Integrand size = 24, antiderivative size = 91 \[ \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^4 \left (a+b x^2\right )^7}{14 b^5}-\frac {a^3 \left (a+b x^2\right )^8}{4 b^5}+\frac {a^2 \left (a+b x^2\right )^9}{3 b^5}-\frac {a \left (a+b x^2\right )^{10}}{5 b^5}+\frac {\left (a+b x^2\right )^{11}}{22 b^5} \] Output:
1/14*a^4*(b*x^2+a)^7/b^5-1/4*a^3*(b*x^2+a)^8/b^5+1/3*a^2*(b*x^2+a)^9/b^5-1 /5*a*(b*x^2+a)^10/b^5+1/22*(b*x^2+a)^11/b^5
Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90 \[ \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6 x^{10}}{10}+\frac {1}{2} a^5 b x^{12}+\frac {15}{14} a^4 b^2 x^{14}+\frac {5}{4} a^3 b^3 x^{16}+\frac {5}{6} a^2 b^4 x^{18}+\frac {3}{10} a b^5 x^{20}+\frac {b^6 x^{22}}{22} \] Input:
Integrate[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
(a^6*x^10)/10 + (a^5*b*x^12)/2 + (15*a^4*b^2*x^14)/14 + (5*a^3*b^3*x^16)/4 + (5*a^2*b^4*x^18)/6 + (3*a*b^5*x^20)/10 + (b^6*x^22)/22
Time = 0.41 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int b^6 x^9 \left (b x^2+a\right )^6dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int x^9 \left (a+b x^2\right )^6dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int x^8 \left (b x^2+a\right )^6dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\left (b x^2+a\right )^{10}}{b^4}-\frac {4 a \left (b x^2+a\right )^9}{b^4}+\frac {6 a^2 \left (b x^2+a\right )^8}{b^4}-\frac {4 a^3 \left (b x^2+a\right )^7}{b^4}+\frac {a^4 \left (b x^2+a\right )^6}{b^4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a^4 \left (a+b x^2\right )^7}{7 b^5}-\frac {a^3 \left (a+b x^2\right )^8}{2 b^5}+\frac {2 a^2 \left (a+b x^2\right )^9}{3 b^5}+\frac {\left (a+b x^2\right )^{11}}{11 b^5}-\frac {2 a \left (a+b x^2\right )^{10}}{5 b^5}\right )\) |
Input:
Int[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
((a^4*(a + b*x^2)^7)/(7*b^5) - (a^3*(a + b*x^2)^8)/(2*b^5) + (2*a^2*(a + b *x^2)^9)/(3*b^5) - (2*a*(a + b*x^2)^10)/(5*b^5) + (a + b*x^2)^11/(11*b^5)) /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 = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {1}{10} a^{6} x^{10}+\frac {1}{2} a^{5} b \,x^{12}+\frac {15}{14} a^{4} b^{2} x^{14}+\frac {5}{4} a^{3} b^{3} x^{16}+\frac {5}{6} a^{2} b^{4} x^{18}+\frac {3}{10} b^{5} a \,x^{20}+\frac {1}{22} b^{6} x^{22}\) | \(69\) |
norman | \(\frac {1}{10} a^{6} x^{10}+\frac {1}{2} a^{5} b \,x^{12}+\frac {15}{14} a^{4} b^{2} x^{14}+\frac {5}{4} a^{3} b^{3} x^{16}+\frac {5}{6} a^{2} b^{4} x^{18}+\frac {3}{10} b^{5} a \,x^{20}+\frac {1}{22} b^{6} x^{22}\) | \(69\) |
risch | \(\frac {1}{10} a^{6} x^{10}+\frac {1}{2} a^{5} b \,x^{12}+\frac {15}{14} a^{4} b^{2} x^{14}+\frac {5}{4} a^{3} b^{3} x^{16}+\frac {5}{6} a^{2} b^{4} x^{18}+\frac {3}{10} b^{5} a \,x^{20}+\frac {1}{22} b^{6} x^{22}\) | \(69\) |
parallelrisch | \(\frac {1}{10} a^{6} x^{10}+\frac {1}{2} a^{5} b \,x^{12}+\frac {15}{14} a^{4} b^{2} x^{14}+\frac {5}{4} a^{3} b^{3} x^{16}+\frac {5}{6} a^{2} b^{4} x^{18}+\frac {3}{10} b^{5} a \,x^{20}+\frac {1}{22} b^{6} x^{22}\) | \(69\) |
gosper | \(\frac {x^{10} \left (210 b^{6} x^{12}+1386 a \,b^{5} x^{10}+3850 a^{2} b^{4} x^{8}+5775 a^{3} b^{3} x^{6}+4950 a^{4} b^{2} x^{4}+2310 a^{5} b \,x^{2}+462 a^{6}\right )}{4620}\) | \(71\) |
