Integrand size = 9, antiderivative size = 181 \[ \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx=\frac {x}{18 a \left (a+b x^2\right )^9}+\frac {17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac {85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac {1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac {2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac {2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac {2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac {12155 x}{98304 a^8 \left (a+b x^2\right )^2}+\frac {12155 x}{65536 a^9 \left (a+b x^2\right )}+\frac {12155 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{19/2} \sqrt {b}} \]
1/18*x/a/(b*x^2+a)^9+17/288*x/a^2/(b*x^2+a)^8+85/1344*x/a^3/(b*x^2+a)^7+11 05/16128*x/a^4/(b*x^2+a)^6+2431/32256*x/a^5/(b*x^2+a)^5+2431/28672*x/a^6/( b*x^2+a)^4+2431/24576*x/a^7/(b*x^2+a)^3+12155/98304*x/a^8/(b*x^2+a)^2+1215 5/65536*x/a^9/(b*x^2+a)+12155/65536*arctan(x*b^(1/2)/a^(1/2))/a^(19/2)/b^( 1/2)
Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {3363003 a^8 x+16759722 a^7 b x^3+44765658 a^6 b^2 x^5+73947042 a^5 b^3 x^7+79659008 a^4 b^4 x^9+56404062 a^3 b^5 x^{11}+25423398 a^2 b^6 x^{13}+6636630 a b^7 x^{15}+765765 b^8 x^{17}}{a^9 \left (a+b x^2\right )^9}+\frac {765765 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{19/2} \sqrt {b}}}{4128768} \]
((3363003*a^8*x + 16759722*a^7*b*x^3 + 44765658*a^6*b^2*x^5 + 73947042*a^5 *b^3*x^7 + 79659008*a^4*b^4*x^9 + 56404062*a^3*b^5*x^11 + 25423398*a^2*b^6 *x^13 + 6636630*a*b^7*x^15 + 765765*b^8*x^17)/(a^9*(a + b*x^2)^9) + (76576 5*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(19/2)*Sqrt[b]))/4128768
Time = 0.27 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.35, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {215, 215, 215, 215, 215, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \int \frac {1}{\left (b x^2+a\right )^9}dx}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \left (\frac {15 \int \frac {1}{\left (b x^2+a\right )^8}dx}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \left (\frac {15 \left (\frac {13 \int \frac {1}{\left (b x^2+a\right )^7}dx}{14 a}+\frac {x}{14 a \left (a+b x^2\right )^7}\right )}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \int \frac {1}{\left (b x^2+a\right )^6}dx}{12 a}+\frac {x}{12 a \left (a+b x^2\right )^6}\right )}{14 a}+\frac {x}{14 a \left (a+b x^2\right )^7}\right )}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \int \frac {1}{\left (b x^2+a\right )^5}dx}{10 a}+\frac {x}{10 a \left (a+b x^2\right )^5}\right )}{12 a}+\frac {x}{12 a \left (a+b x^2\right )^6}\right )}{14 a}+\frac {x}{14 a \left (a+b x^2\right )^7}\right )}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \int \frac {1}{\left (b x^2+a\right )^4}dx}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}\right )}{10 a}+\frac {x}{10 a \left (a+b x^2\right )^5}\right )}{12 a}+\frac {x}{12 a \left (a+b x^2\right )^6}\right )}{14 a}+\frac {x}{14 a \left (a+b x^2\right )^7}\right )}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{\left (b x^2+a\right )^3}dx}{6 a}+\frac {x}{6 a \left (a+b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}\right )}{10 a}+\frac {x}{10 a \left (a+b x^2\right )^5}\right )}{12 a}+\frac {x}{12 a \left (a+b x^2\right )^6}\right )}{14 a}+\frac {x}{14 a \left (a+b x^2\right )^7}\right )}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{\left (b x^2+a\right )^2}dx}{4 a}+\frac {x}{4 a \left (a+b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a+b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}\right )}{10 a}+\frac {x}{10 a \left (a+b x^2\right )^5}\right )}{12 a}+\frac {x}{12 a \left (a+b x^2\right )^6}\right )}{14 a}+\frac {x}{14 a \left (a+b x^2\right )^7}\right )}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{b x^2+a}dx}{2 a}+\frac {x}{2 a \left (a+b x^2\right )}\right )}{4 a}+\frac {x}{4 a \left (a+b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a+b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}\right )}{10 a}+\frac {x}{10 a \left (a+b x^2\right )^5}\right )}{12 a}+\frac {x}{12 a \left (a+b x^2\right )^6}\right )}{14 a}+\frac {x}{14 a \left (a+b x^2\right )^7}\right )}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b x^2\right )}\right )}{4 a}+\frac {x}{4 a \left (a+b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a+b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a+b x^2\right )^4}\right )}{10 a}+\frac {x}{10 a \left (a+b x^2\right )^5}\right )}{12 a}+\frac {x}{12 a \left (a+b x^2\right )^6}\right )}{14 a}+\frac {x}{14 a \left (a+b x^2\right )^7}\right )}{16 a}+\frac {x}{16 a \left (a+b x^2\right )^8}\right )}{18 a}+\frac {x}{18 a \left (a+b x^2\right )^9}\) |
x/(18*a*(a + b*x^2)^9) + (17*(x/(16*a*(a + b*x^2)^8) + (15*(x/(14*a*(a + b *x^2)^7) + (13*(x/(12*a*(a + b*x^2)^6) + (11*(x/(10*a*(a + b*x^2)^5) + (9* (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)*Sq rt[b])))/(4*a)))/(6*a)))/(8*a)))/(10*a)))/(12*a)))/(14*a)))/(16*a)))/(18*a )
3.3.21.3.1 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]
Time = 1.82 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(\frac {\frac {53381 x}{65536 a}+\frac {399041 b \,x^{3}}{98304 a^{2}}+\frac {355283 b^{2} x^{5}}{32768 a^{3}}+\frac {4108169 b^{3} x^{7}}{229376 a^{4}}+\frac {2431 b^{4} x^{9}}{126 a^{5}}+\frac {3133559 b^{5} x^{11}}{229376 a^{6}}+\frac {201773 b^{6} x^{13}}{32768 a^{7}}+\frac {158015 b^{7} x^{15}}{98304 a^{8}}+\frac {12155 b^{8} x^{17}}{65536 a^{9}}}{\left (b \,x^{2}+a \right )^{9}}-\frac {12155 \ln \left (b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, a^{9}}+\frac {12155 \ln \left (-b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, a^{9}}\) | \(150\) |
| default | \(\frac {x}{18 a \left (b \,x^{2}+a \right )^{9}}+\frac {\frac {17 x}{288 a \left (b \,x^{2}+a \right )^{8}}+\frac {17 \left (\frac {15 x}{224 a \left (b \,x^{2}+a \right )^{7}}+\frac {15 \left (\frac {13 x}{168 a \left (b \,x^{2}+a \right )^{6}}+\frac {13 \left (\frac {11 x}{120 a \left (b \,x^{2}+a \right )^{5}}+\frac {11 \left (\frac {9 x}{80 a \left (b \,x^{2}+a \right )^{4}}+\frac {9 \left (\frac {7 x}{48 a \left (b \,x^{2}+a \right )^{3}}+\frac {7 \left (\frac {5 x}{24 a \left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {3 x}{8 a \left (b \,x^{2}+a \right )}+\frac {3 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{12 a}\right )}{14 a}\right )}{16 a}\right )}{18 a}}{a}\) | \(204\) |
(53381/65536*x/a+399041/98304*b/a^2*x^3+355283/32768*b^2/a^3*x^5+4108169/2 29376*b^3/a^4*x^7+2431/126*b^4/a^5*x^9+3133559/229376*b^5/a^6*x^11+201773/ 32768*b^6/a^7*x^13+158015/98304*b^7/a^8*x^15+12155/65536*b^8/a^9*x^17)/(b* x^2+a)^9-12155/131072/(-a*b)^(1/2)/a^9*ln(b*x+(-a*b)^(1/2))+12155/131072/( -a*b)^(1/2)/a^9*ln(-b*x+(-a*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (153) = 306\).
