\(\int \frac {x^{20}}{(a+b x^2)^{10}} \, dx\) [211]

Optimal result

Integrand size = 13, antiderivative size = 211 \[ \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx=\frac {x}{b^{10}}+\frac {a^9 x}{18 b^{10} \left (a+b x^2\right )^9}-\frac {163 a^8 x}{288 b^{10} \left (a+b x^2\right )^8}+\frac {3505 a^7 x}{1344 b^{10} \left (a+b x^2\right )^7}-\frac {115715 a^6 x}{16128 b^{10} \left (a+b x^2\right )^6}+\frac {422803 a^5 x}{32256 b^{10} \left (a+b x^2\right )^5}-\frac {480365 a^4 x}{28672 b^{10} \left (a+b x^2\right )^4}+\frac {379795 a^3 x}{24576 b^{10} \left (a+b x^2\right )^3}-\frac {1050145 a^2 x}{98304 b^{10} \left (a+b x^2\right )^2}+\frac {424415 a x}{65536 b^{10} \left (a+b x^2\right )}-\frac {230945 \sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{21/2}} \] Output:

x/b^10+1/18*a^9*x/b^10/(b*x^2+a)^9-163/288*a^8*x/b^10/(b*x^2+a)^8+3505/134 
4*a^7*x/b^10/(b*x^2+a)^7-115715/16128*a^6*x/b^10/(b*x^2+a)^6+422803/32256* 
a^5*x/b^10/(b*x^2+a)^5-480365/28672*a^4*x/b^10/(b*x^2+a)^4+379795/24576*a^ 
3*x/b^10/(b*x^2+a)^3-1050145/98304*a^2*x/b^10/(b*x^2+a)^2+424415/65536*a*x 
/b^10/(b*x^2+a)-230945/65536*a^(1/2)*arctan(b^(1/2)*x/a^(1/2))/b^(21/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.68 \[ \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {b} x \left (14549535 a^9+126095970 a^8 b x^2+483044562 a^7 b^2 x^4+1071677178 a^6 b^3 x^6+1513521152 a^5 b^4 x^8+1404993798 a^4 b^5 x^{10}[phone] a^3 b^6 x^{12}[phone] a^2 b^7 x^{14}[phone] a b^8 x^{16}+4128768 b^9 x^{18}\right )}{\left (a+b x^2\right )^9}-14549535 \sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4128768 b^{21/2}} \] Input:

Integrate[x^20/(a + b*x^2)^10,x]
 

Output:

((Sqrt[b]*x*(14549535*a^9 + 126095970*a^8*b*x^2 + 483044562*a^7*b^2*x^4 + 
1071677178*a^6*b^3*x^6 + 1513521152*a^5*b^4*x^8 + 1404993798*a^4*b^5*x^10 
+ 850547502*a^3*b^6*x^12 + 318434718*a^2*b^7*x^14 + 63897057*a*b^8*x^16 + 
4128768*b^9*x^18))/(a + b*x^2)^9 - 14549535*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqr 
t[a]])/(4128768*b^(21/2))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.30, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {252, 252, 252, 252, 252, 252, 252, 252, 252, 262, 218}

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.

\(\displaystyle \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \int \frac {x^{18}}{\left (b x^2+a\right )^9}dx}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \left (\frac {17 \int \frac {x^{16}}{\left (b x^2+a\right )^8}dx}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \int \frac {x^{14}}{\left (b x^2+a\right )^7}dx}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \left (\frac {13 \int \frac {x^{12}}{\left (b x^2+a\right )^6}dx}{12 b}-\frac {x^{13}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \int \frac {x^{10}}{\left (b x^2+a\right )^5}dx}{10 b}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{13}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \int \frac {x^8}{\left (b x^2+a\right )^4}dx}{8 b}-\frac {x^9}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{13}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \int \frac {x^6}{\left (b x^2+a\right )^3}dx}{6 b}-\frac {x^7}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^9}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{13}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \int \frac {x^4}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^5}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^7}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^9}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{13}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {x^2}{b x^2+a}dx}{2 b}-\frac {x^3}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^5}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^7}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^9}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{13}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {x}{b}-\frac {a \int \frac {1}{b x^2+a}dx}{b}\right )}{2 b}-\frac {x^3}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^5}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^7}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^9}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{13}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {19 \left (\frac {17 \left (\frac {15 \left (\frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {x}{b}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}\right )}{2 b}-\frac {x^3}{2 b \left (a+b x^2\right )}\right )}{4 b}-\frac {x^5}{4 b \left (a+b x^2\right )^2}\right )}{6 b}-\frac {x^7}{6 b \left (a+b x^2\right )^3}\right )}{8 b}-\frac {x^9}{8 b \left (a+b x^2\right )^4}\right )}{10 b}-\frac {x^{11}}{10 b \left (a+b x^2\right )^5}\right )}{12 b}-\frac {x^{13}}{12 b \left (a+b x^2\right )^6}\right )}{14 b}-\frac {x^{15}}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^{17}}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}\)

