Integrand size = 15, antiderivative size = 140 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx=-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac {10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac {80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac {32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}+\frac {256 b^5 \left (a+b x^2\right )^{11/2}}{969969 a^6 x^{11}} \] Output:
-1/21*(b*x^2+a)^(11/2)/a/x^21+10/399*b*(b*x^2+a)^(11/2)/a^2/x^19-80/6783*b ^2*(b*x^2+a)^(11/2)/a^3/x^17+32/6783*b^3*(b*x^2+a)^(11/2)/a^4/x^15-128/881 79*b^4*(b*x^2+a)^(11/2)/a^5/x^13+256/969969*b^5*(b*x^2+a)^(11/2)/a^6/x^11
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx=\frac {\left (a+b x^2\right )^{11/2} \left (-46189 a^5+24310 a^4 b x^2-11440 a^3 b^2 x^4+4576 a^2 b^3 x^6-1408 a b^4 x^8+256 b^5 x^{10}\right )}{969969 a^6 x^{21}} \] Input:
Integrate[(a + b*x^2)^(9/2)/x^22,x]
Output:
((a + b*x^2)^(11/2)*(-46189*a^5 + 24310*a^4*b*x^2 - 11440*a^3*b^2*x^4 + 45 76*a^2*b^3*x^6 - 1408*a*b^4*x^8 + 256*b^5*x^10))/(969969*a^6*x^21)
Time = 0.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {10 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{20}}dx}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {10 b \left (-\frac {8 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{18}}dx}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{16}}dx}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{14}}dx}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{12}}dx}{13 a}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\right )}{21 a}-\frac {\left (a+b x^2\right )^{11/2}}{21 a x^{21}}\) |
Input:
Int[(a + b*x^2)^(9/2)/x^22,x]
Output:
-1/21*(a + b*x^2)^(11/2)/(a*x^21) - (10*b*(-1/19*(a + b*x^2)^(11/2)/(a*x^1 9) - (8*b*(-1/17*(a + b*x^2)^(11/2)/(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)
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 = 22.40 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.51
| method | result | size |
| gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-256 b^{5} x^{10}+1408 a \,b^{4} x^{8}-4576 a^{2} b^{3} x^{6}+11440 a^{3} b^{2} x^{4}-24310 a^{4} b \,x^{2}+46189 a^{5}\right )}{969969 x^{21} a^{6}}\) | \(72\) |
| pseudoelliptic | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-256 b^{5} x^{10}+1408 a \,b^{4} x^{8}-4576 a^{2} b^{3} x^{6}+11440 a^{3} b^{2} x^{4}-24310 a^{4} b \,x^{2}+46189 a^{5}\right )}{969969 x^{21} a^{6}}\) | \(72\) |
| orering | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-256 b^{5} x^{10}+1408 a \,b^{4} x^{8}-4576 a^{2} b^{3} x^{6}+11440 a^{3} b^{2} x^{4}-24310 a^{4} b \,x^{2}+46189 a^{5}\right )}{969969 x^{21} a^{6}}\) | \(72\) |
| trager | \(-\frac {\left (-256 b^{10} x^{20}+128 a \,b^{9} x^{18}-96 a^{2} b^{8} x^{16}+80 a^{3} b^{7} x^{14}-70 a^{4} b^{6} x^{12}+63 a^{5} b^{5} x^{10}+80773 a^{6} b^{4} x^{8}+271414 a^{7} b^{3} x^{6}+351780 a^{8} b^{2} x^{4}+206635 a^{9} b \,x^{2}+46189 a^{10}\right ) \sqrt {b \,x^{2}+a}}{969969 a^{6} x^{21}}\) | \(127\) |
| risch | \(-\frac {\left (-256 b^{10} x^{20}+128 a \,b^{9} x^{18}-96 a^{2} b^{8} x^{16}+80 a^{3} b^{7} x^{14}-70 a^{4} b^{6} x^{12}+63 a^{5} b^{5} x^{10}+80773 a^{6} b^{4} x^{8}+271414 a^{7} b^{3} x^{6}+351780 a^{8} b^{2} x^{4}+206635 a^{9} b \,x^{2}+46189 a^{10}\right ) \sqrt {b \,x^{2}+a}}{969969 a^{6} x^{21}}\) | \(127\) |
| default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{21 a \,x^{21}}-\frac {10 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{19 a \,x^{19}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{17 a \,x^{17}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{15 a \,x^{15}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 a \,x^{13}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 a^{2} x^{11}}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\) | \(133\) |
Input:
int((b*x^2+a)^(9/2)/x^22,x,method=_RETURNVERBOSE)
Output:
-1/969969*(b*x^2+a)^(11/2)*(-256*b^5*x^10+1408*a*b^4*x^8-4576*a^2*b^3*x^6+ 11440*a^3*b^2*x^4-24310*a^4*b*x^2+46189*a^5)/x^21/a^6
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx=\frac {{\left (256 \, b^{10} x^{20} - 128 \, a b^{9} x^{18} + 96 \, a^{2} b^{8} x^{16} - 80 \, a^{3} b^{7} x^{14} + 70 \, a^{4} b^{6} x^{12} - 63 \, a^{5} b^{5} x^{10} - 80773 \, a^{6} b^{4} x^{8} - 271414 \, a^{7} b^{3} x^{6} - 351780 \, a^{8} b^{2} x^{4} - 206635 \, a^{9} b x^{2} - 46189 \, a^{10}\right )} \sqrt {b x^{2} + a}}{969969 \, a^{6} x^{21}} \] Input:
integrate((b*x^2+a)^(9/2)/x^22,x, algorithm="fricas")
Output:
1/969969*(256*b^10*x^20 - 128*a*b^9*x^18 + 96*a^2*b^8*x^16 - 80*a^3*b^7*x^ 14 + 70*a^4*b^6*x^12 - 63*a^5*b^5*x^10 - 80773*a^6*b^4*x^8 - 271414*a^7*b^ 3*x^6 - 351780*a^8*b^2*x^4 - 206635*a^9*b*x^2 - 46189*a^10)*sqrt(b*x^2 + a )/(a^6*x^21)
Leaf count of result is larger than twice the leaf count of optimal. 1540 vs. \(2 (133) = 266\).
