\(\int \frac {(a+b x^2)^{9/2}}{x^{20}} \, dx\) [448]

Optimal result

Integrand size = 15, antiderivative size = 116 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac {8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac {16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac {64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}-\frac {128 b^4 \left (a+b x^2\right )^{11/2}}{230945 a^5 x^{11}} \] Output:

-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
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.55 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\frac {\left (a+b x^2\right )^{11/2} \left (-12155 a^4+5720 a^3 b x^2-2288 a^2 b^2 x^4+704 a b^3 x^6-128 b^4 x^8\right )}{230945 a^5 x^{19}} \] Input:

Integrate[(a + b*x^2)^(9/2)/x^20,x]
 

Output:

((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)
 

Rubi [A] (verified)

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

\(\displaystyle -\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\right )}{19 a}-\frac {\left (a+b x^2\right )^{11/2}}{19 a x^{19}}\)

Input:

Int[(a + b*x^2)^(9/2)/x^20,x]
 

Output:

-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)
 

Defintions of rubi rules used

rule 242

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]
 

rule 245

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]
 

Maple [A] (verified)

Time = 5.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.53

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (128 b^{4} x^{8}-704 a \,b^{3} x^{6}+2288 a^{2} b^{2} x^{4}-5720 a^{3} b \,x^{2}+12155 a^{4}\right )}{230945 x^{19} a^{5}}\)	\(61\)
pseudoelliptic	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (128 b^{4} x^{8}-704 a \,b^{3} x^{6}+2288 a^{2} b^{2} x^{4}-5720 a^{3} b \,x^{2}+12155 a^{4}\right )}{230945 x^{19} a^{5}}\)	\(61\)
orering	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (128 b^{4} x^{8}-704 a \,b^{3} x^{6}+2288 a^{2} b^{2} x^{4}-5720 a^{3} b \,x^{2}+12155 a^{4}\right )}{230945 x^{19} a^{5}}\)	\(61\)
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	\(-\frac {\left (128 b^{9} x^{18}-64 a \,b^{8} x^{16}+48 a^{2} b^{7} x^{14}-40 a^{3} b^{6} x^{12}+35 a^{4} b^{5} x^{10}+23063 a^{5} b^{4} x^{8}+75086 a^{6} b^{3} x^{6}+95238 a^{7} b^{2} x^{4}+55055 a^{8} b \,x^{2}+12155 a^{9}\right ) \sqrt {b \,x^{2}+a}}{230945 x^{19} a^{5}}\)	\(116\)
risch	\(-\frac {\left (128 b^{9} x^{18}-64 a \,b^{8} x^{16}+48 a^{2} b^{7} x^{14}-40 a^{3} b^{6} x^{12}+35 a^{4} b^{5} x^{10}+23063 a^{5} b^{4} x^{8}+75086 a^{6} b^{3} x^{6}+95238 a^{7} b^{2} x^{4}+55055 a^{8} b \,x^{2}+12155 a^{9}\right ) \sqrt {b \,x^{2}+a}}{230945 x^{19} a^{5}}\)	\(116\)

Input:

int((b*x^2+a)^(9/2)/x^20,x,method=_RETURNVERBOSE)
 

Output:

-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
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=-\frac {{\left (128 \, b^{9} x^{18} - 64 \, a b^{8} x^{16} + 48 \, a^{2} b^{7} x^{14} - 40 \, a^{3} b^{6} x^{12} + 35 \, a^{4} b^{5} x^{10} + 23063 \, a^{5} b^{4} x^{8} + 75086 \, a^{6} b^{3} x^{6} + 95238 \, a^{7} b^{2} x^{4} + 55055 \, a^{8} b x^{2} + 12155 \, a^{9}\right )} \sqrt {b x^{2} + a}}{230945 \, a^{5} x^{19}} \] Input:

integrate((b*x^2+a)^(9/2)/x^20,x, algorithm="fricas")
 

Output:

-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)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1182 vs. \(2 (109) = 218\).

Time = 3.22 (sec) , antiderivative size = 1182, normalized size of antiderivative = 10.19 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\text {Too large to display} \] Input:

integrate((b*x**2+a)**(9/2)/x**20,x)
 

Output:

-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...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=-\frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{230945 \, a^{5} x^{11}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{20995 \, a^{4} x^{13}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{1615 \, a^{3} x^{15}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{323 \, a^{2} x^{17}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{19 \, a x^{19}} \] Input:

integrate((b*x^2+a)^(9/2)/x^20,x, algorithm="maxima")
 

Output:

-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)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (96) = 192\).

