Integrand size = 22, antiderivative size = 161 \[ \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=a^3 A x+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {3}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^5+\frac {1}{7} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^7+\frac {1}{9} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^9+\frac {3}{11} c \left (b^2 B+A b c+a B c\right ) x^{11}+\frac {1}{13} c^2 (3 b B+A c) x^{13}+\frac {1}{15} B c^3 x^{15} \]
a^3*A*x+1/3*a^2*(3*A*b+B*a)*x^3+3/5*a*(a*b*B+A*(a*c+b^2))*x^5+1/7*(3*a*B*( a*c+b^2)+A*(6*a*b*c+b^3))*x^7+1/9*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)*x^ 9+3/11*c*(A*b*c+B*a*c+B*b^2)*x^11+1/13*c^2*(A*c+3*B*b)*x^13+1/15*B*c^3*x^1 5
Time = 0.04 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00 \[ \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=a^3 A x+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {3}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^5+\frac {1}{7} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^7+\frac {1}{9} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^9+\frac {3}{11} c \left (b^2 B+A b c+a B c\right ) x^{11}+\frac {1}{13} c^2 (3 b B+A c) x^{13}+\frac {1}{15} B c^3 x^{15} \]
a^3*A*x + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^5)/5 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^7)/7 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^9)/9 + (3*c*(b^2*B + A*b*c + a*B*c)*x^11)/11 + (c ^2*(3*b*B + A*c)*x^13)/13 + (B*c^3*x^15)/15
Time = 0.32 (sec) , antiderivative size = 161, 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 = {1467, 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.
| \(\displaystyle \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \int \left (a^3 A+a^2 x^2 (a B+3 A b)+3 c x^{10} \left (a B c+A b c+b^2 B\right )+3 a x^4 \left (A \left (a c+b^2\right )+a b B\right )+x^8 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+x^6 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+c^2 x^{12} (A c+3 b B)+B c^3 x^{14}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 A x+\frac {1}{3} a^2 x^3 (a B+3 A b)+\frac {3}{11} c x^{11} \left (a B c+A b c+b^2 B\right )+\frac {3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{9} x^9 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{7} x^7 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{13} c^2 x^{13} (A c+3 b B)+\frac {1}{15} B c^3 x^{15}\) |
a^3*A*x + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^5)/5 + ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^7)/7 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^9)/9 + (3*c*(b^2*B + A*b*c + a*B*c)*x^11)/11 + (c ^2*(3*b*B + A*c)*x^13)/13 + (B*c^3*x^15)/15
3.1.98.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.03
| method | result | size |
| norman | \(a^{3} A x +\left (A \,a^{2} b +\frac {1}{3} B \,a^{3}\right ) x^{3}+\left (\frac {3}{5} A c \,a^{2}+\frac {3}{5} A a \,b^{2}+\frac {3}{5} B \,a^{2} b \right ) x^{5}+\left (\frac {6}{7} A a b c +\frac {1}{7} A \,b^{3}+\frac {3}{7} a^{2} B c +\frac {3}{7} B a \,b^{2}\right ) x^{7}+\left (\frac {1}{3} A a \,c^{2}+\frac {1}{3} A \,b^{2} c +\frac {2}{3} B a b c +\frac {1}{9} B \,b^{3}\right ) x^{9}+\left (\frac {3}{11} A b \,c^{2}+\frac {3}{11} B a \,c^{2}+\frac {3}{11} B \,b^{2} c \right ) x^{11}+\left (\frac {1}{13} A \,c^{3}+\frac {3}{13} B b \,c^{2}\right ) x^{13}+\frac {B \,c^{3} x^{15}}{15}\) | \(166\) |
