Integrand size = 13, antiderivative size = 231 \[ \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx=\frac {7436429 a^2 x}{65536 b^{12}}-\frac {7436429 a x^3}{196608 b^{11}}+\frac {7436429 x^5}{327680 b^{10}}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac {1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac {1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac {7436429 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{25/2}} \]
7436429/65536*a^2*x/b^12-7436429/196608*a*x^3/b^11+7436429/327680*x^5/b^10 -1/18*x^23/b/(b*x^2+a)^9-23/288*x^21/b^2/(b*x^2+a)^8-23/192*x^19/b^3/(b*x^ 2+a)^7-437/2304*x^17/b^4/(b*x^2+a)^6-7429/23040*x^15/b^5/(b*x^2+a)^5-7429/ 12288*x^13/b^6/(b*x^2+a)^4-96577/73728*x^11/b^7/(b*x^2+a)^3-1062347/294912 *x^9/b^8/(b*x^2+a)^2-1062347/65536*x^7/b^9/(b*x^2+a)-7436429/65536*a^(5/2) *arctan(x*b^(1/2)/a^(1/2))/b^(25/2)
Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.72 \[ \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {b} x \left (334639305 a^{11}+2900207310 a^{10} b x^2+11110024926 a^9 b^2 x^4+24648575094 a^8 b^3 x^6+34810986496 a^7 b^4 x^8+32314857354 a^6 b^5 x^{10}+19562592546 a^5 b^6 x^{12}+7323998514 a^4 b^7 x^{14}+1469632311 a^3 b^8 x^{16}+94961664 a^2 b^9 x^{18}-4521984 a b^{10} x^{20}+589824 b^{11} x^{22}\right )}{\left (a+b x^2\right )^9}-334639305 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2949120 b^{25/2}} \]
((Sqrt[b]*x*(334639305*a^11 + 2900207310*a^10*b*x^2 + 11110024926*a^9*b^2* x^4 + 24648575094*a^8*b^3*x^6 + 34810986496*a^7*b^4*x^8 + 32314857354*a^6* b^5*x^10 + 19562592546*a^5*b^6*x^12 + 7323998514*a^4*b^7*x^14 + 1469632311 *a^3*b^8*x^16 + 94961664*a^2*b^9*x^18 - 4521984*a*b^10*x^20 + 589824*b^11* x^22))/(a + b*x^2)^9 - 334639305*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(294 9120*b^(25/2))
Time = 0.39 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.29, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {252, 252, 252, 252, 252, 252, 252, 252, 252, 254, 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.
| \(\displaystyle \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \int \frac {x^{22}}{\left (b x^2+a\right )^9}dx}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \left (\frac {21 \int \frac {x^{20}}{\left (b x^2+a\right )^8}dx}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \int \frac {x^{18}}{\left (b x^2+a\right )^7}dx}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \int \frac {x^{16}}{\left (b x^2+a\right )^6}dx}{12 b}-\frac {x^{17}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \int \frac {x^{14}}{\left (b x^2+a\right )^5}dx}{2 b}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{17}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \int \frac {x^{12}}{\left (b x^2+a\right )^4}dx}{8 b}-\frac {x^{13}}{8 b \left (a+b x^2\right )^4}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{17}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \int \frac {x^{10}}{\left (b x^2+a\right )^3}dx}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{13}}{8 b \left (a+b x^2\right )^4}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{17}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \int \frac {x^8}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^9}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{13}}{8 b \left (a+b x^2\right )^4}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{17}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \int \frac {x^6}{b x^2+a}dx}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^9}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{13}}{8 b \left (a+b x^2\right )^4}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{17}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \int \left (\frac {x^4}{b}-\frac {a x^2}{b^2}-\frac {a^3}{b^3 \left (b x^2+a\right )}+\frac {a^2}{b^3}\right )dx}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^9}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{13}}{8 b \left (a+b x^2\right )^4}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{17}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}+\frac {a^2 x}{b^3}-\frac {a x^3}{3 b^2}+\frac {x^5}{5 b}\right )}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^9}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^{11}}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{13}}{8 b \left (a+b x^2\right )^4}\right )}{2 b}-\frac {x^{15}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{17}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{19}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{21}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}\) |
-1/18*x^23/(b*(a + b*x^2)^9) + (23*(-1/16*x^21/(b*(a + b*x^2)^8) + (21*(-1 /14*x^19/(b*(a + b*x^2)^7) + (19*(-1/12*x^17/(b*(a + b*x^2)^6) + (17*(-1/1 0*x^15/(b*(a + b*x^2)^5) + (3*(-1/8*x^13/(b*(a + b*x^2)^4) + (13*(-1/6*x^1 1/(b*(a + b*x^2)^3) + (11*(-1/4*x^9/(b*(a + b*x^2)^2) + (9*(-1/2*x^7/(b*(a + b*x^2)) + (7*((a^2*x)/b^3 - (a*x^3)/(3*b^2) + x^5/(5*b) - (a^(5/2)*ArcT an[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)))/(2*b)))/(4*b)))/(6*b)))/(8*b)))/(2*b))) /(12*b)))/(14*b)))/(16*b)))/(18*b)
3.3.9.3.1 Defintions of rubi rules used
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]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Time = 1.79 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65
| method | result | size |
| default | \(\frac {\frac {1}{5} b^{2} x^{5}-\frac {10}{3} a b \,x^{3}+55 a^{2} x}{b^{12}}-\frac {a^{3} \left (\frac {-\frac {3831949}{65536} a^{8} x -\frac {48340777}{98304} x^{3} b \,a^{7}-\frac {297702839}{163840} x^{5} b^{2} a^{6}-\frac {631790371}{163840} x^{7} b^{3} a^{5}-\frac {463199}{90} x^{9} b^{4} a^{4}-\frac {725918941}{163840} x^{11} b^{5} a^{3}-\frac {394553929}{163840} x^{13} b^{6} a^{2}-\frac {74539223}{98304} a \,b^{7} x^{15}-\frac {6981491}{65536} b^{8} x^{17}}{\left (b \,x^{2}+a \right )^{9}}+\frac {7436429 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}}\right )}{b^{12}}\) | \(151\) |
