{2} x^{3} x^{m} + 189 \, b^{2} c^{2} x^{5} x^{m} + 756 \, a b c d x^{5} x^{m} + 189 \, a^{2} d^{2} x^{5} x^{m} + 24 \, a^{2} c^{2} m^{3} x x^{m} + 916 \, a b c^{2} m x^{3} x^{m} + 916 \, a^{2} c d m x^{3} x^{m} + 206 \, a^{2} c^{2} m^{2} x x^{m} + 630 \, a b c^{2} x^{3} x^{m} + 630 \, a^{2} c d x^{3} x^{m} + 744 \, a^{2} c^{2} m x x^{m} + 945 \, a^{2} c^{2} x x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="giac")
Output:
(b^2*d^2*m^4*x^9*x^m + 16*b^2*d^2*m^3*x^9*x^m + 2*b^2*c*d*m^4*x^7*x^m + 2* a*b*d^2*m^4*x^7*x^m + 86*b^2*d^2*m^2*x^9*x^m + 36*b^2*c*d*m^3*x^7*x^m + 36 *a*b*d^2*m^3*x^7*x^m + 176*b^2*d^2*m*x^9*x^m + b^2*c^2*m^4*x^5*x^m + 4*a*b *c*d*m^4*x^5*x^m + a^2*d^2*m^4*x^5*x^m + 208*b^2*c*d*m^2*x^7*x^m + 208*a*b *d^2*m^2*x^7*x^m + 105*b^2*d^2*x^9*x^m + 20*b^2*c^2*m^3*x^5*x^m + 80*a*b*c *d*m^3*x^5*x^m + 20*a^2*d^2*m^3*x^5*x^m + 444*b^2*c*d*m*x^7*x^m + 444*a*b* d^2*m*x^7*x^m + 2*a*b*c^2*m^4*x^3*x^m + 2*a^2*c*d*m^4*x^3*x^m + 130*b^2*c^ 2*m^2*x^5*x^m + 520*a*b*c*d*m^2*x^5*x^m + 130*a^2*d^2*m^2*x^5*x^m + 270*b^ 2*c*d*x^7*x^m + 270*a*b*d^2*x^7*x^m + 44*a*b*c^2*m^3*x^3*x^m + 44*a^2*c*d* m^3*x^3*x^m + 300*b^2*c^2*m*x^5*x^m + 1200*a*b*c*d*m*x^5*x^m + 300*a^2*d^2 *m*x^5*x^m + a^2*c^2*m^4*x*x^m + 328*a*b*c^2*m^2*x^3*x^m + 328*a^2*c*d*m^2 *x^3*x^m + 189*b^2*c^2*x^5*x^m + 756*a*b*c*d*x^5*x^m + 189*a^2*d^2*x^5*x^m + 24*a^2*c^2*m^3*x*x^m + 916*a*b*c^2*m*x^3*x^m + 916*a^2*c*d*m*x^3*x^m + 206*a^2*c^2*m^2*x*x^m + 630*a*b*c^2*x^3*x^m + 630*a^2*c*d*x^3*x^m + 744*a^ 2*c^2*m*x*x^m + 945*a^2*c^2*x*x^m)/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 168 9*m + 945)
Time = 1.59 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.77 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {x^m\,x^5\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {b^2\,d^2\,x^m\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^2\,c^2\,x\,x^m\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {2\,a\,c\,x^m\,x^3\,\left (a\,d+b\,c\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {2\,b\,d\,x^m\,x^7\,\left (a\,d+b\,c\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945} \] Input:
int(x^m*(a + b*x^2)^2*(c + d*x^2)^2,x)
Output:
(x^m*x^5*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d)*(300*m + 130*m^2 + 20*m^3 + m^4 + 189))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (b^2*d^2*x^m*x^ 9*(176*m + 86*m^2 + 16*m^3 + m^4 + 105))/(1689*m + 950*m^2 + 230*m^3 + 25* m^4 + m^5 + 945) + (a^2*c^2*x*x^m*(744*m + 206*m^2 + 24*m^3 + m^4 + 945))/ (1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (2*a*c*x^m*x^3*(a*d + b*c)*(458*m + 164*m^2 + 22*m^3 + m^4 + 315))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (2*b*d*x^m*x^7*(a*d + b*c)*(222*m + 104*m^2 + 18*m^ 3 + m^4 + 135))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945)