Input:

Int[x^20/(a + b*x^2)^10,x]
 

Output:

-1/18*x^19/(b*(a + b*x^2)^9) + (19*(-1/16*x^17/(b*(a + b*x^2)^8) + (17*(-1 
/14*x^15/(b*(a + b*x^2)^7) + (15*(-1/12*x^13/(b*(a + b*x^2)^6) + (13*(-1/1 
0*x^11/(b*(a + b*x^2)^5) + (11*(-1/8*x^9/(b*(a + b*x^2)^4) + (9*(-1/6*x^7/ 
(b*(a + b*x^2)^3) + (7*(-1/4*x^5/(b*(a + b*x^2)^2) + (5*(-1/2*x^3/(b*(a + 
b*x^2)) + (3*(x/b - (Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2)))/(2*b)) 
)/(4*b)))/(6*b)))/(8*b)))/(10*b)))/(12*b)))/(14*b)))/(16*b)))/(18*b)
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {x}{b^{10}}-\frac {a \left (\frac {-\frac {165409}{65536} a^{8} x -\frac {2117549}{98304} a^{7} b \,x^{3}-\frac {2654039}{32768} a^{6} b^{2} x^{5}-\frac {40270037}{229376} a^{5} b^{3} x^{7}-\frac {30313}{126} a^{4} b^{4} x^{9}-\frac {49153835}{229376} a^{3} b^{5} x^{11}-\frac {3997865}{32768} a^{2} b^{6} x^{13}-\frac {4042835}{98304} a \,b^{7} x^{15}-\frac {424415}{65536} b^{8} x^{17}}{\left (b \,x^{2}+a \right )^{9}}+\frac {230945 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}}\right )}{b^{10}}\)	\(128\)
risch	\(\frac {x}{b^{10}}+\frac {\frac {165409}{65536} a^{9} x +\frac {2117549}{98304} a^{8} b \,x^{3}+\frac {2654039}{32768} a^{7} b^{2} x^{5}+\frac {40270037}{229376} a^{6} b^{3} x^{7}+\frac {30313}{126} a^{5} b^{4} x^{9}+\frac {49153835}{229376} a^{4} b^{5} x^{11}+\frac {3997865}{32768} a^{3} b^{6} x^{13}+\frac {4042835}{98304} a^{2} b^{7} x^{15}+\frac {424415}{65536} a \,b^{8} x^{17}}{b^{10} \left (b \,x^{2}+a \right )^{9}}+\frac {230945 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x -a \right )}{131072 b^{11}}-\frac {230945 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x -a \right )}{131072 b^{11}}\)	\(160\)

Input:

int(x^20/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
 

Output:

x/b^10-1/b^10*a*((-165409/65536*a^8*x-2117549/98304*a^7*b*x^3-2654039/3276 
8*a^6*b^2*x^5-40270037/229376*a^5*b^3*x^7-30313/126*a^4*b^4*x^9-49153835/2 
29376*a^3*b^5*x^11-3997865/32768*a^2*b^6*x^13-4042835/98304*a*b^7*x^15-424 
415/65536*b^8*x^17)/(b*x^2+a)^9+230945/65536/(a*b)^(1/2)*arctan(b*x/(a*b)^ 
(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 664, normalized size of antiderivative = 3.15 \[ \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {8257536 \, b^{9} x^{19} + 127794114 \, a b^{8} x^{17} + 636869436 \, a^{2} b^{7} x^{15} + 1701095004 \, a^{3} b^{6} x^{13} + 2809987596 \, a^{4} b^{5} x^{11} + 3027042304 \, a^{5} b^{4} x^{9} + 2143354356 \, a^{6} b^{3} x^{7} + 966089124 \, a^{7} b^{2} x^{5} + 252191940 \, a^{8} b x^{3} + 29099070 \, a^{9} x + 14549535 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{8257536 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}}, \frac {4128768 \, b^{9} x^{19} + 63897057 \, a b^{8} x^{17} + 318434718 \, a^{2} b^{7} x^{15} + 850547502 \, a^{3} b^{6} x^{13} + 1404993798 \, a^{4} b^{5} x^{11} + 1513521152 \, a^{5} b^{4} x^{9} + 1071677178 \, a^{6} b^{3} x^{7} + 483044562 \, a^{7} b^{2} x^{5} + 126095970 \, a^{8} b x^{3} + 14549535 \, a^{9} x - 14549535 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{4128768 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}}\right ] \] Input:

integrate(x^20/(b*x^2+a)^10,x, algorithm="fricas")
 

Output:

[1/8257536*(8257536*b^9*x^19 + 127794114*a*b^8*x^17 + 636869436*a^2*b^7*x^ 
15 + 1701095004*a^3*b^6*x^13 + 2809987596*a^4*b^5*x^11 + 3027042304*a^5*b^ 
4*x^9 + 2143354356*a^6*b^3*x^7 + 966089124*a^7*b^2*x^5 + 252191940*a^8*b*x 
^3 + 29099070*a^9*x + 14549535*(b^9*x^18 + 9*a*b^8*x^16 + 36*a^2*b^7*x^14 
+ 84*a^3*b^6*x^12 + 126*a^4*b^5*x^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 
36*a^7*b^2*x^4 + 9*a^8*b*x^2 + a^9)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/ 
b) - a)/(b*x^2 + a)))/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2*b^17*x^14 + 84*a 
^3*b^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6 + 36 
*a^7*b^12*x^4 + 9*a^8*b^11*x^2 + a^9*b^10), 1/4128768*(4128768*b^9*x^19 + 
63897057*a*b^8*x^17 + 318434718*a^2*b^7*x^15 + 850547502*a^3*b^6*x^13 + 14 
04993798*a^4*b^5*x^11 + 1513521152*a^5*b^4*x^9 + 1071677178*a^6*b^3*x^7 + 
483044562*a^7*b^2*x^5 + 126095970*a^8*b*x^3 + 14549535*a^9*x - 14549535*(b 
^9*x^18 + 9*a*b^8*x^16 + 36*a^2*b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x 
^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 36*a^7*b^2*x^4 + 9*a^8*b*x^2 + a^ 
9)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2* 
b^17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a 
^6*b^13*x^6 + 36*a^7*b^12*x^4 + 9*a^8*b^11*x^2 + a^9*b^10)]
 

Sympy [A] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.30 \[ \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx=\frac {230945 \sqrt {- \frac {a}{b^{21}}} \log {\left (- b^{10} \sqrt {- \frac {a}{b^{21}}} + x \right )}}{131072} - \frac {230945 \sqrt {- \frac {a}{b^{21}}} \log {\left (b^{10} \sqrt {- \frac {a}{b^{21}}} + x \right )}}{131072} + \frac {10420767 a^{9} x + 88937058 a^{8} b x^{3} + 334408914 a^{7} b^{2} x^{5} + 724860666 a^{6} b^{3} x^{7} + 993296384 a^{5} b^{4} x^{9} + 884769030 a^{4} b^{5} x^{11} + 503730990 a^{3} b^{6} x^{13} + 169799070 a^{2} b^{7} x^{15} + 26738145 a b^{8} x^{17}}{4128768 a^{9} b^{10} + 37158912 a^{8} b^{11} x^{2} + 148635648 a^{7} b^{12} x^{4} + 346816512 a^{6} b^{13} x^{6} + 520224768 a^{5} b^{14} x^{8} + 520224768 a^{4} b^{15} x^{10} + 346816512 a^{3} b^{16} x^{12} + 148635648 a^{2} b^{17} x^{14} + 37158912 a b^{18} x^{16} + 4128768 b^{19} x^{18}} + \frac {x}{b^{10}} \] Input:

integrate(x**20/(b*x**2+a)**10,x)
 