| risch | \(\frac {x^{5}}{5 b^{10}}-\frac {10 a \,x^{3}}{3 b^{11}}+\frac {55 a^{2} x}{b^{12}}+\frac {\frac {3831949}{65536} a^{11} x +\frac {48340777}{98304} a^{10} b \,x^{3}+\frac {297702839}{163840} b^{2} a^{9} x^{5}+\frac {631790371}{163840} a^{8} b^{3} x^{7}+\frac {463199}{90} a^{7} b^{4} x^{9}+\frac {725918941}{163840} a^{6} b^{5} x^{11}+\frac {394553929}{163840} a^{5} b^{6} x^{13}+\frac {74539223}{98304} a^{4} b^{7} x^{15}+\frac {6981491}{65536} a^{3} b^{8} x^{17}}{b^{12} \left (b \,x^{2}+a \right )^{9}}+\frac {7436429 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right )}{131072 b^{13}}-\frac {7436429 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right )}{131072 b^{13}}\) | \(189\) |
1/b^12*(1/5*b^2*x^5-10/3*a*b*x^3+55*a^2*x)-1/b^12*a^3*((-3831949/65536*a^8 *x-48340777/98304*x^3*b*a^7-297702839/163840*x^5*b^2*a^6-631790371/163840* x^7*b^3*a^5-463199/90*x^9*b^4*a^4-725918941/163840*x^11*b^5*a^3-394553929/ 163840*x^13*b^6*a^2-74539223/98304*a*b^7*x^15-6981491/65536*b^8*x^17)/(b*x ^2+a)^9+7436429/65536/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 718, normalized size of antiderivative = 3.11 \[ \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {1179648 \, b^{11} x^{23} - 9043968 \, a b^{10} x^{21} + 189923328 \, a^{2} b^{9} x^{19} + 2939264622 \, a^{3} b^{8} x^{17} + 14647997028 \, a^{4} b^{7} x^{15} + 39125185092 \, a^{5} b^{6} x^{13} + 64629714708 \, a^{6} b^{5} x^{11} + 69621972992 \, a^{7} b^{4} x^{9} + 49297150188 \, a^{8} b^{3} x^{7} + 22220049852 \, a^{9} b^{2} x^{5} + 5800414620 \, a^{10} b x^{3} + 669278610 \, a^{11} x + 334639305 \, {\left (a^{2} b^{9} x^{18} + 9 \, a^{3} b^{8} x^{16} + 36 \, a^{4} b^{7} x^{14} + 84 \, a^{5} b^{6} x^{12} + 126 \, a^{6} b^{5} x^{10} + 126 \, a^{7} b^{4} x^{8} + 84 \, a^{8} b^{3} x^{6} + 36 \, a^{9} b^{2} x^{4} + 9 \, a^{10} b x^{2} + a^{11}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{5898240 \, {\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}}, \frac {589824 \, b^{11} x^{23} - 4521984 \, a b^{10} x^{21} + 94961664 \, a^{2} b^{9} x^{19} + 1469632311 \, a^{3} b^{8} x^{17} + 7323998514 \, a^{4} b^{7} x^{15} + 19562592546 \, a^{5} b^{6} x^{13} + 32314857354 \, a^{6} b^{5} x^{11} + 34810986496 \, a^{7} b^{4} x^{9} + 24648575094 \, a^{8} b^{3} x^{7} + 11110024926 \, a^{9} b^{2} x^{5} + 2900207310 \, a^{10} b x^{3} + 334639305 \, a^{11} x - 334639305 \, {\left (a^{2} b^{9} x^{18} + 9 \, a^{3} b^{8} x^{16} + 36 \, a^{4} b^{7} x^{14} + 84 \, a^{5} b^{6} x^{12} + 126 \, a^{6} b^{5} x^{10} + 126 \, a^{7} b^{4} x^{8} + 84 \, a^{8} b^{3} x^{6} + 36 \, a^{9} b^{2} x^{4} + 9 \, a^{10} b x^{2} + a^{11}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{2949120 \, {\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}}\right ] \]
[1/5898240*(1179648*b^11*x^23 - 9043968*a*b^10*x^21 + 189923328*a^2*b^9*x^ 19 + 2939264622*a^3*b^8*x^17 + 14647997028*a^4*b^7*x^15 + 39125185092*a^5* b^6*x^13 + 64629714708*a^6*b^5*x^11 + 69621972992*a^7*b^4*x^9 + 4929715018 8*a^8*b^3*x^7 + 22220049852*a^9*b^2*x^5 + 5800414620*a^10*b*x^3 + 66927861 0*a^11*x + 334639305*(a^2*b^9*x^18 + 9*a^3*b^8*x^16 + 36*a^4*b^7*x^14 + 84 *a^5*b^6*x^12 + 126*a^6*b^5*x^10 + 126*a^7*b^4*x^8 + 84*a^8*b^3*x^6 + 36*a ^9*b^2*x^4 + 9*a^10*b*x^2 + a^11)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^21*x^18 + 9*a*b^20*x^16 + 36*a^2*b^19*x^14 + 84*a^3 *b^18*x^12 + 126*a^4*b^17*x^10 + 126*a^5*b^16*x^8 + 84*a^6*b^15*x^6 + 36*a ^7*b^14*x^4 + 9*a^8*b^13*x^2 + a^9*b^12), 1/2949120*(589824*b^11*x^23 - 45 21984*a*b^10*x^21 + 94961664*a^2*b^9*x^19 + 1469632311*a^3*b^8*x^17 + 7323 998514*a^4*b^7*x^15 + 19562592546*a^5*b^6*x^13 + 32314857354*a^6*b^5*x^11 + 34810986496*a^7*b^4*x^9 + 24648575094*a^8*b^3*x^7 + 11110024926*a^9*b^2* x^5 + 2900207310*a^10*b*x^3 + 334639305*a^11*x - 334639305*(a^2*b^9*x^18 + 9*a^3*b^8*x^16 + 36*a^4*b^7*x^14 + 84*a^5*b^6*x^12 + 126*a^6*b^5*x^10 + 1 26*a^7*b^4*x^8 + 84*a^8*b^3*x^6 + 36*a^9*b^2*x^4 + 9*a^10*b*x^2 + a^11)*sq rt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^21*x^18 + 9*a*b^20*x^16 + 36*a^2*b^19* x^14 + 84*a^3*b^18*x^12 + 126*a^4*b^17*x^10 + 126*a^5*b^16*x^8 + 84*a^6*b^ 15*x^6 + 36*a^7*b^14*x^4 + 9*a^8*b^13*x^2 + a^9*b^12)]
Time = 0.99 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.36 \[ \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx=\frac {55 a^{2} x}{b^{12}} - \frac {10 a x^{3}}{3 b^{11}} + \frac {7436429 \sqrt {- \frac {a^{5}}{b^{25}}} \log {\left (x - \frac {b^{12} \sqrt {- \frac {a^{5}}{b^{25}}}}{a^{2}} \right )}}{131072} - \frac {7436429 \sqrt {- \frac {a^{5}}{b^{25}}} \log {\left (x + \frac {b^{12} \sqrt {- \frac {a^{5}}{b^{25}}}}{a^{2}} \right )}}{131072} + \frac {172437705 a^{11} x + 1450223310 a^{10} b x^{3} + 5358651102 a^{9} b^{2} x^{5} + 11372226678 a^{8} b^{3} x^{7} + 15178104832 a^{7} b^{4} x^{9} + 13066540938 a^{6} b^{5} x^{11} + 7101970722 a^{5} b^{6} x^{13} + 2236176690 a^{4} b^{7} x^{15} + 314167095 a^{3} b^{8} x^{17}}{2949120 a^{9} b^{12} + 26542080 a^{8} b^{13} x^{2} + 106168320 a^{7} b^{14} x^{4} + 247726080 a^{6} b^{15} x^{6} + 371589120 a^{5} b^{16} x^{8} + 371589120 a^{4} b^{17} x^{10} + 247726080 a^{3} b^{18} x^{12} + 106168320 a^{2} b^{19} x^{14} + 26542080 a b^{20} x^{16} + 2949120 b^{21} x^{18}} + \frac {x^{5}}{5 b^{10}} \]
55*a**2*x/b**12 - 10*a*x**3/(3*b**11) + 7436429*sqrt(-a**5/b**25)*log(x - b**12*sqrt(-a**5/b**25)/a**2)/131072 - 7436429*sqrt(-a**5/b**25)*log(x + b **12*sqrt(-a**5/b**25)/a**2)/131072 + (172437705*a**11*x + 1450223310*a**1 0*b*x**3 + 5358651102*a**9*b**2*x**5 + 11372226678*a**8*b**3*x**7 + 151781 04832*a**7*b**4*x**9 + 13066540938*a**6*b**5*x**11 + 7101970722*a**5*b**6* x**13 + 2236176690*a**4*b**7*x**15 + 314167095*a**3*b**8*x**17)/(2949120*a **9*b**12 + 26542080*a**8*b**13*x**2 + 106168320*a**7*b**14*x**4 + 2477260 80*a**6*b**15*x**6 + 371589120*a**5*b**16*x**8 + 371589120*a**4*b**17*x**1 0 + 247726080*a**3*b**18*x**12 + 106168320*a**2*b**19*x**14 + 26542080*a*b **20*x**16 + 2949120*b**21*x**18) + x**5/(5*b**10)
Time = 0.30 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.07 \[ \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx=\frac {314167095 \, a^{3} b^{8} x^{17} + 2236176690 \, a^{4} b^{7} x^{15} + 7101970722 \, a^{5} b^{6} x^{13} + 13066540938 \, a^{6} b^{5} x^{11} + 15178104832 \, a^{7} b^{4} x^{9} + 11372226678 \, a^{8} b^{3} x^{7} + 5358651102 \, a^{9} b^{2} x^{5} + 1450223310 \, a^{10} b x^{3} + 172437705 \, a^{11} x}{2949120 \, {\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}} - \frac {7436429 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{12}} + \frac {3 \, b^{2} x^{5} - 50 \, a b x^{3} + 825 \, a^{2} x}{15 \, b^{12}} \]