Time = 0.26 (sec) , antiderivative size = 650, normalized size of antiderivative = 3.59 \[ \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {1531530 \, a b^{9} x^{17} + 13273260 \, a^{2} b^{8} x^{15} + 50846796 \, a^{3} b^{7} x^{13} + 112808124 \, a^{4} b^{6} x^{11} + 159318016 \, a^{5} b^{5} x^{9} + 147894084 \, a^{6} b^{4} x^{7} + 89531316 \, a^{7} b^{3} x^{5} + 33519444 \, a^{8} b^{2} x^{3} + 6726006 \, a^{9} b x - 765765 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{8257536 \, {\left (a^{10} b^{10} x^{18} + 9 \, a^{11} b^{9} x^{16} + 36 \, a^{12} b^{8} x^{14} + 84 \, a^{13} b^{7} x^{12} + 126 \, a^{14} b^{6} x^{10} + 126 \, a^{15} b^{5} x^{8} + 84 \, a^{16} b^{4} x^{6} + 36 \, a^{17} b^{3} x^{4} + 9 \, a^{18} b^{2} x^{2} + a^{19} b\right )}}, \frac {765765 \, a b^{9} x^{17} + 6636630 \, a^{2} b^{8} x^{15} + 25423398 \, a^{3} b^{7} x^{13} + 56404062 \, a^{4} b^{6} x^{11} + 79659008 \, a^{5} b^{5} x^{9} + 73947042 \, a^{6} b^{4} x^{7} + 44765658 \, a^{7} b^{3} x^{5} + 16759722 \, a^{8} b^{2} x^{3} + 3363003 \, a^{9} b x + 765765 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{4128768 \, {\left (a^{10} b^{10} x^{18} + 9 \, a^{11} b^{9} x^{16} + 36 \, a^{12} b^{8} x^{14} + 84 \, a^{13} b^{7} x^{12} + 126 \, a^{14} b^{6} x^{10} + 126 \, a^{15} b^{5} x^{8} + 84 \, a^{16} b^{4} x^{6} + 36 \, a^{17} b^{3} x^{4} + 9 \, a^{18} b^{2} x^{2} + a^{19} b\right )}}\right ] \]
[1/8257536*(1531530*a*b^9*x^17 + 13273260*a^2*b^8*x^15 + 50846796*a^3*b^7* x^13 + 112808124*a^4*b^6*x^11 + 159318016*a^5*b^5*x^9 + 147894084*a^6*b^4* x^7 + 89531316*a^7*b^3*x^5 + 33519444*a^8*b^2*x^3 + 6726006*a^9*b*x - 7657 65*(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*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^10*b^ 10*x^18 + 9*a^11*b^9*x^16 + 36*a^12*b^8*x^14 + 84*a^13*b^7*x^12 + 126*a^14 *b^6*x^10 + 126*a^15*b^5*x^8 + 84*a^16*b^4*x^6 + 36*a^17*b^3*x^4 + 9*a^18* b^2*x^2 + a^19*b), 1/4128768*(765765*a*b^9*x^17 + 6636630*a^2*b^8*x^15 + 2 5423398*a^3*b^7*x^13 + 56404062*a^4*b^6*x^11 + 79659008*a^5*b^5*x^9 + 7394 7042*a^6*b^4*x^7 + 44765658*a^7*b^3*x^5 + 16759722*a^8*b^2*x^3 + 3363003*a ^9*b*x + 765765*(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^10*b^10*x^18 + 9* a^11*b^9*x^16 + 36*a^12*b^8*x^14 + 84*a^13*b^7*x^12 + 126*a^14*b^6*x^10 + 126*a^15*b^5*x^8 + 84*a^16*b^4*x^6 + 36*a^17*b^3*x^4 + 9*a^18*b^2*x^2 + a^ 19*b)]
Time = 0.52 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx=- \frac {12155 \sqrt {- \frac {1}{a^{19} b}} \log {\left (- a^{10} \sqrt {- \frac {1}{a^{19} b}} + x \right )}}{131072} + \frac {12155 \sqrt {- \frac {1}{a^{19} b}} \log {\left (a^{10} \sqrt {- \frac {1}{a^{19} b}} + x \right )}}{131072} + \frac {3363003 a^{8} x + 16759722 a^{7} b x^{3} + 44765658 a^{6} b^{2} x^{5} + 73947042 a^{5} b^{3} x^{7} + 79659008 a^{4} b^{4} x^{9} + 56404062 a^{3} b^{5} x^{11} + 25423398 a^{2} b^{6} x^{13} + 6636630 a b^{7} x^{15} + 765765 b^{8} x^{17}}{4128768 a^{18} + 37158912 a^{17} b x^{2} + 148635648 a^{16} b^{2} x^{4} + 346816512 a^{15} b^{3} x^{6} + 520224768 a^{14} b^{4} x^{8} + 520224768 a^{13} b^{5} x^{10} + 346816512 a^{12} b^{6} x^{12} + 148635648 a^{11} b^{7} x^{14} + 37158912 a^{10} b^{8} x^{16} + 4128768 a^{9} b^{9} x^{18}} \]
