\(\int \frac {x^{16}}{(a+b x^2)^{10}} \, dx\) [213]

Optimal result

Integrand size = 13, antiderivative size = 198 \[ \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}-\frac {5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac {65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac {715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac {143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac {143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac {715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac {715 x}{32768 b^8 \left (a+b x^2\right )^2}+\frac {715 x}{65536 a b^8 \left (a+b x^2\right )}+\frac {715 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{3/2} b^{17/2}} \] Output:

-1/18*x^15/b/(b*x^2+a)^9-5/96*x^13/b^2/(b*x^2+a)^8-65/1344*x^11/b^3/(b*x^2 
+a)^7-715/16128*x^9/b^4/(b*x^2+a)^6-143/3584*x^7/b^5/(b*x^2+a)^5-143/4096* 
x^5/b^6/(b*x^2+a)^4-715/24576*x^3/b^7/(b*x^2+a)^3-715/32768*x/b^8/(b*x^2+a 
)^2+715/65536*x/a/b^8/(b*x^2+a)+715/65536*arctan(b^(1/2)*x/a^(1/2))/a^(3/2 
)/b^(17/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.70 \[ \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {a} \sqrt {b} x \left (-45045 a^8-390390 a^7 b x^2-1495494 a^6 b^2 x^4-3317886 a^5 b^3 x^6-4685824 a^4 b^4 x^8-4349826 a^3 b^5 x^{10}-2633274 a^2 b^6 x^{12}-985866 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4128768 a^{3/2} b^{17/2}} \] Input:

Integrate[x^16/(a + b*x^2)^10,x]
 

Output:

((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 - 390390*a^7*b*x^2 - 1495494*a^6*b^2*x^4 - 
 3317886*a^5*b^3*x^6 - 4685824*a^4*b^4*x^8 - 4349826*a^3*b^5*x^10 - 263327 
4*a^2*b^6*x^12 - 985866*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 4504 
5*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(3/2)*b^(17/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.31, 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, 252, 252, 252, 252, 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.

\(\displaystyle \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {5 \int \frac {x^{14}}{\left (b x^2+a\right )^9}dx}{6 b}-\frac {x^{15}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 218

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

Input:

Int[x^16/(a + b*x^2)^10,x]
 

Output:

-1/18*x^15/(b*(a + b*x^2)^9) + (5*(-1/16*x^13/(b*(a + b*x^2)^8) + (13*(-1/ 
14*x^11/(b*(a + b*x^2)^7) + (11*(-1/12*x^9/(b*(a + b*x^2)^6) + (3*(-1/10*x 
^7/(b*(a + b*x^2)^5) + (7*(-1/8*x^5/(b*(a + b*x^2)^4) + (5*(-1/6*x^3/(b*(a 
 + b*x^2)^3) + (-1/4*x/(b*(a + b*x^2)^2) + (x/(2*a*(a + b*x^2)) + ArcTan[( 
Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]))/(4*b))/(2*b)))/(8*b)))/(10*b)))/( 
4*b)))/(14*b)))/(16*b)))/(6*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 = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.63

method	result	size
default	\(\frac {-\frac {715 a^{7} x}{65536 b^{8}}-\frac {9295 a^{6} x^{3}}{98304 b^{7}}-\frac {11869 a^{5} x^{5}}{32768 b^{6}}-\frac {184327 a^{4} x^{7}}{229376 b^{5}}-\frac {143 a^{3} x^{9}}{126 b^{4}}-\frac {241657 a^{2} x^{11}}{229376 b^{3}}-\frac {20899 a \,x^{13}}{32768 b^{2}}-\frac {23473 x^{15}}{98304 b}+\frac {715 x^{17}}{65536 a}}{\left (b \,x^{2}+a \right )^{9}}+\frac {715 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 a \,b^{8} \sqrt {a b}}\)	\(124\)
risch	\(\frac {-\frac {715 a^{7} x}{65536 b^{8}}-\frac {9295 a^{6} x^{3}}{98304 b^{7}}-\frac {11869 a^{5} x^{5}}{32768 b^{6}}-\frac {184327 a^{4} x^{7}}{229376 b^{5}}-\frac {143 a^{3} x^{9}}{126 b^{4}}-\frac {241657 a^{2} x^{11}}{229376 b^{3}}-\frac {20899 a \,x^{13}}{32768 b^{2}}-\frac {23473 x^{15}}{98304 b}+\frac {715 x^{17}}{65536 a}}{\left (b \,x^{2}+a \right )^{9}}-\frac {715 \ln \left (b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{8} a}+\frac {715 \ln \left (-b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{8} a}\)	\(153\)

