Integrand size = 13, antiderivative size = 227 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx=-\frac {1}{3 a^{10} x^3}+\frac {10 b}{a^{11} x}+\frac {b^2 x}{18 a^3 \left (a+b x^2\right )^9}+\frac {53 b^2 x}{288 a^4 \left (a+b x^2\right )^8}+\frac {79 b^2 x}{192 a^5 \left (a+b x^2\right )^7}+\frac {1795 b^2 x}{2304 a^6 \left (a+b x^2\right )^6}+\frac {6253 b^2 x}{4608 a^7 \left (a+b x^2\right )^5}+\frac {9325 b^2 x}{4096 a^8 \left (a+b x^2\right )^4}+\frac {93947 b^2 x}{24576 a^9 \left (a+b x^2\right )^3}+\frac {666343 b^2 x}{98304 a^{10} \left (a+b x^2\right )^2}+\frac {961255 b^2 x}{65536 a^{11} \left (a+b x^2\right )}+\frac {1616615 b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{23/2}} \] Output:
-1/3/a^10/x^3+10*b/a^11/x+1/18*b^2*x/a^3/(b*x^2+a)^9+53/288*b^2*x/a^4/(b*x ^2+a)^8+79/192*b^2*x/a^5/(b*x^2+a)^7+1795/2304*b^2*x/a^6/(b*x^2+a)^6+6253/ 4608*b^2*x/a^7/(b*x^2+a)^5+9325/4096*b^2*x/a^8/(b*x^2+a)^4+93947/24576*b^2 *x/a^9/(b*x^2+a)^3+666343/98304*b^2*x/a^10/(b*x^2+a)^2+961255/65536*b^2*x/ a^11/(b*x^2+a)+1616615/65536*b^(3/2)*arctan(b^(1/2)*x/a^(1/2))/a^(23/2)
Time = 0.05 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {a} \left (-196608 a^{10}+4128768 a^9 b x^2+63897057 a^8 b^2 x^4+318434718 a^7 b^3 x^6+850547502 a^6 b^4 x^8+1404993798 a^5 b^5 x^{10}+1513521152 a^4 b^6 x^{12}+1071677178 a^3 b^7 x^{14}+483044562 a^2 b^8 x^{16}+126095970 a b^9 x^{18}+14549535 b^{10} x^{20}\right )}{x^3 \left (a+b x^2\right )^9}+14549535 b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{589824 a^{23/2}} \] Input:
Integrate[1/(x^4*(a + b*x^2)^10),x]
Output:
((Sqrt[a]*(-196608*a^10 + 4128768*a^9*b*x^2 + 63897057*a^8*b^2*x^4 + 31843 4718*a^7*b^3*x^6 + 850547502*a^6*b^4*x^8 + 1404993798*a^5*b^5*x^10 + 15135 21152*a^4*b^6*x^12 + 1071677178*a^3*b^7*x^14 + 483044562*a^2*b^8*x^16 + 12 6095970*a*b^9*x^18 + 14549535*b^10*x^20))/(x^3*(a + b*x^2)^9) + 14549535*b ^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(589824*a^(23/2))
Time = 0.33 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.30, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {253, 253, 253, 253, 253, 253, 253, 253, 253, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \int \frac {1}{x^4 \left (b x^2+a\right )^9}dx}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \left (\frac {19 \int \frac {1}{x^4 \left (b x^2+a\right )^8}dx}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \int \frac {1}{x^4 \left (b x^2+a\right )^7}dx}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \int \frac {1}{x^4 \left (b x^2+a\right )^6}dx}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \int \frac {1}{x^4 \left (b x^2+a\right )^5}dx}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\right )}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \int \frac {1}{x^4 \left (b x^2+a\right )^4}dx}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\right )}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \int \frac {1}{x^4 \left (b x^2+a\right )^3}dx}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\right )}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \int \frac {1}{x^4 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\right )}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^4 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\right )}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\right )}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\right )}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {7 \left (\frac {19 \left (\frac {17 \left (\frac {5 \left (\frac {13 \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{2 a}+\frac {1}{2 a x^3 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^3 \left (a+b x^2\right )^2}\right )}{2 a}+\frac {1}{6 a x^3 \left (a+b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x^3 \left (a+b x^2\right )^4}\right )}{10 a}+\frac {1}{10 a x^3 \left (a+b x^2\right )^5}\right )}{4 a}+\frac {1}{12 a x^3 \left (a+b x^2\right )^6}\right )}{14 a}+\frac {1}{14 a x^3 \left (a+b x^2\right )^7}\right )}{16 a}+\frac {1}{16 a x^3 \left (a+b x^2\right )^8}\right )}{6 a}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}\) |
Input:
Int[1/(x^4*(a + b*x^2)^10),x]
Output:
1/(18*a*x^3*(a + b*x^2)^9) + (7*(1/(16*a*x^3*(a + b*x^2)^8) + (19*(1/(14*a *x^3*(a + b*x^2)^7) + (17*(1/(12*a*x^3*(a + b*x^2)^6) + (5*(1/(10*a*x^3*(a + b*x^2)^5) + (13*(1/(8*a*x^3*(a + b*x^2)^4) + (11*(1/(6*a*x^3*(a + b*x^2 )^3) + (3*(1/(4*a*x^3*(a + b*x^2)^2) + (7*(1/(2*a*x^3*(a + b*x^2)) + (5*(- 1/3*1/(a*x^3) - (b*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^( 3/2)))/a))/(2*a)))/(4*a)))/(2*a)))/(8*a)))/(10*a)))/(4*a)))/(14*a)))/(16*a )))/(6*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 = 141, normalized size of antiderivative = 0.62
| method | result | size |
| default | \(\frac {b^{2} \left (\frac {\frac {1987865}{65536} a^{8} x +\frac {20435525}{98304} a^{7} b \,x^{3}+\frac {21103775}{32768} a^{6} b^{2} x^{5}+\frac {38143787}{32768} a^{5} b^{3} x^{7}+\frac {24013}{18} a^{4} b^{4} x^{9}+\frac {32405717}{32768} a^{3} b^{5} x^{11}+\frac {15137633}{32768} a^{2} b^{6} x^{13}+\frac {12201403}{98304} a \,b^{7} x^{15}+\frac {961255}{65536} b^{8} x^{17}}{\left (b \,x^{2}+a \right )^{9}}+\frac {1616615 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}}\right )}{a^{11}}-\frac {1}{3 a^{10} x^{3}}+\frac {10 b}{a^{11} x}\) | \(141\) |