Output:

230945*sqrt(-a/b**21)*log(-b**10*sqrt(-a/b**21) + x)/131072 - 230945*sqrt( 
-a/b**21)*log(b**10*sqrt(-a/b**21) + x)/131072 + (10420767*a**9*x + 889370 
58*a**8*b*x**3 + 334408914*a**7*b**2*x**5 + 724860666*a**6*b**3*x**7 + 993 
296384*a**5*b**4*x**9 + 884769030*a**4*b**5*x**11 + 503730990*a**3*b**6*x* 
*13 + 169799070*a**2*b**7*x**15 + 26738145*a*b**8*x**17)/(4128768*a**9*b** 
10 + 37158912*a**8*b**11*x**2 + 148635648*a**7*b**12*x**4 + 346816512*a**6 
*b**13*x**6 + 520224768*a**5*b**14*x**8 + 520224768*a**4*b**15*x**10 + 346 
816512*a**3*b**16*x**12 + 148635648*a**2*b**17*x**14 + 37158912*a*b**18*x* 
*16 + 4128768*b**19*x**18) + x/b**10
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.05 \[ \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx=\frac {26738145 \, a b^{8} x^{17} + 169799070 \, a^{2} b^{7} x^{15} + 503730990 \, a^{3} b^{6} x^{13} + 884769030 \, a^{4} b^{5} x^{11} + 993296384 \, a^{5} b^{4} x^{9} + 724860666 \, a^{6} b^{3} x^{7} + 334408914 \, a^{7} b^{2} x^{5} + 88937058 \, a^{8} b x^{3} + 10420767 \, a^{9} x}{4128768 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}} - \frac {230945 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{10}} + \frac {x}{b^{10}} \] Input:

integrate(x^20/(b*x^2+a)^10,x, algorithm="maxima")
 

Output:

1/4128768*(26738145*a*b^8*x^17 + 169799070*a^2*b^7*x^15 + 503730990*a^3*b^ 
6*x^13 + 884769030*a^4*b^5*x^11 + 993296384*a^5*b^4*x^9 + 724860666*a^6*b^ 
3*x^7 + 334408914*a^7*b^2*x^5 + 88937058*a^8*b*x^3 + 10420767*a^9*x)/(b^19 
*x^18 + 9*a*b^18*x^16 + 36*a^2*b^17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15 
*x^10 + 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6 + 36*a^7*b^12*x^4 + 9*a^8*b^11* 
x^2 + a^9*b^10) - 230945/65536*a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^10) + 
x/b^10
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.62 \[ \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {230945 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{10}} + \frac {x}{b^{10}} + \frac {26738145 \, a b^{8} x^{17} + 169799070 \, a^{2} b^{7} x^{15} + 503730990 \, a^{3} b^{6} x^{13} + 884769030 \, a^{4} b^{5} x^{11} + 993296384 \, a^{5} b^{4} x^{9} + 724860666 \, a^{6} b^{3} x^{7} + 334408914 \, a^{7} b^{2} x^{5} + 88937058 \, a^{8} b x^{3} + 10420767 \, a^{9} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} b^{10}} \] Input:

integrate(x^20/(b*x^2+a)^10,x, algorithm="giac")
 

Output:

-230945/65536*a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^10) + x/b^10 + 1/412876 
8*(26738145*a*b^8*x^17 + 169799070*a^2*b^7*x^15 + 503730990*a^3*b^6*x^13 + 
 884769030*a^4*b^5*x^11 + 993296384*a^5*b^4*x^9 + 724860666*a^6*b^3*x^7 + 
334408914*a^7*b^2*x^5 + 88937058*a^8*b*x^3 + 10420767*a^9*x)/((b*x^2 + a)^ 
9*b^10)
 