orering | \(\frac {x^{10} \left (210 b^{6} x^{12}+1386 a \,b^{5} x^{10}+3850 a^{2} b^{4} x^{8}+5775 a^{3} b^{3} x^{6}+4950 a^{4} b^{2} x^{4}+2310 a^{5} b \,x^{2}+462 a^{6}\right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}{4620 \left (b \,x^{2}+a \right )^{6}}\) | \(100\) |
Input:
int(x^9*(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/10*a^6*x^10+1/2*a^5*b*x^12+15/14*a^4*b^2*x^14+5/4*a^3*b^3*x^16+5/6*a^2*b ^4*x^18+3/10*b^5*a*x^20+1/22*b^6*x^22
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{22} \, b^{6} x^{22} + \frac {3}{10} \, a b^{5} x^{20} + \frac {5}{6} \, a^{2} b^{4} x^{18} + \frac {5}{4} \, a^{3} b^{3} x^{16} + \frac {15}{14} \, a^{4} b^{2} x^{14} + \frac {1}{2} \, a^{5} b x^{12} + \frac {1}{10} \, a^{6} x^{10} \] Input:
integrate(x^9*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/22*b^6*x^22 + 3/10*a*b^5*x^20 + 5/6*a^2*b^4*x^18 + 5/4*a^3*b^3*x^16 + 15 /14*a^4*b^2*x^14 + 1/2*a^5*b*x^12 + 1/10*a^6*x^10
Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.86 \[ \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^{6} x^{10}}{10} + \frac {a^{5} b x^{12}}{2} + \frac {15 a^{4} b^{2} x^{14}}{14} + \frac {5 a^{3} b^{3} x^{16}}{4} + \frac {5 a^{2} b^{4} x^{18}}{6} + \frac {3 a b^{5} x^{20}}{10} + \frac {b^{6} x^{22}}{22} \] Input:
integrate(x**9*(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
a**6*x**10/10 + a**5*b*x**12/2 + 15*a**4*b**2*x**14/14 + 5*a**3*b**3*x**16 /4 + 5*a**2*b**4*x**18/6 + 3*a*b**5*x**20/10 + b**6*x**22/22
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{22} \, b^{6} x^{22} + \frac {3}{10} \, a b^{5} x^{20} + \frac {5}{6} \, a^{2} b^{4} x^{18} + \frac {5}{4} \, a^{3} b^{3} x^{16} + \frac {15}{14} \, a^{4} b^{2} x^{14} + \frac {1}{2} \, a^{5} b x^{12} + \frac {1}{10} \, a^{6} x^{10} \] Input:
integrate(x^9*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/22*b^6*x^22 + 3/10*a*b^5*x^20 + 5/6*a^2*b^4*x^18 + 5/4*a^3*b^3*x^16 + 15 /14*a^4*b^2*x^14 + 1/2*a^5*b*x^12 + 1/10*a^6*x^10
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{22} \, b^{6} x^{22} + \frac {3}{10} \, a b^{5} x^{20} + \frac {5}{6} \, a^{2} b^{4} x^{18} + \frac {5}{4} \, a^{3} b^{3} x^{16} + \frac {15}{14} \, a^{4} b^{2} x^{14} + \frac {1}{2} \, a^{5} b x^{12} + \frac {1}{10} \, a^{6} x^{10} \] Input:
integrate(x^9*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
1/22*b^6*x^22 + 3/10*a*b^5*x^20 + 5/6*a^2*b^4*x^18 + 5/4*a^3*b^3*x^16 + 15 /14*a^4*b^2*x^14 + 1/2*a^5*b*x^12 + 1/10*a^6*x^10
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75 \[ \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6\,x^{10}}{10}+\frac {a^5\,b\,x^{12}}{2}+\frac {15\,a^4\,b^2\,x^{14}}{14}+\frac {5\,a^3\,b^3\,x^{16}}{4}+\frac {5\,a^2\,b^4\,x^{18}}{6}+\frac {3\,a\,b^5\,x^{20}}{10}+\frac {b^6\,x^{22}}{22} \] Input:
int(x^9*(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
(a^6*x^10)/10 + (b^6*x^22)/22 + (a^5*b*x^12)/2 + (3*a*b^5*x^20)/10 + (15*a ^4*b^2*x^14)/14 + (5*a^3*b^3*x^16)/4 + (5*a^2*b^4*x^18)/6
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77 \[ \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {x^{10} \left (210 b^{6} x^{12}+1386 a \,b^{5} x^{10}+3850 a^{2} b^{4} x^{8}+5775 a^{3} b^{3} x^{6}+4950 a^{4} b^{2} x^{4}+2310 a^{5} b \,x^{2}+462 a^{6}\right )}{4620} \] Input:
int(x^9*(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(x**10*(462*a**6 + 2310*a**5*b*x**2 + 4950*a**4*b**2*x**4 + 5775*a**3*b**3 *x**6 + 3850*a**2*b**4*x**8 + 1386*a*b**5*x**10 + 210*b**6*x**12))/4620