Integrand size = 13, antiderivative size = 108 \[ \int x^4 \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^5}{5}+\frac {8}{7} a^7 b x^7+\frac {28}{9} a^6 b^2 x^9+\frac {56}{11} a^5 b^3 x^{11}+\frac {70}{13} a^4 b^4 x^{13}+\frac {56}{15} a^3 b^5 x^{15}+\frac {28}{17} a^2 b^6 x^{17}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{21}}{21} \] Output:
1/5*a^8*x^5+8/7*a^7*b*x^7+28/9*a^6*b^2*x^9+56/11*a^5*b^3*x^11+70/13*a^4*b^ 4*x^13+56/15*a^3*b^5*x^15+28/17*a^2*b^6*x^17+8/19*a*b^7*x^19+1/21*b^8*x^21
Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^2\right )^8 \, dx=\frac {a^8 x^5}{5}+\frac {8}{7} a^7 b x^7+\frac {28}{9} a^6 b^2 x^9+\frac {56}{11} a^5 b^3 x^{11}+\frac {70}{13} a^4 b^4 x^{13}+\frac {56}{15} a^3 b^5 x^{15}+\frac {28}{17} a^2 b^6 x^{17}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{21}}{21} \] Input:
Integrate[x^4*(a + b*x^2)^8,x]
Output:
(a^8*x^5)/5 + (8*a^7*b*x^7)/7 + (28*a^6*b^2*x^9)/9 + (56*a^5*b^3*x^11)/11 + (70*a^4*b^4*x^13)/13 + (56*a^3*b^5*x^15)/15 + (28*a^2*b^6*x^17)/17 + (8* a*b^7*x^19)/19 + (b^8*x^21)/21
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.
\(\displaystyle \int x^4 \left (a+b x^2\right )^8 \, dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (a^8 x^4+8 a^7 b x^6+28 a^6 b^2 x^8+56 a^5 b^3 x^{10}+70 a^4 b^4 x^{12}+56 a^3 b^5 x^{14}+28 a^2 b^6 x^{16}+8 a b^7 x^{18}+b^8 x^{20}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^8 x^5}{5}+\frac {8}{7} a^7 b x^7+\frac {28}{9} a^6 b^2 x^9+\frac {56}{11} a^5 b^3 x^{11}+\frac {70}{13} a^4 b^4 x^{13}+\frac {56}{15} a^3 b^5 x^{15}+\frac {28}{17} a^2 b^6 x^{17}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{21}}{21}\) |
Input:
Int[x^4*(a + b*x^2)^8,x]
Output:
(a^8*x^5)/5 + (8*a^7*b*x^7)/7 + (28*a^6*b^2*x^9)/9 + (56*a^5*b^3*x^11)/11 + (70*a^4*b^4*x^13)/13 + (56*a^3*b^5*x^15)/15 + (28*a^2*b^6*x^17)/17 + (8* a*b^7*x^19)/19 + (b^8*x^21)/21
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 | \(\frac {1}{5} a^{8} x^{5}+\frac {8}{7} a^{7} b \,x^{7}+\frac {28}{9} a^{6} b^{2} x^{9}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {56}{15} a^{3} b^{5} x^{15}+\frac {28}{17} a^{2} b^{6} x^{17}+\frac {8}{19} a \,b^{7} x^{19}+\frac {1}{21} b^{8} x^{21}\) | \(91\) |
default | \(\frac {1}{5} a^{8} x^{5}+\frac {8}{7} a^{7} b \,x^{7}+\frac {28}{9} a^{6} b^{2} x^{9}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {56}{15} a^{3} b^{5} x^{15}+\frac {28}{17} a^{2} b^{6} x^{17}+\frac {8}{19} a \,b^{7} x^{19}+\frac {1}{21} b^{8} x^{21}\) | \(91\) |
