\(\int \frac {x^8}{(a+b x^2)^{10}} \, dx\) [217]

Optimal result

Integrand size = 13, antiderivative size = 202 \[ \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx=-\frac {x^7}{18 b \left (a+b x^2\right )^9}-\frac {7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac {5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac {5 x}{2304 b^4 \left (a+b x^2\right )^6}+\frac {x}{4608 a b^4 \left (a+b x^2\right )^5}+\frac {x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac {7 x}{24576 a^3 b^4 \left (a+b x^2\right )^3}+\frac {35 x}{98304 a^4 b^4 \left (a+b x^2\right )^2}+\frac {35 x}{65536 a^5 b^4 \left (a+b x^2\right )}+\frac {35 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{11/2} b^{9/2}} \] Output:

-1/18*x^7/b/(b*x^2+a)^9-7/288*x^5/b^2/(b*x^2+a)^8-5/576*x^3/b^3/(b*x^2+a)^ 
7-5/2304*x/b^4/(b*x^2+a)^6+1/4608*x/a/b^4/(b*x^2+a)^5+1/4096*x/a^2/b^4/(b* 
x^2+a)^4+7/24576*x/a^3/b^4/(b*x^2+a)^3+35/98304*x/a^4/b^4/(b*x^2+a)^2+35/6 
5536*x/a^5/b^4/(b*x^2+a)+35/65536*arctan(b^(1/2)*x/a^(1/2))/a^(11/2)/b^(9/ 
2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.68 \[ \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {a} \sqrt {b} x \left (-315 a^8-2730 a^7 b x^2-10458 a^6 b^2 x^4-23202 a^5 b^3 x^6+32768 a^4 b^4 x^8+23202 a^3 b^5 x^{10}+10458 a^2 b^6 x^{12}+2730 a b^7 x^{14}+315 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+315 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{589824 a^{11/2} b^{9/2}} \] Input:

Integrate[x^8/(a + b*x^2)^10,x]
 

Output:

((Sqrt[a]*Sqrt[b]*x*(-315*a^8 - 2730*a^7*b*x^2 - 10458*a^6*b^2*x^4 - 23202 
*a^5*b^3*x^6 + 32768*a^4*b^4*x^8 + 23202*a^3*b^5*x^10 + 10458*a^2*b^6*x^12 
 + 2730*a*b^7*x^14 + 315*b^8*x^16))/(a + b*x^2)^9 + 315*ArcTan[(Sqrt[b]*x) 
/Sqrt[a]])/(589824*a^(11/2)*b^(9/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {252, 252, 252, 252, 215, 215, 215, 215, 215, 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.

\(\displaystyle \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {7 \int \frac {x^6}{\left (b x^2+a\right )^9}dx}{18 b}-\frac {x^7}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {7 \left (\frac {5 \int \frac {x^4}{\left (b x^2+a\right )^8}dx}{16 b}-\frac {x^5}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^7}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \int \frac {x^2}{\left (b x^2+a\right )^7}dx}{14 b}-\frac {x^3}{14 b \left (a+b x^2\right )^7}\right )}{16 b}-\frac {x^5}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^7}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 218

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

Input:

Int[x^8/(a + b*x^2)^10,x]
 

Output:

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

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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]
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.60

method	result	size
default	\(\frac {-\frac {35 a^{3} x}{65536 b^{4}}-\frac {455 a^{2} x^{3}}{98304 b^{3}}-\frac {581 a \,x^{5}}{32768 b^{2}}-\frac {1289 x^{7}}{32768 b}+\frac {x^{9}}{18 a}+\frac {1289 b \,x^{11}}{32768 a^{2}}+\frac {581 b^{2} x^{13}}{32768 a^{3}}+\frac {455 b^{3} x^{15}}{98304 a^{4}}+\frac {35 b^{4} x^{17}}{65536 a^{5}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {35 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 a^{5} b^{4} \sqrt {a b}}\)	\(122\)
risch	\(\frac {-\frac {35 a^{3} x}{65536 b^{4}}-\frac {455 a^{2} x^{3}}{98304 b^{3}}-\frac {581 a \,x^{5}}{32768 b^{2}}-\frac {1289 x^{7}}{32768 b}+\frac {x^{9}}{18 a}+\frac {1289 b \,x^{11}}{32768 a^{2}}+\frac {581 b^{2} x^{13}}{32768 a^{3}}+\frac {455 b^{3} x^{15}}{98304 a^{4}}+\frac {35 b^{4} x^{17}}{65536 a^{5}}}{\left (b \,x^{2}+a \right )^{9}}-\frac {35 \ln \left (b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{4} a^{5}}+\frac {35 \ln \left (-b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{4} a^{5}}\)	\(151\)

