Integrand size = 15, antiderivative size = 116 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx=-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac {8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac {64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{45045 a^5 x^7} \]
-1/15*(b*x^2+a)^(7/2)/a/x^15+8/195*b*(b*x^2+a)^(7/2)/a^2/x^13-16/715*b^2*( b*x^2+a)^(7/2)/a^3/x^11+64/6435*b^3*(b*x^2+a)^(7/2)/a^4/x^9-128/45045*b^4* (b*x^2+a)^(7/2)/a^5/x^7
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.55 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx=\frac {\left (a+b x^2\right )^{7/2} \left (-3003 a^4+1848 a^3 b x^2-1008 a^2 b^2 x^4+448 a b^3 x^6-128 b^4 x^8\right )}{45045 a^5 x^{15}} \]
((a + b*x^2)^(7/2)*(-3003*a^4 + 1848*a^3*b*x^2 - 1008*a^2*b^2*x^4 + 448*a* b^3*x^6 - 128*b^4*x^8))/(45045*a^5*x^15)
Time = 0.22 (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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {8 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{14}}dx}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {8 b \left (-\frac {6 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{12}}dx}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{10}}dx}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^8}dx}{9 a}-\frac {\left (a+b x^2\right )^{7/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac {\left (a+b x^2\right )^{7/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
-1/15*(a + b*x^2)^(7/2)/(a*x^15) - (8*b*(-1/13*(a + b*x^2)^(7/2)/(a*x^13) - (6*b*(-1/11*(a + b*x^2)^(7/2)/(a*x^11) - (4*b*(-1/9*(a + b*x^2)^(7/2)/(a *x^9) + (2*b*(a + b*x^2)^(7/2))/(63*a^2*x^7)))/(11*a)))/(13*a)))/(15*a)
3.5.7.3.1 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 = 2.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.53
| method | result | size |
| gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (128 x^{8} b^{4}-448 a \,b^{3} x^{6}+1008 a^{2} x^{4} b^{2}-1848 a^{3} b \,x^{2}+3003 a^{4}\right )}{45045 x^{15} a^{5}}\) | \(61\) |
| pseudoelliptic | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (128 x^{8} b^{4}-448 a \,b^{3} x^{6}+1008 a^{2} x^{4} b^{2}-1848 a^{3} b \,x^{2}+3003 a^{4}\right )}{45045 x^{15} a^{5}}\) | \(61\) |
| trager | \(-\frac {\left (128 b^{7} x^{14}-64 a \,b^{6} x^{12}+48 a^{2} b^{5} x^{10}-40 a^{3} b^{4} x^{8}+35 a^{4} b^{3} x^{6}+4473 a^{5} b^{2} x^{4}+7161 a^{6} b \,x^{2}+3003 a^{7}\right ) \sqrt {b \,x^{2}+a}}{45045 x^{15} a^{5}}\) | \(94\) |
| risch | \(-\frac {\left (128 b^{7} x^{14}-64 a \,b^{6} x^{12}+48 a^{2} b^{5} x^{10}-40 a^{3} b^{4} x^{8}+35 a^{4} b^{3} x^{6}+4473 a^{5} b^{2} x^{4}+7161 a^{6} b \,x^{2}+3003 a^{7}\right ) \sqrt {b \,x^{2}+a}}{45045 x^{15} a^{5}}\) | \(94\) |
| default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{15 a \,x^{15}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 a \,x^{13}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 a \,x^{11}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\right )}{15 a}\) | \(109\) |
-1/45045*(b*x^2+a)^(7/2)*(128*b^4*x^8-448*a*b^3*x^6+1008*a^2*b^2*x^4-1848* a^3*b*x^2+3003*a^4)/x^15/a^5
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx=-\frac {{\left (128 \, b^{7} x^{14} - 64 \, a b^{6} x^{12} + 48 \, a^{2} b^{5} x^{10} - 40 \, a^{3} b^{4} x^{8} + 35 \, a^{4} b^{3} x^{6} + 4473 \, a^{5} b^{2} x^{4} + 7161 \, a^{6} b x^{2} + 3003 \, a^{7}\right )} \sqrt {b x^{2} + a}}{45045 \, a^{5} x^{15}} \]
-1/45045*(128*b^7*x^14 - 64*a*b^6*x^12 + 48*a^2*b^5*x^10 - 40*a^3*b^4*x^8 + 35*a^4*b^3*x^6 + 4473*a^5*b^2*x^4 + 7161*a^6*b*x^2 + 3003*a^7)*sqrt(b*x^ 2 + a)/(a^5*x^15)
Leaf count of result is larger than twice the leaf count of optimal. 1012 vs. \(2 (109) = 218\).
