Integrand size = 22, antiderivative size = 103 \[ \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx=a^3 A x+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {1}{5} a \left (3 A b^2+a (3 b B+a C)\right ) x^5+\frac {1}{7} b \left (A b^2+3 a (b B+a C)\right ) x^7+\frac {1}{9} b^2 (b B+3 a C) x^9+\frac {1}{11} b^3 C x^{11} \] Output:
a^3*A*x+1/3*a^2*(3*A*b+B*a)*x^3+1/5*a*(3*A*b^2+a*(3*B*b+C*a))*x^5+1/7*b*(A *b^2+3*a*(B*b+C*a))*x^7+1/9*b^2*(B*b+3*C*a)*x^9+1/11*b^3*C*x^11
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx=a^3 A x+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {1}{5} a \left (3 A b^2+3 a b B+a^2 C\right ) x^5+\frac {1}{7} b \left (A b^2+3 a b B+3 a^2 C\right ) x^7+\frac {1}{9} b^2 (b B+3 a C) x^9+\frac {1}{11} b^3 C x^{11} \] Input:
Integrate[(a + b*x^2)^3*(A + B*x^2 + C*x^4),x]
Output:
a^3*A*x + (a^2*(3*A*b + a*B)*x^3)/3 + (a*(3*A*b^2 + 3*a*b*B + a^2*C)*x^5)/ 5 + (b*(A*b^2 + 3*a*b*B + 3*a^2*C)*x^7)/7 + (b^2*(b*B + 3*a*C)*x^9)/9 + (b ^3*C*x^11)/11
Time = 0.28 (sec) , antiderivative size = 103, 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.
\(\displaystyle \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx\) |
\(\Big \downarrow \) 1467 |
\(\displaystyle \int \left (a^3 A+a^2 x^2 (a B+3 A b)+b x^6 \left (3 a (a C+b B)+A b^2\right )+a x^4 \left (a (a C+3 b B)+3 A b^2\right )+b^2 x^8 (3 a C+b B)+b^3 C x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^3 A x+\frac {1}{3} a^2 x^3 (a B+3 A b)+\frac {1}{7} b x^7 \left (3 a (a C+b B)+A b^2\right )+\frac {1}{5} a x^5 \left (a (a C+3 b B)+3 A b^2\right )+\frac {1}{9} b^2 x^9 (3 a C+b B)+\frac {1}{11} b^3 C x^{11}\) |
Input:
Int[(a + b*x^2)^3*(A + B*x^2 + C*x^4),x]
Output:
a^3*A*x + (a^2*(3*A*b + a*B)*x^3)/3 + (a*(3*A*b^2 + a*(3*b*B + a*C))*x^5)/ 5 + (b*(A*b^2 + 3*a*(b*B + a*C))*x^7)/7 + (b^2*(b*B + 3*a*C)*x^9)/9 + (b^3 *C*x^11)/11
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 = 0.48 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\frac {b^{3} C \,x^{11}}{11}+\left (\frac {1}{9} B \,b^{3}+\frac {1}{3} a C \,b^{2}\right ) x^{9}+\left (\frac {1}{7} b^{3} A +\frac {3}{7} a \,b^{2} B +\frac {3}{7} a^{2} b C \right ) x^{7}+\left (\frac {3}{5} a \,b^{2} A +\frac {3}{5} a^{2} b B +\frac {1}{5} C \,a^{3}\right ) x^{5}+\left (a^{2} b A +\frac {1}{3} a^{3} B \right ) x^{3}+a^{3} A x\) | \(102\) |
default | \(\frac {b^{3} C \,x^{11}}{11}+\frac {\left (B \,b^{3}+3 a C \,b^{2}\right ) x^{9}}{9}+\frac {\left (b^{3} A +3 a \,b^{2} B +3 a^{2} b C \right ) x^{7}}{7}+\frac {\left (3 a \,b^{2} A +3 a^{2} b B +C \,a^{3}\right ) x^{5}}{5}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{3}}{3}+a^{3} A x\) | \(103\) |