Time = 0.14 (sec) , antiderivative size = 567, normalized size of antiderivative = 5.20 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {x^{m} x \left (b^{2} d^{2} m^{4} x^{8}+16 b^{2} d^{2} m^{3} x^{8}+2 a b \,d^{2} m^{4} x^{6}+2 b^{2} c d \,m^{4} x^{6}+86 b^{2} d^{2} m^{2} x^{8}+36 a b \,d^{2} m^{3} x^{6}+36 b^{2} c d \,m^{3} x^{6}+176 b^{2} d^{2} m \,x^{8}+a^{2} d^{2} m^{4} x^{4}+4 a b c d \,m^{4} x^{4}+208 a b \,d^{2} m^{2} x^{6}+b^{2} c^{2} m^{4} x^{4}+208 b^{2} c d \,m^{2} x^{6}+105 b^{2} d^{2} x^{8}+20 a^{2} d^{2} m^{3} x^{4}+80 a b c d \,m^{3} x^{4}+444 a b \,d^{2} m \,x^{6}+20 b^{2} c^{2} m^{3} x^{4}+444 b^{2} c d m \,x^{6}+2 a^{2} c d \,m^{4} x^{2}+130 a^{2} d^{2} m^{2} x^{4}+2 a b \,c^{2} m^{4} x^{2}+520 a b c d \,m^{2} x^{4}+270 a b \,d^{2} x^{6}+130 b^{2} c^{2} m^{2} x^{4}+270 b^{2} c d \,x^{6}+44 a^{2} c d \,m^{3} x^{2}+300 a^{2} d^{2} m \,x^{4}+44 a b \,c^{2} m^{3} x^{2}+1200 a b c d m \,x^{4}+300 b^{2} c^{2} m \,x^{4}+a^{2} c^{2} m^{4}+328 a^{2} c d \,m^{2} x^{2}+189 a^{2} d^{2} x^{4}+328 a b \,c^{2} m^{2} x^{2}+756 a b c d \,x^{4}+189 b^{2} c^{2} x^{4}+24 a^{2} c^{2} m^{3}+916 a^{2} c d m \,x^{2}+916 a b \,c^{2} m \,x^{2}+206 a^{2} c^{2} m^{2}+630 a^{2} c d \,x^{2}+630 a b \,c^{2} x^{2}+744 a^{2} c^{2} m +945 a^{2} c^{2}\right )}{m^{5}+25 m^{4}+230 m^{3}+950 m^{2}+1689 m +945} \] Input:
int(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x)
Output:
(x**m*x*(a**2*c**2*m**4 + 24*a**2*c**2*m**3 + 206*a**2*c**2*m**2 + 744*a** 2*c**2*m + 945*a**2*c**2 + 2*a**2*c*d*m**4*x**2 + 44*a**2*c*d*m**3*x**2 + 328*a**2*c*d*m**2*x**2 + 916*a**2*c*d*m*x**2 + 630*a**2*c*d*x**2 + a**2*d* *2*m**4*x**4 + 20*a**2*d**2*m**3*x**4 + 130*a**2*d**2*m**2*x**4 + 300*a**2 *d**2*m*x**4 + 189*a**2*d**2*x**4 + 2*a*b*c**2*m**4*x**2 + 44*a*b*c**2*m** 3*x**2 + 328*a*b*c**2*m**2*x**2 + 916*a*b*c**2*m*x**2 + 630*a*b*c**2*x**2 + 4*a*b*c*d*m**4*x**4 + 80*a*b*c*d*m**3*x**4 + 520*a*b*c*d*m**2*x**4 + 120 0*a*b*c*d*m*x**4 + 756*a*b*c*d*x**4 + 2*a*b*d**2*m**4*x**6 + 36*a*b*d**2*m **3*x**6 + 208*a*b*d**2*m**2*x**6 + 444*a*b*d**2*m*x**6 + 270*a*b*d**2*x** 6 + b**2*c**2*m**4*x**4 + 20*b**2*c**2*m**3*x**4 + 130*b**2*c**2*m**2*x**4 + 300*b**2*c**2*m*x**4 + 189*b**2*c**2*x**4 + 2*b**2*c*d*m**4*x**6 + 36*b **2*c*d*m**3*x**6 + 208*b**2*c*d*m**2*x**6 + 444*b**2*c*d*m*x**6 + 270*b** 2*c*d*x**6 + b**2*d**2*m**4*x**8 + 16*b**2*d**2*m**3*x**8 + 86*b**2*d**2*m **2*x**8 + 176*b**2*d**2*m*x**8 + 105*b**2*d**2*x**8))/(m**5 + 25*m**4 + 2 30*m**3 + 950*m**2 + 1689*m + 945)