1/2949120*(314167095*a^3*b^8*x^17 + 2236176690*a^4*b^7*x^15 + 7101970722*a ^5*b^6*x^13 + 13066540938*a^6*b^5*x^11 + 15178104832*a^7*b^4*x^9 + 1137222 6678*a^8*b^3*x^7 + 5358651102*a^9*b^2*x^5 + 1450223310*a^10*b*x^3 + 172437 705*a^11*x)/(b^21*x^18 + 9*a*b^20*x^16 + 36*a^2*b^19*x^14 + 84*a^3*b^18*x^ 12 + 126*a^4*b^17*x^10 + 126*a^5*b^16*x^8 + 84*a^6*b^15*x^6 + 36*a^7*b^14* x^4 + 9*a^8*b^13*x^2 + a^9*b^12) - 7436429/65536*a^3*arctan(b*x/sqrt(a*b)) /(sqrt(a*b)*b^12) + 1/15*(3*b^2*x^5 - 50*a*b*x^3 + 825*a^2*x)/b^12
Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.70 \[ \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {7436429 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{12}} + \frac {314167095 \, a^{3} b^{8} x^{17} + 2236176690 \, a^{4} b^{7} x^{15} + 7101970722 \, a^{5} b^{6} x^{13} + 13066540938 \, a^{6} b^{5} x^{11} + 15178104832 \, a^{7} b^{4} x^{9} + 11372226678 \, a^{8} b^{3} x^{7} + 5358651102 \, a^{9} b^{2} x^{5} + 1450223310 \, a^{10} b x^{3} + 172437705 \, a^{11} x}{2949120 \, {\left (b x^{2} + a\right )}^{9} b^{12}} + \frac {3 \, b^{40} x^{5} - 50 \, a b^{39} x^{3} + 825 \, a^{2} b^{38} x}{15 \, b^{50}} \]
-7436429/65536*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^12) + 1/2949120*(314 167095*a^3*b^8*x^17 + 2236176690*a^4*b^7*x^15 + 7101970722*a^5*b^6*x^13 + 13066540938*a^6*b^5*x^11 + 15178104832*a^7*b^4*x^9 + 11372226678*a^8*b^3*x ^7 + 5358651102*a^9*b^2*x^5 + 1450223310*a^10*b*x^3 + 172437705*a^11*x)/(( b*x^2 + a)^9*b^12) + 1/15*(3*b^40*x^5 - 50*a*b^39*x^3 + 825*a^2*b^38*x)/b^ 50
Time = 4.85 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.04 \[ \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {3831949\,a^{11}\,x}{65536}+\frac {48340777\,a^{10}\,b\,x^3}{98304}+\frac {297702839\,a^9\,b^2\,x^5}{163840}+\frac {631790371\,a^8\,b^3\,x^7}{163840}+\frac {463199\,a^7\,b^4\,x^9}{90}+\frac {725918941\,a^6\,b^5\,x^{11}}{163840}+\frac {394553929\,a^5\,b^6\,x^{13}}{163840}+\frac {74539223\,a^4\,b^7\,x^{15}}{98304}+\frac {6981491\,a^3\,b^8\,x^{17}}{65536}}{a^9\,b^{12}+9\,a^8\,b^{13}\,x^2+36\,a^7\,b^{14}\,x^4+84\,a^6\,b^{15}\,x^6+126\,a^5\,b^{16}\,x^8+126\,a^4\,b^{17}\,x^{10}+84\,a^3\,b^{18}\,x^{12}+36\,a^2\,b^{19}\,x^{14}+9\,a\,b^{20}\,x^{16}+b^{21}\,x^{18}}+\frac {x^5}{5\,b^{10}}-\frac {10\,a\,x^3}{3\,b^{11}}+\frac {55\,a^2\,x}{b^{12}}-\frac {7436429\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,b^{25/2}} \]
((3831949*a^11*x)/65536 + (48340777*a^10*b*x^3)/98304 + (297702839*a^9*b^2 *x^5)/163840 + (631790371*a^8*b^3*x^7)/163840 + (463199*a^7*b^4*x^9)/90 + (725918941*a^6*b^5*x^11)/163840 + (394553929*a^5*b^6*x^13)/163840 + (74539 223*a^4*b^7*x^15)/98304 + (6981491*a^3*b^8*x^17)/65536)/(a^9*b^12 + b^21*x ^18 + 9*a*b^20*x^16 + 9*a^8*b^13*x^2 + 36*a^7*b^14*x^4 + 84*a^6*b^15*x^6 + 126*a^5*b^16*x^8 + 126*a^4*b^17*x^10 + 84*a^3*b^18*x^12 + 36*a^2*b^19*x^1 4) + x^5/(5*b^10) - (10*a*x^3)/(3*b^11) + (55*a^2*x)/b^12 - (7436429*a^(5/ 2)*atan((b^(1/2)*x)/a^(1/2)))/(65536*b^(25/2))