Integrand size = 15, antiderivative size = 164 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx=-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac {40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac {320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac {512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac {1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}} \]
-1/23*(b*x^2+a)^(11/2)/a/x^23+4/161*b*(b*x^2+a)^(11/2)/a^2/x^21-40/3059*b^ 2*(b*x^2+a)^(11/2)/a^3/x^19+320/52003*b^3*(b*x^2+a)^(11/2)/a^4/x^17-128/52 003*b^4*(b*x^2+a)^(11/2)/a^5/x^15+512/676039*b^5*(b*x^2+a)^(11/2)/a^6/x^13 -1024/7436429*b^6*(b*x^2+a)^(11/2)/a^7/x^11
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.52 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx=\frac {\left (a+b x^2\right )^{11/2} \left (-323323 a^6+184756 a^5 b x^2-97240 a^4 b^2 x^4+45760 a^3 b^3 x^6-18304 a^2 b^4 x^8+5632 a b^5 x^{10}-1024 b^6 x^{12}\right )}{7436429 a^7 x^{23}} \]
((a + b*x^2)^(11/2)*(-323323*a^6 + 184756*a^5*b*x^2 - 97240*a^4*b^2*x^4 + 45760*a^3*b^3*x^6 - 18304*a^2*b^4*x^8 + 5632*a*b^5*x^10 - 1024*b^6*x^12))/ (7436429*a^7*x^23)
Time = 0.26 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {245, 245, 245, 245, 245, 245, 242}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {12 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{22}}dx}{23 a}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {12 b \left (-\frac {10 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{20}}dx}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\right )}{23 a}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {12 b \left (-\frac {10 b \left (-\frac {8 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{18}}dx}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\right )}{23 a}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {12 b \left (-\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{16}}dx}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\right )}{23 a}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {12 b \left (-\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{14}}dx}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\right )}{23 a}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {12 b \left (-\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{12}}dx}{13 a}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\right )}{23 a}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {12 b \left (-\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\right )}{23 a}-\frac {\left (a+b x^2\right )^{11/2}}{23 a x^{23}}\) |
-1/23*(a + b*x^2)^(11/2)/(a*x^23) - (12*b*(-1/21*(a + b*x^2)^(11/2)/(a*x^2 1) - (10*b*(-1/19*(a + b*x^2)^(11/2)/(a*x^19) - (8*b*(-1/17*(a + b*x^2)^(1 1/2)/(a*x^17) - (6*b*(-1/15*(a + b*x^2)^(11/2)/(a*x^15) - (4*b*(-1/13*(a + b*x^2)^(11/2)/(a*x^13) + (2*b*(a + b*x^2)^(11/2))/(143*a^2*x^11)))/(15*a) ))/(17*a)))/(19*a)))/(21*a)))/(23*a)
3.5.40.3.1 Defintions of rubi rules used
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] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Time = 31.50 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.51
| method | result | size |
| gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (1024 b^{6} x^{12}-5632 a \,b^{5} x^{10}+18304 a^{2} b^{4} x^{8}-45760 a^{3} b^{3} x^{6}+97240 a^{4} b^{2} x^{4}-184756 a^{5} b \,x^{2}+323323 a^{6}\right )}{7436429 x^{23} a^{7}}\) | \(83\) |
| pseudoelliptic | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (1024 b^{6} x^{12}-5632 a \,b^{5} x^{10}+18304 a^{2} b^{4} x^{8}-45760 a^{3} b^{3} x^{6}+97240 a^{4} b^{2} x^{4}-184756 a^{5} b \,x^{2}+323323 a^{6}\right )}{7436429 x^{23} a^{7}}\) | \(83\) |
| trager | \(-\frac {\left (1024 b^{11} x^{22}-512 a \,b^{10} x^{20}+384 a^{2} b^{9} x^{18}-320 a^{3} b^{8} x^{16}+280 a^{4} b^{7} x^{14}-252 a^{5} b^{6} x^{12}+231 a^{6} b^{5} x^{10}+530959 a^{7} b^{4} x^{8}+1826110 x^{6} a^{8} b^{3}+2406690 a^{9} b^{2} x^{4}+1431859 a^{10} b \,x^{2}+323323 a^{11}\right ) \sqrt {b \,x^{2}+a}}{7436429 a^{7} x^{23}}\) | \(138\) |