| gosper | \(a^{3} A x +x^{3} A \,a^{2} b +\frac {1}{3} x^{3} B \,a^{3}+\frac {3}{5} x^{5} A c \,a^{2}+\frac {3}{5} x^{5} A a \,b^{2}+\frac {3}{5} x^{5} B \,a^{2} b +\frac {6}{7} x^{7} A a b c +\frac {1}{7} x^{7} A \,b^{3}+\frac {3}{7} x^{7} a^{2} B c +\frac {3}{7} x^{7} B a \,b^{2}+\frac {1}{3} x^{9} A a \,c^{2}+\frac {1}{3} x^{9} A \,b^{2} c +\frac {2}{3} x^{9} B a b c +\frac {1}{9} x^{9} B \,b^{3}+\frac {3}{11} x^{11} A b \,c^{2}+\frac {3}{11} x^{11} B a \,c^{2}+\frac {3}{11} x^{11} B \,b^{2} c +\frac {1}{13} x^{13} A \,c^{3}+\frac {3}{13} x^{13} B b \,c^{2}+\frac {1}{15} B \,c^{3} x^{15}\) | \(190\) |
| risch | \(a^{3} A x +x^{3} A \,a^{2} b +\frac {1}{3} x^{3} B \,a^{3}+\frac {3}{5} x^{5} A c \,a^{2}+\frac {3}{5} x^{5} A a \,b^{2}+\frac {3}{5} x^{5} B \,a^{2} b +\frac {6}{7} x^{7} A a b c +\frac {1}{7} x^{7} A \,b^{3}+\frac {3}{7} x^{7} a^{2} B c +\frac {3}{7} x^{7} B a \,b^{2}+\frac {1}{3} x^{9} A a \,c^{2}+\frac {1}{3} x^{9} A \,b^{2} c +\frac {2}{3} x^{9} B a b c +\frac {1}{9} x^{9} B \,b^{3}+\frac {3}{11} x^{11} A b \,c^{2}+\frac {3}{11} x^{11} B a \,c^{2}+\frac {3}{11} x^{11} B \,b^{2} c +\frac {1}{13} x^{13} A \,c^{3}+\frac {3}{13} x^{13} B b \,c^{2}+\frac {1}{15} B \,c^{3} x^{15}\) | \(190\) |
| parallelrisch | \(a^{3} A x +x^{3} A \,a^{2} b +\frac {1}{3} x^{3} B \,a^{3}+\frac {3}{5} x^{5} A c \,a^{2}+\frac {3}{5} x^{5} A a \,b^{2}+\frac {3}{5} x^{5} B \,a^{2} b +\frac {6}{7} x^{7} A a b c +\frac {1}{7} x^{7} A \,b^{3}+\frac {3}{7} x^{7} a^{2} B c +\frac {3}{7} x^{7} B a \,b^{2}+\frac {1}{3} x^{9} A a \,c^{2}+\frac {1}{3} x^{9} A \,b^{2} c +\frac {2}{3} x^{9} B a b c +\frac {1}{9} x^{9} B \,b^{3}+\frac {3}{11} x^{11} A b \,c^{2}+\frac {3}{11} x^{11} B a \,c^{2}+\frac {3}{11} x^{11} B \,b^{2} c +\frac {1}{13} x^{13} A \,c^{3}+\frac {3}{13} x^{13} B b \,c^{2}+\frac {1}{15} B \,c^{3} x^{15}\) | \(190\) |
| default | \(\frac {B \,c^{3} x^{15}}{15}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{13}}{13}+\frac {\left (3 A b \,c^{2}+B \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{11}}{11}+\frac {\left (A \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{9}}{9}+\frac {\left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )\right ) x^{7}}{7}+\frac {\left (A \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+3 B \,a^{2} b \right ) x^{5}}{5}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) x^{3}}{3}+a^{3} A x\) | \(223\) |
a^3*A*x+(A*a^2*b+1/3*B*a^3)*x^3+(3/5*A*c*a^2+3/5*A*a*b^2+3/5*B*a^2*b)*x^5+ (6/7*A*a*b*c+1/7*A*b^3+3/7*a^2*B*c+3/7*B*a*b^2)*x^7+(1/3*A*a*c^2+1/3*A*b^2 *c+2/3*B*a*b*c+1/9*B*b^3)*x^9+(3/11*A*b*c^2+3/11*B*a*c^2+3/11*B*b^2*c)*x^1 1+(1/13*A*c^3+3/13*B*b*c^2)*x^13+1/15*B*c^3*x^15
Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01 \[ \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{15} \, B c^{3} x^{15} + \frac {1}{13} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{13} + \frac {3}{11} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{9} + \frac {1}{7} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{7} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + A a^{3} x + \frac {1}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \]
1/15*B*c^3*x^15 + 1/13*(3*B*b*c^2 + A*c^3)*x^13 + 3/11*(B*b^2*c + (B*a + A *b)*c^2)*x^11 + 1/9*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^9 + 1/7* (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^7 + 3/5*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^5 + A*a^3*x + 1/3*(B*a^3 + 3*A*a^2*b)*x^3
Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.24 \[ \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=A a^{3} x + \frac {B c^{3} x^{15}}{15} + x^{13} \left (\frac {A c^{3}}{13} + \frac {3 B b c^{2}}{13}\right ) + x^{11} \cdot \left (\frac {3 A b c^{2}}{11} + \frac {3 B a c^{2}}{11} + \frac {3 B b^{2} c}{11}\right ) + x^{9} \left (\frac {A a c^{2}}{3} + \frac {A b^{2} c}{3} + \frac {2 B a b c}{3} + \frac {B b^{3}}{9}\right ) + x^{7} \cdot \left (\frac {6 A a b c}{7} + \frac {A b^{3}}{7} + \frac {3 B a^{2} c}{7} + \frac {3 B a b^{2}}{7}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} c}{5} + \frac {3 A a b^{2}}{5} + \frac {3 B a^{2} b}{5}\right ) + x^{3} \left (A a^{2} b + \frac {B a^{3}}{3}\right ) \]
A*a**3*x + B*c**3*x**15/15 + x**13*(A*c**3/13 + 3*B*b*c**2/13) + x**11*(3* A*b*c**2/11 + 3*B*a*c**2/11 + 3*B*b**2*c/11) + x**9*(A*a*c**2/3 + A*b**2*c /3 + 2*B*a*b*c/3 + B*b**3/9) + x**7*(6*A*a*b*c/7 + A*b**3/7 + 3*B*a**2*c/7 + 3*B*a*b**2/7) + x**5*(3*A*a**2*c/5 + 3*A*a*b**2/5 + 3*B*a**2*b/5) + x** 3*(A*a**2*b + B*a**3/3)
Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01 \[ \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{15} \, B c^{3} x^{15} + \frac {1}{13} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{13} + \frac {3}{11} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{9} + \frac {1}{7} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{7} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + A a^{3} x + \frac {1}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \]
1/15*B*c^3*x^15 + 1/13*(3*B*b*c^2 + A*c^3)*x^13 + 3/11*(B*b^2*c + (B*a + A *b)*c^2)*x^11 + 1/9*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^9 + 1/7* (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^7 + 3/5*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^5 + A*a^3*x + 1/3*(B*a^3 + 3*A*a^2*b)*x^3
Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.17 \[ \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{15} \, B c^{3} x^{15} + \frac {3}{13} \, B b c^{2} x^{13} + \frac {1}{13} \, A c^{3} x^{13} + \frac {3}{11} \, B b^{2} c x^{11} + \frac {3}{11} \, B a c^{2} x^{11} + \frac {3}{11} \, A b c^{2} x^{11} + \frac {1}{9} \, B b^{3} x^{9} + \frac {2}{3} \, B a b c x^{9} + \frac {1}{3} \, A b^{2} c x^{9} + \frac {1}{3} \, A a c^{2} x^{9} + \frac {3}{7} \, B a b^{2} x^{7} + \frac {1}{7} \, A b^{3} x^{7} + \frac {3}{7} \, B a^{2} c x^{7} + \frac {6}{7} \, A a b c x^{7} + \frac {3}{5} \, B a^{2} b x^{5} + \frac {3}{5} \, A a b^{2} x^{5} + \frac {3}{5} \, A a^{2} c x^{5} + \frac {1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + A a^{3} x \]
1/15*B*c^3*x^15 + 3/13*B*b*c^2*x^13 + 1/13*A*c^3*x^13 + 3/11*B*b^2*c*x^11 + 3/11*B*a*c^2*x^11 + 3/11*A*b*c^2*x^11 + 1/9*B*b^3*x^9 + 2/3*B*a*b*c*x^9 + 1/3*A*b^2*c*x^9 + 1/3*A*a*c^2*x^9 + 3/7*B*a*b^2*x^7 + 1/7*A*b^3*x^7 + 3/ 7*B*a^2*c*x^7 + 6/7*A*a*b*c*x^7 + 3/5*B*a^2*b*x^5 + 3/5*A*a*b^2*x^5 + 3/5* A*a^2*c*x^5 + 1/3*B*a^3*x^3 + A*a^2*b*x^3 + A*a^3*x
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.02 \[ \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx=x^7\,\left (\frac {3\,B\,c\,a^2}{7}+\frac {3\,B\,a\,b^2}{7}+\frac {6\,A\,c\,a\,b}{7}+\frac {A\,b^3}{7}\right )+x^9\,\left (\frac {B\,b^3}{9}+\frac {A\,b^2\,c}{3}+\frac {2\,B\,a\,b\,c}{3}+\frac {A\,a\,c^2}{3}\right )+x^3\,\left (\frac {B\,a^3}{3}+A\,b\,a^2\right )+x^{13}\,\left (\frac {A\,c^3}{13}+\frac {3\,B\,b\,c^2}{13}\right )+x^5\,\left (\frac {3\,B\,a^2\,b}{5}+\frac {3\,A\,c\,a^2}{5}+\frac {3\,A\,a\,b^2}{5}\right )+x^{11}\,\left (\frac {3\,B\,b^2\,c}{11}+\frac {3\,A\,b\,c^2}{11}+\frac {3\,B\,a\,c^2}{11}\right )+\frac {B\,c^3\,x^{15}}{15}+A\,a^3\,x \]
x^7*((A*b^3)/7 + (3*B*a*b^2)/7 + (3*B*a^2*c)/7 + (6*A*a*b*c)/7) + x^9*((B* b^3)/9 + (A*a*c^2)/3 + (A*b^2*c)/3 + (2*B*a*b*c)/3) + x^3*((B*a^3)/3 + A*a ^2*b) + x^13*((A*c^3)/13 + (3*B*b*c^2)/13) + x^5*((3*A*a*b^2)/5 + (3*A*a^2 *c)/5 + (3*B*a^2*b)/5) + x^11*((3*A*b*c^2)/11 + (3*B*a*c^2)/11 + (3*B*b^2* c)/11) + (B*c^3*x^15)/15 + A*a^3*x