Integrand size = 13, antiderivative size = 225 \[ \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {10 a x}{b^{11}}+\frac {x^3}{3 b^{10}}-\frac {a^{10} x}{18 b^{11} \left (a+b x^2\right )^9}+\frac {181 a^9 x}{288 b^{11} \left (a+b x^2\right )^8}-\frac {625 a^8 x}{192 b^{11} \left (a+b x^2\right )^7}+\frac {23555 a^7 x}{2304 b^{11} \left (a+b x^2\right )^6}-\frac {100243 a^6 x}{4608 b^{11} \left (a+b x^2\right )^5}+\frac {136301 a^5 x}{4096 b^{11} \left (a+b x^2\right )^4}-\frac {938245 a^4 x}{24576 b^{11} \left (a+b x^2\right )^3}+\frac {3418855 a^3 x}{98304 b^{11} \left (a+b x^2\right )^2}-\frac {1987865 a^2 x}{65536 b^{11} \left (a+b x^2\right )}+\frac {1616615 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{23/2}} \] Output:
-10*a*x/b^11+1/3*x^3/b^10-1/18*a^10*x/b^11/(b*x^2+a)^9+181/288*a^9*x/b^11/ (b*x^2+a)^8-625/192*a^8*x/b^11/(b*x^2+a)^7+23555/2304*a^7*x/b^11/(b*x^2+a) ^6-100243/4608*a^6*x/b^11/(b*x^2+a)^5+136301/4096*a^5*x/b^11/(b*x^2+a)^4-9 38245/24576*a^4*x/b^11/(b*x^2+a)^3+3418855/98304*a^3*x/b^11/(b*x^2+a)^2-19 87865/65536*a^2*x/b^11/(b*x^2+a)+1616615/65536*a^(3/2)*arctan(b^(1/2)*x/a^ (1/2))/b^(23/2)
Time = 0.05 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.69 \[ \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {b} x \left (-14549535 a^{10}-126095970 a^9 b x^2-483044562 a^8 b^2 x^4-1071677178 a^7 b^3 x^6-1513521152 a^6 b^4 x^8-1404993798 a^5 b^5 x^{10}-850547502 a^4 b^6 x^{12}-318434718 a^3 b^7 x^{14}-63897057 a^2 b^8 x^{16}-4128768 a b^9 x^{18}+196608 b^{10} x^{20}\right )}{\left (a+b x^2\right )^9}+14549535 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{589824 b^{23/2}} \] Input:
Integrate[x^22/(a + b*x^2)^10,x]
Output:
((Sqrt[b]*x*(-14549535*a^10 - 126095970*a^9*b*x^2 - 483044562*a^8*b^2*x^4 - 1071677178*a^7*b^3*x^6 - 1513521152*a^6*b^4*x^8 - 1404993798*a^5*b^5*x^1 0 - 850547502*a^4*b^6*x^12 - 318434718*a^3*b^7*x^14 - 63897057*a^2*b^8*x^1 6 - 4128768*a*b^9*x^18 + 196608*b^10*x^20))/(a + b*x^2)^9 + 14549535*a^(3/ 2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(589824*b^(23/2))
Time = 0.38 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.27, 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.