| risch | \(\frac {-\frac {1}{3 a}+\frac {7 b \,x^{2}}{a^{2}}+\frac {7099673 b^{2} x^{4}}{65536 a^{3}}+\frac {53072453 b^{3} x^{6}}{98304 a^{4}}+\frac {47252639 b^{4} x^{8}}{32768 a^{5}}+\frac {78055211 b^{5} x^{10}}{32768 a^{6}}+\frac {46189 b^{6} x^{12}}{18 a^{7}}+\frac {59537621 b^{7} x^{14}}{32768 a^{8}}+\frac {26835809 b^{8} x^{16}}{32768 a^{9}}+\frac {21015995 b^{9} x^{18}}{98304 a^{10}}+\frac {1616615 b^{10} x^{20}}{65536 a^{11}}}{x^{3} \left (b \,x^{2}+a \right )^{9}}+\frac {1616615 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{23} \textit {\_Z}^{2}+b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{23}+2 b^{3}\right ) x -a^{12} b \textit {\_R} \right )\right )}{131072}\) | \(173\) |
Input:
int(1/x^4/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
Output:
b^2/a^11*((1987865/65536*a^8*x+20435525/98304*a^7*b*x^3+21103775/32768*a^6 *b^2*x^5+38143787/32768*a^5*b^3*x^7+24013/18*a^4*b^4*x^9+32405717/32768*a^ 3*b^5*x^11+15137633/32768*a^2*b^6*x^13+12201403/98304*a*b^7*x^15+961255/65 536*b^8*x^17)/(b*x^2+a)^9+1616615/65536/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2) ))-1/3/a^10/x^3+10*b/a^11/x
Time = 0.09 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.08 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx=\left [\frac {29099070 \, b^{10} x^{20} + 252191940 \, a b^{9} x^{18} + 966089124 \, a^{2} b^{8} x^{16} + 2143354356 \, a^{3} b^{7} x^{14} + 3027042304 \, a^{4} b^{6} x^{12} + 2809987596 \, a^{5} b^{5} x^{10} + 1701095004 \, a^{6} b^{4} x^{8} + 636869436 \, a^{7} b^{3} x^{6} + 127794114 \, a^{8} b^{2} x^{4} + 8257536 \, a^{9} b x^{2} - 393216 \, a^{10} + 14549535 \, {\left (b^{10} x^{21} + 9 \, a b^{9} x^{19} + 36 \, a^{2} b^{8} x^{17} + 84 \, a^{3} b^{7} x^{15} + 126 \, a^{4} b^{6} x^{13} + 126 \, a^{5} b^{5} x^{11} + 84 \, a^{6} b^{4} x^{9} + 36 \, a^{7} b^{3} x^{7} + 9 \, a^{8} b^{2} x^{5} + a^{9} b x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{1179648 \, {\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}}, \frac {14549535 \, b^{10} x^{20} + 126095970 \, a b^{9} x^{18} + 483044562 \, a^{2} b^{8} x^{16} + 1071677178 \, a^{3} b^{7} x^{14} + 1513521152 \, a^{4} b^{6} x^{12} + 1404993798 \, a^{5} b^{5} x^{10} + 850547502 \, a^{6} b^{4} x^{8} + 318434718 \, a^{7} b^{3} x^{6} + 63897057 \, a^{8} b^{2} x^{4} + 4128768 \, a^{9} b x^{2} - 196608 \, a^{10} + 14549535 \, {\left (b^{10} x^{21} + 9 \, a b^{9} x^{19} + 36 \, a^{2} b^{8} x^{17} + 84 \, a^{3} b^{7} x^{15} + 126 \, a^{4} b^{6} x^{13} + 126 \, a^{5} b^{5} x^{11} + 84 \, a^{6} b^{4} x^{9} + 36 \, a^{7} b^{3} x^{7} + 9 \, a^{8} b^{2} x^{5} + a^{9} b x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{589824 \, {\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}}\right ] \] Input:
integrate(1/x^4/(b*x^2+a)^10,x, algorithm="fricas")
Output:
[1/1179648*(29099070*b^10*x^20 + 252191940*a*b^9*x^18 + 966089124*a^2*b^8* x^16 + 2143354356*a^3*b^7*x^14 + 3027042304*a^4*b^6*x^12 + 2809987596*a^5* b^5*x^10 + 1701095004*a^6*b^4*x^8 + 636869436*a^7*b^3*x^6 + 127794114*a^8* b^2*x^4 + 8257536*a^9*b*x^2 - 393216*a^10 + 14549535*(b^10*x^21 + 9*a*b^9* x^19 + 36*a^2*b^8*x^17 + 84*a^3*b^7*x^15 + 126*a^4*b^6*x^13 + 126*a^5*b^5* x^11 + 84*a^6*b^4*x^9 + 36*a^7*b^3*x^7 + 9*a^8*b^2*x^5 + a^9*b*x^3)*sqrt(- b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^11*b^9*x^21 + 9*a ^12*b^8*x^19 + 36*a^13*b^7*x^17 + 84*a^14*b^6*x^15 + 126*a^15*b^5*x^13 + 1 26*a^16*b^4*x^11 + 84*a^17*b^3*x^9 + 36*a^18*b^2*x^7 + 9*a^19*b*x^5 + a^20 *x^3), 1/589824*(14549535*b^10*x^20 + 126095970*a*b^9*x^18 + 483044562*a^2 *b^8*x^16 + 1071677178*a^3*b^7*x^14 + 1513521152*a^4*b^6*x^12 + 1404993798 *a^5*b^5*x^10 + 850547502*a^6*b^4*x^8 + 318434718*a^7*b^3*x^6 + 63897057*a ^8*b^2*x^4 + 4128768*a^9*b*x^2 - 196608*a^10 + 14549535*(b^10*x^21 + 9*a*b ^9*x^19 + 36*a^2*b^8*x^17 + 84*a^3*b^7*x^15 + 126*a^4*b^6*x^13 + 126*a^5*b ^5*x^11 + 84*a^6*b^4*x^9 + 36*a^7*b^3*x^7 + 9*a^8*b^2*x^5 + a^9*b*x^3)*sqr t(b/a)*arctan(x*sqrt(b/a)))/(a^11*b^9*x^21 + 9*a^12*b^8*x^19 + 36*a^13*b^7 *x^17 + 84*a^14*b^6*x^15 + 126*a^15*b^5*x^13 + 126*a^16*b^4*x^11 + 84*a^17 *b^3*x^9 + 36*a^18*b^2*x^7 + 9*a^19*b*x^5 + a^20*x^3)]
Time = 0.76 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx=- \frac {1616615 \sqrt {- \frac {b^{3}}{a^{23}}} \log {\left (- \frac {a^{12} \sqrt {- \frac {b^{3}}{a^{23}}}}{b^{2}} + x \right )}}{131072} + \frac {1616615 \sqrt {- \frac {b^{3}}{a^{23}}} \log {\left (\frac {a^{12} \sqrt {- \frac {b^{3}}{a^{23}}}}{b^{2}} + x \right )}}{131072} + \frac {- 196608 a^{10} + 4128768 a^{9} b x^{2} + 63897057 a^{8} b^{2} x^{4} + 318434718 a^{7} b^{3} x^{6} + 850547502 a^{6} b^{4} x^{8} + 1404993798 a^{5} b^{5} x^{10} + 1513521152 a^{4} b^{6} x^{12} + 1071677178 a^{3} b^{7} x^{14} + 483044562 a^{2} b^{8} x^{16} + 126095970 a b^{9} x^{18} + 14549535 b^{10} x^{20}}{589824 a^{20} x^{3} + 5308416 a^{19} b x^{5} + 21233664 a^{18} b^{2} x^{7} + 49545216 a^{17} b^{3} x^{9} + 74317824 a^{16} b^{4} x^{11} + 74317824 a^{15} b^{5} x^{13} + 49545216 a^{14} b^{6} x^{15} + 21233664 a^{13} b^{7} x^{17} + 5308416 a^{12} b^{8} x^{19} + 589824 a^{11} b^{9} x^{21}} \] Input:
integrate(1/x**4/(b*x**2+a)**10,x)
Output:
-1616615*sqrt(-b**3/a**23)*log(-a**12*sqrt(-b**3/a**23)/b**2 + x)/131072 + 1616615*sqrt(-b**3/a**23)*log(a**12*sqrt(-b**3/a**23)/b**2 + x)/131072 + (-196608*a**10 + 4128768*a**9*b*x**2 + 63897057*a**8*b**2*x**4 + 318434718 *a**7*b**3*x**6 + 850547502*a**6*b**4*x**8 + 1404993798*a**5*b**5*x**10 + 1513521152*a**4*b**6*x**12 + 1071677178*a**3*b**7*x**14 + 483044562*a**2*b **8*x**16 + 126095970*a*b**9*x**18 + 14549535*b**10*x**20)/(589824*a**20*x **3 + 5308416*a**19*b*x**5 + 21233664*a**18*b**2*x**7 + 49545216*a**17*b** 3*x**9 + 74317824*a**16*b**4*x**11 + 74317824*a**15*b**5*x**13 + 49545216* a**14*b**6*x**15 + 21233664*a**13*b**7*x**17 + 5308416*a**12*b**8*x**19 + 589824*a**11*b**9*x**21)