norman | \(\frac {1}{5} a^{8} x^{5}+\frac {8}{7} a^{7} b \,x^{7}+\frac {28}{9} a^{6} b^{2} x^{9}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {56}{15} a^{3} b^{5} x^{15}+\frac {28}{17} a^{2} b^{6} x^{17}+\frac {8}{19} a \,b^{7} x^{19}+\frac {1}{21} b^{8} x^{21}\) | \(91\) |
risch | \(\frac {1}{5} a^{8} x^{5}+\frac {8}{7} a^{7} b \,x^{7}+\frac {28}{9} a^{6} b^{2} x^{9}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {56}{15} a^{3} b^{5} x^{15}+\frac {28}{17} a^{2} b^{6} x^{17}+\frac {8}{19} a \,b^{7} x^{19}+\frac {1}{21} b^{8} x^{21}\) | \(91\) |
parallelrisch | \(\frac {1}{5} a^{8} x^{5}+\frac {8}{7} a^{7} b \,x^{7}+\frac {28}{9} a^{6} b^{2} x^{9}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {56}{15} a^{3} b^{5} x^{15}+\frac {28}{17} a^{2} b^{6} x^{17}+\frac {8}{19} a \,b^{7} x^{19}+\frac {1}{21} b^{8} x^{21}\) | \(91\) |
orering | \(\frac {x^{5} \left (692835 b^{8} x^{16}+6126120 a \,b^{7} x^{14}+23963940 a^{2} b^{6} x^{12}+54318264 a^{3} b^{5} x^{10}+78343650 a^{4} b^{4} x^{8}+74070360 a^{5} b^{3} x^{6}+45265220 a^{6} b^{2} x^{4}+16628040 a^{7} b \,x^{2}+2909907 a^{8}\right )}{14549535}\) | \(93\) |
Input:
int(x^4*(b*x^2+a)^8,x,method=_RETURNVERBOSE)
Output:
1/5*a^8*x^5+8/7*a^7*b*x^7+28/9*a^6*b^2*x^9+56/11*a^5*b^3*x^11+70/13*a^4*b^ 4*x^13+56/15*a^3*b^5*x^15+28/17*a^2*b^6*x^17+8/19*a*b^7*x^19+1/21*b^8*x^21
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^4 \left (a+b x^2\right )^8 \, dx=\frac {1}{21} \, b^{8} x^{21} + \frac {8}{19} \, a b^{7} x^{19} + \frac {28}{17} \, a^{2} b^{6} x^{17} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {28}{9} \, a^{6} b^{2} x^{9} + \frac {8}{7} \, a^{7} b x^{7} + \frac {1}{5} \, a^{8} x^{5} \] Input:
integrate(x^4*(b*x^2+a)^8,x, algorithm="fricas")
Output:
1/21*b^8*x^21 + 8/19*a*b^7*x^19 + 28/17*a^2*b^6*x^17 + 56/15*a^3*b^5*x^15 + 70/13*a^4*b^4*x^13 + 56/11*a^5*b^3*x^11 + 28/9*a^6*b^2*x^9 + 8/7*a^7*b*x ^7 + 1/5*a^8*x^5
Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int x^4 \left (a+b x^2\right )^8 \, dx=\frac {a^{8} x^{5}}{5} + \frac {8 a^{7} b x^{7}}{7} + \frac {28 a^{6} b^{2} x^{9}}{9} + \frac {56 a^{5} b^{3} x^{11}}{11} + \frac {70 a^{4} b^{4} x^{13}}{13} + \frac {56 a^{3} b^{5} x^{15}}{15} + \frac {28 a^{2} b^{6} x^{17}}{17} + \frac {8 a b^{7} x^{19}}{19} + \frac {b^{8} x^{21}}{21} \] Input:
integrate(x**4*(b*x**2+a)**8,x)
Output:
a**8*x**5/5 + 8*a**7*b*x**7/7 + 28*a**6*b**2*x**9/9 + 56*a**5*b**3*x**11/1 1 + 70*a**4*b**4*x**13/13 + 56*a**3*b**5*x**15/15 + 28*a**2*b**6*x**17/17 + 8*a*b**7*x**19/19 + b**8*x**21/21
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^4 \left (a+b x^2\right )^8 \, dx=\frac {1}{21} \, b^{8} x^{21} + \frac {8}{19} \, a b^{7} x^{19} + \frac {28}{17} \, a^{2} b^{6} x^{17} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {28}{9} \, a^{6} b^{2} x^{9} + \frac {8}{7} \, a^{7} b x^{7} + \frac {1}{5} \, a^{8} x^{5} \] Input:
integrate(x^4*(b*x^2+a)^8,x, algorithm="maxima")
Output:
1/21*b^8*x^21 + 8/19*a*b^7*x^19 + 28/17*a^2*b^6*x^17 + 56/15*a^3*b^5*x^15 + 70/13*a^4*b^4*x^13 + 56/11*a^5*b^3*x^11 + 28/9*a^6*b^2*x^9 + 8/7*a^7*b*x ^7 + 1/5*a^8*x^5
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^4 \left (a+b x^2\right )^8 \, dx=\frac {1}{21} \, b^{8} x^{21} + \frac {8}{19} \, a b^{7} x^{19} + \frac {28}{17} \, a^{2} b^{6} x^{17} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {28}{9} \, a^{6} b^{2} x^{9} + \frac {8}{7} \, a^{7} b x^{7} + \frac {1}{5} \, a^{8} x^{5} \] Input:
integrate(x^4*(b*x^2+a)^8,x, algorithm="giac")
Output:
1/21*b^8*x^21 + 8/19*a*b^7*x^19 + 28/17*a^2*b^6*x^17 + 56/15*a^3*b^5*x^15 + 70/13*a^4*b^4*x^13 + 56/11*a^5*b^3*x^11 + 28/9*a^6*b^2*x^9 + 8/7*a^7*b*x ^7 + 1/5*a^8*x^5
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^4 \left (a+b x^2\right )^8 \, dx=\frac {a^8\,x^5}{5}+\frac {8\,a^7\,b\,x^7}{7}+\frac {28\,a^6\,b^2\,x^9}{9}+\frac {56\,a^5\,b^3\,x^{11}}{11}+\frac {70\,a^4\,b^4\,x^{13}}{13}+\frac {56\,a^3\,b^5\,x^{15}}{15}+\frac {28\,a^2\,b^6\,x^{17}}{17}+\frac {8\,a\,b^7\,x^{19}}{19}+\frac {b^8\,x^{21}}{21} \] Input:
int(x^4*(a + b*x^2)^8,x)
Output:
(a^8*x^5)/5 + (b^8*x^21)/21 + (8*a^7*b*x^7)/7 + (8*a*b^7*x^19)/19 + (28*a^ 6*b^2*x^9)/9 + (56*a^5*b^3*x^11)/11 + (70*a^4*b^4*x^13)/13 + (56*a^3*b^5*x ^15)/15 + (28*a^2*b^6*x^17)/17
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int x^4 \left (a+b x^2\right )^8 \, dx=\frac {x^{5} \left (692835 b^{8} x^{16}+6126120 a \,b^{7} x^{14}+23963940 a^{2} b^{6} x^{12}+54318264 a^{3} b^{5} x^{10}+78343650 a^{4} b^{4} x^{8}+74070360 a^{5} b^{3} x^{6}+45265220 a^{6} b^{2} x^{4}+16628040 a^{7} b \,x^{2}+2909907 a^{8}\right )}{14549535} \] Input:
int(x^4*(b*x^2+a)^8,x)
Output:
(x**5*(2909907*a**8 + 16628040*a**7*b*x**2 + 45265220*a**6*b**2*x**4 + 740 70360*a**5*b**3*x**6 + 78343650*a**4*b**4*x**8 + 54318264*a**3*b**5*x**10 + 23963940*a**2*b**6*x**12 + 6126120*a*b**7*x**14 + 692835*b**8*x**16))/14 549535