Time = 3.76 (sec) , antiderivative size = 1540, normalized size of antiderivative = 11.00 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**2+a)**(9/2)/x**22,x)
Output:
-46189*a**15*b**(51/2)*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 48 49845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x* *26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 437580*a**14*b **(53/2)*x**2*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a** 10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 484 9845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 1846845*a**13*b**(55/2) *x**4*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26 *x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a** 7*b**29*x**28 + 969969*a**6*b**30*x**30) - 4558554*a**12*b**(57/2)*x**6*sq rt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29* x**28 + 969969*a**6*b**30*x**30) - 7252938*a**11*b**(59/2)*x**8*sqrt(a/(b* x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690 *a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 7715232*a**10*b**(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b **27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969* a**6*b**30*x**30) - 5487650*a**9*b**(63/2)*x**12*sqrt(a/(b*x**2) + 1)/(969 969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x** 24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*...
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx=\frac {256 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{5}}{969969 \, a^{6} x^{11}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{88179 \, a^{5} x^{13}} + \frac {32 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{6783 \, a^{4} x^{15}} - \frac {80 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{6783 \, a^{3} x^{17}} + \frac {10 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{399 \, a^{2} x^{19}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{21 \, a x^{21}} \] Input:
integrate((b*x^2+a)^(9/2)/x^22,x, algorithm="maxima")
Output:
256/969969*(b*x^2 + a)^(11/2)*b^5/(a^6*x^11) - 128/88179*(b*x^2 + a)^(11/2 )*b^4/(a^5*x^13) + 32/6783*(b*x^2 + a)^(11/2)*b^3/(a^4*x^15) - 80/6783*(b* x^2 + a)^(11/2)*b^2/(a^3*x^17) + 10/399*(b*x^2 + a)^(11/2)*b/(a^2*x^19) - 1/21*(b*x^2 + a)^(11/2)/(a*x^21)
Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (116) = 232\).
Time = 0.15 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.11 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx=\frac {512 \, {\left (646646 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{30} b^{\frac {21}{2}} + 4157010 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{28} a b^{\frac {21}{2}} + 14549535 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} a^{2} b^{\frac {21}{2}} + 30715685 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a^{3} b^{\frac {21}{2}} + 44618574 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{4} b^{\frac {21}{2}} + 44265858 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{5} b^{\frac {21}{2}} + 31009615 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{6} b^{\frac {21}{2}} + 14346045 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{7} b^{\frac {21}{2}} + 4273290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{8} b^{\frac {21}{2}} + 592382 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{9} b^{\frac {21}{2}} + 20349 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{10} b^{\frac {21}{2}} - 5985 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{11} b^{\frac {21}{2}} + 1330 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{12} b^{\frac {21}{2}} - 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{13} b^{\frac {21}{2}} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{14} b^{\frac {21}{2}} - a^{15} b^{\frac {21}{2}}\right )}}{969969 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{21}} \] Input:
integrate((b*x^2+a)^(9/2)/x^22,x, algorithm="giac")
Output:
512/969969*(646646*(sqrt(b)*x - sqrt(b*x^2 + a))^30*b^(21/2) + 4157010*(sq rt(b)*x - sqrt(b*x^2 + a))^28*a*b^(21/2) + 14549535*(sqrt(b)*x - sqrt(b*x^ 2 + a))^26*a^2*b^(21/2) + 30715685*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a^3*b^ (21/2) + 44618574*(sqrt(b)*x - sqrt(b*x^2 + a))^22*a^4*b^(21/2) + 44265858 *(sqrt(b)*x - sqrt(b*x^2 + a))^20*a^5*b^(21/2) + 31009615*(sqrt(b)*x - sqr t(b*x^2 + a))^18*a^6*b^(21/2) + 14346045*(sqrt(b)*x - sqrt(b*x^2 + a))^16* a^7*b^(21/2) + 4273290*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^8*b^(21/2) + 592 382*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^9*b^(21/2) + 20349*(sqrt(b)*x - sqr t(b*x^2 + a))^10*a^10*b^(21/2) - 5985*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^11 *b^(21/2) + 1330*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^12*b^(21/2) - 210*(sqrt (b)*x - sqrt(b*x^2 + a))^4*a^13*b^(21/2) + 21*(sqrt(b)*x - sqrt(b*x^2 + a) )^2*a^14*b^(21/2) - a^15*b^(21/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^2 1
Time = 5.80 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx=\frac {10\,b^6\,\sqrt {b\,x^2+a}}{138567\,a^2\,x^9}-\frac {1049\,b^4\,\sqrt {b\,x^2+a}}{12597\,x^{13}}-\frac {1898\,a\,b^3\,\sqrt {b\,x^2+a}}{6783\,x^{15}}-\frac {85\,a^3\,b\,\sqrt {b\,x^2+a}}{399\,x^{19}}-\frac {3\,b^5\,\sqrt {b\,x^2+a}}{46189\,a\,x^{11}}-\frac {a^4\,\sqrt {b\,x^2+a}}{21\,x^{21}}-\frac {80\,b^7\,\sqrt {b\,x^2+a}}{969969\,a^3\,x^7}+\frac {32\,b^8\,\sqrt {b\,x^2+a}}{323323\,a^4\,x^5}-\frac {128\,b^9\,\sqrt {b\,x^2+a}}{969969\,a^5\,x^3}+\frac {256\,b^{10}\,\sqrt {b\,x^2+a}}{969969\,a^6\,x}-\frac {820\,a^2\,b^2\,\sqrt {b\,x^2+a}}{2261\,x^{17}} \] Input:
int((a + b*x^2)^(9/2)/x^22,x)
Output:
(10*b^6*(a + b*x^2)^(1/2))/(138567*a^2*x^9) - (1049*b^4*(a + b*x^2)^(1/2)) /(12597*x^13) - (1898*a*b^3*(a + b*x^2)^(1/2))/(6783*x^15) - (85*a^3*b*(a + b*x^2)^(1/2))/(399*x^19) - (3*b^5*(a + b*x^2)^(1/2))/(46189*a*x^11) - (a ^4*(a + b*x^2)^(1/2))/(21*x^21) - (80*b^7*(a + b*x^2)^(1/2))/(969969*a^3*x ^7) + (32*b^8*(a + b*x^2)^(1/2))/(323323*a^4*x^5) - (128*b^9*(a + b*x^2)^( 1/2))/(969969*a^5*x^3) + (256*b^10*(a + b*x^2)^(1/2))/(969969*a^6*x) - (82 0*a^2*b^2*(a + b*x^2)^(1/2))/(2261*x^17)
Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx=\frac {-46189 \sqrt {b \,x^{2}+a}\, a^{10}-206635 \sqrt {b \,x^{2}+a}\, a^{9} b \,x^{2}-351780 \sqrt {b \,x^{2}+a}\, a^{8} b^{2} x^{4}-271414 \sqrt {b \,x^{2}+a}\, a^{7} b^{3} x^{6}-80773 \sqrt {b \,x^{2}+a}\, a^{6} b^{4} x^{8}-63 \sqrt {b \,x^{2}+a}\, a^{5} b^{5} x^{10}+70 \sqrt {b \,x^{2}+a}\, a^{4} b^{6} x^{12}-80 \sqrt {b \,x^{2}+a}\, a^{3} b^{7} x^{14}+96 \sqrt {b \,x^{2}+a}\, a^{2} b^{8} x^{16}-128 \sqrt {b \,x^{2}+a}\, a \,b^{9} x^{18}+256 \sqrt {b \,x^{2}+a}\, b^{10} x^{20}-256 \sqrt {b}\, b^{10} x^{21}}{969969 a^{6} x^{21}} \] Input:
int((b*x^2+a)^(9/2)/x^22,x)
Output:
( - 46189*sqrt(a + b*x**2)*a**10 - 206635*sqrt(a + b*x**2)*a**9*b*x**2 - 3 51780*sqrt(a + b*x**2)*a**8*b**2*x**4 - 271414*sqrt(a + b*x**2)*a**7*b**3* x**6 - 80773*sqrt(a + b*x**2)*a**6*b**4*x**8 - 63*sqrt(a + b*x**2)*a**5*b* *5*x**10 + 70*sqrt(a + b*x**2)*a**4*b**6*x**12 - 80*sqrt(a + b*x**2)*a**3* b**7*x**14 + 96*sqrt(a + b*x**2)*a**2*b**8*x**16 - 128*sqrt(a + b*x**2)*a* b**9*x**18 + 256*sqrt(a + b*x**2)*b**10*x**20 - 256*sqrt(b)*b**10*x**21)/( 969969*a**6*x**21)