Integrand size = 13, antiderivative size = 240 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx=-\frac {1}{5 a^{10} x^5}+\frac {10 b}{3 a^{11} x^3}-\frac {55 b^2}{a^{12} x}-\frac {b^3 x}{18 a^4 \left (a+b x^2\right )^9}-\frac {71 b^3 x}{288 a^5 \left (a+b x^2\right )^8}-\frac {133 b^3 x}{192 a^6 \left (a+b x^2\right )^7}-\frac {3649 b^3 x}{2304 a^7 \left (a+b x^2\right )^6}-\frac {74699 b^3 x}{23040 a^8 \left (a+b x^2\right )^5}-\frac {128459 b^3 x}{20480 a^9 \left (a+b x^2\right )^4}-\frac {1472653 b^3 x}{122880 a^{10} \left (a+b x^2\right )^3}-\frac {2357389 b^3 x}{98304 a^{11} \left (a+b x^2\right )^2}-\frac {3831949 b^3 x}{65536 a^{12} \left (a+b x^2\right )}-\frac {7436429 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{25/2}} \] Output:
-1/5/a^10/x^5+10/3*b/a^11/x^3-55*b^2/a^12/x-1/18*b^3*x/a^4/(b*x^2+a)^9-71/ 288*b^3*x/a^5/(b*x^2+a)^8-133/192*b^3*x/a^6/(b*x^2+a)^7-3649/2304*b^3*x/a^ 7/(b*x^2+a)^6-74699/23040*b^3*x/a^8/(b*x^2+a)^5-128459/20480*b^3*x/a^9/(b* x^2+a)^4-1472653/122880*b^3*x/a^10/(b*x^2+a)^3-2357389/98304*b^3*x/a^11/(b *x^2+a)^2-3831949/65536*b^3*x/a^12/(b*x^2+a)-7436429/65536*b^(5/2)*arctan( b^(1/2)*x/a^(1/2))/a^(25/2)
Time = 0.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx=\frac {-\frac {\sqrt {a} \left (589824 a^{11}-4521984 a^{10} b x^2+94961664 a^9 b^2 x^4+1469632311 a^8 b^3 x^6+7323998514 a^7 b^4 x^8+19562592546 a^6 b^5 x^{10}+32314857354 a^5 b^6 x^{12}+34810986496 a^4 b^7 x^{14}+24648575094 a^3 b^8 x^{16}+11110024926 a^2 b^9 x^{18}+2900207310 a b^{10} x^{20}+334639305 b^{11} x^{22}\right )}{x^5 \left (a+b x^2\right )^9}-334639305 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2949120 a^{25/2}} \] Input:
Integrate[1/(x^6*(a + b*x^2)^10),x]
Output:
(-((Sqrt[a]*(589824*a^11 - 4521984*a^10*b*x^2 + 94961664*a^9*b^2*x^4 + 146 9632311*a^8*b^3*x^6 + 7323998514*a^7*b^4*x^8 + 19562592546*a^6*b^5*x^10 + 32314857354*a^5*b^6*x^12 + 34810986496*a^4*b^7*x^14 + 24648575094*a^3*b^8* x^16 + 11110024926*a^2*b^9*x^18 + 2900207310*a*b^10*x^20 + 334639305*b^11* x^22))/(x^5*(a + b*x^2)^9)) - 334639305*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a] ])/(2949120*a^(25/2))
Time = 0.35 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.30, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {253, 253, 253, 253, 253, 253, 253, 253, 253, 264, 264, 264, 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.