Input:

int(x^16/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
 

Output:

(-715/65536*a^7/b^8*x-9295/98304*a^6/b^7*x^3-11869/32768*a^5/b^6*x^5-18432 
7/229376*a^4/b^5*x^7-143/126*a^3/b^4*x^9-241657/229376*a^2/b^3*x^11-20899/ 
32768*a/b^2*x^13-23473/98304/b*x^15+715/65536/a*x^17)/(b*x^2+a)^9+715/6553 
6/a/b^8/(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 = 3.30 \[ \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {90090 \, a b^{9} x^{17} - 1971732 \, a^{2} b^{8} x^{15} - 5266548 \, a^{3} b^{7} x^{13} - 8699652 \, a^{4} b^{6} x^{11} - 9371648 \, a^{5} b^{5} x^{9} - 6635772 \, a^{6} b^{4} x^{7} - 2990988 \, a^{7} b^{3} x^{5} - 780780 \, a^{8} b^{2} x^{3} - 90090 \, a^{9} b x - 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{8257536 \, {\left (a^{2} b^{18} x^{18} + 9 \, a^{3} b^{17} x^{16} + 36 \, a^{4} b^{16} x^{14} + 84 \, a^{5} b^{15} x^{12} + 126 \, a^{6} b^{14} x^{10} + 126 \, a^{7} b^{13} x^{8} + 84 \, a^{8} b^{12} x^{6} + 36 \, a^{9} b^{11} x^{4} + 9 \, a^{10} b^{10} x^{2} + a^{11} b^{9}\right )}}, \frac {45045 \, a b^{9} x^{17} - 985866 \, a^{2} b^{8} x^{15} - 2633274 \, a^{3} b^{7} x^{13} - 4349826 \, a^{4} b^{6} x^{11} - 4685824 \, a^{5} b^{5} x^{9} - 3317886 \, a^{6} b^{4} x^{7} - 1495494 \, a^{7} b^{3} x^{5} - 390390 \, a^{8} b^{2} x^{3} - 45045 \, a^{9} b x + 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{4128768 \, {\left (a^{2} b^{18} x^{18} + 9 \, a^{3} b^{17} x^{16} + 36 \, a^{4} b^{16} x^{14} + 84 \, a^{5} b^{15} x^{12} + 126 \, a^{6} b^{14} x^{10} + 126 \, a^{7} b^{13} x^{8} + 84 \, a^{8} b^{12} x^{6} + 36 \, a^{9} b^{11} x^{4} + 9 \, a^{10} b^{10} x^{2} + a^{11} b^{9}\right )}}\right ] \] Input:

integrate(x^16/(b*x^2+a)^10,x, algorithm="fricas")
 

Output:

[1/8257536*(90090*a*b^9*x^17 - 1971732*a^2*b^8*x^15 - 5266548*a^3*b^7*x^13 
 - 8699652*a^4*b^6*x^11 - 9371648*a^5*b^5*x^9 - 6635772*a^6*b^4*x^7 - 2990 
988*a^7*b^3*x^5 - 780780*a^8*b^2*x^3 - 90090*a^9*b*x - 45045*(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*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^2*b^18*x^18 + 9*a^3*b 
^17*x^16 + 36*a^4*b^16*x^14 + 84*a^5*b^15*x^12 + 126*a^6*b^14*x^10 + 126*a 
^7*b^13*x^8 + 84*a^8*b^12*x^6 + 36*a^9*b^11*x^4 + 9*a^10*b^10*x^2 + a^11*b 
^9), 1/4128768*(45045*a*b^9*x^17 - 985866*a^2*b^8*x^15 - 2633274*a^3*b^7*x 
^13 - 4349826*a^4*b^6*x^11 - 4685824*a^5*b^5*x^9 - 3317886*a^6*b^4*x^7 - 1 
495494*a^7*b^3*x^5 - 390390*a^8*b^2*x^3 - 45045*a^9*b*x + 45045*(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 + 12 
6*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^2*b^18*x^18 + 9*a^3*b^17*x^16 + 36*a^4*b^16 
*x^14 + 84*a^5*b^15*x^12 + 126*a^6*b^14*x^10 + 126*a^7*b^13*x^8 + 84*a^8*b 
^12*x^6 + 36*a^9*b^11*x^4 + 9*a^10*b^10*x^2 + a^11*b^9)]
 

