Integrand size = 13, antiderivative size = 110 \[ \int x^{11} \left (a+b x^2\right )^8 \, dx=-\frac {a^5 \left (a+b x^2\right )^9}{18 b^6}+\frac {a^4 \left (a+b x^2\right )^{10}}{4 b^6}-\frac {5 a^3 \left (a+b x^2\right )^{11}}{11 b^6}+\frac {5 a^2 \left (a+b x^2\right )^{12}}{12 b^6}-\frac {5 a \left (a+b x^2\right )^{13}}{26 b^6}+\frac {\left (a+b x^2\right )^{14}}{28 b^6} \] Output:
-1/18*a^5*(b*x^2+a)^9/b^6+1/4*a^4*(b*x^2+a)^10/b^6-5/11*a^3*(b*x^2+a)^11/b ^6+5/12*a^2*(b*x^2+a)^12/b^6-5/26*a*(b*x^2+a)^13/b^6+1/28*(b*x^2+a)^14/b^6
Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98 \[ \int x^{11} \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^{12}}{12}+\frac {4}{7} a^7 b x^{14}+\frac {7}{4} a^6 b^2 x^{16}+\frac {28}{9} a^5 b^3 x^{18}+\frac {7}{2} a^4 b^4 x^{20}+\frac {28}{11} a^3 b^5 x^{22}+\frac {7}{6} a^2 b^6 x^{24}+\frac {4}{13} a b^7 x^{26}+\frac {b^8 x^{28}}{28} \] Input:
Integrate[x^11*(a + b*x^2)^8,x]
Output:
(a^8*x^12)/12 + (4*a^7*b*x^14)/7 + (7*a^6*b^2*x^16)/4 + (28*a^5*b^3*x^18)/ 9 + (7*a^4*b^4*x^20)/2 + (28*a^3*b^5*x^22)/11 + (7*a^2*b^6*x^24)/6 + (4*a* b^7*x^26)/13 + (b^8*x^28)/28
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, 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.
\(\displaystyle \int x^{11} \left (a+b x^2\right )^8 \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int x^{10} \left (b x^2+a\right )^8dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\left (b x^2+a\right )^{13}}{b^5}-\frac {5 a \left (b x^2+a\right )^{12}}{b^5}+\frac {10 a^2 \left (b x^2+a\right )^{11}}{b^5}-\frac {10 a^3 \left (b x^2+a\right )^{10}}{b^5}+\frac {5 a^4 \left (b x^2+a\right )^9}{b^5}-\frac {a^5 \left (b x^2+a\right )^8}{b^5}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^5 \left (a+b x^2\right )^9}{9 b^6}+\frac {a^4 \left (a+b x^2\right )^{10}}{2 b^6}-\frac {10 a^3 \left (a+b x^2\right )^{11}}{11 b^6}+\frac {5 a^2 \left (a+b x^2\right )^{12}}{6 b^6}+\frac {\left (a+b x^2\right )^{14}}{14 b^6}-\frac {5 a \left (a+b x^2\right )^{13}}{13 b^6}\right )\) |
Input:
Int[x^11*(a + b*x^2)^8,x]
Output:
(-1/9*(a^5*(a + b*x^2)^9)/b^6 + (a^4*(a + b*x^2)^10)/(2*b^6) - (10*a^3*(a + b*x^2)^11)/(11*b^6) + (5*a^2*(a + b*x^2)^12)/(6*b^6) - (5*a*(a + b*x^2)^ 13)/(13*b^6) + (a + b*x^2)^14/(14*b^6))/2
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 = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {4}{7} a^{7} b \,x^{14}+\frac {7}{4} a^{6} b^{2} x^{16}+\frac {4}{13} a \,b^{7} x^{26}+\frac {28}{11} a^{3} b^{5} x^{22}+\frac {1}{28} b^{8} x^{28}+\frac {7}{2} a^{4} b^{4} x^{20}+\frac {1}{12} a^{8} x^{12}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {7}{6} a^{2} b^{6} x^{24}\) | \(91\) |
