Integrand size = 24, antiderivative size = 82 \[ \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6 x^{12}}{12}+\frac {3}{7} a^5 b x^{14}+\frac {15}{16} a^4 b^2 x^{16}+\frac {10}{9} a^3 b^3 x^{18}+\frac {3}{4} a^2 b^4 x^{20}+\frac {3}{11} a b^5 x^{22}+\frac {b^6 x^{24}}{24} \] Output:
1/12*a^6*x^12+3/7*a^5*b*x^14+15/16*a^4*b^2*x^16+10/9*a^3*b^3*x^18+3/4*a^2* b^4*x^20+3/11*a*b^5*x^22+1/24*b^6*x^24
Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6 x^{12}}{12}+\frac {3}{7} a^5 b x^{14}+\frac {15}{16} a^4 b^2 x^{16}+\frac {10}{9} a^3 b^3 x^{18}+\frac {3}{4} a^2 b^4 x^{20}+\frac {3}{11} a b^5 x^{22}+\frac {b^6 x^{24}}{24} \] Input:
Integrate[x^11*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
(a^6*x^12)/12 + (3*a^5*b*x^14)/7 + (15*a^4*b^2*x^16)/16 + (10*a^3*b^3*x^18 )/9 + (3*a^2*b^4*x^20)/4 + (3*a*b^5*x^22)/11 + (b^6*x^24)/24
Time = 0.40 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1380, 27, 243, 49, 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.
\(\displaystyle \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int b^6 x^{11} \left (b x^2+a\right )^6dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int x^{11} \left (a+b x^2\right )^6dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int x^{10} \left (b x^2+a\right )^6dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (b^6 x^{22}+6 a b^5 x^{20}+15 a^2 b^4 x^{18}+20 a^3 b^3 x^{16}+15 a^4 b^2 x^{14}+6 a^5 b x^{12}+a^6 x^{10}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a^6 x^{12}}{6}+\frac {6}{7} a^5 b x^{14}+\frac {15}{8} a^4 b^2 x^{16}+\frac {20}{9} a^3 b^3 x^{18}+\frac {3}{2} a^2 b^4 x^{20}+\frac {6}{11} a b^5 x^{22}+\frac {b^6 x^{24}}{12}\right )\) |
Input:
Int[x^11*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
((a^6*x^12)/6 + (6*a^5*b*x^14)/7 + (15*a^4*b^2*x^16)/8 + (20*a^3*b^3*x^18) /9 + (3*a^2*b^4*x^20)/2 + (6*a*b^5*x^22)/11 + (b^6*x^24)/12)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {1}{12} a^{6} x^{12}+\frac {3}{7} a^{5} b \,x^{14}+\frac {15}{16} a^{4} b^{2} x^{16}+\frac {10}{9} a^{3} b^{3} x^{18}+\frac {3}{4} a^{2} b^{4} x^{20}+\frac {3}{11} a \,b^{5} x^{22}+\frac {1}{24} b^{6} x^{24}\) | \(69\) |
norman | \(\frac {1}{12} a^{6} x^{12}+\frac {3}{7} a^{5} b \,x^{14}+\frac {15}{16} a^{4} b^{2} x^{16}+\frac {10}{9} a^{3} b^{3} x^{18}+\frac {3}{4} a^{2} b^{4} x^{20}+\frac {3}{11} a \,b^{5} x^{22}+\frac {1}{24} b^{6} x^{24}\) | \(69\) |
risch | \(\frac {1}{12} a^{6} x^{12}+\frac {3}{7} a^{5} b \,x^{14}+\frac {15}{16} a^{4} b^{2} x^{16}+\frac {10}{9} a^{3} b^{3} x^{18}+\frac {3}{4} a^{2} b^{4} x^{20}+\frac {3}{11} a \,b^{5} x^{22}+\frac {1}{24} b^{6} x^{24}\) | \(69\) |
parallelrisch | \(\frac {1}{12} a^{6} x^{12}+\frac {3}{7} a^{5} b \,x^{14}+\frac {15}{16} a^{4} b^{2} x^{16}+\frac {10}{9} a^{3} b^{3} x^{18}+\frac {3}{4} a^{2} b^{4} x^{20}+\frac {3}{11} a \,b^{5} x^{22}+\frac {1}{24} b^{6} x^{24}\) | \(69\) |
gosper | \(\frac {x^{12} \left (462 b^{6} x^{12}+3024 a \,b^{5} x^{10}+8316 a^{2} b^{4} x^{8}+12320 a^{3} b^{3} x^{6}+10395 a^{4} b^{2} x^{4}+4752 a^{5} b \,x^{2}+924 a^{6}\right )}{11088}\) | \(71\) |