| risch | \(-\frac {\left (1024 b^{11} x^{22}-512 a \,b^{10} x^{20}+384 a^{2} b^{9} x^{18}-320 a^{3} b^{8} x^{16}+280 a^{4} b^{7} x^{14}-252 a^{5} b^{6} x^{12}+231 a^{6} b^{5} x^{10}+530959 a^{7} b^{4} x^{8}+1826110 x^{6} a^{8} b^{3}+2406690 a^{9} b^{2} x^{4}+1431859 a^{10} b \,x^{2}+323323 a^{11}\right ) \sqrt {b \,x^{2}+a}}{7436429 a^{7} x^{23}}\) | \(138\) |
| default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{23 a \,x^{23}}-\frac {12 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{21 a \,x^{21}}-\frac {10 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{19 a \,x^{19}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{17 a \,x^{17}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{15 a \,x^{15}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 a \,x^{13}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 a^{2} x^{11}}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\) | \(157\) |
-1/7436429*(b*x^2+a)^(11/2)*(1024*b^6*x^12-5632*a*b^5*x^10+18304*a^2*b^4*x ^8-45760*a^3*b^3*x^6+97240*a^4*b^2*x^4-184756*a^5*b*x^2+323323*a^6)/x^23/a ^7
Time = 0.57 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx=-\frac {{\left (1024 \, b^{11} x^{22} - 512 \, a b^{10} x^{20} + 384 \, a^{2} b^{9} x^{18} - 320 \, a^{3} b^{8} x^{16} + 280 \, a^{4} b^{7} x^{14} - 252 \, a^{5} b^{6} x^{12} + 231 \, a^{6} b^{5} x^{10} + 530959 \, a^{7} b^{4} x^{8} + 1826110 \, a^{8} b^{3} x^{6} + 2406690 \, a^{9} b^{2} x^{4} + 1431859 \, a^{10} b x^{2} + 323323 \, a^{11}\right )} \sqrt {b x^{2} + a}}{7436429 \, a^{7} x^{23}} \]
-1/7436429*(1024*b^11*x^22 - 512*a*b^10*x^20 + 384*a^2*b^9*x^18 - 320*a^3* b^8*x^16 + 280*a^4*b^7*x^14 - 252*a^5*b^6*x^12 + 231*a^6*b^5*x^10 + 530959 *a^7*b^4*x^8 + 1826110*a^8*b^3*x^6 + 2406690*a^9*b^2*x^4 + 1431859*a^10*b* x^2 + 323323*a^11)*sqrt(b*x^2 + a)/(a^7*x^23)
Leaf count of result is larger than twice the leaf count of optimal. 1950 vs. \(2 (156) = 312\).
Time = 4.46 (sec) , antiderivative size = 1950, normalized size of antiderivative = 11.89 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx=\text {Too large to display} \]
-323323*a**17*b**(73/2)*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10 *b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 74 36429*a**7*b**42*x**34) - 3371797*a**16*b**(75/2)*x**2*sqrt(a/(b*x**2) + 1 )/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**1 1*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 15847689*a**15*b* *(77/2)*x**4*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a* *12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**2 8 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7* b**42*x**34) - 44210595*a**14*b**(79/2)*x**6*sqrt(a/(b*x**2) + 1)/(7436429 *a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x* *26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574* a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 81074994*a**13*b**(81/2)*x* *8*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37* x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546 435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**3 4) - 102129258*a**12*b**(83/2)*x**10*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b **36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 14 8728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*...
Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx=-\frac {1024 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{6}}{7436429 \, a^{7} x^{11}} + \frac {512 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{5}}{676039 \, a^{6} x^{13}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{52003 \, a^{5} x^{15}} + \frac {320 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{52003 \, a^{4} x^{17}} - \frac {40 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{3059 \, a^{3} x^{19}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{161 \, a^{2} x^{21}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{23 \, a x^{23}} \]
-1024/7436429*(b*x^2 + a)^(11/2)*b^6/(a^7*x^11) + 512/676039*(b*x^2 + a)^( 11/2)*b^5/(a^6*x^13) - 128/52003*(b*x^2 + a)^(11/2)*b^4/(a^5*x^15) + 320/5 2003*(b*x^2 + a)^(11/2)*b^3/(a^4*x^17) - 40/3059*(b*x^2 + a)^(11/2)*b^2/(a ^3*x^19) + 4/161*(b*x^2 + a)^(11/2)*b/(a^2*x^21) - 1/23*(b*x^2 + a)^(11/2) /(a*x^23)
Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (136) = 272\).
