Integrand size = 13, antiderivative size = 108 \[ \int x^8 \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^9}{9}+\frac {8}{11} a^7 b x^{11}+\frac {28}{13} a^6 b^2 x^{13}+\frac {56}{15} a^5 b^3 x^{15}+\frac {70}{17} a^4 b^4 x^{17}+\frac {56}{19} a^3 b^5 x^{19}+\frac {4}{3} a^2 b^6 x^{21}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{25}}{25} \] Output:
1/9*a^8*x^9+8/11*a^7*b*x^11+28/13*a^6*b^2*x^13+56/15*a^5*b^3*x^15+70/17*a^ 4*b^4*x^17+56/19*a^3*b^5*x^19+4/3*a^2*b^6*x^21+8/23*a*b^7*x^23+1/25*b^8*x^ 25
Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int x^8 \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^9}{9}+\frac {8}{11} a^7 b x^{11}+\frac {28}{13} a^6 b^2 x^{13}+\frac {56}{15} a^5 b^3 x^{15}+\frac {70}{17} a^4 b^4 x^{17}+\frac {56}{19} a^3 b^5 x^{19}+\frac {4}{3} a^2 b^6 x^{21}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{25}}{25} \] Input:
Integrate[x^8*(a + b*x^2)^8,x]
Output:
(a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15) /15 + (70*a^4*b^4*x^17)/17 + (56*a^3*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + ( 8*a*b^7*x^23)/23 + (b^8*x^25)/25
Time = 0.22 (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.
\(\displaystyle \int x^8 \left (a+b x^2\right )^8 \, dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (a^8 x^8+8 a^7 b x^{10}+28 a^6 b^2 x^{12}+56 a^5 b^3 x^{14}+70 a^4 b^4 x^{16}+56 a^3 b^5 x^{18}+28 a^2 b^6 x^{20}+8 a b^7 x^{22}+b^8 x^{24}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^8 x^9}{9}+\frac {8}{11} a^7 b x^{11}+\frac {28}{13} a^6 b^2 x^{13}+\frac {56}{15} a^5 b^3 x^{15}+\frac {70}{17} a^4 b^4 x^{17}+\frac {56}{19} a^3 b^5 x^{19}+\frac {4}{3} a^2 b^6 x^{21}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{25}}{25}\) |
Input:
Int[x^8*(a + b*x^2)^8,x]
Output:
(a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15) /15 + (70*a^4*b^4*x^17)/17 + (56*a^3*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + ( 8*a*b^7*x^23)/23 + (b^8*x^25)/25
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 = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\) | \(91\) |
default | \(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\) | \(91\) |
norman | \(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\) | \(91\) |
risch | \(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\) | \(91\) |
parallelrisch | \(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\) | \(91\) |
orering | \(\frac {x^{9} \left (9561123 b^{8} x^{16}+83140200 a \,b^{7} x^{14}+318704100 a^{2} b^{6} x^{12}+704503800 a^{3} b^{5} x^{10}+984233250 a^{4} b^{4} x^{8}+892371480 a^{5} b^{3} x^{6}+514829700 a^{6} b^{2} x^{4}+173838600 a^{7} b \,x^{2}+26558675 a^{8}\right )}{239028075}\) | \(93\) |
Input:
int(x^8*(b*x^2+a)^8,x,method=_RETURNVERBOSE)
Output:
1/9*a^8*x^9+8/11*a^7*b*x^11+28/13*a^6*b^2*x^13+56/15*a^5*b^3*x^15+70/17*a^ 4*b^4*x^17+56/19*a^3*b^5*x^19+4/3*a^2*b^6*x^21+8/23*a*b^7*x^23+1/25*b^8*x^ 25
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^8 \left (a+b x^2\right )^8 \, dx=\frac {1}{25} \, b^{8} x^{25} + \frac {8}{23} \, a b^{7} x^{23} + \frac {4}{3} \, a^{2} b^{6} x^{21} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {70}{17} \, a^{4} b^{4} x^{17} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {28}{13} \, a^{6} b^{2} x^{13} + \frac {8}{11} \, a^{7} b x^{11} + \frac {1}{9} \, a^{8} x^{9} \] Input:
integrate(x^8*(b*x^2+a)^8,x, algorithm="fricas")