-12155*sqrt(-1/(a**19*b))*log(-a**10*sqrt(-1/(a**19*b)) + x)/131072 + 1215 5*sqrt(-1/(a**19*b))*log(a**10*sqrt(-1/(a**19*b)) + x)/131072 + (3363003*a **8*x + 16759722*a**7*b*x**3 + 44765658*a**6*b**2*x**5 + 73947042*a**5*b** 3*x**7 + 79659008*a**4*b**4*x**9 + 56404062*a**3*b**5*x**11 + 25423398*a** 2*b**6*x**13 + 6636630*a*b**7*x**15 + 765765*b**8*x**17)/(4128768*a**18 + 37158912*a**17*b*x**2 + 148635648*a**16*b**2*x**4 + 346816512*a**15*b**3*x **6 + 520224768*a**14*b**4*x**8 + 520224768*a**13*b**5*x**10 + 346816512*a **12*b**6*x**12 + 148635648*a**11*b**7*x**14 + 37158912*a**10*b**8*x**16 + 4128768*a**9*b**9*x**18)
Time = 0.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx=\frac {765765 \, b^{8} x^{17} + 6636630 \, a b^{7} x^{15} + 25423398 \, a^{2} b^{6} x^{13} + 56404062 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 73947042 \, a^{5} b^{3} x^{7} + 44765658 \, a^{6} b^{2} x^{5} + 16759722 \, a^{7} b x^{3} + 3363003 \, a^{8} x}{4128768 \, {\left (a^{9} b^{9} x^{18} + 9 \, a^{10} b^{8} x^{16} + 36 \, a^{11} b^{7} x^{14} + 84 \, a^{12} b^{6} x^{12} + 126 \, a^{13} b^{5} x^{10} + 126 \, a^{14} b^{4} x^{8} + 84 \, a^{15} b^{3} x^{6} + 36 \, a^{16} b^{2} x^{4} + 9 \, a^{17} b x^{2} + a^{18}\right )}} + \frac {12155 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{9}} \]
1/4128768*(765765*b^8*x^17 + 6636630*a*b^7*x^15 + 25423398*a^2*b^6*x^13 + 56404062*a^3*b^5*x^11 + 79659008*a^4*b^4*x^9 + 73947042*a^5*b^3*x^7 + 4476 5658*a^6*b^2*x^5 + 16759722*a^7*b*x^3 + 3363003*a^8*x)/(a^9*b^9*x^18 + 9*a ^10*b^8*x^16 + 36*a^11*b^7*x^14 + 84*a^12*b^6*x^12 + 126*a^13*b^5*x^10 + 1 26*a^14*b^4*x^8 + 84*a^15*b^3*x^6 + 36*a^16*b^2*x^4 + 9*a^17*b*x^2 + a^18) + 12155/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^9)
Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx=\frac {12155 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{9}} + \frac {765765 \, b^{8} x^{17} + 6636630 \, a b^{7} x^{15} + 25423398 \, a^{2} b^{6} x^{13} + 56404062 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 73947042 \, a^{5} b^{3} x^{7} + 44765658 \, a^{6} b^{2} x^{5} + 16759722 \, a^{7} b x^{3} + 3363003 \, a^{8} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} a^{9}} \]
12155/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^9) + 1/4128768*(765765*b^8* x^17 + 6636630*a*b^7*x^15 + 25423398*a^2*b^6*x^13 + 56404062*a^3*b^5*x^11 + 79659008*a^4*b^4*x^9 + 73947042*a^5*b^3*x^7 + 44765658*a^6*b^2*x^5 + 167 59722*a^7*b*x^3 + 3363003*a^8*x)/((b*x^2 + a)^9*a^9)
Time = 4.61 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {53381\,x}{65536\,a}+\frac {399041\,b\,x^3}{98304\,a^2}+\frac {355283\,b^2\,x^5}{32768\,a^3}+\frac {4108169\,b^3\,x^7}{229376\,a^4}+\frac {2431\,b^4\,x^9}{126\,a^5}+\frac {3133559\,b^5\,x^{11}}{229376\,a^6}+\frac {201773\,b^6\,x^{13}}{32768\,a^7}+\frac {158015\,b^7\,x^{15}}{98304\,a^8}+\frac {12155\,b^8\,x^{17}}{65536\,a^9}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {12155\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{19/2}\,\sqrt {b}} \]
((53381*x)/(65536*a) + (399041*b*x^3)/(98304*a^2) + (355283*b^2*x^5)/(3276 8*a^3) + (4108169*b^3*x^7)/(229376*a^4) + (2431*b^4*x^9)/(126*a^5) + (3133 559*b^5*x^11)/(229376*a^6) + (201773*b^6*x^13)/(32768*a^7) + (158015*b^7*x ^15)/(98304*a^8) + (12155*b^8*x^17)/(65536*a^9))/(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 + 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) + (12155*atan((b^(1/ 2)*x)/a^(1/2)))/(65536*a^(19/2)*b^(1/2))