gosper | \(\frac {1}{11} b^{3} C \,x^{11}+\frac {1}{9} b^{3} B \,x^{9}+\frac {1}{3} x^{9} a C \,b^{2}+\frac {1}{7} A \,b^{3} x^{7}+\frac {3}{7} x^{7} a \,b^{2} B +\frac {3}{7} x^{7} a^{2} b C +\frac {3}{5} a A \,b^{2} x^{5}+\frac {3}{5} x^{5} a^{2} b B +\frac {1}{5} x^{5} C \,a^{3}+a^{2} A b \,x^{3}+\frac {1}{3} x^{3} a^{3} B +a^{3} A x\) | \(112\) |
risch | \(\frac {1}{11} b^{3} C \,x^{11}+\frac {1}{9} b^{3} B \,x^{9}+\frac {1}{3} x^{9} a C \,b^{2}+\frac {1}{7} A \,b^{3} x^{7}+\frac {3}{7} x^{7} a \,b^{2} B +\frac {3}{7} x^{7} a^{2} b C +\frac {3}{5} a A \,b^{2} x^{5}+\frac {3}{5} x^{5} a^{2} b B +\frac {1}{5} x^{5} C \,a^{3}+a^{2} A b \,x^{3}+\frac {1}{3} x^{3} a^{3} B +a^{3} A x\) | \(112\) |
parallelrisch | \(\frac {1}{11} b^{3} C \,x^{11}+\frac {1}{9} b^{3} B \,x^{9}+\frac {1}{3} x^{9} a C \,b^{2}+\frac {1}{7} A \,b^{3} x^{7}+\frac {3}{7} x^{7} a \,b^{2} B +\frac {3}{7} x^{7} a^{2} b C +\frac {3}{5} a A \,b^{2} x^{5}+\frac {3}{5} x^{5} a^{2} b B +\frac {1}{5} x^{5} C \,a^{3}+a^{2} A b \,x^{3}+\frac {1}{3} x^{3} a^{3} B +a^{3} A x\) | \(112\) |
orering | \(\frac {x \left (315 b^{3} C \,x^{10}+385 b^{3} B \,x^{8}+1155 C a \,b^{2} x^{8}+495 A \,b^{3} x^{6}+1485 B a \,b^{2} x^{6}+1485 C \,a^{2} b \,x^{6}+2079 a A \,b^{2} x^{4}+2079 B \,a^{2} b \,x^{4}+693 C \,a^{3} x^{4}+3465 a^{2} A b \,x^{2}+1155 B \,a^{3} x^{2}+3465 a^{3} A \right )}{3465}\) | \(116\) |
Input:
int((b*x^2+a)^3*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/11*b^3*C*x^11+(1/9*B*b^3+1/3*a*C*b^2)*x^9+(1/7*b^3*A+3/7*a*b^2*B+3/7*a^2 *b*C)*x^7+(3/5*a*b^2*A+3/5*a^2*b*B+1/5*C*a^3)*x^5+(a^2*b*A+1/3*a^3*B)*x^3+ a^3*A*x
Time = 0.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx=\frac {1}{11} \, C b^{3} x^{11} + \frac {1}{9} \, {\left (3 \, C a b^{2} + B b^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} x^{7} + \frac {1}{5} \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} x^{5} + A a^{3} x + \frac {1}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \] Input:
integrate((b*x^2+a)^3*(C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
1/11*C*b^3*x^11 + 1/9*(3*C*a*b^2 + B*b^3)*x^9 + 1/7*(3*C*a^2*b + 3*B*a*b^2 + A*b^3)*x^7 + 1/5*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*x^5 + A*a^3*x + 1/3*(B *a^3 + 3*A*a^2*b)*x^3
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx=A a^{3} x + \frac {C b^{3} x^{11}}{11} + x^{9} \left (\frac {B b^{3}}{9} + \frac {C a b^{2}}{3}\right ) + x^{7} \left (\frac {A b^{3}}{7} + \frac {3 B a b^{2}}{7} + \frac {3 C a^{2} b}{7}\right ) + x^{5} \cdot \left (\frac {3 A a b^{2}}{5} + \frac {3 B a^{2} b}{5} + \frac {C a^{3}}{5}\right ) + x^{3} \left (A a^{2} b + \frac {B a^{3}}{3}\right ) \] Input:
integrate((b*x**2+a)**3*(C*x**4+B*x**2+A),x)
Output:
A*a**3*x + C*b**3*x**11/11 + x**9*(B*b**3/9 + C*a*b**2/3) + x**7*(A*b**3/7 + 3*B*a*b**2/7 + 3*C*a**2*b/7) + x**5*(3*A*a*b**2/5 + 3*B*a**2*b/5 + C*a* *3/5) + x**3*(A*a**2*b + B*a**3/3)
Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx=\frac {1}{11} \, C b^{3} x^{11} + \frac {1}{9} \, {\left (3 \, C a b^{2} + B b^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} x^{7} + \frac {1}{5} \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} x^{5} + A a^{3} x + \frac {1}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \] Input:
integrate((b*x^2+a)^3*(C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
1/11*C*b^3*x^11 + 1/9*(3*C*a*b^2 + B*b^3)*x^9 + 1/7*(3*C*a^2*b + 3*B*a*b^2 + A*b^3)*x^7 + 1/5*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*x^5 + A*a^3*x + 1/3*(B *a^3 + 3*A*a^2*b)*x^3
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08 \[ \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx=\frac {1}{11} \, C b^{3} x^{11} + \frac {1}{3} \, C a b^{2} x^{9} + \frac {1}{9} \, B b^{3} x^{9} + \frac {3}{7} \, C a^{2} b x^{7} + \frac {3}{7} \, B a b^{2} x^{7} + \frac {1}{7} \, A b^{3} x^{7} + \frac {1}{5} \, C a^{3} x^{5} + \frac {3}{5} \, B a^{2} b x^{5} + \frac {3}{5} \, A a b^{2} x^{5} + \frac {1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + A a^{3} x \] Input:
integrate((b*x^2+a)^3*(C*x^4+B*x^2+A),x, algorithm="giac")
Output:
1/11*C*b^3*x^11 + 1/3*C*a*b^2*x^9 + 1/9*B*b^3*x^9 + 3/7*C*a^2*b*x^7 + 3/7* B*a*b^2*x^7 + 1/7*A*b^3*x^7 + 1/5*C*a^3*x^5 + 3/5*B*a^2*b*x^5 + 3/5*A*a*b^ 2*x^5 + 1/3*B*a^3*x^3 + A*a^2*b*x^3 + A*a^3*x
Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx=x^5\,\left (\frac {C\,a^3}{5}+\frac {3\,B\,a^2\,b}{5}+\frac {3\,A\,a\,b^2}{5}\right )+x^7\,\left (\frac {3\,C\,a^2\,b}{7}+\frac {3\,B\,a\,b^2}{7}+\frac {A\,b^3}{7}\right )+x^3\,\left (\frac {B\,a^3}{3}+A\,b\,a^2\right )+x^9\,\left (\frac {B\,b^3}{9}+\frac {C\,a\,b^2}{3}\right )+\frac {C\,b^3\,x^{11}}{11}+A\,a^3\,x \] Input:
int((a + b*x^2)^3*(A + B*x^2 + C*x^4),x)
Output:
x^5*((C*a^3)/5 + (3*A*a*b^2)/5 + (3*B*a^2*b)/5) + x^7*((A*b^3)/7 + (3*B*a* b^2)/7 + (3*C*a^2*b)/7) + x^3*((B*a^3)/3 + A*a^2*b) + x^9*((B*b^3)/9 + (C* a*b^2)/3) + (C*b^3*x^11)/11 + A*a^3*x
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82 \[ \int \left (a+b x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx=\frac {x \left (315 b^{3} c \,x^{10}+1155 a \,b^{2} c \,x^{8}+385 b^{4} x^{8}+1485 a^{2} b c \,x^{6}+1980 a \,b^{3} x^{6}+693 a^{3} c \,x^{4}+4158 a^{2} b^{2} x^{4}+4620 a^{3} b \,x^{2}+3465 a^{4}\right )}{3465} \] Input:
int((b*x^2+a)^3*(C*x^4+B*x^2+A),x)
Output:
(x*(3465*a**4 + 4620*a**3*b*x**2 + 693*a**3*c*x**4 + 4158*a**2*b**2*x**4 + 1485*a**2*b*c*x**6 + 1980*a*b**3*x**6 + 1155*a*b**2*c*x**8 + 385*b**4*x** 8 + 315*b**3*c*x**10))/3465