orering | \(\frac {x^{12} \left (462 b^{6} x^{12}+3024 a \,b^{5} x^{10}+8316 a^{2} b^{4} x^{8}+12320 a^{3} b^{3} x^{6}+10395 a^{4} b^{2} x^{4}+4752 a^{5} b \,x^{2}+924 a^{6}\right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}{11088 \left (b \,x^{2}+a \right )^{6}}\) | \(100\) |
Input:
int(x^11*(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/12*a^6*x^12+3/7*a^5*b*x^14+15/16*a^4*b^2*x^16+10/9*a^3*b^3*x^18+3/4*a^2* b^4*x^20+3/11*a*b^5*x^22+1/24*b^6*x^24
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{24} \, b^{6} x^{24} + \frac {3}{11} \, a b^{5} x^{22} + \frac {3}{4} \, a^{2} b^{4} x^{20} + \frac {10}{9} \, a^{3} b^{3} x^{18} + \frac {15}{16} \, a^{4} b^{2} x^{16} + \frac {3}{7} \, a^{5} b x^{14} + \frac {1}{12} \, a^{6} x^{12} \] Input:
integrate(x^11*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/24*b^6*x^24 + 3/11*a*b^5*x^22 + 3/4*a^2*b^4*x^20 + 10/9*a^3*b^3*x^18 + 1 5/16*a^4*b^2*x^16 + 3/7*a^5*b*x^14 + 1/12*a^6*x^12
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^{6} x^{12}}{12} + \frac {3 a^{5} b x^{14}}{7} + \frac {15 a^{4} b^{2} x^{16}}{16} + \frac {10 a^{3} b^{3} x^{18}}{9} + \frac {3 a^{2} b^{4} x^{20}}{4} + \frac {3 a b^{5} x^{22}}{11} + \frac {b^{6} x^{24}}{24} \] Input:
integrate(x**11*(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
a**6*x**12/12 + 3*a**5*b*x**14/7 + 15*a**4*b**2*x**16/16 + 10*a**3*b**3*x* *18/9 + 3*a**2*b**4*x**20/4 + 3*a*b**5*x**22/11 + b**6*x**24/24
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{24} \, b^{6} x^{24} + \frac {3}{11} \, a b^{5} x^{22} + \frac {3}{4} \, a^{2} b^{4} x^{20} + \frac {10}{9} \, a^{3} b^{3} x^{18} + \frac {15}{16} \, a^{4} b^{2} x^{16} + \frac {3}{7} \, a^{5} b x^{14} + \frac {1}{12} \, a^{6} x^{12} \] Input:
integrate(x^11*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/24*b^6*x^24 + 3/11*a*b^5*x^22 + 3/4*a^2*b^4*x^20 + 10/9*a^3*b^3*x^18 + 1 5/16*a^4*b^2*x^16 + 3/7*a^5*b*x^14 + 1/12*a^6*x^12
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{24} \, b^{6} x^{24} + \frac {3}{11} \, a b^{5} x^{22} + \frac {3}{4} \, a^{2} b^{4} x^{20} + \frac {10}{9} \, a^{3} b^{3} x^{18} + \frac {15}{16} \, a^{4} b^{2} x^{16} + \frac {3}{7} \, a^{5} b x^{14} + \frac {1}{12} \, a^{6} x^{12} \] Input:
integrate(x^11*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
1/24*b^6*x^24 + 3/11*a*b^5*x^22 + 3/4*a^2*b^4*x^20 + 10/9*a^3*b^3*x^18 + 1 5/16*a^4*b^2*x^16 + 3/7*a^5*b*x^14 + 1/12*a^6*x^12
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6\,x^{12}}{12}+\frac {3\,a^5\,b\,x^{14}}{7}+\frac {15\,a^4\,b^2\,x^{16}}{16}+\frac {10\,a^3\,b^3\,x^{18}}{9}+\frac {3\,a^2\,b^4\,x^{20}}{4}+\frac {3\,a\,b^5\,x^{22}}{11}+\frac {b^6\,x^{24}}{24} \] Input:
int(x^11*(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
(a^6*x^12)/12 + (b^6*x^24)/24 + (3*a^5*b*x^14)/7 + (3*a*b^5*x^22)/11 + (15 *a^4*b^2*x^16)/16 + (10*a^3*b^3*x^18)/9 + (3*a^2*b^4*x^20)/4
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {x^{12} \left (462 b^{6} x^{12}+3024 a \,b^{5} x^{10}+8316 a^{2} b^{4} x^{8}+12320 a^{3} b^{3} x^{6}+10395 a^{4} b^{2} x^{4}+4752 a^{5} b \,x^{2}+924 a^{6}\right )}{11088} \] Input:
int(x^11*(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(x**12*(924*a**6 + 4752*a**5*b*x**2 + 10395*a**4*b**2*x**4 + 12320*a**3*b* *3*x**6 + 8316*a**2*b**4*x**8 + 3024*a*b**5*x**10 + 462*b**6*x**12))/11088