| \(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \int \frac {1}{x^6 \left (b x^2+a\right )^9}dx}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {21 \int \frac {1}{x^6 \left (b x^2+a\right )^8}dx}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \int \frac {1}{x^6 \left (b x^2+a\right )^7}dx}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \int \frac {1}{x^6 \left (b x^2+a\right )^6}dx}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \int \frac {1}{x^6 \left (b x^2+a\right )^5}dx}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \int \frac {1}{x^6 \left (b x^2+a\right )^4}dx}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \int \frac {1}{x^6 \left (b x^2+a\right )^3}dx}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \int \frac {1}{x^6 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \int \frac {1}{x^6 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \int \frac {1}{x^4 \left (b x^2+a\right )}dx}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {23 \left (\frac {21 \left (\frac {19 \left (\frac {17 \left (\frac {3 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x^5 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^5 \left (a+b x^2\right )^4}\right )}{2 a}+\frac {1}{10 a x^5 \left (a+b x^2\right )^5}\right )}{12 a}+\frac {1}{12 a x^5 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^5 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^5 \left (a+b x^2\right )^8}\right )}{18 a}+\frac {1}{18 a x^5 \left (a+b x^2\right )^9}\) |
Input:
Int[1/(x^6*(a + b*x^2)^10),x]
Output:
1/(18*a*x^5*(a + b*x^2)^9) + (23*(1/(16*a*x^5*(a + b*x^2)^8) + (21*(1/(14* a*x^5*(a + b*x^2)^7) + (19*(1/(12*a*x^5*(a + b*x^2)^6) + (17*(1/(10*a*x^5* (a + b*x^2)^5) + (3*(1/(8*a*x^5*(a + b*x^2)^4) + (13*(1/(6*a*x^5*(a + b*x^ 2)^3) + (11*(1/(4*a*x^5*(a + b*x^2)^2) + (9*(1/(2*a*x^5*(a + b*x^2)) + (7* (-1/5*1/(a*x^5) - (b*(-1/3*1/(a*x^3) - (b*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(S qrt[b]*x)/Sqrt[a]])/a^(3/2)))/a))/a))/(2*a)))/(4*a)))/(6*a)))/(8*a)))/(2*a )))/(12*a)))/(14*a)))/(16*a)))/(18*a)
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*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Time = 0.39 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(-\frac {b^{3} \left (\frac {\frac {6981491}{65536} a^{8} x +\frac {74539223}{98304} a^{7} b \,x^{3}+\frac {394553929}{163840} a^{6} b^{2} x^{5}+\frac {725918941}{163840} a^{5} b^{3} x^{7}+\frac {463199}{90} a^{4} b^{4} x^{9}+\frac {631790371}{163840} a^{3} b^{5} x^{11}+\frac {297702839}{163840} a^{2} b^{6} x^{13}+\frac {48340777}{98304} a \,b^{7} x^{15}+\frac {3831949}{65536} b^{8} x^{17}}{\left (b \,x^{2}+a \right )^{9}}+\frac {7436429 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}}\right )}{a^{12}}-\frac {1}{5 a^{10} x^{5}}+\frac {10 b}{3 a^{11} x^{3}}-\frac {55 b^{2}}{a^{12} x}\) | \(153\) |
| risch | \(\frac {-\frac {1}{5 a}+\frac {23 b \,x^{2}}{15 a^{2}}-\frac {161 b^{2} x^{4}}{5 a^{3}}-\frac {163292479 b^{3} x^{6}}{327680 a^{4}}-\frac {1220666419 b^{4} x^{8}}{491520 a^{5}}-\frac {1086810697 b^{5} x^{10}}{163840 a^{6}}-\frac {1795269853 b^{6} x^{12}}{163840 a^{7}}-\frac {1062347 b^{7} x^{14}}{90 a^{8}}-\frac {1369365283 b^{8} x^{16}}{163840 a^{9}}-\frac {617223607 b^{9} x^{18}}{163840 a^{10}}-\frac {96673577 b^{10} x^{20}}{98304 a^{11}}-\frac {7436429 b^{11} x^{22}}{65536 a^{12}}}{x^{5} \left (b \,x^{2}+a \right )^{9}}+\frac {7436429 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{25} \textit {\_Z}^{2}+b^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{25}+2 b^{5}\right ) x +a^{13} b^{2} \textit {\_R} \right )\right )}{131072}\) | \(185\) |