Sympy [A] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.46 \[ \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx=- \frac {715 \sqrt {- \frac {1}{a^{3} b^{17}}} \log {\left (- a^{2} b^{8} \sqrt {- \frac {1}{a^{3} b^{17}}} + x \right )}}{131072} + \frac {715 \sqrt {- \frac {1}{a^{3} b^{17}}} \log {\left (a^{2} b^{8} \sqrt {- \frac {1}{a^{3} b^{17}}} + x \right )}}{131072} + \frac {- 45045 a^{8} x - 390390 a^{7} b x^{3} - 1495494 a^{6} b^{2} x^{5} - 3317886 a^{5} b^{3} x^{7} - 4685824 a^{4} b^{4} x^{9} - 4349826 a^{3} b^{5} x^{11} - 2633274 a^{2} b^{6} x^{13} - 985866 a b^{7} x^{15} + 45045 b^{8} x^{17}}{4128768 a^{10} b^{8} + 37158912 a^{9} b^{9} x^{2} + 148635648 a^{8} b^{10} x^{4} + 346816512 a^{7} b^{11} x^{6} + 520224768 a^{6} b^{12} x^{8} + 520224768 a^{5} b^{13} x^{10} + 346816512 a^{4} b^{14} x^{12} + 148635648 a^{3} b^{15} x^{14} + 37158912 a^{2} b^{16} x^{16} + 4128768 a b^{17} x^{18}} \] Input:

integrate(x**16/(b*x**2+a)**10,x)
 

Output:

-715*sqrt(-1/(a**3*b**17))*log(-a**2*b**8*sqrt(-1/(a**3*b**17)) + x)/13107 
2 + 715*sqrt(-1/(a**3*b**17))*log(a**2*b**8*sqrt(-1/(a**3*b**17)) + x)/131 
072 + (-45045*a**8*x - 390390*a**7*b*x**3 - 1495494*a**6*b**2*x**5 - 33178 
86*a**5*b**3*x**7 - 4685824*a**4*b**4*x**9 - 4349826*a**3*b**5*x**11 - 263 
3274*a**2*b**6*x**13 - 985866*a*b**7*x**15 + 45045*b**8*x**17)/(4128768*a* 
*10*b**8 + 37158912*a**9*b**9*x**2 + 148635648*a**8*b**10*x**4 + 346816512 
*a**7*b**11*x**6 + 520224768*a**6*b**12*x**8 + 520224768*a**5*b**13*x**10 
+ 346816512*a**4*b**14*x**12 + 148635648*a**3*b**15*x**14 + 37158912*a**2* 
b**16*x**16 + 4128768*a*b**17*x**18)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.11 \[ \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx=\frac {45045 \, b^{8} x^{17} - 985866 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \, {\left (a b^{17} x^{18} + 9 \, a^{2} b^{16} x^{16} + 36 \, a^{3} b^{15} x^{14} + 84 \, a^{4} b^{14} x^{12} + 126 \, a^{5} b^{13} x^{10} + 126 \, a^{6} b^{12} x^{8} + 84 \, a^{7} b^{11} x^{6} + 36 \, a^{8} b^{10} x^{4} + 9 \, a^{9} b^{9} x^{2} + a^{10} b^{8}\right )}} + \frac {715 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a b^{8}} \] Input:

integrate(x^16/(b*x^2+a)^10,x, algorithm="maxima")
 

Output:

1/4128768*(45045*b^8*x^17 - 985866*a*b^7*x^15 - 2633274*a^2*b^6*x^13 - 434 
9826*a^3*b^5*x^11 - 4685824*a^4*b^4*x^9 - 3317886*a^5*b^3*x^7 - 1495494*a^ 
6*b^2*x^5 - 390390*a^7*b*x^3 - 45045*a^8*x)/(a*b^17*x^18 + 9*a^2*b^16*x^16 
 + 36*a^3*b^15*x^14 + 84*a^4*b^14*x^12 + 126*a^5*b^13*x^10 + 126*a^6*b^12* 
x^8 + 84*a^7*b^11*x^6 + 36*a^8*b^10*x^4 + 9*a^9*b^9*x^2 + a^10*b^8) + 715/ 
65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^8)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.65 \[ \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx=\frac {715 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a b^{8}} + \frac {45045 \, b^{8} x^{17} - 985866 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} a b^{8}} \] Input:

integrate(x^16/(b*x^2+a)^10,x, algorithm="giac")
 

