Integrand size = 13, antiderivative size = 129 \[ \int x^{13} \left (a+b x^2\right )^8 \, dx=\frac {a^6 \left (a+b x^2\right )^9}{18 b^7}-\frac {3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac {15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac {5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac {15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}-\frac {3 a \left (a+b x^2\right )^{14}}{14 b^7}+\frac {\left (a+b x^2\right )^{15}}{30 b^7} \]
1/18*a^6*(b*x^2+a)^9/b^7-3/10*a^5*(b*x^2+a)^10/b^7+15/22*a^4*(b*x^2+a)^11/ b^7-5/6*a^3*(b*x^2+a)^12/b^7+15/26*a^2*(b*x^2+a)^13/b^7-3/14*a*(b*x^2+a)^1 4/b^7+1/30*(b*x^2+a)^15/b^7
Time = 0.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int x^{13} \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^{14}}{14}+\frac {1}{2} a^7 b x^{16}+\frac {14}{9} a^6 b^2 x^{18}+\frac {14}{5} a^5 b^3 x^{20}+\frac {35}{11} a^4 b^4 x^{22}+\frac {7}{3} a^3 b^5 x^{24}+\frac {14}{13} a^2 b^6 x^{26}+\frac {2}{7} a b^7 x^{28}+\frac {b^8 x^{30}}{30} \]
(a^8*x^14)/14 + (a^7*b*x^16)/2 + (14*a^6*b^2*x^18)/9 + (14*a^5*b^3*x^20)/5 + (35*a^4*b^4*x^22)/11 + (7*a^3*b^5*x^24)/3 + (14*a^2*b^6*x^26)/13 + (2*a *b^7*x^28)/7 + (b^8*x^30)/30
Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {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.
\(\displaystyle \int x^{13} \left (a+b x^2\right )^8 \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int x^{12} \left (b x^2+a\right )^8dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\left (b x^2+a\right )^{14}}{b^6}-\frac {6 a \left (b x^2+a\right )^{13}}{b^6}+\frac {15 a^2 \left (b x^2+a\right )^{12}}{b^6}-\frac {20 a^3 \left (b x^2+a\right )^{11}}{b^6}+\frac {15 a^4 \left (b x^2+a\right )^{10}}{b^6}-\frac {6 a^5 \left (b x^2+a\right )^9}{b^6}+\frac {a^6 \left (b x^2+a\right )^8}{b^6}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a^6 \left (a+b x^2\right )^9}{9 b^7}-\frac {3 a^5 \left (a+b x^2\right )^{10}}{5 b^7}+\frac {15 a^4 \left (a+b x^2\right )^{11}}{11 b^7}-\frac {5 a^3 \left (a+b x^2\right )^{12}}{3 b^7}+\frac {15 a^2 \left (a+b x^2\right )^{13}}{13 b^7}+\frac {\left (a+b x^2\right )^{15}}{15 b^7}-\frac {3 a \left (a+b x^2\right )^{14}}{7 b^7}\right )\) |
((a^6*(a + b*x^2)^9)/(9*b^7) - (3*a^5*(a + b*x^2)^10)/(5*b^7) + (15*a^4*(a + b*x^2)^11)/(11*b^7) - (5*a^3*(a + b*x^2)^12)/(3*b^7) + (15*a^2*(a + b*x ^2)^13)/(13*b^7) - (3*a*(a + b*x^2)^14)/(7*b^7) + (a + b*x^2)^15/(15*b^7)) /2
3.1.85.3.1 Defintions of rubi rules used
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]
Time = 1.69 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(\frac {14}{13} a^{2} b^{6} x^{26}+\frac {2}{7} a \,b^{7} x^{28}+\frac {1}{30} b^{8} x^{30}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {14}{5} a^{5} b^{3} x^{20}+\frac {35}{11} a^{4} b^{4} x^{22}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {1}{14} a^{8} x^{14}+\frac {1}{2} a^{7} b \,x^{16}\) | \(91\) |
default | \(\frac {14}{13} a^{2} b^{6} x^{26}+\frac {2}{7} a \,b^{7} x^{28}+\frac {1}{30} b^{8} x^{30}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {14}{5} a^{5} b^{3} x^{20}+\frac {35}{11} a^{4} b^{4} x^{22}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {1}{14} a^{8} x^{14}+\frac {1}{2} a^{7} b \,x^{16}\) | \(91\) |
norman | \(\frac {14}{13} a^{2} b^{6} x^{26}+\frac {2}{7} a \,b^{7} x^{28}+\frac {1}{30} b^{8} x^{30}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {14}{5} a^{5} b^{3} x^{20}+\frac {35}{11} a^{4} b^{4} x^{22}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {1}{14} a^{8} x^{14}+\frac {1}{2} a^{7} b \,x^{16}\) | \(91\) |