Time = 0.13 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx=\frac {14549535 \, b^{10} x^{20} + 126095970 \, a b^{9} x^{18} + 483044562 \, a^{2} b^{8} x^{16} + 1071677178 \, a^{3} b^{7} x^{14} + 1513521152 \, a^{4} b^{6} x^{12} + 1404993798 \, a^{5} b^{5} x^{10} + 850547502 \, a^{6} b^{4} x^{8} + 318434718 \, a^{7} b^{3} x^{6} + 63897057 \, a^{8} b^{2} x^{4} + 4128768 \, a^{9} b x^{2} - 196608 \, a^{10}}{589824 \, {\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}} + \frac {1616615 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{11}} \] Input:
integrate(1/x^4/(b*x^2+a)^10,x, algorithm="maxima")
Output:
1/589824*(14549535*b^10*x^20 + 126095970*a*b^9*x^18 + 483044562*a^2*b^8*x^ 16 + 1071677178*a^3*b^7*x^14 + 1513521152*a^4*b^6*x^12 + 1404993798*a^5*b^ 5*x^10 + 850547502*a^6*b^4*x^8 + 318434718*a^7*b^3*x^6 + 63897057*a^8*b^2* x^4 + 4128768*a^9*b*x^2 - 196608*a^10)/(a^11*b^9*x^21 + 9*a^12*b^8*x^19 + 36*a^13*b^7*x^17 + 84*a^14*b^6*x^15 + 126*a^15*b^5*x^13 + 126*a^16*b^4*x^1 1 + 84*a^17*b^3*x^9 + 36*a^18*b^2*x^7 + 9*a^19*b*x^5 + a^20*x^3) + 1616615 /65536*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^11)
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx=\frac {1616615 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{11}} + \frac {30 \, b x^{2} - a}{3 \, a^{11} x^{3}} + \frac {8651295 \, b^{10} x^{17} + 73208418 \, a b^{9} x^{15} + 272477394 \, a^{2} b^{8} x^{13} + 583302906 \, a^{3} b^{7} x^{11} + 786857984 \, a^{4} b^{6} x^{9} + 686588166 \, a^{5} b^{5} x^{7} + 379867950 \, a^{6} b^{4} x^{5} + 122613150 \, a^{7} b^{3} x^{3} + 17890785 \, a^{8} b^{2} x}{589824 \, {\left (b x^{2} + a\right )}^{9} a^{11}} \] Input:
integrate(1/x^4/(b*x^2+a)^10,x, algorithm="giac")
Output:
1616615/65536*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^11) + 1/3*(30*b*x^2 - a)/(a^11*x^3) + 1/589824*(8651295*b^10*x^17 + 73208418*a*b^9*x^15 + 27247 7394*a^2*b^8*x^13 + 583302906*a^3*b^7*x^11 + 786857984*a^4*b^6*x^9 + 68658 8166*a^5*b^5*x^7 + 379867950*a^6*b^4*x^5 + 122613150*a^7*b^3*x^3 + 1789078 5*a^8*b^2*x)/((b*x^2 + a)^9*a^11)
Time = 0.66 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx=\frac {\frac {7\,b\,x^2}{a^2}-\frac {1}{3\,a}+\frac {7099673\,b^2\,x^4}{65536\,a^3}+\frac {53072453\,b^3\,x^6}{98304\,a^4}+\frac {47252639\,b^4\,x^8}{32768\,a^5}+\frac {78055211\,b^5\,x^{10}}{32768\,a^6}+\frac {46189\,b^6\,x^{12}}{18\,a^7}+\frac {59537621\,b^7\,x^{14}}{32768\,a^8}+\frac {26835809\,b^8\,x^{16}}{32768\,a^9}+\frac {21015995\,b^9\,x^{18}}{98304\,a^{10}}+\frac {1616615\,b^{10}\,x^{20}}{65536\,a^{11}}}{a^9\,x^3+9\,a^8\,b\,x^5+36\,a^7\,b^2\,x^7+84\,a^6\,b^3\,x^9+126\,a^5\,b^4\,x^{11}+126\,a^4\,b^5\,x^{13}+84\,a^3\,b^6\,x^{15}+36\,a^2\,b^7\,x^{17}+9\,a\,b^8\,x^{19}+b^9\,x^{21}}+\frac {1616615\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{23/2}} \] Input:
int(1/(x^4*(a + b*x^2)^10),x)
Output:
((7*b*x^2)/a^2 - 1/(3*a) + (7099673*b^2*x^4)/(65536*a^3) + (53072453*b^3*x ^6)/(98304*a^4) + (47252639*b^4*x^8)/(32768*a^5) + (78055211*b^5*x^10)/(32 768*a^6) + (46189*b^6*x^12)/(18*a^7) + (59537621*b^7*x^14)/(32768*a^8) + ( 26835809*b^8*x^16)/(32768*a^9) + (21015995*b^9*x^18)/(98304*a^10) + (16166 15*b^10*x^20)/(65536*a^11))/(a^9*x^3 + b^9*x^21 + 9*a^8*b*x^5 + 9*a*b^8*x^ 19 + 36*a^7*b^2*x^7 + 84*a^6*b^3*x^9 + 126*a^5*b^4*x^11 + 126*a^4*b^5*x^13 + 84*a^3*b^6*x^15 + 36*a^2*b^7*x^17) + (1616615*b^(3/2)*atan((b^(1/2)*x)/ a^(1/2)))/(65536*a^(23/2))
Time = 0.23 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.11 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{10}} \, dx=\frac {14549535 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{9} b \,x^{3}+130945815 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{8} b^{2} x^{5}+523783260 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{7} b^{3} x^{7}+1222160940 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6} b^{4} x^{9}+1833241410 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{5} x^{11}+1833241410 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{6} x^{13}+1222160940 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{7} x^{15}+523783260 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{8} x^{17}+130945815 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{9} x^{19}+14549535 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{10} x^{21}-196608 a^{11}+4128768 a^{10} b \,x^{2}+63897057 a^{9} b^{2} x^{4}+318434718 a^{8} b^{3} x^{6}+850547502 a^{7} b^{4} x^{8}+1404993798 a^{6} b^{5} x^{10}+1513521152 a^{5} b^{6} x^{12}+1071677178 a^{4} b^{7} x^{14}+483044562 a^{3} b^{8} x^{16}+126095970 a^{2} b^{9} x^{18}+14549535 a \,b^{10} x^{20}}{589824 a^{12} x^{3} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:
int(1/x^4/(b*x^2+a)^10,x)
Output:
(14549535*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**9*b*x**3 + 1309 45815*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**8*b**2*x**5 + 52378 3260*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**7*b**3*x**7 + 122216 0940*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**6*b**4*x**9 + 183324 1410*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**5*x**11 + 18332 41410*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**6*x**13 + 1222 160940*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**7*x**15 + 523 783260*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**8*x**17 + 130 945815*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**9*x**19 + 145495 35*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**10*x**21 - 196608*a**1 1 + 4128768*a**10*b*x**2 + 63897057*a**9*b**2*x**4 + 318434718*a**8*b**3*x **6 + 850547502*a**7*b**4*x**8 + 1404993798*a**6*b**5*x**10 + 1513521152*a **5*b**6*x**12 + 1071677178*a**4*b**7*x**14 + 483044562*a**3*b**8*x**16 + 126095970*a**2*b**9*x**18 + 14549535*a*b**10*x**20)/(589824*a**12*x**3*(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))