Output:

715/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^8) + 1/4128768*(45045*b^8*x 
^17 - 985866*a*b^7*x^15 - 2633274*a^2*b^6*x^13 - 4349826*a^3*b^5*x^11 - 46 
85824*a^4*b^4*x^9 - 3317886*a^5*b^3*x^7 - 1495494*a^6*b^2*x^5 - 390390*a^7 
*b*x^3 - 45045*a^8*x)/((b*x^2 + a)^9*a*b^8)
 

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.05 \[ \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx=\frac {715\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{3/2}\,b^{17/2}}-\frac {\frac {23473\,x^{15}}{98304\,b}-\frac {715\,x^{17}}{65536\,a}+\frac {20899\,a\,x^{13}}{32768\,b^2}+\frac {715\,a^7\,x}{65536\,b^8}+\frac {241657\,a^2\,x^{11}}{229376\,b^3}+\frac {143\,a^3\,x^9}{126\,b^4}+\frac {184327\,a^4\,x^7}{229376\,b^5}+\frac {11869\,a^5\,x^5}{32768\,b^6}+\frac {9295\,a^6\,x^3}{98304\,b^7}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \] Input:

int(x^16/(a + b*x^2)^10,x)
 

Output:

(715*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^(3/2)*b^(17/2)) - ((23473*x^15)/( 
98304*b) - (715*x^17)/(65536*a) + (20899*a*x^13)/(32768*b^2) + (715*a^7*x) 
/(65536*b^8) + (241657*a^2*x^11)/(229376*b^3) + (143*a^3*x^9)/(126*b^4) + 
(184327*a^4*x^7)/(229376*b^5) + (11869*a^5*x^5)/(32768*b^6) + (9295*a^6*x^ 
3)/(98304*b^7))/(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^1 
2 + 36*a^2*b^7*x^14)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.30 \[ \int \frac {x^{16}}{\left (a+b x^2\right )^{10}} \, dx=\frac {45045 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{9}+405405 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{8} b \,x^{2}+1621620 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{7} b^{2} x^{4}+3783780 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6} b^{3} x^{6}+5675670 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{4} x^{8}+5675670 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{5} x^{10}+3783780 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{6} x^{12}+1621620 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{7} x^{14}+405405 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{8} x^{16}+45045 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{9} x^{18}-45045 a^{9} b x -390390 a^{8} b^{2} x^{3}-1495494 a^{7} b^{3} x^{5}-3317886 a^{6} b^{4} x^{7}-4685824 a^{5} b^{5} x^{9}-4349826 a^{4} b^{6} x^{11}-2633274 a^{3} b^{7} x^{13}-985866 a^{2} b^{8} x^{15}+45045 a \,b^{9} x^{17}}{4128768 a^{2} b^{9} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:

int(x^16/(b*x^2+a)^10,x)
 

Output:

(45045*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**9 + 405405*sqrt(b) 
*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**8*b*x**2 + 1621620*sqrt(b)*sqrt( 
a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**7*b**2*x**4 + 3783780*sqrt(b)*sqrt(a)* 
atan((b*x)/(sqrt(b)*sqrt(a)))*a**6*b**3*x**6 + 5675670*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**5*b**4*x**8 + 5675670*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a**4*b**5*x**10 + 3783780*sqrt(b)*sqrt(a)*atan((b* 
x)/(sqrt(b)*sqrt(a)))*a**3*b**6*x**12 + 1621620*sqrt(b)*sqrt(a)*atan((b*x) 
/(sqrt(b)*sqrt(a)))*a**2*b**7*x**14 + 405405*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a*b**8*x**16 + 45045*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*b**9*x**18 - 45045*a**9*b*x - 390390*a**8*b**2*x**3 - 1495494*a* 
*7*b**3*x**5 - 3317886*a**6*b**4*x**7 - 4685824*a**5*b**5*x**9 - 4349826*a 
**4*b**6*x**11 - 2633274*a**3*b**7*x**13 - 985866*a**2*b**8*x**15 + 45045* 
a*b**9*x**17)/(4128768*a**2*b**9*(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))