Input:
int(1/x^6/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
Output:
-b^3/a^12*((6981491/65536*a^8*x+74539223/98304*a^7*b*x^3+394553929/163840* a^6*b^2*x^5+725918941/163840*a^5*b^3*x^7+463199/90*a^4*b^4*x^9+631790371/1 63840*a^3*b^5*x^11+297702839/163840*a^2*b^6*x^13+48340777/98304*a*b^7*x^15 +3831949/65536*b^8*x^17)/(b*x^2+a)^9+7436429/65536/(a*b)^(1/2)*arctan(b*x/ (a*b)^(1/2)))-1/5/a^10/x^5+10/3*b/a^11/x^3-55*b^2/a^12/x
Time = 0.09 (sec) , antiderivative size = 726, normalized size of antiderivative = 3.02 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx =\text {Too large to display} \] Input:
integrate(1/x^6/(b*x^2+a)^10,x, algorithm="fricas")
Output:
[-1/5898240*(669278610*b^11*x^22 + 5800414620*a*b^10*x^20 + 22220049852*a^ 2*b^9*x^18 + 49297150188*a^3*b^8*x^16 + 69621972992*a^4*b^7*x^14 + 6462971 4708*a^5*b^6*x^12 + 39125185092*a^6*b^5*x^10 + 14647997028*a^7*b^4*x^8 + 2 939264622*a^8*b^3*x^6 + 189923328*a^9*b^2*x^4 - 9043968*a^10*b*x^2 + 11796 48*a^11 - 334639305*(b^11*x^23 + 9*a*b^10*x^21 + 36*a^2*b^9*x^19 + 84*a^3* b^8*x^17 + 126*a^4*b^7*x^15 + 126*a^5*b^6*x^13 + 84*a^6*b^5*x^11 + 36*a^7* b^4*x^9 + 9*a^8*b^3*x^7 + a^9*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt( -b/a) - a)/(b*x^2 + a)))/(a^12*b^9*x^23 + 9*a^13*b^8*x^21 + 36*a^14*b^7*x^ 19 + 84*a^15*b^6*x^17 + 126*a^16*b^5*x^15 + 126*a^17*b^4*x^13 + 84*a^18*b^ 3*x^11 + 36*a^19*b^2*x^9 + 9*a^20*b*x^7 + a^21*x^5), -1/2949120*(334639305 *b^11*x^22 + 2900207310*a*b^10*x^20 + 11110024926*a^2*b^9*x^18 + 246485750 94*a^3*b^8*x^16 + 34810986496*a^4*b^7*x^14 + 32314857354*a^5*b^6*x^12 + 19 562592546*a^6*b^5*x^10 + 7323998514*a^7*b^4*x^8 + 1469632311*a^8*b^3*x^6 + 94961664*a^9*b^2*x^4 - 4521984*a^10*b*x^2 + 589824*a^11 + 334639305*(b^11 *x^23 + 9*a*b^10*x^21 + 36*a^2*b^9*x^19 + 84*a^3*b^8*x^17 + 126*a^4*b^7*x^ 15 + 126*a^5*b^6*x^13 + 84*a^6*b^5*x^11 + 36*a^7*b^4*x^9 + 9*a^8*b^3*x^7 + a^9*b^2*x^5)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^12*b^9*x^23 + 9*a^13*b^8*x ^21 + 36*a^14*b^7*x^19 + 84*a^15*b^6*x^17 + 126*a^16*b^5*x^15 + 126*a^17*b ^4*x^13 + 84*a^18*b^3*x^11 + 36*a^19*b^2*x^9 + 9*a^20*b*x^7 + a^21*x^5)]
Time = 0.78 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx=\frac {7436429 \sqrt {- \frac {b^{5}}{a^{25}}} \log {\left (- \frac {a^{13} \sqrt {- \frac {b^{5}}{a^{25}}}}{b^{3}} + x \right )}}{131072} - \frac {7436429 \sqrt {- \frac {b^{5}}{a^{25}}} \log {\left (\frac {a^{13} \sqrt {- \frac {b^{5}}{a^{25}}}}{b^{3}} + x \right )}}{131072} + \frac {- 589824 a^{11} + 4521984 a^{10} b x^{2} - 94961664 a^{9} b^{2} x^{4} - 1469632311 a^{8} b^{3} x^{6} - 7323998514 a^{7} b^{4} x^{8} - 19562592546 a^{6} b^{5} x^{10} - 32314857354 a^{5} b^{6} x^{12} - 34810986496 a^{4} b^{7} x^{14} - 24648575094 a^{3} b^{8} x^{16} - 11110024926 a^{2} b^{9} x^{18} - 2900207310 a b^{10} x^{20} - 334639305 b^{11} x^{22}}{2949120 a^{21} x^{5} + 26542080 a^{20} b x^{7} + 106168320 a^{19} b^{2} x^{9} + 247726080 a^{18} b^{3} x^{11} + 371589120 a^{17} b^{4} x^{13} + 371589120 a^{16} b^{5} x^{15} + 247726080 a^{15} b^{6} x^{17} + 106168320 a^{14} b^{7} x^{19} + 26542080 a^{13} b^{8} x^{21} + 2949120 a^{12} b^{9} x^{23}} \] Input:
integrate(1/x**6/(b*x**2+a)**10,x)
Output:
