Integrand size = 13, antiderivative size = 106 \[ \int x^2 \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^3}{3}+\frac {8}{5} a^7 b x^5+4 a^6 b^2 x^7+\frac {56}{9} a^5 b^3 x^9+\frac {70}{11} a^4 b^4 x^{11}+\frac {56}{13} a^3 b^5 x^{13}+\frac {28}{15} a^2 b^6 x^{15}+\frac {8}{17} a b^7 x^{17}+\frac {b^8 x^{19}}{19} \] Output:
1/3*a^8*x^3+8/5*a^7*b*x^5+4*a^6*b^2*x^7+56/9*a^5*b^3*x^9+70/11*a^4*b^4*x^1 1+56/13*a^3*b^5*x^13+28/15*a^2*b^6*x^15+8/17*a*b^7*x^17+1/19*b^8*x^19
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^3}{3}+\frac {8}{5} a^7 b x^5+4 a^6 b^2 x^7+\frac {56}{9} a^5 b^3 x^9+\frac {70}{11} a^4 b^4 x^{11}+\frac {56}{13} a^3 b^5 x^{13}+\frac {28}{15} a^2 b^6 x^{15}+\frac {8}{17} a b^7 x^{17}+\frac {b^8 x^{19}}{19} \] Input:
Integrate[x^2*(a + b*x^2)^8,x]
Output:
(a^8*x^3)/3 + (8*a^7*b*x^5)/5 + 4*a^6*b^2*x^7 + (56*a^5*b^3*x^9)/9 + (70*a ^4*b^4*x^11)/11 + (56*a^3*b^5*x^13)/13 + (28*a^2*b^6*x^15)/15 + (8*a*b^7*x ^17)/17 + (b^8*x^19)/19
Time = 0.21 (sec) , antiderivative size = 106, 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.
\(\displaystyle \int x^2 \left (a+b x^2\right )^8 \, dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (a^8 x^2+8 a^7 b x^4+28 a^6 b^2 x^6+56 a^5 b^3 x^8+70 a^4 b^4 x^{10}+56 a^3 b^5 x^{12}+28 a^2 b^6 x^{14}+8 a b^7 x^{16}+b^8 x^{18}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^8 x^3}{3}+\frac {8}{5} a^7 b x^5+4 a^6 b^2 x^7+\frac {56}{9} a^5 b^3 x^9+\frac {70}{11} a^4 b^4 x^{11}+\frac {56}{13} a^3 b^5 x^{13}+\frac {28}{15} a^2 b^6 x^{15}+\frac {8}{17} a b^7 x^{17}+\frac {b^8 x^{19}}{19}\) |
Input:
Int[x^2*(a + b*x^2)^8,x]
Output:
(a^8*x^3)/3 + (8*a^7*b*x^5)/5 + 4*a^6*b^2*x^7 + (56*a^5*b^3*x^9)/9 + (70*a ^4*b^4*x^11)/11 + (56*a^3*b^5*x^13)/13 + (28*a^2*b^6*x^15)/15 + (8*a*b^7*x ^17)/17 + (b^8*x^19)/19
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 = 0.86
method | result | size |
gosper | \(\frac {1}{3} a^{8} x^{3}+\frac {8}{5} a^{7} b \,x^{5}+4 a^{6} b^{2} x^{7}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {70}{11} a^{4} b^{4} x^{11}+\frac {56}{13} a^{3} b^{5} x^{13}+\frac {28}{15} a^{2} b^{6} x^{15}+\frac {8}{17} a \,b^{7} x^{17}+\frac {1}{19} b^{8} x^{19}\) | \(91\) |
default | \(\frac {1}{3} a^{8} x^{3}+\frac {8}{5} a^{7} b \,x^{5}+4 a^{6} b^{2} x^{7}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {70}{11} a^{4} b^{4} x^{11}+\frac {56}{13} a^{3} b^{5} x^{13}+\frac {28}{15} a^{2} b^{6} x^{15}+\frac {8}{17} a \,b^{7} x^{17}+\frac {1}{19} b^{8} x^{19}\) | \(91\) |
