Integrand size = 13, antiderivative size = 201 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {x^9}{18 b \left (a+b x^2\right )^9}-\frac {x^7}{32 b^2 \left (a+b x^2\right )^8}-\frac {x^5}{64 b^3 \left (a+b x^2\right )^7}-\frac {5 x^3}{768 b^4 \left (a+b x^2\right )^6}-\frac {x}{512 b^5 \left (a+b x^2\right )^5}+\frac {x}{4096 a b^5 \left (a+b x^2\right )^4}+\frac {7 x}{24576 a^2 b^5 \left (a+b x^2\right )^3}+\frac {35 x}{98304 a^3 b^5 \left (a+b x^2\right )^2}+\frac {35 x}{65536 a^4 b^5 \left (a+b x^2\right )}+\frac {35 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{9/2} b^{11/2}} \]
-1/18*x^9/b/(b*x^2+a)^9-1/32*x^7/b^2/(b*x^2+a)^8-1/64*x^5/b^3/(b*x^2+a)^7- 5/768*x^3/b^4/(b*x^2+a)^6-1/512*x/b^5/(b*x^2+a)^5+1/4096*x/a/b^5/(b*x^2+a) ^4+7/24576*x/a^2/b^5/(b*x^2+a)^3+35/98304*x/a^3/b^5/(b*x^2+a)^2+35/65536*x /a^4/b^5/(b*x^2+a)+35/65536*arctan(x*b^(1/2)/a^(1/2))/a^(9/2)/b^(11/2)
Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {a} \sqrt {b} x \left (-315 a^8-2730 a^7 b x^2-10458 a^6 b^2 x^4-23202 a^5 b^3 x^6-32768 a^4 b^4 x^8+23202 a^3 b^5 x^{10}+10458 a^2 b^6 x^{12}+2730 a b^7 x^{14}+315 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+315 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{589824 a^{9/2} b^{11/2}} \]
((Sqrt[a]*Sqrt[b]*x*(-315*a^8 - 2730*a^7*b*x^2 - 10458*a^6*b^2*x^4 - 23202 *a^5*b^3*x^6 - 32768*a^4*b^4*x^8 + 23202*a^3*b^5*x^10 + 10458*a^2*b^6*x^12 + 2730*a*b^7*x^14 + 315*b^8*x^16))/(a + b*x^2)^9 + 315*ArcTan[(Sqrt[b]*x) /Sqrt[a]])/(589824*a^(9/2)*b^(11/2))
Time = 0.29 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.26, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {252, 252, 252, 252, 252, 215, 215, 215, 215, 218}
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.
| \(\displaystyle \int \frac {x^{10}}{\left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\int \frac {x^8}{\left (b x^2+a\right )^9}dx}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {7 \int \frac {x^6}{\left (b x^2+a\right )^8}dx}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {7 \left (\frac {5 \int \frac {x^4}{\left (b x^2+a\right )^7}dx}{14 b}-\frac {x^5}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {7 \left (\frac {5 \left (\frac {\int \frac {x^2}{\left (b x^2+a\right )^6}dx}{4 b}-\frac {x^3}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^5}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {7 \left (\frac {5 \left (\frac {\frac {\int \frac {1}{\left (b x^2+a\right )^5}dx}{10 b}-\frac {x}{10 b \left (a+b x^2\right )^5}}{4 b}-\frac {x^3}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^5}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {7 \left (\frac {5 \left (\frac {\frac {\frac {7 \int \frac {1}{\left (b x^2+a\right )^4}dx}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}}{10 b}-\frac {x}{10 b \left (a+b x^2\right )^5}}{4 b}-\frac {x^3}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^5}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {7 \left (\frac {5 \left (\frac {\frac {\frac {7 \left (\frac {5 \int \frac {1}{\left (b x^2+a\right )^3}dx}{6 a}+\frac {x}{6 a \left (a+b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}}{10 b}-\frac {x}{10 b \left (a+b x^2\right )^5}}{4 b}-\frac {x^3}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^5}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {7 \left (\frac {5 \left (\frac {\frac {\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{\left (b x^2+a\right )^2}dx}{4 a}+\frac {x}{4 a \left (a+b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a+b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}}{10 b}-\frac {x}{10 b \left (a+b x^2\right )^5}}{4 b}-\frac {x^3}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^5}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {7 \left (\frac {5 \left (\frac {\frac {\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{b x^2+a}dx}{2 a}+\frac {x}{2 a \left (a+b x^2\right )}\right )}{4 a}+\frac {x}{4 a \left (a+b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a+b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}}{10 b}-\frac {x}{10 b \left (a+b x^2\right )^5}}{4 b}-\frac {x^3}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^5}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {7 \left (\frac {5 \left (\frac {\frac {\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b x^2\right )}\right )}{4 a}+\frac {x}{4 a \left (a+b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a+b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}}{10 b}-\frac {x}{10 b \left (a+b x^2\right )^5}}{4 b}-\frac {x^3}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^5}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^7}{16 b \left (a+b x^2\right )^8}}{2 b}-\frac {x^9}{18 b \left (a+b x^2\right )^9}\) |
-1/18*x^9/(b*(a + b*x^2)^9) + (-1/16*x^7/(b*(a + b*x^2)^8) + (7*(-1/14*x^5 /(b*(a + b*x^2)^7) + (5*(-1/12*x^3/(b*(a + b*x^2)^6) + (-1/10*x/(b*(a + b* x^2)^5) + (x/(8*a*(a + b*x^2)^4) + (7*(x/(6*a*(a + b*x^2)^3) + (5*(x/(4*a* (a + b*x^2)^2) + (3*(x/(2*a*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2* a^(3/2)*Sqrt[b])))/(4*a)))/(6*a)))/(8*a))/(10*b))/(4*b)))/(14*b)))/(16*b)) /(2*b)
3.3.16.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Time = 1.80 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(\frac {-\frac {35 a^{4} x}{65536 b^{5}}-\frac {455 a^{3} x^{3}}{98304 b^{4}}-\frac {581 a^{2} x^{5}}{32768 b^{3}}-\frac {1289 a \,x^{7}}{32768 b^{2}}-\frac {x^{9}}{18 b}+\frac {1289 x^{11}}{32768 a}+\frac {581 b \,x^{13}}{32768 a^{2}}+\frac {455 b^{2} x^{15}}{98304 a^{3}}+\frac {35 b^{3} x^{17}}{65536 a^{4}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {35 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 a^{4} b^{5} \sqrt {a b}}\) | \(122\) |
| risch | \(\frac {-\frac {35 a^{4} x}{65536 b^{5}}-\frac {455 a^{3} x^{3}}{98304 b^{4}}-\frac {581 a^{2} x^{5}}{32768 b^{3}}-\frac {1289 a \,x^{7}}{32768 b^{2}}-\frac {x^{9}}{18 b}+\frac {1289 x^{11}}{32768 a}+\frac {581 b \,x^{13}}{32768 a^{2}}+\frac {455 b^{2} x^{15}}{98304 a^{3}}+\frac {35 b^{3} x^{17}}{65536 a^{4}}}{\left (b \,x^{2}+a \right )^{9}}-\frac {35 \ln \left (b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{5} a^{4}}+\frac {35 \ln \left (-b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{5} a^{4}}\) | \(151\) |
(-35/65536*a^4*x/b^5-455/98304*a^3/b^4*x^3-581/32768*a^2/b^3*x^5-1289/3276 8*a/b^2*x^7-1/18*x^9/b+1289/32768/a*x^11+581/32768*b/a^2*x^13+455/98304*b^ 2/a^3*x^15+35/65536*b^3/a^4*x^17)/(b*x^2+a)^9+35/65536/a^4/b^5/(a*b)^(1/2) *arctan(b*x/(a*b)^(1/2))