Output:
1/25*b^8*x^25 + 8/23*a*b^7*x^23 + 4/3*a^2*b^6*x^21 + 56/19*a^3*b^5*x^19 + 70/17*a^4*b^4*x^17 + 56/15*a^5*b^3*x^15 + 28/13*a^6*b^2*x^13 + 8/11*a^7*b* x^11 + 1/9*a^8*x^9
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int x^8 \left (a+b x^2\right )^8 \, dx=\frac {a^{8} x^{9}}{9} + \frac {8 a^{7} b x^{11}}{11} + \frac {28 a^{6} b^{2} x^{13}}{13} + \frac {56 a^{5} b^{3} x^{15}}{15} + \frac {70 a^{4} b^{4} x^{17}}{17} + \frac {56 a^{3} b^{5} x^{19}}{19} + \frac {4 a^{2} b^{6} x^{21}}{3} + \frac {8 a b^{7} x^{23}}{23} + \frac {b^{8} x^{25}}{25} \] Input:
integrate(x**8*(b*x**2+a)**8,x)
Output:
a**8*x**9/9 + 8*a**7*b*x**11/11 + 28*a**6*b**2*x**13/13 + 56*a**5*b**3*x** 15/15 + 70*a**4*b**4*x**17/17 + 56*a**3*b**5*x**19/19 + 4*a**2*b**6*x**21/ 3 + 8*a*b**7*x**23/23 + b**8*x**25/25
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^8 \left (a+b x^2\right )^8 \, dx=\frac {1}{25} \, b^{8} x^{25} + \frac {8}{23} \, a b^{7} x^{23} + \frac {4}{3} \, a^{2} b^{6} x^{21} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {70}{17} \, a^{4} b^{4} x^{17} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {28}{13} \, a^{6} b^{2} x^{13} + \frac {8}{11} \, a^{7} b x^{11} + \frac {1}{9} \, a^{8} x^{9} \] Input:
integrate(x^8*(b*x^2+a)^8,x, algorithm="maxima")
Output:
1/25*b^8*x^25 + 8/23*a*b^7*x^23 + 4/3*a^2*b^6*x^21 + 56/19*a^3*b^5*x^19 + 70/17*a^4*b^4*x^17 + 56/15*a^5*b^3*x^15 + 28/13*a^6*b^2*x^13 + 8/11*a^7*b* x^11 + 1/9*a^8*x^9
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^8 \left (a+b x^2\right )^8 \, dx=\frac {1}{25} \, b^{8} x^{25} + \frac {8}{23} \, a b^{7} x^{23} + \frac {4}{3} \, a^{2} b^{6} x^{21} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {70}{17} \, a^{4} b^{4} x^{17} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {28}{13} \, a^{6} b^{2} x^{13} + \frac {8}{11} \, a^{7} b x^{11} + \frac {1}{9} \, a^{8} x^{9} \] Input:
integrate(x^8*(b*x^2+a)^8,x, algorithm="giac")
Output:
1/25*b^8*x^25 + 8/23*a*b^7*x^23 + 4/3*a^2*b^6*x^21 + 56/19*a^3*b^5*x^19 + 70/17*a^4*b^4*x^17 + 56/15*a^5*b^3*x^15 + 28/13*a^6*b^2*x^13 + 8/11*a^7*b* x^11 + 1/9*a^8*x^9
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^8 \left (a+b x^2\right )^8 \, dx=\frac {a^8\,x^9}{9}+\frac {8\,a^7\,b\,x^{11}}{11}+\frac {28\,a^6\,b^2\,x^{13}}{13}+\frac {56\,a^5\,b^3\,x^{15}}{15}+\frac {70\,a^4\,b^4\,x^{17}}{17}+\frac {56\,a^3\,b^5\,x^{19}}{19}+\frac {4\,a^2\,b^6\,x^{21}}{3}+\frac {8\,a\,b^7\,x^{23}}{23}+\frac {b^8\,x^{25}}{25} \] Input:
int(x^8*(a + b*x^2)^8,x)
Output:
(a^8*x^9)/9 + (b^8*x^25)/25 + (8*a^7*b*x^11)/11 + (8*a*b^7*x^23)/23 + (28* a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)/15 + (70*a^4*b^4*x^17)/17 + (56*a^3*b ^5*x^19)/19 + (4*a^2*b^6*x^21)/3
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int x^8 \left (a+b x^2\right )^8 \, dx=\frac {x^{9} \left (9561123 b^{8} x^{16}+83140200 a \,b^{7} x^{14}+318704100 a^{2} b^{6} x^{12}+704503800 a^{3} b^{5} x^{10}+984233250 a^{4} b^{4} x^{8}+892371480 a^{5} b^{3} x^{6}+514829700 a^{6} b^{2} x^{4}+173838600 a^{7} b \,x^{2}+26558675 a^{8}\right )}{239028075} \] Input:
int(x^8*(b*x^2+a)^8,x)
Output:
(x**9*(26558675*a**8 + 173838600*a**7*b*x**2 + 514829700*a**6*b**2*x**4 + 892371480*a**5*b**3*x**6 + 984233250*a**4*b**4*x**8 + 704503800*a**3*b**5* x**10 + 318704100*a**2*b**6*x**12 + 83140200*a*b**7*x**14 + 9561123*b**8*x **16))/239028075