| \(\displaystyle \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \int \frac {x^{20}}{\left (b x^2+a\right )^9}dx}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \left (\frac {19 \int \frac {x^{18}}{\left (b x^2+a\right )^8}dx}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \int \frac {x^{16}}{\left (b x^2+a\right )^7}dx}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \int \frac {x^{14}}{\left (b x^2+a\right )^6}dx}{4 b}-\frac {x^{15}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \int \frac {x^{12}}{\left (b x^2+a\right )^5}dx}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\right )}{4 b}-\frac {x^{15}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \int \frac {x^{10}}{\left (b x^2+a\right )^4}dx}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\right )}{4 b}-\frac {x^{15}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \int \frac {x^8}{\left (b x^2+a\right )^3}dx}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\right )}{4 b}-\frac {x^{15}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \int \frac {x^6}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\right )}{4 b}-\frac {x^{15}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \frac {x^4}{b x^2+a}dx}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\right )}{4 b}-\frac {x^{15}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \left (\frac {a^2}{b^2 \left (b x^2+a\right )}-\frac {a}{b^2}+\frac {x^2}{b}\right )dx}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\right )}{4 b}-\frac {x^{15}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a x}{b^2}+\frac {x^3}{3 b}\right )}{2 b}-\frac {x^5}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^7}{4 b \left (a+b x^2\right )^2}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^{11}}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}\right )}{4 b}-\frac {x^{15}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{17}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{19}}{16 b \left (a+b x^2\right )^8}\right )}{6 b}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}\) |
Input:
Int[x^22/(a + b*x^2)^10,x]
Output:
-1/18*x^21/(b*(a + b*x^2)^9) + (7*(-1/16*x^19/(b*(a + b*x^2)^8) + (19*(-1/ 14*x^17/(b*(a + b*x^2)^7) + (17*(-1/12*x^15/(b*(a + b*x^2)^6) + (5*(-1/10* x^13/(b*(a + b*x^2)^5) + (13*(-1/8*x^11/(b*(a + b*x^2)^4) + (11*(-1/6*x^9/ (b*(a + b*x^2)^3) + (3*(-1/4*x^7/(b*(a + b*x^2)^2) + (7*(-1/2*x^5/(b*(a + b*x^2)) + (5*(-((a*x)/b^2) + x^3/(3*b) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[ a]])/b^(5/2)))/(2*b)))/(4*b)))/(2*b)))/(8*b)))/(10*b)))/(4*b)))/(14*b)))/( 16*b)))/(6*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]
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 = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.62
| method | result | size |
| default | \(-\frac {-\frac {1}{3} b \,x^{3}+10 a x}{b^{11}}+\frac {a^{2} \left (\frac {-\frac {961255}{65536} a^{8} x -\frac {12201403}{98304} a^{7} b \,x^{3}-\frac {15137633}{32768} a^{6} b^{2} x^{5}-\frac {32405717}{32768} a^{5} b^{3} x^{7}-\frac {24013}{18} a^{4} b^{4} x^{9}-\frac {38143787}{32768} a^{3} b^{5} x^{11}-\frac {21103775}{32768} a^{2} b^{6} x^{13}-\frac {20435525}{98304} a \,b^{7} x^{15}-\frac {1987865}{65536} b^{8} x^{17}}{\left (b \,x^{2}+a \right )^{9}}+\frac {1616615 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}}\right )}{b^{11}}\) | \(140\) |