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.03 \[ \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {165409\,a^9\,x}{65536}+\frac {2117549\,a^8\,b\,x^3}{98304}+\frac {2654039\,a^7\,b^2\,x^5}{32768}+\frac {40270037\,a^6\,b^3\,x^7}{229376}+\frac {30313\,a^5\,b^4\,x^9}{126}+\frac {49153835\,a^4\,b^5\,x^{11}}{229376}+\frac {3997865\,a^3\,b^6\,x^{13}}{32768}+\frac {4042835\,a^2\,b^7\,x^{15}}{98304}+\frac {424415\,a\,b^8\,x^{17}}{65536}}{a^9\,b^{10}+9\,a^8\,b^{11}\,x^2+36\,a^7\,b^{12}\,x^4+84\,a^6\,b^{13}\,x^6+126\,a^5\,b^{14}\,x^8+126\,a^4\,b^{15}\,x^{10}+84\,a^3\,b^{16}\,x^{12}+36\,a^2\,b^{17}\,x^{14}+9\,a\,b^{18}\,x^{16}+b^{19}\,x^{18}}+\frac {x}{b^{10}}-\frac {230945\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,b^{21/2}} \] Input:

int(x^20/(a + b*x^2)^10,x)
 

Output:

((165409*a^9*x)/65536 + (2117549*a^8*b*x^3)/98304 + (424415*a*b^8*x^17)/65 
536 + (2654039*a^7*b^2*x^5)/32768 + (40270037*a^6*b^3*x^7)/229376 + (30313 
*a^5*b^4*x^9)/126 + (49153835*a^4*b^5*x^11)/229376 + (3997865*a^3*b^6*x^13 
)/32768 + (4042835*a^2*b^7*x^15)/98304)/(a^9*b^10 + b^19*x^18 + 9*a*b^18*x 
^16 + 9*a^8*b^11*x^2 + 36*a^7*b^12*x^4 + 84*a^6*b^13*x^6 + 126*a^5*b^14*x^ 
8 + 126*a^4*b^15*x^10 + 84*a^3*b^16*x^12 + 36*a^2*b^17*x^14) + x/b^10 - (2 
30945*a^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/(65536*b^(21/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.18 \[ \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx=\frac {-14549535 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{9}-130945815 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{8} b \,x^{2}-523783260 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{7} b^{2} x^{4}-1222160940 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6} b^{3} x^{6}-1833241410 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{4} x^{8}-1833241410 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{5} x^{10}-1222160940 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{6} x^{12}-523783260 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{7} x^{14}-130945815 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{8} x^{16}-14549535 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{9} x^{18}[phone] a^{9} b x [phone] a^{8} b^{2} x^{3}[phone] a^{7} b^{3} x^{5}[phone] a^{6} b^{4} x^{7}[phone] a^{5} b^{5} x^{9}[phone] a^{4} b^{6} x^{11}[phone] a^{3} b^{7} x^{13}[phone] a^{2} b^{8} x^{15}[phone] a \,b^{9} x^{17}+4128768 b^{10} x^{19}}{4128768 b^{11} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:

int(x^20/(b*x^2+a)^10,x)
 

Output:

( - 14549535*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**9 - 13094581 
5*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**8*b*x**2 - 523783260*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**7*b**2*x**4 - 1222160940*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**6*b**3*x**6 - 1833241410*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**4*x**8 - 1833241410*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b**5*x**10 - 1222160940*s 
qrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**6*x**12 - 523783260*s 
qrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**7*x**14 - 130945815*s 
qrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**8*x**16 - 14549535*sqrt( 
b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**9*x**18 + 14549535*a**9*b*x + 
126095970*a**8*b**2*x**3 + 483044562*a**7*b**3*x**5 + 1071677178*a**6*b**4 
*x**7 + 1513521152*a**5*b**5*x**9 + 1404993798*a**4*b**6*x**11 + 850547502 
*a**3*b**7*x**13 + 318434718*a**2*b**8*x**15 + 63897057*a*b**9*x**17 + 412 
8768*b**10*x**19)/(4128768*b**11*(a**9 + 9*a**8*b*x**2 + 36*a**7*b**2*x**4 
 + 84*a**6*b**3*x**6 + 126*a**5*b**4*x**8 + 126*a**4*b**5*x**10 + 84*a**3* 
b**6*x**12 + 36*a**2*b**7*x**14 + 9*a*b**8*x**16 + b**9*x**18))