Time = 1.76 (sec) , antiderivative size = 1012, normalized size of antiderivative = 8.72 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx=- \frac {3003 a^{11} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {19173 a^{10} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {51135 a^{9} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {72905 a^{8} b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {58585 a^{7} b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {25151 a^{6} b^{\frac {43}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {4501 a^{5} b^{\frac {45}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {35 a^{4} b^{\frac {47}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {280 a^{3} b^{\frac {49}{2}} x^{16} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {560 a^{2} b^{\frac {51}{2}} x^{18} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {448 a b^{\frac {53}{2}} x^{20} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {128 b^{\frac {55}{2}} x^{22} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} \]
-3003*a**11*b**(33/2)*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 18018 0*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 4 5045*a**5*b**20*x**22) - 19173*a**10*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/( 45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 51135*a**9*b**(37/2 )*x**4*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x* *16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20 *x**22) - 72905*a**8*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16 *x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b **19*x**20 + 45045*a**5*b**20*x**22) - 58585*a**7*b**(41/2)*x**8*sqrt(a/(b *x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a** 7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 25151* a**6*b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180 *a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45 045*a**5*b**20*x**22) - 4501*a**5*b**(45/2)*x**12*sqrt(a/(b*x**2) + 1)/(45 045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 35*a**4*b**(47/2)*x** 14*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x** 22) - 280*a**3*b**(49/2)*x**16*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x...
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx=-\frac {128 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}{45045 \, a^{5} x^{7}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}}{6435 \, a^{4} x^{9}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{715 \, a^{3} x^{11}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{195 \, a^{2} x^{13}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{15 \, a x^{15}} \]
-128/45045*(b*x^2 + a)^(7/2)*b^4/(a^5*x^7) + 64/6435*(b*x^2 + a)^(7/2)*b^3 /(a^4*x^9) - 16/715*(b*x^2 + a)^(7/2)*b^2/(a^3*x^11) + 8/195*(b*x^2 + a)^( 7/2)*b/(a^2*x^13) - 1/15*(b*x^2 + a)^(7/2)/(a*x^15)
Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (96) = 192\).
Time = 0.33 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.59 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx=\frac {256 \, {\left (18018 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} b^{\frac {15}{2}} + 60060 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a b^{\frac {15}{2}} + 115830 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{2} b^{\frac {15}{2}} + 109395 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{3} b^{\frac {15}{2}} + 65065 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{4} b^{\frac {15}{2}} + 15015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{5} b^{\frac {15}{2}} + 1365 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{6} b^{\frac {15}{2}} - 455 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{7} b^{\frac {15}{2}} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{8} b^{\frac {15}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{9} b^{\frac {15}{2}} + a^{10} b^{\frac {15}{2}}\right )}}{45045 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{15}} \]
256/45045*(18018*(sqrt(b)*x - sqrt(b*x^2 + a))^20*b^(15/2) + 60060*(sqrt(b )*x - sqrt(b*x^2 + a))^18*a*b^(15/2) + 115830*(sqrt(b)*x - sqrt(b*x^2 + a) )^16*a^2*b^(15/2) + 109395*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^3*b^(15/2) + 65065*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^4*b^(15/2) + 15015*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^5*b^(15/2) + 1365*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^ 6*b^(15/2) - 455*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^7*b^(15/2) + 105*(sqrt( b)*x - sqrt(b*x^2 + a))^4*a^8*b^(15/2) - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^ 2*a^9*b^(15/2) + a^10*b^(15/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^15
Time = 6.55 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx=\frac {8\,b^4\,\sqrt {b\,x^2+a}}{9009\,a^2\,x^7}-\frac {71\,b^2\,\sqrt {b\,x^2+a}}{715\,x^{11}}-\frac {b^3\,\sqrt {b\,x^2+a}}{1287\,a\,x^9}-\frac {a^2\,\sqrt {b\,x^2+a}}{15\,x^{15}}-\frac {16\,b^5\,\sqrt {b\,x^2+a}}{15015\,a^3\,x^5}+\frac {64\,b^6\,\sqrt {b\,x^2+a}}{45045\,a^4\,x^3}-\frac {128\,b^7\,\sqrt {b\,x^2+a}}{45045\,a^5\,x}-\frac {31\,a\,b\,\sqrt {b\,x^2+a}}{195\,x^{13}} \]
(8*b^4*(a + b*x^2)^(1/2))/(9009*a^2*x^7) - (71*b^2*(a + b*x^2)^(1/2))/(715 *x^11) - (b^3*(a + b*x^2)^(1/2))/(1287*a*x^9) - (a^2*(a + b*x^2)^(1/2))/(1 5*x^15) - (16*b^5*(a + b*x^2)^(1/2))/(15015*a^3*x^5) + (64*b^6*(a + b*x^2) ^(1/2))/(45045*a^4*x^3) - (128*b^7*(a + b*x^2)^(1/2))/(45045*a^5*x) - (31* a*b*(a + b*x^2)^(1/2))/(195*x^13)