Integrand size = 20, antiderivative size = 95 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {a^2 (A b-a B) \left (a+b x^2\right )^6}{12 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^7}{14 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^8}{16 b^4}+\frac {B \left (a+b x^2\right )^9}{18 b^4} \] Output:
1/12*a^2*(A*b-B*a)*(b*x^2+a)^6/b^4-1/14*a*(2*A*b-3*B*a)*(b*x^2+a)^7/b^4+1/ 16*(A*b-3*B*a)*(b*x^2+a)^8/b^4+1/18*B*(b*x^2+a)^9/b^4
Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {x^6 \left (168 a^5 A+126 a^4 (5 A b+a B) x^2+504 a^3 b (2 A b+a B) x^4+840 a^2 b^2 (A b+a B) x^6+360 a b^3 (A b+2 a B) x^8+63 b^4 (A b+5 a B) x^{10}+56 b^5 B x^{12}\right )}{1008} \] Input:
Integrate[x^5*(a + b*x^2)^5*(A + B*x^2),x]
Output:
(x^6*(168*a^5*A + 126*a^4*(5*A*b + a*B)*x^2 + 504*a^3*b*(2*A*b + a*B)*x^4 + 840*a^2*b^2*(A*b + a*B)*x^6 + 360*a*b^3*(A*b + 2*a*B)*x^8 + 63*b^4*(A*b + 5*a*B)*x^10 + 56*b^5*B*x^12))/1008
Time = 0.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {354, 85, 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.
\(\displaystyle \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int x^4 \left (b x^2+a\right )^5 \left (B x^2+A\right )dx^2\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \frac {1}{2} \int \left (\frac {B \left (b x^2+a\right )^8}{b^3}+\frac {(A b-3 a B) \left (b x^2+a\right )^7}{b^3}+\frac {a (3 a B-2 A b) \left (b x^2+a\right )^6}{b^3}-\frac {a^2 (a B-A b) \left (b x^2+a\right )^5}{b^3}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a^2 \left (a+b x^2\right )^6 (A b-a B)}{6 b^4}+\frac {\left (a+b x^2\right )^8 (A b-3 a B)}{8 b^4}-\frac {a \left (a+b x^2\right )^7 (2 A b-3 a B)}{7 b^4}+\frac {B \left (a+b x^2\right )^9}{9 b^4}\right )\) |
Input:
Int[x^5*(a + b*x^2)^5*(A + B*x^2),x]
Output:
((a^2*(A*b - a*B)*(a + b*x^2)^6)/(6*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^2)^ 7)/(7*b^4) + ((A*b - 3*a*B)*(a + b*x^2)^8)/(8*b^4) + (B*(a + b*x^2)^9)/(9* b^4))/2
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.40 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.26
method | result | size |
norman | \(\frac {a^{5} A \,x^{6}}{6}+\left (\frac {5}{8} a^{4} b A +\frac {1}{8} a^{5} B \right ) x^{8}+\left (a^{3} b^{2} A +\frac {1}{2} a^{4} b B \right ) x^{10}+\left (\frac {5}{6} a^{2} b^{3} A +\frac {5}{6} a^{3} b^{2} B \right ) x^{12}+\left (\frac {5}{14} a \,b^{4} A +\frac {5}{7} a^{2} b^{3} B \right ) x^{14}+\left (\frac {1}{16} b^{5} A +\frac {5}{16} a \,b^{4} B \right ) x^{16}+\frac {b^{5} B \,x^{18}}{18}\) | \(120\) |
default | \(\frac {b^{5} B \,x^{18}}{18}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{16}}{16}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{14}}{14}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{12}}{12}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{10}}{10}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{8}}{8}+\frac {a^{5} A \,x^{6}}{6}\) | \(124\) |