Time = 0.33 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.82 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx=\frac {2048 \, {\left (4249388 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{32} b^{\frac {23}{2}} + 28683369 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{30} a b^{\frac {23}{2}} + 100922965 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{28} a^{2} b^{\frac {23}{2}} + 215656441 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} a^{3} b^{\frac {23}{2}} + 313006057 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a^{4} b^{\frac {23}{2}} + 311653979 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{5} b^{\frac {23}{2}} + 216800507 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{6} b^{\frac {23}{2}} + 100105775 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{7} b^{\frac {23}{2}} + 29173683 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{8} b^{\frac {23}{2}} + 4004231 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{9} b^{\frac {23}{2}} + 100947 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{10} b^{\frac {23}{2}} - 33649 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{11} b^{\frac {23}{2}} + 8855 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{12} b^{\frac {23}{2}} - 1771 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{13} b^{\frac {23}{2}} + 253 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{14} b^{\frac {23}{2}} - 23 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{15} b^{\frac {23}{2}} + a^{16} b^{\frac {23}{2}}\right )}}{7436429 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{23}} \]
2048/7436429*(4249388*(sqrt(b)*x - sqrt(b*x^2 + a))^32*b^(23/2) + 28683369 *(sqrt(b)*x - sqrt(b*x^2 + a))^30*a*b^(23/2) + 100922965*(sqrt(b)*x - sqrt (b*x^2 + a))^28*a^2*b^(23/2) + 215656441*(sqrt(b)*x - sqrt(b*x^2 + a))^26* a^3*b^(23/2) + 313006057*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a^4*b^(23/2) + 3 11653979*(sqrt(b)*x - sqrt(b*x^2 + a))^22*a^5*b^(23/2) + 216800507*(sqrt(b )*x - sqrt(b*x^2 + a))^20*a^6*b^(23/2) + 100105775*(sqrt(b)*x - sqrt(b*x^2 + a))^18*a^7*b^(23/2) + 29173683*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^8*b^( 23/2) + 4004231*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^9*b^(23/2) + 100947*(sq rt(b)*x - sqrt(b*x^2 + a))^12*a^10*b^(23/2) - 33649*(sqrt(b)*x - sqrt(b*x^ 2 + a))^10*a^11*b^(23/2) + 8855*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^12*b^(23 /2) - 1771*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^13*b^(23/2) + 253*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^14*b^(23/2) - 23*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^ 15*b^(23/2) + a^16*b^(23/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^23
Time = 10.91 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{24}} \, dx=\frac {36\,b^6\,\sqrt {b\,x^2+a}}{1062347\,a^2\,x^{11}}-\frac {3713\,b^4\,\sqrt {b\,x^2+a}}{52003\,x^{15}}-\frac {12770\,a\,b^3\,\sqrt {b\,x^2+a}}{52003\,x^{17}}-\frac {31\,a^3\,b\,\sqrt {b\,x^2+a}}{161\,x^{21}}-\frac {3\,b^5\,\sqrt {b\,x^2+a}}{96577\,a\,x^{13}}-\frac {a^4\,\sqrt {b\,x^2+a}}{23\,x^{23}}-\frac {40\,b^7\,\sqrt {b\,x^2+a}}{1062347\,a^3\,x^9}+\frac {320\,b^8\,\sqrt {b\,x^2+a}}{7436429\,a^4\,x^7}-\frac {384\,b^9\,\sqrt {b\,x^2+a}}{7436429\,a^5\,x^5}+\frac {512\,b^{10}\,\sqrt {b\,x^2+a}}{7436429\,a^6\,x^3}-\frac {1024\,b^{11}\,\sqrt {b\,x^2+a}}{7436429\,a^7\,x}-\frac {990\,a^2\,b^2\,\sqrt {b\,x^2+a}}{3059\,x^{19}} \]
(36*b^6*(a + b*x^2)^(1/2))/(1062347*a^2*x^11) - (3713*b^4*(a + b*x^2)^(1/2 ))/(52003*x^15) - (12770*a*b^3*(a + b*x^2)^(1/2))/(52003*x^17) - (31*a^3*b *(a + b*x^2)^(1/2))/(161*x^21) - (3*b^5*(a + b*x^2)^(1/2))/(96577*a*x^13) - (a^4*(a + b*x^2)^(1/2))/(23*x^23) - (40*b^7*(a + b*x^2)^(1/2))/(1062347* a^3*x^9) + (320*b^8*(a + b*x^2)^(1/2))/(7436429*a^4*x^7) - (384*b^9*(a + b *x^2)^(1/2))/(7436429*a^5*x^5) + (512*b^10*(a + b*x^2)^(1/2))/(7436429*a^6 *x^3) - (1024*b^11*(a + b*x^2)^(1/2))/(7436429*a^7*x) - (990*a^2*b^2*(a + b*x^2)^(1/2))/(3059*x^19)