| risch | \(\frac {x^{3}}{3 b^{10}}-\frac {10 a x}{b^{11}}+\frac {-\frac {961255}{65536} a^{10} x -\frac {12201403}{98304} a^{9} b \,x^{3}-\frac {15137633}{32768} a^{8} b^{2} x^{5}-\frac {32405717}{32768} a^{7} b^{3} x^{7}-\frac {24013}{18} a^{6} b^{4} x^{9}-\frac {38143787}{32768} a^{5} b^{5} x^{11}-\frac {21103775}{32768} a^{4} b^{6} x^{13}-\frac {20435525}{98304} a^{3} b^{7} x^{15}-\frac {1987865}{65536} a^{2} b^{8} x^{17}}{b^{11} \left (b \,x^{2}+a \right )^{9}}+\frac {1616615 \sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x +a \right )}{131072 b^{12}}-\frac {1616615 \sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x +a \right )}{131072 b^{12}}\) | \(170\) |
Input:
int(x^22/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
Output:
-1/b^11*(-1/3*b*x^3+10*a*x)+1/b^11*a^2*((-961255/65536*a^8*x-12201403/9830 4*a^7*b*x^3-15137633/32768*a^6*b^2*x^5-32405717/32768*a^5*b^3*x^7-24013/18 *a^4*b^4*x^9-38143787/32768*a^3*b^5*x^11-21103775/32768*a^2*b^6*x^13-20435 525/98304*a*b^7*x^15-1987865/65536*b^8*x^17)/(b*x^2+a)^9+1616615/65536/(a* b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 692, normalized size of antiderivative = 3.08 \[ \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {393216 \, b^{10} x^{21} - 8257536 \, a b^{9} x^{19} - 127794114 \, a^{2} b^{8} x^{17} - 636869436 \, a^{3} b^{7} x^{15} - 1701095004 \, a^{4} b^{6} x^{13} - 2809987596 \, a^{5} b^{5} x^{11} - 3027042304 \, a^{6} b^{4} x^{9} - 2143354356 \, a^{7} b^{3} x^{7} - 966089124 \, a^{8} b^{2} x^{5} - 252191940 \, a^{9} b x^{3} - 29099070 \, a^{10} x + 14549535 \, {\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{1179648 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}}, \frac {196608 \, b^{10} x^{21} - 4128768 \, a b^{9} x^{19} - 63897057 \, a^{2} b^{8} x^{17} - 318434718 \, a^{3} b^{7} x^{15} - 850547502 \, a^{4} b^{6} x^{13} - 1404993798 \, a^{5} b^{5} x^{11} - 1513521152 \, a^{6} b^{4} x^{9} - 1071677178 \, a^{7} b^{3} x^{7} - 483044562 \, a^{8} b^{2} x^{5} - 126095970 \, a^{9} b x^{3} - 14549535 \, a^{10} x + 14549535 \, {\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{589824 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}}\right ] \] Input:
integrate(x^22/(b*x^2+a)^10,x, algorithm="fricas")
Output:
[1/1179648*(393216*b^10*x^21 - 8257536*a*b^9*x^19 - 127794114*a^2*b^8*x^17 - 636869436*a^3*b^7*x^15 - 1701095004*a^4*b^6*x^13 - 2809987596*a^5*b^5*x ^11 - 3027042304*a^6*b^4*x^9 - 2143354356*a^7*b^3*x^7 - 966089124*a^8*b^2* x^5 - 252191940*a^9*b*x^3 - 29099070*a^10*x + 14549535*(a*b^9*x^18 + 9*a^2 *b^8*x^16 + 36*a^3*b^7*x^14 + 84*a^4*b^6*x^12 + 126*a^5*b^5*x^10 + 126*a^6 *b^4*x^8 + 84*a^7*b^3*x^6 + 36*a^8*b^2*x^4 + 9*a^9*b*x^2 + a^10)*sqrt(-a/b )*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^20*x^18 + 9*a*b^19*x ^16 + 36*a^2*b^18*x^14 + 84*a^3*b^17*x^12 + 126*a^4*b^16*x^10 + 126*a^5*b^ 15*x^8 + 84*a^6*b^14*x^6 + 36*a^7*b^13*x^4 + 9*a^8*b^12*x^2 + a^9*b^11), 1 /589824*(196608*b^10*x^21 - 4128768*a*b^9*x^19 - 63897057*a^2*b^8*x^17 - 3 18434718*a^3*b^7*x^15 - 850547502*a^4*b^6*x^13 - 1404993798*a^5*b^5*x^11 - 1513521152*a^6*b^4*x^9 - 1071677178*a^7*b^3*x^7 - 483044562*a^8*b^2*x^5 - 126095970*a^9*b*x^3 - 14549535*a^10*x + 14549535*(a*b^9*x^18 + 9*a^2*b^8* x^16 + 36*a^3*b^7*x^14 + 84*a^4*b^6*x^12 + 126*a^5*b^5*x^10 + 126*a^6*b^4* x^8 + 84*a^7*b^3*x^6 + 36*a^8*b^2*x^4 + 9*a^9*b*x^2 + a^10)*sqrt(a/b)*arct an(b*x*sqrt(a/b)/a))/(b^20*x^18 + 9*a*b^19*x^16 + 36*a^2*b^18*x^14 + 84*a^ 3*b^17*x^12 + 126*a^4*b^16*x^10 + 126*a^5*b^15*x^8 + 84*a^6*b^14*x^6 + 36* a^7*b^13*x^4 + 9*a^8*b^12*x^2 + a^9*b^11)]