Time = 0.14 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.52 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\frac {256 \, {\left (92378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{28} b^{\frac {19}{2}} + 554268 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} a b^{\frac {19}{2}} + 1939938 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a^{2} b^{\frac {19}{2}} + 4018443 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{3} b^{\frac {19}{2}} + 5866003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{4} b^{\frac {19}{2}} + 5773625 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{5} b^{\frac {19}{2}} + 4094025 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{6} b^{\frac {19}{2}} + 1889550 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{7} b^{\frac {19}{2}} + 581400 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{8} b^{\frac {19}{2}} + 80750 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{9} b^{\frac {19}{2}} + 3876 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{10} b^{\frac {19}{2}} - 969 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{11} b^{\frac {19}{2}} + 171 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{12} b^{\frac {19}{2}} - 19 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{13} b^{\frac {19}{2}} + a^{14} b^{\frac {19}{2}}\right )}}{230945 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{19}} \] Input:

integrate((b*x^2+a)^(9/2)/x^20,x, algorithm="giac")
 

Output:

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
 

Mupad [B] (verification not implemented)

Time = 4.92 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\frac {8\,b^6\,\sqrt {b\,x^2+a}}{46189\,a^2\,x^7}-\frac {23063\,b^4\,\sqrt {b\,x^2+a}}{230945\,x^{11}}-\frac {6826\,a\,b^3\,\sqrt {b\,x^2+a}}{20995\,x^{13}}-\frac {77\,a^3\,b\,\sqrt {b\,x^2+a}}{323\,x^{17}}-\frac {7\,b^5\,\sqrt {b\,x^2+a}}{46189\,a\,x^9}-\frac {a^4\,\sqrt {b\,x^2+a}}{19\,x^{19}}-\frac {48\,b^7\,\sqrt {b\,x^2+a}}{230945\,a^3\,x^5}+\frac {64\,b^8\,\sqrt {b\,x^2+a}}{230945\,a^4\,x^3}-\frac {128\,b^9\,\sqrt {b\,x^2+a}}{230945\,a^5\,x}-\frac {666\,a^2\,b^2\,\sqrt {b\,x^2+a}}{1615\,x^{15}} \] Input:

int((a + b*x^2)^(9/2)/x^20,x)
 

Output:

(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)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx=\frac {-12155 \sqrt {b \,x^{2}+a}\, a^{9}-55055 \sqrt {b \,x^{2}+a}\, a^{8} b \,x^{2}-95238 \sqrt {b \,x^{2}+a}\, a^{7} b^{2} x^{4}-75086 \sqrt {b \,x^{2}+a}\, a^{6} b^{3} x^{6}-23063 \sqrt {b \,x^{2}+a}\, a^{5} b^{4} x^{8}-35 \sqrt {b \,x^{2}+a}\, a^{4} b^{5} x^{10}+40 \sqrt {b \,x^{2}+a}\, a^{3} b^{6} x^{12}-48 \sqrt {b \,x^{2}+a}\, a^{2} b^{7} x^{14}+64 \sqrt {b \,x^{2}+a}\, a \,b^{8} x^{16}-128 \sqrt {b \,x^{2}+a}\, b^{9} x^{18}+128 \sqrt {b}\, b^{9} x^{19}}{230945 a^{5} x^{19}} \] Input:

int((b*x^2+a)^(9/2)/x^20,x)
 

Output:

( - 12155*sqrt(a + b*x**2)*a**9 - 55055*sqrt(a + b*x**2)*a**8*b*x**2 - 952 
38*sqrt(a + b*x**2)*a**7*b**2*x**4 - 75086*sqrt(a + b*x**2)*a**6*b**3*x**6 
 - 23063*sqrt(a + b*x**2)*a**5*b**4*x**8 - 35*sqrt(a + b*x**2)*a**4*b**5*x 
**10 + 40*sqrt(a + b*x**2)*a**3*b**6*x**12 - 48*sqrt(a + b*x**2)*a**2*b**7 
*x**14 + 64*sqrt(a + b*x**2)*a*b**8*x**16 - 128*sqrt(a + b*x**2)*b**9*x**1 
8 + 128*sqrt(b)*b**9*x**19)/(230945*a**5*x**19)