gosper | \(\frac {1}{6} a^{5} A \,x^{6}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {5}{6} x^{12} a^{2} b^{3} A +\frac {5}{6} x^{12} a^{3} b^{2} B +\frac {5}{14} x^{14} a \,b^{4} A +\frac {5}{7} x^{14} a^{2} b^{3} B +\frac {1}{16} x^{16} b^{5} A +\frac {5}{16} x^{16} a \,b^{4} B +\frac {1}{18} b^{5} B \,x^{18}\) | \(125\) |
risch | \(\frac {1}{6} a^{5} A \,x^{6}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {5}{6} x^{12} a^{2} b^{3} A +\frac {5}{6} x^{12} a^{3} b^{2} B +\frac {5}{14} x^{14} a \,b^{4} A +\frac {5}{7} x^{14} a^{2} b^{3} B +\frac {1}{16} x^{16} b^{5} A +\frac {5}{16} x^{16} a \,b^{4} B +\frac {1}{18} b^{5} B \,x^{18}\) | \(125\) |
parallelrisch | \(\frac {1}{6} a^{5} A \,x^{6}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {5}{6} x^{12} a^{2} b^{3} A +\frac {5}{6} x^{12} a^{3} b^{2} B +\frac {5}{14} x^{14} a \,b^{4} A +\frac {5}{7} x^{14} a^{2} b^{3} B +\frac {1}{16} x^{16} b^{5} A +\frac {5}{16} x^{16} a \,b^{4} B +\frac {1}{18} b^{5} B \,x^{18}\) | \(125\) |
orering | \(\frac {x^{6} \left (56 b^{5} B \,x^{12}+63 A \,b^{5} x^{10}+315 B a \,b^{4} x^{10}+360 a A \,b^{4} x^{8}+720 B \,a^{2} b^{3} x^{8}+840 a^{2} A \,b^{3} x^{6}+840 B \,a^{3} b^{2} x^{6}+1008 a^{3} A \,b^{2} x^{4}+504 B \,a^{4} b \,x^{4}+630 a^{4} A b \,x^{2}+126 B \,a^{5} x^{2}+168 a^{5} A \right )}{1008}\) | \(128\) |
Input:
int(x^5*(b*x^2+a)^5*(B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/6*a^5*A*x^6+(5/8*a^4*b*A+1/8*a^5*B)*x^8+(a^3*b^2*A+1/2*a^4*b*B)*x^10+(5/ 6*a^2*b^3*A+5/6*a^3*b^2*B)*x^12+(5/14*a*b^4*A+5/7*a^2*b^3*B)*x^14+(1/16*b^ 5*A+5/16*a*b^4*B)*x^16+1/18*b^5*B*x^18
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {1}{16} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{16} + \frac {5}{14} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{14} + \frac {5}{6} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{10} + \frac {1}{8} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{8} \] Input:
integrate(x^5*(b*x^2+a)^5*(B*x^2+A),x, algorithm="fricas")
Output:
1/18*B*b^5*x^18 + 1/16*(5*B*a*b^4 + A*b^5)*x^16 + 5/14*(2*B*a^2*b^3 + A*a* b^4)*x^14 + 5/6*(B*a^3*b^2 + A*a^2*b^3)*x^12 + 1/6*A*a^5*x^6 + 1/2*(B*a^4* b + 2*A*a^3*b^2)*x^10 + 1/8*(B*a^5 + 5*A*a^4*b)*x^8
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {A a^{5} x^{6}}{6} + \frac {B b^{5} x^{18}}{18} + x^{16} \left (\frac {A b^{5}}{16} + \frac {5 B a b^{4}}{16}\right ) + x^{14} \cdot \left (\frac {5 A a b^{4}}{14} + \frac {5 B a^{2} b^{3}}{7}\right ) + x^{12} \cdot \left (\frac {5 A a^{2} b^{3}}{6} + \frac {5 B a^{3} b^{2}}{6}\right ) + x^{10} \left (A a^{3} b^{2} + \frac {B a^{4} b}{2}\right ) + x^{8} \cdot \left (\frac {5 A a^{4} b}{8} + \frac {B a^{5}}{8}\right ) \] Input:
integrate(x**5*(b*x**2+a)**5*(B*x**2+A),x)
Output:
A*a**5*x**6/6 + B*b**5*x**18/18 + x**16*(A*b**5/16 + 5*B*a*b**4/16) + x**1 4*(5*A*a*b**4/14 + 5*B*a**2*b**3/7) + x**12*(5*A*a**2*b**3/6 + 5*B*a**3*b* *2/6) + x**10*(A*a**3*b**2 + B*a**4*b/2) + x**8*(5*A*a**4*b/8 + B*a**5/8)
Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {1}{16} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{16} + \frac {5}{14} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{14} + \frac {5}{6} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{10} + \frac {1}{8} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{8} \] Input:
integrate(x^5*(b*x^2+a)^5*(B*x^2+A),x, algorithm="maxima")
Output:
1/18*B*b^5*x^18 + 1/16*(5*B*a*b^4 + A*b^5)*x^16 + 5/14*(2*B*a^2*b^3 + A*a* b^4)*x^14 + 5/6*(B*a^3*b^2 + A*a^2*b^3)*x^12 + 1/6*A*a^5*x^6 + 1/2*(B*a^4* b + 2*A*a^3*b^2)*x^10 + 1/8*(B*a^5 + 5*A*a^4*b)*x^8
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {5}{16} \, B a b^{4} x^{16} + \frac {1}{16} \, A b^{5} x^{16} + \frac {5}{7} \, B a^{2} b^{3} x^{14} + \frac {5}{14} \, A a b^{4} x^{14} + \frac {5}{6} \, B a^{3} b^{2} x^{12} + \frac {5}{6} \, A a^{2} b^{3} x^{12} + \frac {1}{2} \, B a^{4} b x^{10} + A a^{3} b^{2} x^{10} + \frac {1}{8} \, B a^{5} x^{8} + \frac {5}{8} \, A a^{4} b x^{8} + \frac {1}{6} \, A a^{5} x^{6} \] Input:
integrate(x^5*(b*x^2+a)^5*(B*x^2+A),x, algorithm="giac")
Output:
1/18*B*b^5*x^18 + 5/16*B*a*b^4*x^16 + 1/16*A*b^5*x^16 + 5/7*B*a^2*b^3*x^14 + 5/14*A*a*b^4*x^14 + 5/6*B*a^3*b^2*x^12 + 5/6*A*a^2*b^3*x^12 + 1/2*B*a^4 *b*x^10 + A*a^3*b^2*x^10 + 1/8*B*a^5*x^8 + 5/8*A*a^4*b*x^8 + 1/6*A*a^5*x^6
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=x^8\,\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )+x^{16}\,\left (\frac {A\,b^5}{16}+\frac {5\,B\,a\,b^4}{16}\right )+\frac {A\,a^5\,x^6}{6}+\frac {B\,b^5\,x^{18}}{18}+\frac {5\,a^2\,b^2\,x^{12}\,\left (A\,b+B\,a\right )}{6}+\frac {a^3\,b\,x^{10}\,\left (2\,A\,b+B\,a\right )}{2}+\frac {5\,a\,b^3\,x^{14}\,\left (A\,b+2\,B\,a\right )}{14} \] Input:
int(x^5*(A + B*x^2)*(a + b*x^2)^5,x)
Output:
x^8*((B*a^5)/8 + (5*A*a^4*b)/8) + x^16*((A*b^5)/16 + (5*B*a*b^4)/16) + (A* a^5*x^6)/6 + (B*b^5*x^18)/18 + (5*a^2*b^2*x^12*(A*b + B*a))/6 + (a^3*b*x^1 0*(2*A*b + B*a))/2 + (5*a*b^3*x^14*(A*b + 2*B*a))/14
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {x^{6} \left (28 b^{6} x^{12}+189 a \,b^{5} x^{10}+540 a^{2} b^{4} x^{8}+840 a^{3} b^{3} x^{6}+756 a^{4} b^{2} x^{4}+378 a^{5} b \,x^{2}+84 a^{6}\right )}{504} \] Input:
int(x^5*(b*x^2+a)^5*(B*x^2+A),x)
Output:
(x**6*(84*a**6 + 378*a**5*b*x**2 + 756*a**4*b**2*x**4 + 840*a**3*b**3*x**6 + 540*a**2*b**4*x**8 + 189*a*b**5*x**10 + 28*b**6*x**12))/504