norman | \(\frac {1}{3} a^{8} x^{3}+\frac {8}{5} a^{7} b \,x^{5}+4 a^{6} b^{2} x^{7}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {70}{11} a^{4} b^{4} x^{11}+\frac {56}{13} a^{3} b^{5} x^{13}+\frac {28}{15} a^{2} b^{6} x^{15}+\frac {8}{17} a \,b^{7} x^{17}+\frac {1}{19} b^{8} x^{19}\) | \(91\) |
risch | \(\frac {1}{3} a^{8} x^{3}+\frac {8}{5} a^{7} b \,x^{5}+4 a^{6} b^{2} x^{7}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {70}{11} a^{4} b^{4} x^{11}+\frac {56}{13} a^{3} b^{5} x^{13}+\frac {28}{15} a^{2} b^{6} x^{15}+\frac {8}{17} a \,b^{7} x^{17}+\frac {1}{19} b^{8} x^{19}\) | \(91\) |
parallelrisch | \(\frac {1}{3} a^{8} x^{3}+\frac {8}{5} a^{7} b \,x^{5}+4 a^{6} b^{2} x^{7}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {70}{11} a^{4} b^{4} x^{11}+\frac {56}{13} a^{3} b^{5} x^{13}+\frac {28}{15} a^{2} b^{6} x^{15}+\frac {8}{17} a \,b^{7} x^{17}+\frac {1}{19} b^{8} x^{19}\) | \(91\) |
orering | \(\frac {x^{3} \left (109395 b^{8} x^{16}+978120 a \,b^{7} x^{14}+3879876 a^{2} b^{6} x^{12}+8953560 a^{3} b^{5} x^{10}+13226850 a^{4} b^{4} x^{8}+12932920 a^{5} b^{3} x^{6}+8314020 a^{6} b^{2} x^{4}+3325608 a^{7} b \,x^{2}+692835 a^{8}\right )}{2078505}\) | \(93\) |
Input:
int(x^2*(b*x^2+a)^8,x,method=_RETURNVERBOSE)
Output:
1/3*a^8*x^3+8/5*a^7*b*x^5+4*a^6*b^2*x^7+56/9*a^5*b^3*x^9+70/11*a^4*b^4*x^1 1+56/13*a^3*b^5*x^13+28/15*a^2*b^6*x^15+8/17*a*b^7*x^17+1/19*b^8*x^19
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a+b x^2\right )^8 \, dx=\frac {1}{19} \, b^{8} x^{19} + \frac {8}{17} \, a b^{7} x^{17} + \frac {28}{15} \, a^{2} b^{6} x^{15} + \frac {56}{13} \, a^{3} b^{5} x^{13} + \frac {70}{11} \, a^{4} b^{4} x^{11} + \frac {56}{9} \, a^{5} b^{3} x^{9} + 4 \, a^{6} b^{2} x^{7} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{3} \, a^{8} x^{3} \] Input:
integrate(x^2*(b*x^2+a)^8,x, algorithm="fricas")
Output:
1/19*b^8*x^19 + 8/17*a*b^7*x^17 + 28/15*a^2*b^6*x^15 + 56/13*a^3*b^5*x^13 + 70/11*a^4*b^4*x^11 + 56/9*a^5*b^3*x^9 + 4*a^6*b^2*x^7 + 8/5*a^7*b*x^5 + 1/3*a^8*x^3
Time = 0.02 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int x^2 \left (a+b x^2\right )^8 \, dx=\frac {a^{8} x^{3}}{3} + \frac {8 a^{7} b x^{5}}{5} + 4 a^{6} b^{2} x^{7} + \frac {56 a^{5} b^{3} x^{9}}{9} + \frac {70 a^{4} b^{4} x^{11}}{11} + \frac {56 a^{3} b^{5} x^{13}}{13} + \frac {28 a^{2} b^{6} x^{15}}{15} + \frac {8 a b^{7} x^{17}}{17} + \frac {b^{8} x^{19}}{19} \] Input:
integrate(x**2*(b*x**2+a)**8,x)
Output:
a**8*x**3/3 + 8*a**7*b*x**5/5 + 4*a**6*b**2*x**7 + 56*a**5*b**3*x**9/9 + 7 0*a**4*b**4*x**11/11 + 56*a**3*b**5*x**13/13 + 28*a**2*b**6*x**15/15 + 8*a *b**7*x**17/17 + b**8*x**19/19
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a+b x^2\right )^8 \, dx=\frac {1}{19} \, b^{8} x^{19} + \frac {8}{17} \, a b^{7} x^{17} + \frac {28}{15} \, a^{2} b^{6} x^{15} + \frac {56}{13} \, a^{3} b^{5} x^{13} + \frac {70}{11} \, a^{4} b^{4} x^{11} + \frac {56}{9} \, a^{5} b^{3} x^{9} + 4 \, a^{6} b^{2} x^{7} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{3} \, a^{8} x^{3} \] Input:
integrate(x^2*(b*x^2+a)^8,x, algorithm="maxima")
Output:
1/19*b^8*x^19 + 8/17*a*b^7*x^17 + 28/15*a^2*b^6*x^15 + 56/13*a^3*b^5*x^13 + 70/11*a^4*b^4*x^11 + 56/9*a^5*b^3*x^9 + 4*a^6*b^2*x^7 + 8/5*a^7*b*x^5 + 1/3*a^8*x^3
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a+b x^2\right )^8 \, dx=\frac {1}{19} \, b^{8} x^{19} + \frac {8}{17} \, a b^{7} x^{17} + \frac {28}{15} \, a^{2} b^{6} x^{15} + \frac {56}{13} \, a^{3} b^{5} x^{13} + \frac {70}{11} \, a^{4} b^{4} x^{11} + \frac {56}{9} \, a^{5} b^{3} x^{9} + 4 \, a^{6} b^{2} x^{7} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{3} \, a^{8} x^{3} \] Input:
integrate(x^2*(b*x^2+a)^8,x, algorithm="giac")
Output:
1/19*b^8*x^19 + 8/17*a*b^7*x^17 + 28/15*a^2*b^6*x^15 + 56/13*a^3*b^5*x^13 + 70/11*a^4*b^4*x^11 + 56/9*a^5*b^3*x^9 + 4*a^6*b^2*x^7 + 8/5*a^7*b*x^5 + 1/3*a^8*x^3
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a+b x^2\right )^8 \, dx=\frac {a^8\,x^3}{3}+\frac {8\,a^7\,b\,x^5}{5}+4\,a^6\,b^2\,x^7+\frac {56\,a^5\,b^3\,x^9}{9}+\frac {70\,a^4\,b^4\,x^{11}}{11}+\frac {56\,a^3\,b^5\,x^{13}}{13}+\frac {28\,a^2\,b^6\,x^{15}}{15}+\frac {8\,a\,b^7\,x^{17}}{17}+\frac {b^8\,x^{19}}{19} \] Input:
int(x^2*(a + b*x^2)^8,x)
Output:
(a^8*x^3)/3 + (b^8*x^19)/19 + (8*a^7*b*x^5)/5 + (8*a*b^7*x^17)/17 + 4*a^6* b^2*x^7 + (56*a^5*b^3*x^9)/9 + (70*a^4*b^4*x^11)/11 + (56*a^3*b^5*x^13)/13 + (28*a^2*b^6*x^15)/15
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int x^2 \left (a+b x^2\right )^8 \, dx=\frac {x^{3} \left (109395 b^{8} x^{16}+978120 a \,b^{7} x^{14}+3879876 a^{2} b^{6} x^{12}+8953560 a^{3} b^{5} x^{10}+13226850 a^{4} b^{4} x^{8}+12932920 a^{5} b^{3} x^{6}+8314020 a^{6} b^{2} x^{4}+3325608 a^{7} b \,x^{2}+692835 a^{8}\right )}{2078505} \] Input:
int(x^2*(b*x^2+a)^8,x)
Output:
(x**3*(692835*a**8 + 3325608*a**7*b*x**2 + 8314020*a**6*b**2*x**4 + 129329 20*a**5*b**3*x**6 + 13226850*a**4*b**4*x**8 + 8953560*a**3*b**5*x**10 + 38 79876*a**2*b**6*x**12 + 978120*a*b**7*x**14 + 109395*b**8*x**16))/2078505