Time = 0.25 (sec) , antiderivative size = 654, normalized size of antiderivative = 3.25 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {630 \, a b^{9} x^{17} + 5460 \, a^{2} b^{8} x^{15} + 20916 \, a^{3} b^{7} x^{13} + 46404 \, a^{4} b^{6} x^{11} - 65536 \, a^{5} b^{5} x^{9} - 46404 \, a^{6} b^{4} x^{7} - 20916 \, a^{7} b^{3} x^{5} - 5460 \, a^{8} b^{2} x^{3} - 630 \, a^{9} b x - 315 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{1179648 \, {\left (a^{5} b^{15} x^{18} + 9 \, a^{6} b^{14} x^{16} + 36 \, a^{7} b^{13} x^{14} + 84 \, a^{8} b^{12} x^{12} + 126 \, a^{9} b^{11} x^{10} + 126 \, a^{10} b^{10} x^{8} + 84 \, a^{11} b^{9} x^{6} + 36 \, a^{12} b^{8} x^{4} + 9 \, a^{13} b^{7} x^{2} + a^{14} b^{6}\right )}}, \frac {315 \, a b^{9} x^{17} + 2730 \, a^{2} b^{8} x^{15} + 10458 \, a^{3} b^{7} x^{13} + 23202 \, a^{4} b^{6} x^{11} - 32768 \, a^{5} b^{5} x^{9} - 23202 \, a^{6} b^{4} x^{7} - 10458 \, a^{7} b^{3} x^{5} - 2730 \, a^{8} b^{2} x^{3} - 315 \, a^{9} b x + 315 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{589824 \, {\left (a^{5} b^{15} x^{18} + 9 \, a^{6} b^{14} x^{16} + 36 \, a^{7} b^{13} x^{14} + 84 \, a^{8} b^{12} x^{12} + 126 \, a^{9} b^{11} x^{10} + 126 \, a^{10} b^{10} x^{8} + 84 \, a^{11} b^{9} x^{6} + 36 \, a^{12} b^{8} x^{4} + 9 \, a^{13} b^{7} x^{2} + a^{14} b^{6}\right )}}\right ] \]
[1/1179648*(630*a*b^9*x^17 + 5460*a^2*b^8*x^15 + 20916*a^3*b^7*x^13 + 4640 4*a^4*b^6*x^11 - 65536*a^5*b^5*x^9 - 46404*a^6*b^4*x^7 - 20916*a^7*b^3*x^5 - 5460*a^8*b^2*x^3 - 630*a^9*b*x - 315*(b^9*x^18 + 9*a*b^8*x^16 + 36*a^2* b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x^10 + 126*a^5*b^4*x^8 + 84*a^6*b ^3*x^6 + 36*a^7*b^2*x^4 + 9*a^8*b*x^2 + a^9)*sqrt(-a*b)*log((b*x^2 - 2*sqr t(-a*b)*x - a)/(b*x^2 + a)))/(a^5*b^15*x^18 + 9*a^6*b^14*x^16 + 36*a^7*b^1 3*x^14 + 84*a^8*b^12*x^12 + 126*a^9*b^11*x^10 + 126*a^10*b^10*x^8 + 84*a^1 1*b^9*x^6 + 36*a^12*b^8*x^4 + 9*a^13*b^7*x^2 + a^14*b^6), 1/589824*(315*a* b^9*x^17 + 2730*a^2*b^8*x^15 + 10458*a^3*b^7*x^13 + 23202*a^4*b^6*x^11 - 3 2768*a^5*b^5*x^9 - 23202*a^6*b^4*x^7 - 10458*a^7*b^3*x^5 - 2730*a^8*b^2*x^ 3 - 315*a^9*b*x + 315*(b^9*x^18 + 9*a*b^8*x^16 + 36*a^2*b^7*x^14 + 84*a^3* b^6*x^12 + 126*a^4*b^5*x^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 36*a^7*b^ 2*x^4 + 9*a^8*b*x^2 + a^9)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^5*b^15*x^18 + 9*a^6*b^14*x^16 + 36*a^7*b^13*x^14 + 84*a^8*b^12*x^12 + 126*a^9*b^11*x^ 10 + 126*a^10*b^10*x^8 + 84*a^11*b^9*x^6 + 36*a^12*b^8*x^4 + 9*a^13*b^7*x^ 2 + a^14*b^6)]
Time = 0.59 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.45 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{10}} \, dx=- \frac {35 \sqrt {- \frac {1}{a^{9} b^{11}}} \log {\left (- a^{5} b^{5} \sqrt {- \frac {1}{a^{9} b^{11}}} + x \right )}}{131072} + \frac {35 \sqrt {- \frac {1}{a^{9} b^{11}}} \log {\left (a^{5} b^{5} \sqrt {- \frac {1}{a^{9} b^{11}}} + x \right )}}{131072} + \frac {- 315 a^{8} x - 2730 a^{7} b x^{3} - 10458 a^{6} b^{2} x^{5} - 23202 a^{5} b^{3} x^{7} - 32768 a^{4} b^{4} x^{9} + 23202 a^{3} b^{5} x^{11} + 10458 a^{2} b^{6} x^{13} + 2730 a b^{7} x^{15} + 315 b^{8} x^{17}}{589824 a^{13} b^{5} + 5308416 a^{12} b^{6} x^{2} + 21233664 a^{11} b^{7} x^{4} + 49545216 a^{10} b^{8} x^{6} + 74317824 a^{9} b^{9} x^{8} + 74317824 a^{8} b^{10} x^{10} + 49545216 a^{7} b^{11} x^{12} + 21233664 a^{6} b^{12} x^{14} + 5308416 a^{5} b^{13} x^{16} + 589824 a^{4} b^{14} x^{18}} \]