7436429*sqrt(-b**5/a**25)*log(-a**13*sqrt(-b**5/a**25)/b**3 + x)/131072 - 7436429*sqrt(-b**5/a**25)*log(a**13*sqrt(-b**5/a**25)/b**3 + x)/131072 + ( -589824*a**11 + 4521984*a**10*b*x**2 - 94961664*a**9*b**2*x**4 - 146963231 1*a**8*b**3*x**6 - 7323998514*a**7*b**4*x**8 - 19562592546*a**6*b**5*x**10 - 32314857354*a**5*b**6*x**12 - 34810986496*a**4*b**7*x**14 - 24648575094 *a**3*b**8*x**16 - 11110024926*a**2*b**9*x**18 - 2900207310*a*b**10*x**20 - 334639305*b**11*x**22)/(2949120*a**21*x**5 + 26542080*a**20*b*x**7 + 106 168320*a**19*b**2*x**9 + 247726080*a**18*b**3*x**11 + 371589120*a**17*b**4 *x**13 + 371589120*a**16*b**5*x**15 + 247726080*a**15*b**6*x**17 + 1061683 20*a**14*b**7*x**19 + 26542080*a**13*b**8*x**21 + 2949120*a**12*b**9*x**23 )
Time = 0.13 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx=-\frac {334639305 \, b^{11} x^{22} + 2900207310 \, a b^{10} x^{20} + 11110024926 \, a^{2} b^{9} x^{18} + 24648575094 \, a^{3} b^{8} x^{16} + 34810986496 \, a^{4} b^{7} x^{14} + 32314857354 \, a^{5} b^{6} x^{12} + 19562592546 \, a^{6} b^{5} x^{10} + 7323998514 \, a^{7} b^{4} x^{8} + 1469632311 \, a^{8} b^{3} x^{6} + 94961664 \, a^{9} b^{2} x^{4} - 4521984 \, a^{10} b x^{2} + 589824 \, a^{11}}{2949120 \, {\left (a^{12} b^{9} x^{23} + 9 \, a^{13} b^{8} x^{21} + 36 \, a^{14} b^{7} x^{19} + 84 \, a^{15} b^{6} x^{17} + 126 \, a^{16} b^{5} x^{15} + 126 \, a^{17} b^{4} x^{13} + 84 \, a^{18} b^{3} x^{11} + 36 \, a^{19} b^{2} x^{9} + 9 \, a^{20} b x^{7} + a^{21} x^{5}\right )}} - \frac {7436429 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{12}} \] Input:
integrate(1/x^6/(b*x^2+a)^10,x, algorithm="maxima")
Output:
-1/2949120*(334639305*b^11*x^22 + 2900207310*a*b^10*x^20 + 11110024926*a^2 *b^9*x^18 + 24648575094*a^3*b^8*x^16 + 34810986496*a^4*b^7*x^14 + 32314857 354*a^5*b^6*x^12 + 19562592546*a^6*b^5*x^10 + 7323998514*a^7*b^4*x^8 + 146 9632311*a^8*b^3*x^6 + 94961664*a^9*b^2*x^4 - 4521984*a^10*b*x^2 + 589824*a ^11)/(a^12*b^9*x^23 + 9*a^13*b^8*x^21 + 36*a^14*b^7*x^19 + 84*a^15*b^6*x^1 7 + 126*a^16*b^5*x^15 + 126*a^17*b^4*x^13 + 84*a^18*b^3*x^11 + 36*a^19*b^2 *x^9 + 9*a^20*b*x^7 + a^21*x^5) - 7436429/65536*b^3*arctan(b*x/sqrt(a*b))/ (sqrt(a*b)*a^12)
Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx=-\frac {7436429 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{12}} - \frac {825 \, b^{2} x^{4} - 50 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{12} x^{5}} - \frac {172437705 \, b^{11} x^{17} + 1450223310 \, a b^{10} x^{15} + 5358651102 \, a^{2} b^{9} x^{13} + 11372226678 \, a^{3} b^{8} x^{11} + 15178104832 \, a^{4} b^{7} x^{9} + 13066540938 \, a^{5} b^{6} x^{7} + 7101970722 \, a^{6} b^{5} x^{5} + 2236176690 \, a^{7} b^{4} x^{3} + 314167095 \, a^{8} b^{3} x}{2949120 \, {\left (b x^{2} + a\right )}^{9} a^{12}} \] Input:
integrate(1/x^6/(b*x^2+a)^10,x, algorithm="giac")
Output:
-7436429/65536*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^12) - 1/15*(825*b^2* x^4 - 50*a*b*x^2 + 3*a^2)/(a^12*x^5) - 1/2949120*(172437705*b^11*x^17 + 14 50223310*a*b^10*x^15 + 5358651102*a^2*b^9*x^13 + 11372226678*a^3*b^8*x^11 + 15178104832*a^4*b^7*x^9 + 13066540938*a^5*b^6*x^7 + 7101970722*a^6*b^5*x ^5 + 2236176690*a^7*b^4*x^3 + 314167095*a^8*b^3*x)/((b*x^2 + a)^9*a^12)
Time = 1.28 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx=-\frac {\frac {1}{5\,a}-\frac {23\,b\,x^2}{15\,a^2}+\frac {161\,b^2\,x^4}{5\,a^3}+\frac {163292479\,b^3\,x^6}{327680\,a^4}+\frac {1220666419\,b^4\,x^8}{491520\,a^5}+\frac {1086810697\,b^5\,x^{10}}{163840\,a^6}+\frac {1795269853\,b^6\,x^{12}}{163840\,a^7}+\frac {1062347\,b^7\,x^{14}}{90\,a^8}+\frac {1369365283\,b^8\,x^{16}}{163840\,a^9}+\frac {617223607\,b^9\,x^{18}}{163840\,a^{10}}+\frac {96673577\,b^{10}\,x^{20}}{98304\,a^{11}}+\frac {7436429\,b^{11}\,x^{22}}{65536\,a^{12}}}{a^9\,x^5+9\,a^8\,b\,x^7+36\,a^7\,b^2\,x^9+84\,a^6\,b^3\,x^{11}+126\,a^5\,b^4\,x^{13}+126\,a^4\,b^5\,x^{15}+84\,a^3\,b^6\,x^{17}+36\,a^2\,b^7\,x^{19}+9\,a\,b^8\,x^{21}+b^9\,x^{23}}-\frac {7436429\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{25/2}} \] Input:
int(1/(x^6*(a + b*x^2)^10),x)
Output:
- (1/(5*a) - (23*b*x^2)/(15*a^2) + (161*b^2*x^4)/(5*a^3) + (163292479*b^3* x^6)/(327680*a^4) + (1220666419*b^4*x^8)/(491520*a^5) + (1086810697*b^5*x^ 10)/(163840*a^6) + (1795269853*b^6*x^12)/(163840*a^7) + (1062347*b^7*x^14) /(90*a^8) + (1369365283*b^8*x^16)/(163840*a^9) + (617223607*b^9*x^18)/(163 840*a^10) + (96673577*b^10*x^20)/(98304*a^11) + (7436429*b^11*x^22)/(65536 *a^12))/(a^9*x^5 + b^9*x^23 + 9*a^8*b*x^7 + 9*a*b^8*x^21 + 36*a^7*b^2*x^9 + 84*a^6*b^3*x^11 + 126*a^5*b^4*x^13 + 126*a^4*b^5*x^15 + 84*a^3*b^6*x^17 + 36*a^2*b^7*x^19) - (7436429*b^(5/2)*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^ (25/2))
Time = 0.25 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.05 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{10}} \, dx=\frac {-334639305 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{9} b^{2} x^{5}-3011753745 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{8} b^{3} x^{7}-12047014980 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{7} b^{4} x^{9}-28109701620 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6} b^{5} x^{11}-42164552430 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{6} x^{13}-42164552430 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{7} x^{15}-28109701620 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{8} x^{17}-12047014980 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{9} x^{19}-3011753745 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{10} x^{21}-334639305 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{11} x^{23}-589824 a^{12}+4521984 a^{11} b \,x^{2}-94961664 a^{10} b^{2} x^{4}-1469632311 a^{9} b^{3} x^{6}-7323998514 a^{8} b^{4} x^{8}-19562592546 a^{7} b^{5} x^{10}-32314857354 a^{6} b^{6} x^{12}-34810986496 a^{5} b^{7} x^{14}-24648575094 a^{4} b^{8} x^{16}-11110024926 a^{3} b^{9} x^{18}-2900207310 a^{2} b^{10} x^{20}-334639305 a \,b^{11} x^{22}}{2949120 a^{13} x^{5} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:
int(1/x^6/(b*x^2+a)^10,x)
Output:
( - 334639305*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**9*b**2*x**5 - 3011753745*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**8*b**3*x**7 - 12047014980*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**7*b**4*x** 9 - 28109701620*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**6*b**5*x* *11 - 42164552430*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**6* x**13 - 42164552430*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b** 7*x**15 - 28109701620*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b **8*x**17 - 12047014980*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2 *b**9*x**19 - 3011753745*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b **10*x**21 - 334639305*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**11 *x**23 - 589824*a**12 + 4521984*a**11*b*x**2 - 94961664*a**10*b**2*x**4 - 1469632311*a**9*b**3*x**6 - 7323998514*a**8*b**4*x**8 - 19562592546*a**7*b **5*x**10 - 32314857354*a**6*b**6*x**12 - 34810986496*a**5*b**7*x**14 - 24 648575094*a**4*b**8*x**16 - 11110024926*a**3*b**9*x**18 - 2900207310*a**2* b**10*x**20 - 334639305*a*b**11*x**22)/(2949120*a**13*x**5*(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))