Integrand size = 13, antiderivative size = 108 \[ \int x^6 \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^7}{7}+\frac {8}{9} a^7 b x^9+\frac {28}{11} a^6 b^2 x^{11}+\frac {56}{13} a^5 b^3 x^{13}+\frac {14}{3} a^4 b^4 x^{15}+\frac {56}{17} a^3 b^5 x^{17}+\frac {28}{19} a^2 b^6 x^{19}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{23}}{23} \] Output:
1/7*a^8*x^7+8/9*a^7*b*x^9+28/11*a^6*b^2*x^11+56/13*a^5*b^3*x^13+14/3*a^4*b ^4*x^15+56/17*a^3*b^5*x^17+28/19*a^2*b^6*x^19+8/21*a*b^7*x^21+1/23*b^8*x^2 3
Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int x^6 \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^7}{7}+\frac {8}{9} a^7 b x^9+\frac {28}{11} a^6 b^2 x^{11}+\frac {56}{13} a^5 b^3 x^{13}+\frac {14}{3} a^4 b^4 x^{15}+\frac {56}{17} a^3 b^5 x^{17}+\frac {28}{19} a^2 b^6 x^{19}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{23}}{23} \] Input:
Integrate[x^6*(a + b*x^2)^8,x]
Output:
(a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/1 3 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8 *a*b^7*x^21)/21 + (b^8*x^23)/23
Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {244, 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.
\(\displaystyle \int x^6 \left (a+b x^2\right )^8 \, dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (a^8 x^6+8 a^7 b x^8+28 a^6 b^2 x^{10}+56 a^5 b^3 x^{12}+70 a^4 b^4 x^{14}+56 a^3 b^5 x^{16}+28 a^2 b^6 x^{18}+8 a b^7 x^{20}+b^8 x^{22}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^8 x^7}{7}+\frac {8}{9} a^7 b x^9+\frac {28}{11} a^6 b^2 x^{11}+\frac {56}{13} a^5 b^3 x^{13}+\frac {14}{3} a^4 b^4 x^{15}+\frac {56}{17} a^3 b^5 x^{17}+\frac {28}{19} a^2 b^6 x^{19}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{23}}{23}\) |
Input:
Int[x^6*(a + b*x^2)^8,x]
Output:
(a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/1 3 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8 *a*b^7*x^21)/21 + (b^8*x^23)/23
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\) | \(91\) |
default | \(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\) | \(91\) |
norman | \(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\) | \(91\) |
risch | \(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\) | \(91\) |
parallelrisch | \(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\) | \(91\) |
orering | \(\frac {x^{7} \left (2909907 b^{8} x^{16}+25496328 a \,b^{7} x^{14}+98630532 a^{2} b^{6} x^{12}+220468248 a^{3} b^{5} x^{10}+312330018 a^{4} b^{4} x^{8}+288304632 a^{5} b^{3} x^{6}+170361828 a^{6} b^{2} x^{4}+59491432 a^{7} b \,x^{2}+9561123 a^{8}\right )}{66927861}\) | \(93\) |
Input:
int(x^6*(b*x^2+a)^8,x,method=_RETURNVERBOSE)
Output:
1/7*a^8*x^7+8/9*a^7*b*x^9+28/11*a^6*b^2*x^11+56/13*a^5*b^3*x^13+14/3*a^4*b ^4*x^15+56/17*a^3*b^5*x^17+28/19*a^2*b^6*x^19+8/21*a*b^7*x^21+1/23*b^8*x^2 3
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^6 \left (a+b x^2\right )^8 \, dx=\frac {1}{23} \, b^{8} x^{23} + \frac {8}{21} \, a b^{7} x^{21} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {56}{17} \, a^{3} b^{5} x^{17} + \frac {14}{3} \, a^{4} b^{4} x^{15} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{7} \, a^{8} x^{7} \] Input:
integrate(x^6*(b*x^2+a)^8,x, algorithm="fricas")