-35*sqrt(-1/(a**9*b**11))*log(-a**5*b**5*sqrt(-1/(a**9*b**11)) + x)/131072 + 35*sqrt(-1/(a**9*b**11))*log(a**5*b**5*sqrt(-1/(a**9*b**11)) + x)/13107 2 + (-315*a**8*x - 2730*a**7*b*x**3 - 10458*a**6*b**2*x**5 - 23202*a**5*b* *3*x**7 - 32768*a**4*b**4*x**9 + 23202*a**3*b**5*x**11 + 10458*a**2*b**6*x **13 + 2730*a*b**7*x**15 + 315*b**8*x**17)/(589824*a**13*b**5 + 5308416*a* *12*b**6*x**2 + 21233664*a**11*b**7*x**4 + 49545216*a**10*b**8*x**6 + 7431 7824*a**9*b**9*x**8 + 74317824*a**8*b**10*x**10 + 49545216*a**7*b**11*x**1 2 + 21233664*a**6*b**12*x**14 + 5308416*a**5*b**13*x**16 + 589824*a**4*b** 14*x**18)
Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.10 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{10}} \, dx=\frac {315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} - 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x}{589824 \, {\left (a^{4} b^{14} x^{18} + 9 \, a^{5} b^{13} x^{16} + 36 \, a^{6} b^{12} x^{14} + 84 \, a^{7} b^{11} x^{12} + 126 \, a^{8} b^{10} x^{10} + 126 \, a^{9} b^{9} x^{8} + 84 \, a^{10} b^{8} x^{6} + 36 \, a^{11} b^{7} x^{4} + 9 \, a^{12} b^{6} x^{2} + a^{13} b^{5}\right )}} + \frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{4} b^{5}} \]
1/589824*(315*b^8*x^17 + 2730*a*b^7*x^15 + 10458*a^2*b^6*x^13 + 23202*a^3* b^5*x^11 - 32768*a^4*b^4*x^9 - 23202*a^5*b^3*x^7 - 10458*a^6*b^2*x^5 - 273 0*a^7*b*x^3 - 315*a^8*x)/(a^4*b^14*x^18 + 9*a^5*b^13*x^16 + 36*a^6*b^12*x^ 14 + 84*a^7*b^11*x^12 + 126*a^8*b^10*x^10 + 126*a^9*b^9*x^8 + 84*a^10*b^8* x^6 + 36*a^11*b^7*x^4 + 9*a^12*b^6*x^2 + a^13*b^5) + 35/65536*arctan(b*x/s qrt(a*b))/(sqrt(a*b)*a^4*b^5)
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.64 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{10}} \, dx=\frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{4} b^{5}} + \frac {315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} - 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x}{589824 \, {\left (b x^{2} + a\right )}^{9} a^{4} b^{5}} \]
35/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4*b^5) + 1/589824*(315*b^8*x^1 7 + 2730*a*b^7*x^15 + 10458*a^2*b^6*x^13 + 23202*a^3*b^5*x^11 - 32768*a^4* b^4*x^9 - 23202*a^5*b^3*x^7 - 10458*a^6*b^2*x^5 - 2730*a^7*b*x^3 - 315*a^8 *x)/((b*x^2 + a)^9*a^4*b^5)
Time = 4.65 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.02 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{10}} \, dx=\frac {35\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{9/2}\,b^{11/2}}-\frac {\frac {x^9}{18\,b}-\frac {1289\,x^{11}}{32768\,a}+\frac {1289\,a\,x^7}{32768\,b^2}+\frac {35\,a^4\,x}{65536\,b^5}-\frac {581\,b\,x^{13}}{32768\,a^2}+\frac {581\,a^2\,x^5}{32768\,b^3}+\frac {455\,a^3\,x^3}{98304\,b^4}-\frac {455\,b^2\,x^{15}}{98304\,a^3}-\frac {35\,b^3\,x^{17}}{65536\,a^4}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \]
(35*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^(9/2)*b^(11/2)) - (x^9/(18*b) - (1 289*x^11)/(32768*a) + (1289*a*x^7)/(32768*b^2) + (35*a^4*x)/(65536*b^5) - (581*b*x^13)/(32768*a^2) + (581*a^2*x^5)/(32768*b^3) + (455*a^3*x^3)/(9830 4*b^4) - (455*b^2*x^15)/(98304*a^3) - (35*b^3*x^17)/(65536*a^4))/(a^9 + b^ 9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 36*a^7*b^2*x^4 + 84*a^6*b^3*x^6 + 12 6*a^5*b^4*x^8 + 126*a^4*b^5*x^10 + 84*a^3*b^6*x^12 + 36*a^2*b^7*x^14)