Input:

int(x^8/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
 

Output:

(-35/65536*a^3*x/b^4-455/98304*a^2*x^3/b^3-581/32768*a/b^2*x^5-1289/32768/ 
b*x^7+1/18/a*x^9+1289/32768*b/a^2*x^11+581/32768*b^2/a^3*x^13+455/98304*b^ 
3/a^4*x^15+35/65536*b^4/a^5*x^17)/(b*x^2+a)^9+35/65536/a^5/b^4/(a*b)^(1/2) 
*arctan(b*x/(a*b)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 654, normalized size of antiderivative = 3.24 \[ \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {630 \, a b^{9} x^{17} + 5460 \, a^{2} b^{8} x^{15} + 20916 \, a^{3} b^{7} x^{13} + 46404 \, a^{4} b^{6} x^{11} + 65536 \, a^{5} b^{5} x^{9} - 46404 \, a^{6} b^{4} x^{7} - 20916 \, a^{7} b^{3} x^{5} - 5460 \, a^{8} b^{2} x^{3} - 630 \, a^{9} b x - 315 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{1179648 \, {\left (a^{6} b^{14} x^{18} + 9 \, a^{7} b^{13} x^{16} + 36 \, a^{8} b^{12} x^{14} + 84 \, a^{9} b^{11} x^{12} + 126 \, a^{10} b^{10} x^{10} + 126 \, a^{11} b^{9} x^{8} + 84 \, a^{12} b^{8} x^{6} + 36 \, a^{13} b^{7} x^{4} + 9 \, a^{14} b^{6} x^{2} + a^{15} b^{5}\right )}}, \frac {315 \, a b^{9} x^{17} + 2730 \, a^{2} b^{8} x^{15} + 10458 \, a^{3} b^{7} x^{13} + 23202 \, a^{4} b^{6} x^{11} + 32768 \, a^{5} b^{5} x^{9} - 23202 \, a^{6} b^{4} x^{7} - 10458 \, a^{7} b^{3} x^{5} - 2730 \, a^{8} b^{2} x^{3} - 315 \, a^{9} b x + 315 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{589824 \, {\left (a^{6} b^{14} x^{18} + 9 \, a^{7} b^{13} x^{16} + 36 \, a^{8} b^{12} x^{14} + 84 \, a^{9} b^{11} x^{12} + 126 \, a^{10} b^{10} x^{10} + 126 \, a^{11} b^{9} x^{8} + 84 \, a^{12} b^{8} x^{6} + 36 \, a^{13} b^{7} x^{4} + 9 \, a^{14} b^{6} x^{2} + a^{15} b^{5}\right )}}\right ] \] Input:

integrate(x^8/(b*x^2+a)^10,x, algorithm="fricas")
 

Output:

[1/1179648*(630*a*b^9*x^17 + 5460*a^2*b^8*x^15 + 20916*a^3*b^7*x^13 + 4640 
4*a^4*b^6*x^11 + 65536*a^5*b^5*x^9 - 46404*a^6*b^4*x^7 - 20916*a^7*b^3*x^5 
 - 5460*a^8*b^2*x^3 - 630*a^9*b*x - 315*(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*sqr 
t(-a*b)*x - a)/(b*x^2 + a)))/(a^6*b^14*x^18 + 9*a^7*b^13*x^16 + 36*a^8*b^1 
2*x^14 + 84*a^9*b^11*x^12 + 126*a^10*b^10*x^10 + 126*a^11*b^9*x^8 + 84*a^1 
2*b^8*x^6 + 36*a^13*b^7*x^4 + 9*a^14*b^6*x^2 + a^15*b^5), 1/589824*(315*a* 
b^9*x^17 + 2730*a^2*b^8*x^15 + 10458*a^3*b^7*x^13 + 23202*a^4*b^6*x^11 + 3 
2768*a^5*b^5*x^9 - 23202*a^6*b^4*x^7 - 10458*a^7*b^3*x^5 - 2730*a^8*b^2*x^ 
3 - 315*a^9*b*x + 315*(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(sqrt(a*b)*x/a))/(a^6*b^14*x^18 
 + 9*a^7*b^13*x^16 + 36*a^8*b^12*x^14 + 84*a^9*b^11*x^12 + 126*a^10*b^10*x 
^10 + 126*a^11*b^9*x^8 + 84*a^12*b^8*x^6 + 36*a^13*b^7*x^4 + 9*a^14*b^6*x^ 
2 + a^15*b^5)]
 