default | \(\frac {4}{7} a^{7} b \,x^{14}+\frac {7}{4} a^{6} b^{2} x^{16}+\frac {4}{13} a \,b^{7} x^{26}+\frac {28}{11} a^{3} b^{5} x^{22}+\frac {1}{28} b^{8} x^{28}+\frac {7}{2} a^{4} b^{4} x^{20}+\frac {1}{12} a^{8} x^{12}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {7}{6} a^{2} b^{6} x^{24}\) | \(91\) |
norman | \(\frac {4}{7} a^{7} b \,x^{14}+\frac {7}{4} a^{6} b^{2} x^{16}+\frac {4}{13} a \,b^{7} x^{26}+\frac {28}{11} a^{3} b^{5} x^{22}+\frac {1}{28} b^{8} x^{28}+\frac {7}{2} a^{4} b^{4} x^{20}+\frac {1}{12} a^{8} x^{12}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {7}{6} a^{2} b^{6} x^{24}\) | \(91\) |
risch | \(\frac {4}{7} a^{7} b \,x^{14}+\frac {7}{4} a^{6} b^{2} x^{16}+\frac {4}{13} a \,b^{7} x^{26}+\frac {28}{11} a^{3} b^{5} x^{22}+\frac {1}{28} b^{8} x^{28}+\frac {7}{2} a^{4} b^{4} x^{20}+\frac {1}{12} a^{8} x^{12}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {7}{6} a^{2} b^{6} x^{24}\) | \(91\) |
parallelrisch | \(\frac {4}{7} a^{7} b \,x^{14}+\frac {7}{4} a^{6} b^{2} x^{16}+\frac {4}{13} a \,b^{7} x^{26}+\frac {28}{11} a^{3} b^{5} x^{22}+\frac {1}{28} b^{8} x^{28}+\frac {7}{2} a^{4} b^{4} x^{20}+\frac {1}{12} a^{8} x^{12}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {7}{6} a^{2} b^{6} x^{24}\) | \(91\) |
orering | \(\frac {x^{12} \left (1287 b^{8} x^{16}+11088 a \,b^{7} x^{14}+42042 a^{2} b^{6} x^{12}+91728 a^{3} b^{5} x^{10}+126126 a^{4} b^{4} x^{8}+112112 a^{5} b^{3} x^{6}+63063 a^{6} b^{2} x^{4}+20592 a^{7} b \,x^{2}+3003 a^{8}\right )}{36036}\) | \(93\) |
Input:
int(x^11*(b*x^2+a)^8,x,method=_RETURNVERBOSE)
Output:
4/7*a^7*b*x^14+7/4*a^6*b^2*x^16+4/13*a*b^7*x^26+28/11*a^3*b^5*x^22+1/28*b^ 8*x^28+7/2*a^4*b^4*x^20+1/12*a^8*x^12+28/9*a^5*b^3*x^18+7/6*a^2*b^6*x^24
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int x^{11} \left (a+b x^2\right )^8 \, dx=\frac {1}{28} \, b^{8} x^{28} + \frac {4}{13} \, a b^{7} x^{26} + \frac {7}{6} \, a^{2} b^{6} x^{24} + \frac {28}{11} \, a^{3} b^{5} x^{22} + \frac {7}{2} \, a^{4} b^{4} x^{20} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {7}{4} \, a^{6} b^{2} x^{16} + \frac {4}{7} \, a^{7} b x^{14} + \frac {1}{12} \, a^{8} x^{12} \] Input:
integrate(x^11*(b*x^2+a)^8,x, algorithm="fricas")
Output:
1/28*b^8*x^28 + 4/13*a*b^7*x^26 + 7/6*a^2*b^6*x^24 + 28/11*a^3*b^5*x^22 + 7/2*a^4*b^4*x^20 + 28/9*a^5*b^3*x^18 + 7/4*a^6*b^2*x^16 + 4/7*a^7*b*x^14 + 1/12*a^8*x^12
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97 \[ \int x^{11} \left (a+b x^2\right )^8 \, dx=\frac {a^{8} x^{12}}{12} + \frac {4 a^{7} b x^{14}}{7} + \frac {7 a^{6} b^{2} x^{16}}{4} + \frac {28 a^{5} b^{3} x^{18}}{9} + \frac {7 a^{4} b^{4} x^{20}}{2} + \frac {28 a^{3} b^{5} x^{22}}{11} + \frac {7 a^{2} b^{6} x^{24}}{6} + \frac {4 a b^{7} x^{26}}{13} + \frac {b^{8} x^{28}}{28} \] Input:
integrate(x**11*(b*x**2+a)**8,x)
Output:
a**8*x**12/12 + 4*a**7*b*x**14/7 + 7*a**6*b**2*x**16/4 + 28*a**5*b**3*x**1 8/9 + 7*a**4*b**4*x**20/2 + 28*a**3*b**5*x**22/11 + 7*a**2*b**6*x**24/6 + 4*a*b**7*x**26/13 + b**8*x**28/28
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int x^{11} \left (a+b x^2\right )^8 \, dx=\frac {1}{28} \, b^{8} x^{28} + \frac {4}{13} \, a b^{7} x^{26} + \frac {7}{6} \, a^{2} b^{6} x^{24} + \frac {28}{11} \, a^{3} b^{5} x^{22} + \frac {7}{2} \, a^{4} b^{4} x^{20} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {7}{4} \, a^{6} b^{2} x^{16} + \frac {4}{7} \, a^{7} b x^{14} + \frac {1}{12} \, a^{8} x^{12} \] Input:
integrate(x^11*(b*x^2+a)^8,x, algorithm="maxima")
Output:
1/28*b^8*x^28 + 4/13*a*b^7*x^26 + 7/6*a^2*b^6*x^24 + 28/11*a^3*b^5*x^22 + 7/2*a^4*b^4*x^20 + 28/9*a^5*b^3*x^18 + 7/4*a^6*b^2*x^16 + 4/7*a^7*b*x^14 + 1/12*a^8*x^12
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int x^{11} \left (a+b x^2\right )^8 \, dx=\frac {1}{28} \, b^{8} x^{28} + \frac {4}{13} \, a b^{7} x^{26} + \frac {7}{6} \, a^{2} b^{6} x^{24} + \frac {28}{11} \, a^{3} b^{5} x^{22} + \frac {7}{2} \, a^{4} b^{4} x^{20} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {7}{4} \, a^{6} b^{2} x^{16} + \frac {4}{7} \, a^{7} b x^{14} + \frac {1}{12} \, a^{8} x^{12} \] Input:
integrate(x^11*(b*x^2+a)^8,x, algorithm="giac")
Output:
1/28*b^8*x^28 + 4/13*a*b^7*x^26 + 7/6*a^2*b^6*x^24 + 28/11*a^3*b^5*x^22 + 7/2*a^4*b^4*x^20 + 28/9*a^5*b^3*x^18 + 7/4*a^6*b^2*x^16 + 4/7*a^7*b*x^14 + 1/12*a^8*x^12
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int x^{11} \left (a+b x^2\right )^8 \, dx=\frac {a^8\,x^{12}}{12}+\frac {4\,a^7\,b\,x^{14}}{7}+\frac {7\,a^6\,b^2\,x^{16}}{4}+\frac {28\,a^5\,b^3\,x^{18}}{9}+\frac {7\,a^4\,b^4\,x^{20}}{2}+\frac {28\,a^3\,b^5\,x^{22}}{11}+\frac {7\,a^2\,b^6\,x^{24}}{6}+\frac {4\,a\,b^7\,x^{26}}{13}+\frac {b^8\,x^{28}}{28} \] Input:
int(x^11*(a + b*x^2)^8,x)
Output:
(a^8*x^12)/12 + (b^8*x^28)/28 + (4*a^7*b*x^14)/7 + (4*a*b^7*x^26)/13 + (7* a^6*b^2*x^16)/4 + (28*a^5*b^3*x^18)/9 + (7*a^4*b^4*x^20)/2 + (28*a^3*b^5*x ^22)/11 + (7*a^2*b^6*x^24)/6
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int x^{11} \left (a+b x^2\right )^8 \, dx=\frac {x^{12} \left (1287 b^{8} x^{16}+11088 a \,b^{7} x^{14}+42042 a^{2} b^{6} x^{12}+91728 a^{3} b^{5} x^{10}+126126 a^{4} b^{4} x^{8}+112112 a^{5} b^{3} x^{6}+63063 a^{6} b^{2} x^{4}+20592 a^{7} b \,x^{2}+3003 a^{8}\right )}{36036} \] Input:
int(x^11*(b*x^2+a)^8,x)
Output:
(x**12*(3003*a**8 + 20592*a**7*b*x**2 + 63063*a**6*b**2*x**4 + 112112*a**5 *b**3*x**6 + 126126*a**4*b**4*x**8 + 91728*a**3*b**5*x**10 + 42042*a**2*b* *6*x**12 + 11088*a*b**7*x**14 + 1287*b**8*x**16))/36036