Integrand size = 15, antiderivative size = 116 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac {8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac {16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac {64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{230945 a^5 x^{11}} \]
-1/19*(b*x^2+a)^(11/2)/a/x^19+8/323*b*(b*x^2+a)^(11/2)/a^2/x^17-16/1615*b^ 2*(b*x^2+a)^(11/2)/a^3/x^15+64/20995*b^3*(b*x^2+a)^(11/2)/a^4/x^13-128/230 945*b^4*(b*x^2+a)^(11/2)/a^5/x^11
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.55 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\frac {\left (a+b x^2\right )^{11/2} \left (-12155 a^4+5720 a^3 b x^2-2288 a^2 b^2 x^4+704 a b^3 x^6-128 b^4 x^8\right )}{230945 a^5 x^{19}} \]
((a + b*x^2)^(11/2)*(-12155*a^4 + 5720*a^3*b*x^2 - 2288*a^2*b^2*x^4 + 704* a*b^3*x^6 - 128*b^4*x^8))/(230945*a^5*x^19)
Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {8 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{18}}dx}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {8 b \left (-\frac {6 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{16}}dx}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{14}}dx}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{12}}dx}{13 a}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\) |
-1/19*(a + b*x^2)^(11/2)/(a*x^19) - (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)
3.5.38.3.1 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 = 5.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (128 x^{8} b^{4}-704 a \,b^{3} x^{6}+2288 a^{2} x^{4} b^{2}-5720 a^{3} b \,x^{2}+12155 a^{4}\right )}{230945 x^{19} a^{5}}\) | \(61\) |
pseudoelliptic | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (128 x^{8} b^{4}-704 a \,b^{3} x^{6}+2288 a^{2} x^{4} b^{2}-5720 a^{3} b \,x^{2}+12155 a^{4}\right )}{230945 x^{19} a^{5}}\) | \(61\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{19 a \,x^{19}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{17 a \,x^{17}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{15 a \,x^{15}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 a \,x^{13}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 a^{2} x^{11}}\right )}{15 a}\right )}{17 a}\right )}{19 a}\) | \(109\) |
trager | \(-\frac {\left (128 b^{9} x^{18}-64 a \,x^{16} b^{8}+48 a^{2} x^{14} b^{7}-40 a^{3} x^{12} b^{6}+35 a^{4} x^{10} b^{5}+23063 a^{5} x^{8} b^{4}+75086 a^{6} x^{6} b^{3}+95238 a^{7} x^{4} b^{2}+55055 a^{8} b \,x^{2}+12155 a^{9}\right ) \sqrt {b \,x^{2}+a}}{230945 a^{5} x^{19}}\) | \(116\) |
risch | \(-\frac {\left (128 b^{9} x^{18}-64 a \,x^{16} b^{8}+48 a^{2} x^{14} b^{7}-40 a^{3} x^{12} b^{6}+35 a^{4} x^{10} b^{5}+23063 a^{5} x^{8} b^{4}+75086 a^{6} x^{6} b^{3}+95238 a^{7} x^{4} b^{2}+55055 a^{8} b \,x^{2}+12155 a^{9}\right ) \sqrt {b \,x^{2}+a}}{230945 a^{5} x^{19}}\) | \(116\) |
-1/230945*(b*x^2+a)^(11/2)*(128*b^4*x^8-704*a*b^3*x^6+2288*a^2*b^2*x^4-572 0*a^3*b*x^2+12155*a^4)/x^19/a^5
Time = 0.39 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=-\frac {{\left (128 \, b^{9} x^{18} - 64 \, a b^{8} x^{16} + 48 \, a^{2} b^{7} x^{14} - 40 \, a^{3} b^{6} x^{12} + 35 \, a^{4} b^{5} x^{10} + 23063 \, a^{5} b^{4} x^{8} + 75086 \, a^{6} b^{3} x^{6} + 95238 \, a^{7} b^{2} x^{4} + 55055 \, a^{8} b x^{2} + 12155 \, a^{9}\right )} \sqrt {b x^{2} + a}}{230945 \, a^{5} x^{19}} \]
-1/230945*(128*b^9*x^18 - 64*a*b^8*x^16 + 48*a^2*b^7*x^14 - 40*a^3*b^6*x^1 2 + 35*a^4*b^5*x^10 + 23063*a^5*b^4*x^8 + 75086*a^6*b^3*x^6 + 95238*a^7*b^ 2*x^4 + 55055*a^8*b*x^2 + 12155*a^9)*sqrt(b*x^2 + a)/(a^5*x^19)
Leaf count of result is larger than twice the leaf count of optimal. 1182 vs. \(2 (109) = 218\).
Time = 2.99 (sec) , antiderivative size = 1182, normalized size of antiderivative = 10.19 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\text {Too large to display} \]
-12155*a**13*b**(33/2)*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923 780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 103675*a**12*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**1 8*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 388388*a**1 1*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a* *8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 2309 45*a**5*b**20*x**26) - 834988*a**10*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(2 30945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**2 2 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 1127210*a**9*b**( 41/2)*x**8*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b** 17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a** 5*b**20*x**26) - 978810*a**8*b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(230945* a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 92 3780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 534060*a**7*b**(45/2)*x **12*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x** 20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**2 0*x**26) - 167436*a**6*b**(47/2)*x**14*sqrt(a/(b*x**2) + 1)/(230945*a**9*b **16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a **6*b**19*x**24 + 230945*a**5*b**20*x**26) - 23091*a**5*b**(49/2)*x**16...
Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=-\frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{230945 \, a^{5} x^{11}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{20995 \, a^{4} x^{13}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{1615 \, a^{3} x^{15}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{323 \, a^{2} x^{17}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{19 \, a x^{19}} \]
-128/230945*(b*x^2 + a)^(11/2)*b^4/(a^5*x^11) + 64/20995*(b*x^2 + a)^(11/2 )*b^3/(a^4*x^13) - 16/1615*(b*x^2 + a)^(11/2)*b^2/(a^3*x^15) + 8/323*(b*x^ 2 + a)^(11/2)*b/(a^2*x^17) - 1/19*(b*x^2 + a)^(11/2)/(a*x^19)
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (96) = 192\).
Time = 0.31 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.52 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\frac {256 \, {\left (92378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{28} b^{\frac {19}{2}} + 554268 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} a b^{\frac {19}{2}} + 1939938 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a^{2} b^{\frac {19}{2}} + 4018443 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{3} b^{\frac {19}{2}} + 5866003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{4} b^{\frac {19}{2}} + 5773625 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{5} b^{\frac {19}{2}} + 4094025 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{6} b^{\frac {19}{2}} + 1889550 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{7} b^{\frac {19}{2}} + 581400 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{8} b^{\frac {19}{2}} + 80750 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{9} b^{\frac {19}{2}} + 3876 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{10} b^{\frac {19}{2}} - 969 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{11} b^{\frac {19}{2}} + 171 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{12} b^{\frac {19}{2}} - 19 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{13} b^{\frac {19}{2}} + a^{14} b^{\frac {19}{2}}\right )}}{230945 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{19}} \]
256/230945*(92378*(sqrt(b)*x - sqrt(b*x^2 + a))^28*b^(19/2) + 554268*(sqrt (b)*x - sqrt(b*x^2 + a))^26*a*b^(19/2) + 1939938*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a^2*b^(19/2) + 4018443*(sqrt(b)*x - sqrt(b*x^2 + a))^22*a^3*b^(19/ 2) + 5866003*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a^4*b^(19/2) + 5773625*(sqrt (b)*x - sqrt(b*x^2 + a))^18*a^5*b^(19/2) + 4094025*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^6*b^(19/2) + 1889550*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^7*b^(1 9/2) + 581400*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^8*b^(19/2) + 80750*(sqrt( b)*x - sqrt(b*x^2 + a))^10*a^9*b^(19/2) + 3876*(sqrt(b)*x - sqrt(b*x^2 + a ))^8*a^10*b^(19/2) - 969*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^11*b^(19/2) + 1 71*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^12*b^(19/2) - 19*(sqrt(b)*x - sqrt(b* x^2 + a))^2*a^13*b^(19/2) + a^14*b^(19/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^ 2 - a)^19
Time = 9.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\frac {8\,b^6\,\sqrt {b\,x^2+a}}{46189\,a^2\,x^7}-\frac {23063\,b^4\,\sqrt {b\,x^2+a}}{230945\,x^{11}}-\frac {6826\,a\,b^3\,\sqrt {b\,x^2+a}}{20995\,x^{13}}-\frac {77\,a^3\,b\,\sqrt {b\,x^2+a}}{323\,x^{17}}-\frac {7\,b^5\,\sqrt {b\,x^2+a}}{46189\,a\,x^9}-\frac {a^4\,\sqrt {b\,x^2+a}}{19\,x^{19}}-\frac {48\,b^7\,\sqrt {b\,x^2+a}}{230945\,a^3\,x^5}+\frac {64\,b^8\,\sqrt {b\,x^2+a}}{230945\,a^4\,x^3}-\frac {128\,b^9\,\sqrt {b\,x^2+a}}{230945\,a^5\,x}-\frac {666\,a^2\,b^2\,\sqrt {b\,x^2+a}}{1615\,x^{15}} \]
(8*b^6*(a + b*x^2)^(1/2))/(46189*a^2*x^7) - (23063*b^4*(a + b*x^2)^(1/2))/ (230945*x^11) - (6826*a*b^3*(a + b*x^2)^(1/2))/(20995*x^13) - (77*a^3*b*(a + b*x^2)^(1/2))/(323*x^17) - (7*b^5*(a + b*x^2)^(1/2))/(46189*a*x^9) - (a ^4*(a + b*x^2)^(1/2))/(19*x^19) - (48*b^7*(a + b*x^2)^(1/2))/(230945*a^3*x ^5) + (64*b^8*(a + b*x^2)^(1/2))/(230945*a^4*x^3) - (128*b^9*(a + b*x^2)^( 1/2))/(230945*a^5*x) - (666*a^2*b^2*(a + b*x^2)^(1/2))/(1615*x^15)