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.44 \[ \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx=- \frac {35 \sqrt {- \frac {1}{a^{11} b^{9}}} \log {\left (- a^{6} b^{4} \sqrt {- \frac {1}{a^{11} b^{9}}} + x \right )}}{131072} + \frac {35 \sqrt {- \frac {1}{a^{11} b^{9}}} \log {\left (a^{6} b^{4} \sqrt {- \frac {1}{a^{11} b^{9}}} + x \right )}}{131072} + \frac {- 315 a^{8} x - 2730 a^{7} b x^{3} - 10458 a^{6} b^{2} x^{5} - 23202 a^{5} b^{3} x^{7} + 32768 a^{4} b^{4} x^{9} + 23202 a^{3} b^{5} x^{11} + 10458 a^{2} b^{6} x^{13} + 2730 a b^{7} x^{15} + 315 b^{8} x^{17}}{589824 a^{14} b^{4} + 5308416 a^{13} b^{5} x^{2} + 21233664 a^{12} b^{6} x^{4} + 49545216 a^{11} b^{7} x^{6} + 74317824 a^{10} b^{8} x^{8} + 74317824 a^{9} b^{9} x^{10} + 49545216 a^{8} b^{10} x^{12} + 21233664 a^{7} b^{11} x^{14} + 5308416 a^{6} b^{12} x^{16} + 589824 a^{5} b^{13} x^{18}} \] Input:

integrate(x**8/(b*x**2+a)**10,x)
 

Output:

-35*sqrt(-1/(a**11*b**9))*log(-a**6*b**4*sqrt(-1/(a**11*b**9)) + x)/131072 
 + 35*sqrt(-1/(a**11*b**9))*log(a**6*b**4*sqrt(-1/(a**11*b**9)) + x)/13107 
2 + (-315*a**8*x - 2730*a**7*b*x**3 - 10458*a**6*b**2*x**5 - 23202*a**5*b* 
*3*x**7 + 32768*a**4*b**4*x**9 + 23202*a**3*b**5*x**11 + 10458*a**2*b**6*x 
**13 + 2730*a*b**7*x**15 + 315*b**8*x**17)/(589824*a**14*b**4 + 5308416*a* 
*13*b**5*x**2 + 21233664*a**12*b**6*x**4 + 49545216*a**11*b**7*x**6 + 7431 
7824*a**10*b**8*x**8 + 74317824*a**9*b**9*x**10 + 49545216*a**8*b**10*x**1 
2 + 21233664*a**7*b**11*x**14 + 5308416*a**6*b**12*x**16 + 589824*a**5*b** 
13*x**18)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.09 \[ \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx=\frac {315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} + 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x}{589824 \, {\left (a^{5} b^{13} x^{18} + 9 \, a^{6} b^{12} x^{16} + 36 \, a^{7} b^{11} x^{14} + 84 \, a^{8} b^{10} x^{12} + 126 \, a^{9} b^{9} x^{10} + 126 \, a^{10} b^{8} x^{8} + 84 \, a^{11} b^{7} x^{6} + 36 \, a^{12} b^{6} x^{4} + 9 \, a^{13} b^{5} x^{2} + a^{14} b^{4}\right )}} + \frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{5} b^{4}} \] Input:

integrate(x^8/(b*x^2+a)^10,x, algorithm="maxima")
 

Output:

1/589824*(315*b^8*x^17 + 2730*a*b^7*x^15 + 10458*a^2*b^6*x^13 + 23202*a^3* 
b^5*x^11 + 32768*a^4*b^4*x^9 - 23202*a^5*b^3*x^7 - 10458*a^6*b^2*x^5 - 273 
0*a^7*b*x^3 - 315*a^8*x)/(a^5*b^13*x^18 + 9*a^6*b^12*x^16 + 36*a^7*b^11*x^ 
14 + 84*a^8*b^10*x^12 + 126*a^9*b^9*x^10 + 126*a^10*b^8*x^8 + 84*a^11*b^7* 
x^6 + 36*a^12*b^6*x^4 + 9*a^13*b^5*x^2 + a^14*b^4) + 35/65536*arctan(b*x/s 
qrt(a*b))/(sqrt(a*b)*a^5*b^4)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.63 \[ \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx=\frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{5} b^{4}} + \frac {315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} + 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x}{589824 \, {\left (b x^{2} + a\right )}^{9} a^{5} b^{4}} \] Input:

integrate(x^8/(b*x^2+a)^10,x, algorithm="giac")
 

Output:

35/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5*b^4) + 1/589824*(315*b^8*x^1 
7 + 2730*a*b^7*x^15 + 10458*a^2*b^6*x^13 + 23202*a^3*b^5*x^11 + 32768*a^4* 
b^4*x^9 - 23202*a^5*b^3*x^7 - 10458*a^6*b^2*x^5 - 2730*a^7*b*x^3 - 315*a^8 
*x)/((b*x^2 + a)^9*a^5*b^4)
 

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.01 \[ \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {x^9}{18\,a}-\frac {1289\,x^7}{32768\,b}-\frac {581\,a\,x^5}{32768\,b^2}-\frac {35\,a^3\,x}{65536\,b^4}+\frac {1289\,b\,x^{11}}{32768\,a^2}-\frac {455\,a^2\,x^3}{98304\,b^3}+\frac {581\,b^2\,x^{13}}{32768\,a^3}+\frac {455\,b^3\,x^{15}}{98304\,a^4}+\frac {35\,b^4\,x^{17}}{65536\,a^5}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {35\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{11/2}\,b^{9/2}} \] Input:

int(x^8/(a + b*x^2)^10,x)
 

Output:

(x^9/(18*a) - (1289*x^7)/(32768*b) - (581*a*x^5)/(32768*b^2) - (35*a^3*x)/ 
(65536*b^4) + (1289*b*x^11)/(32768*a^2) - (455*a^2*x^3)/(98304*b^3) + (581 
*b^2*x^13)/(32768*a^3) + (455*b^3*x^15)/(98304*a^4) + (35*b^4*x^17)/(65536 
*a^5))/(a^9 + b^9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 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) + (35*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^(11/2)*b^(9/2))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.26 \[ \int \frac {x^8}{\left (a+b x^2\right )^{10}} \, dx=\frac {315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{9}+2835 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{8} b \,x^{2}+11340 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{7} b^{2} x^{4}+26460 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6} b^{3} x^{6}+39690 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{4} x^{8}+39690 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{5} x^{10}+26460 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{6} x^{12}+11340 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{7} x^{14}+2835 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{8} x^{16}+315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{9} x^{18}-315 a^{9} b x -2730 a^{8} b^{2} x^{3}-10458 a^{7} b^{3} x^{5}-23202 a^{6} b^{4} x^{7}+32768 a^{5} b^{5} x^{9}+23202 a^{4} b^{6} x^{11}+10458 a^{3} b^{7} x^{13}+2730 a^{2} b^{8} x^{15}+315 a \,b^{9} x^{17}}{589824 a^{6} b^{5} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:

int(x^8/(b*x^2+a)^10,x)
 

Output:

(315*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**9 + 2835*sqrt(b)*sqr 
t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**8*b*x**2 + 11340*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**7*b**2*x**4 + 26460*sqrt(b)*sqrt(a)*atan((b* 
x)/(sqrt(b)*sqrt(a)))*a**6*b**3*x**6 + 39690*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a**5*b**4*x**8 + 39690*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b 
)*sqrt(a)))*a**4*b**5*x**10 + 26460*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a**3*b**6*x**12 + 11340*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a 
)))*a**2*b**7*x**14 + 2835*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a 
*b**8*x**16 + 315*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**9*x**18 
 - 315*a**9*b*x - 2730*a**8*b**2*x**3 - 10458*a**7*b**3*x**5 - 23202*a**6* 
b**4*x**7 + 32768*a**5*b**5*x**9 + 23202*a**4*b**6*x**11 + 10458*a**3*b**7 
*x**13 + 2730*a**2*b**8*x**15 + 315*a*b**9*x**17)/(589824*a**6*b**5*(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))