risch | \(\frac {14}{13} a^{2} b^{6} x^{26}+\frac {2}{7} a \,b^{7} x^{28}+\frac {1}{30} b^{8} x^{30}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {14}{5} a^{5} b^{3} x^{20}+\frac {35}{11} a^{4} b^{4} x^{22}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {1}{14} a^{8} x^{14}+\frac {1}{2} a^{7} b \,x^{16}\) | \(91\) |
parallelrisch | \(\frac {14}{13} a^{2} b^{6} x^{26}+\frac {2}{7} a \,b^{7} x^{28}+\frac {1}{30} b^{8} x^{30}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {14}{5} a^{5} b^{3} x^{20}+\frac {35}{11} a^{4} b^{4} x^{22}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {1}{14} a^{8} x^{14}+\frac {1}{2} a^{7} b \,x^{16}\) | \(91\) |
14/13*a^2*b^6*x^26+2/7*a*b^7*x^28+1/30*b^8*x^30+14/9*a^6*b^2*x^18+14/5*a^5 *b^3*x^20+35/11*a^4*b^4*x^22+7/3*a^3*b^5*x^24+1/14*a^8*x^14+1/2*a^7*b*x^16
Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int x^{13} \left (a+b x^2\right )^8 \, dx=\frac {1}{30} \, b^{8} x^{30} + \frac {2}{7} \, a b^{7} x^{28} + \frac {14}{13} \, a^{2} b^{6} x^{26} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {35}{11} \, a^{4} b^{4} x^{22} + \frac {14}{5} \, a^{5} b^{3} x^{20} + \frac {14}{9} \, a^{6} b^{2} x^{18} + \frac {1}{2} \, a^{7} b x^{16} + \frac {1}{14} \, a^{8} x^{14} \]
1/30*b^8*x^30 + 2/7*a*b^7*x^28 + 14/13*a^2*b^6*x^26 + 7/3*a^3*b^5*x^24 + 3 5/11*a^4*b^4*x^22 + 14/5*a^5*b^3*x^20 + 14/9*a^6*b^2*x^18 + 1/2*a^7*b*x^16 + 1/14*a^8*x^14
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81 \[ \int x^{13} \left (a+b x^2\right )^8 \, dx=\frac {a^{8} x^{14}}{14} + \frac {a^{7} b x^{16}}{2} + \frac {14 a^{6} b^{2} x^{18}}{9} + \frac {14 a^{5} b^{3} x^{20}}{5} + \frac {35 a^{4} b^{4} x^{22}}{11} + \frac {7 a^{3} b^{5} x^{24}}{3} + \frac {14 a^{2} b^{6} x^{26}}{13} + \frac {2 a b^{7} x^{28}}{7} + \frac {b^{8} x^{30}}{30} \]
a**8*x**14/14 + a**7*b*x**16/2 + 14*a**6*b**2*x**18/9 + 14*a**5*b**3*x**20 /5 + 35*a**4*b**4*x**22/11 + 7*a**3*b**5*x**24/3 + 14*a**2*b**6*x**26/13 + 2*a*b**7*x**28/7 + b**8*x**30/30
Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int x^{13} \left (a+b x^2\right )^8 \, dx=\frac {1}{30} \, b^{8} x^{30} + \frac {2}{7} \, a b^{7} x^{28} + \frac {14}{13} \, a^{2} b^{6} x^{26} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {35}{11} \, a^{4} b^{4} x^{22} + \frac {14}{5} \, a^{5} b^{3} x^{20} + \frac {14}{9} \, a^{6} b^{2} x^{18} + \frac {1}{2} \, a^{7} b x^{16} + \frac {1}{14} \, a^{8} x^{14} \]
1/30*b^8*x^30 + 2/7*a*b^7*x^28 + 14/13*a^2*b^6*x^26 + 7/3*a^3*b^5*x^24 + 3 5/11*a^4*b^4*x^22 + 14/5*a^5*b^3*x^20 + 14/9*a^6*b^2*x^18 + 1/2*a^7*b*x^16 + 1/14*a^8*x^14
Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int x^{13} \left (a+b x^2\right )^8 \, dx=\frac {1}{30} \, b^{8} x^{30} + \frac {2}{7} \, a b^{7} x^{28} + \frac {14}{13} \, a^{2} b^{6} x^{26} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {35}{11} \, a^{4} b^{4} x^{22} + \frac {14}{5} \, a^{5} b^{3} x^{20} + \frac {14}{9} \, a^{6} b^{2} x^{18} + \frac {1}{2} \, a^{7} b x^{16} + \frac {1}{14} \, a^{8} x^{14} \]
1/30*b^8*x^30 + 2/7*a*b^7*x^28 + 14/13*a^2*b^6*x^26 + 7/3*a^3*b^5*x^24 + 3 5/11*a^4*b^4*x^22 + 14/5*a^5*b^3*x^20 + 14/9*a^6*b^2*x^18 + 1/2*a^7*b*x^16 + 1/14*a^8*x^14
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int x^{13} \left (a+b x^2\right )^8 \, dx=\frac {a^8\,x^{14}}{14}+\frac {a^7\,b\,x^{16}}{2}+\frac {14\,a^6\,b^2\,x^{18}}{9}+\frac {14\,a^5\,b^3\,x^{20}}{5}+\frac {35\,a^4\,b^4\,x^{22}}{11}+\frac {7\,a^3\,b^5\,x^{24}}{3}+\frac {14\,a^2\,b^6\,x^{26}}{13}+\frac {2\,a\,b^7\,x^{28}}{7}+\frac {b^8\,x^{30}}{30} \]