Output:
1/23*b^8*x^23 + 8/21*a*b^7*x^21 + 28/19*a^2*b^6*x^19 + 56/17*a^3*b^5*x^17 + 14/3*a^4*b^4*x^15 + 56/13*a^5*b^3*x^13 + 28/11*a^6*b^2*x^11 + 8/9*a^7*b* x^9 + 1/7*a^8*x^7
Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int x^6 \left (a+b x^2\right )^8 \, dx=\frac {a^{8} x^{7}}{7} + \frac {8 a^{7} b x^{9}}{9} + \frac {28 a^{6} b^{2} x^{11}}{11} + \frac {56 a^{5} b^{3} x^{13}}{13} + \frac {14 a^{4} b^{4} x^{15}}{3} + \frac {56 a^{3} b^{5} x^{17}}{17} + \frac {28 a^{2} b^{6} x^{19}}{19} + \frac {8 a b^{7} x^{21}}{21} + \frac {b^{8} x^{23}}{23} \] Input:
integrate(x**6*(b*x**2+a)**8,x)
Output:
a**8*x**7/7 + 8*a**7*b*x**9/9 + 28*a**6*b**2*x**11/11 + 56*a**5*b**3*x**13 /13 + 14*a**4*b**4*x**15/3 + 56*a**3*b**5*x**17/17 + 28*a**2*b**6*x**19/19 + 8*a*b**7*x**21/21 + b**8*x**23/23
Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^6 \left (a+b x^2\right )^8 \, dx=\frac {1}{23} \, b^{8} x^{23} + \frac {8}{21} \, a b^{7} x^{21} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {56}{17} \, a^{3} b^{5} x^{17} + \frac {14}{3} \, a^{4} b^{4} x^{15} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{7} \, a^{8} x^{7} \] Input:
integrate(x^6*(b*x^2+a)^8,x, algorithm="maxima")
Output:
1/23*b^8*x^23 + 8/21*a*b^7*x^21 + 28/19*a^2*b^6*x^19 + 56/17*a^3*b^5*x^17 + 14/3*a^4*b^4*x^15 + 56/13*a^5*b^3*x^13 + 28/11*a^6*b^2*x^11 + 8/9*a^7*b* x^9 + 1/7*a^8*x^7
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^6 \left (a+b x^2\right )^8 \, dx=\frac {1}{23} \, b^{8} x^{23} + \frac {8}{21} \, a b^{7} x^{21} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {56}{17} \, a^{3} b^{5} x^{17} + \frac {14}{3} \, a^{4} b^{4} x^{15} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{7} \, a^{8} x^{7} \] Input:
integrate(x^6*(b*x^2+a)^8,x, algorithm="giac")
Output:
1/23*b^8*x^23 + 8/21*a*b^7*x^21 + 28/19*a^2*b^6*x^19 + 56/17*a^3*b^5*x^17 + 14/3*a^4*b^4*x^15 + 56/13*a^5*b^3*x^13 + 28/11*a^6*b^2*x^11 + 8/9*a^7*b* x^9 + 1/7*a^8*x^7
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^6 \left (a+b x^2\right )^8 \, dx=\frac {a^8\,x^7}{7}+\frac {8\,a^7\,b\,x^9}{9}+\frac {28\,a^6\,b^2\,x^{11}}{11}+\frac {56\,a^5\,b^3\,x^{13}}{13}+\frac {14\,a^4\,b^4\,x^{15}}{3}+\frac {56\,a^3\,b^5\,x^{17}}{17}+\frac {28\,a^2\,b^6\,x^{19}}{19}+\frac {8\,a\,b^7\,x^{21}}{21}+\frac {b^8\,x^{23}}{23} \] Input:
int(x^6*(a + b*x^2)^8,x)
Output:
(a^8*x^7)/7 + (b^8*x^23)/23 + (8*a^7*b*x^9)/9 + (8*a*b^7*x^21)/21 + (28*a^ 6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/13 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^5* x^17)/17 + (28*a^2*b^6*x^19)/19
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int x^6 \left (a+b x^2\right )^8 \, dx=\frac {x^{7} \left (2909907 b^{8} x^{16}+25496328 a \,b^{7} x^{14}+98630532 a^{2} b^{6} x^{12}+220468248 a^{3} b^{5} x^{10}+312330018 a^{4} b^{4} x^{8}+288304632 a^{5} b^{3} x^{6}+170361828 a^{6} b^{2} x^{4}+59491432 a^{7} b \,x^{2}+9561123 a^{8}\right )}{66927861} \] Input:
int(x^6*(b*x^2+a)^8,x)
Output:
(x**7*(9561123*a**8 + 59491432*a**7*b*x**2 + 170361828*a**6*b**2*x**4 + 28 8304632*a**5*b**3*x**6 + 312330018*a**4*b**4*x**8 + 220468248*a**3*b**5*x* *10 + 98630532*a**2*b**6*x**12 + 25496328*a*b**7*x**14 + 2909907*b**8*x**1 6))/66927861