Time = 1.07 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.33 \[ \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx=- \frac {10 a x}{b^{11}} - \frac {1616615 \sqrt {- \frac {a^{3}}{b^{23}}} \log {\left (x - \frac {b^{11} \sqrt {- \frac {a^{3}}{b^{23}}}}{a} \right )}}{131072} + \frac {1616615 \sqrt {- \frac {a^{3}}{b^{23}}} \log {\left (x + \frac {b^{11} \sqrt {- \frac {a^{3}}{b^{23}}}}{a} \right )}}{131072} + \frac {- 8651295 a^{10} x - 73208418 a^{9} b x^{3} - 272477394 a^{8} b^{2} x^{5} - 583302906 a^{7} b^{3} x^{7} - 786857984 a^{6} b^{4} x^{9} - 686588166 a^{5} b^{5} x^{11} - 379867950 a^{4} b^{6} x^{13} - 122613150 a^{3} b^{7} x^{15} - 17890785 a^{2} b^{8} x^{17}}{589824 a^{9} b^{11} + 5308416 a^{8} b^{12} x^{2} + 21233664 a^{7} b^{13} x^{4} + 49545216 a^{6} b^{14} x^{6} + 74317824 a^{5} b^{15} x^{8} + 74317824 a^{4} b^{16} x^{10} + 49545216 a^{3} b^{17} x^{12} + 21233664 a^{2} b^{18} x^{14} + 5308416 a b^{19} x^{16} + 589824 b^{20} x^{18}} + \frac {x^{3}}{3 b^{10}} \] Input:
integrate(x**22/(b*x**2+a)**10,x)
Output:
-10*a*x/b**11 - 1616615*sqrt(-a**3/b**23)*log(x - b**11*sqrt(-a**3/b**23)/ a)/131072 + 1616615*sqrt(-a**3/b**23)*log(x + b**11*sqrt(-a**3/b**23)/a)/1 31072 + (-8651295*a**10*x - 73208418*a**9*b*x**3 - 272477394*a**8*b**2*x** 5 - 583302906*a**7*b**3*x**7 - 786857984*a**6*b**4*x**9 - 686588166*a**5*b **5*x**11 - 379867950*a**4*b**6*x**13 - 122613150*a**3*b**7*x**15 - 178907 85*a**2*b**8*x**17)/(589824*a**9*b**11 + 5308416*a**8*b**12*x**2 + 2123366 4*a**7*b**13*x**4 + 49545216*a**6*b**14*x**6 + 74317824*a**5*b**15*x**8 + 74317824*a**4*b**16*x**10 + 49545216*a**3*b**17*x**12 + 21233664*a**2*b**1 8*x**14 + 5308416*a*b**19*x**16 + 589824*b**20*x**18) + x**3/(3*b**10)
Time = 0.12 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.05 \[ \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {17890785 \, a^{2} b^{8} x^{17} + 122613150 \, a^{3} b^{7} x^{15} + 379867950 \, a^{4} b^{6} x^{13} + 686588166 \, a^{5} b^{5} x^{11} + 786857984 \, a^{6} b^{4} x^{9} + 583302906 \, a^{7} b^{3} x^{7} + 272477394 \, a^{8} b^{2} x^{5} + 73208418 \, a^{9} b x^{3} + 8651295 \, a^{10} x}{589824 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} + \frac {1616615 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{11}} + \frac {b x^{3} - 30 \, a x}{3 \, b^{11}} \] Input:
integrate(x^22/(b*x^2+a)^10,x, algorithm="maxima")
Output:
-1/589824*(17890785*a^2*b^8*x^17 + 122613150*a^3*b^7*x^15 + 379867950*a^4* b^6*x^13 + 686588166*a^5*b^5*x^11 + 786857984*a^6*b^4*x^9 + 583302906*a^7* b^3*x^7 + 272477394*a^8*b^2*x^5 + 73208418*a^9*b*x^3 + 8651295*a^10*x)/(b^ 20*x^18 + 9*a*b^19*x^16 + 36*a^2*b^18*x^14 + 84*a^3*b^17*x^12 + 126*a^4*b^ 16*x^10 + 126*a^5*b^15*x^8 + 84*a^6*b^14*x^6 + 36*a^7*b^13*x^4 + 9*a^8*b^1 2*x^2 + a^9*b^11) + 1616615/65536*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^1 1) + 1/3*(b*x^3 - 30*a*x)/b^11
Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx=\frac {1616615 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{11}} - \frac {17890785 \, a^{2} b^{8} x^{17} + 122613150 \, a^{3} b^{7} x^{15} + 379867950 \, a^{4} b^{6} x^{13} + 686588166 \, a^{5} b^{5} x^{11} + 786857984 \, a^{6} b^{4} x^{9} + 583302906 \, a^{7} b^{3} x^{7} + 272477394 \, a^{8} b^{2} x^{5} + 73208418 \, a^{9} b x^{3} + 8651295 \, a^{10} x}{589824 \, {\left (b x^{2} + a\right )}^{9} b^{11}} + \frac {b^{20} x^{3} - 30 \, a b^{19} x}{3 \, b^{30}} \] Input:
integrate(x^22/(b*x^2+a)^10,x, algorithm="giac")
Output:
1616615/65536*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^11) - 1/589824*(17890 785*a^2*b^8*x^17 + 122613150*a^3*b^7*x^15 + 379867950*a^4*b^6*x^13 + 68658 8166*a^5*b^5*x^11 + 786857984*a^6*b^4*x^9 + 583302906*a^7*b^3*x^7 + 272477 394*a^8*b^2*x^5 + 73208418*a^9*b*x^3 + 8651295*a^10*x)/((b*x^2 + a)^9*b^11 ) + 1/3*(b^20*x^3 - 30*a*b^19*x)/b^30
Time = 0.49 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.03 \[ \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx=\frac {x^3}{3\,b^{10}}-\frac {\frac {961255\,a^{10}\,x}{65536}+\frac {12201403\,a^9\,b\,x^3}{98304}+\frac {15137633\,a^8\,b^2\,x^5}{32768}+\frac {32405717\,a^7\,b^3\,x^7}{32768}+\frac {24013\,a^6\,b^4\,x^9}{18}+\frac {38143787\,a^5\,b^5\,x^{11}}{32768}+\frac {21103775\,a^4\,b^6\,x^{13}}{32768}+\frac {20435525\,a^3\,b^7\,x^{15}}{98304}+\frac {1987865\,a^2\,b^8\,x^{17}}{65536}}{a^9\,b^{11}+9\,a^8\,b^{12}\,x^2+36\,a^7\,b^{13}\,x^4+84\,a^6\,b^{14}\,x^6+126\,a^5\,b^{15}\,x^8+126\,a^4\,b^{16}\,x^{10}+84\,a^3\,b^{17}\,x^{12}+36\,a^2\,b^{18}\,x^{14}+9\,a\,b^{19}\,x^{16}+b^{20}\,x^{18}}+\frac {1616615\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,b^{23/2}}-\frac {10\,a\,x}{b^{11}} \] Input:
int(x^22/(a + b*x^2)^10,x)
Output:
x^3/(3*b^10) - ((961255*a^10*x)/65536 + (12201403*a^9*b*x^3)/98304 + (1513 7633*a^8*b^2*x^5)/32768 + (32405717*a^7*b^3*x^7)/32768 + (24013*a^6*b^4*x^ 9)/18 + (38143787*a^5*b^5*x^11)/32768 + (21103775*a^4*b^6*x^13)/32768 + (2 0435525*a^3*b^7*x^15)/98304 + (1987865*a^2*b^8*x^17)/65536)/(a^9*b^11 + b^ 20*x^18 + 9*a*b^19*x^16 + 9*a^8*b^12*x^2 + 36*a^7*b^13*x^4 + 84*a^6*b^14*x ^6 + 126*a^5*b^15*x^8 + 126*a^4*b^16*x^10 + 84*a^3*b^17*x^12 + 36*a^2*b^18 *x^14) + (1616615*a^(3/2)*atan((b^(1/2)*x)/a^(1/2)))/(65536*b^(23/2)) - (1 0*a*x)/b^11
Time = 0.22 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.11 \[ \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx=\frac {14549535 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{10}+130945815 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{9} b \,x^{2}+523783260 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{8} b^{2} x^{4}+1222160940 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{7} b^{3} x^{6}+1833241410 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6} b^{4} x^{8}+1833241410 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{5} x^{10}+1222160940 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{6} x^{12}+523783260 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{7} x^{14}+130945815 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{8} x^{16}+14549535 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{9} x^{18}-14549535 a^{10} b x -126095970 a^{9} b^{2} x^{3}-483044562 a^{8} b^{3} x^{5}-1071677178 a^{7} b^{4} x^{7}-1513521152 a^{6} b^{5} x^{9}-1404993798 a^{5} b^{6} x^{11}-850547502 a^{4} b^{7} x^{13}-318434718 a^{3} b^{8} x^{15}-63897057 a^{2} b^{9} x^{17}-4128768 a \,b^{10} x^{19}+196608 b^{11} x^{21}}{589824 b^{12} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:
int(x^22/(b*x^2+a)^10,x)
Output:
(14549535*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**10 + 130945815* sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**9*b*x**2 + 523783260*sqrt (b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**8*b**2*x**4 + 1222160940*sqrt (b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**7*b**3*x**6 + 1833241410*sqrt (b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**6*b**4*x**8 + 1833241410*sqrt (b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**5*x**10 + 1222160940*sqr t(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**6*x**12 + 523783260*sqr t(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**7*x**14 + 130945815*sqr t(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**8*x**16 + 14549535*sqrt (b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**9*x**18 - 14549535*a**10*b* x - 126095970*a**9*b**2*x**3 - 483044562*a**8*b**3*x**5 - 1071677178*a**7* b**4*x**7 - 1513521152*a**6*b**5*x**9 - 1404993798*a**5*b**6*x**11 - 85054 7502*a**4*b**7*x**13 - 318434718*a**3*b**8*x**15 - 63897057*a**2*b**9*x**1 7 - 4128768*a*b**10*x**19 + 196608*b**11*x**21)/(589824*b**12*(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 + 1 26*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))