\(\int \frac {x^4}{(a+b x^2)^{10}} \, dx\) [219]

Optimal result

Integrand size = 13, antiderivative size = 204 \[ \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx=-\frac {x^3}{18 b \left (a+b x^2\right )^9}-\frac {x}{96 b^2 \left (a+b x^2\right )^8}+\frac {x}{1344 a b^2 \left (a+b x^2\right )^7}+\frac {13 x}{16128 a^2 b^2 \left (a+b x^2\right )^6}+\frac {143 x}{161280 a^3 b^2 \left (a+b x^2\right )^5}+\frac {143 x}{143360 a^4 b^2 \left (a+b x^2\right )^4}+\frac {143 x}{122880 a^5 b^2 \left (a+b x^2\right )^3}+\frac {143 x}{98304 a^6 b^2 \left (a+b x^2\right )^2}+\frac {143 x}{65536 a^7 b^2 \left (a+b x^2\right )}+\frac {143 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{15/2} b^{5/2}} \] Output:

-1/18*x^3/b/(b*x^2+a)^9-1/96*x/b^2/(b*x^2+a)^8+1/1344*x/a/b^2/(b*x^2+a)^7+ 
13/16128*x/a^2/b^2/(b*x^2+a)^6+143/161280*x/a^3/b^2/(b*x^2+a)^5+143/143360 
*x/a^4/b^2/(b*x^2+a)^4+143/122880*x/a^5/b^2/(b*x^2+a)^3+143/98304*x/a^6/b^ 
2/(b*x^2+a)^2+143/65536*x/a^7/b^2/(b*x^2+a)+143/65536*arctan(b^(1/2)*x/a^( 
1/2))/a^(15/2)/b^(5/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.68 \[ \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {a} \sqrt {b} x \left (-45045 a^8-390390 a^7 b x^2+2633274 a^6 b^2 x^4+4349826 a^5 b^3 x^6+4685824 a^4 b^4 x^8+3317886 a^3 b^5 x^{10}+1495494 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{20643840 a^{15/2} b^{5/2}} \] Input:

Integrate[x^4/(a + b*x^2)^10,x]
 

Output:

((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 - 390390*a^7*b*x^2 + 2633274*a^6*b^2*x^4 + 
 4349826*a^5*b^3*x^6 + 4685824*a^4*b^4*x^8 + 3317886*a^3*b^5*x^10 + 149549 
4*a^2*b^6*x^12 + 390390*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 4504 
5*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(20643840*a^(15/2)*b^(5/2))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {252, 252, 215, 215, 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.

\(\displaystyle \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\int \frac {x^2}{\left (b x^2+a\right )^9}dx}{6 b}-\frac {x^3}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 218

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

Input:

Int[x^4/(a + b*x^2)^10,x]
 

Output:

-1/18*x^3/(b*(a + b*x^2)^9) + (-1/16*x/(b*(a + b*x^2)^8) + (x/(14*a*(a + b 
*x^2)^7) + (13*(x/(12*a*(a + b*x^2)^6) + (11*(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)*Sq 
rt[b])))/(4*a)))/(6*a)))/(8*a)))/(10*a)))/(12*a)))/(14*a))/(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 = 122, normalized size of antiderivative = 0.60

method	result	size
default	\(\frac {-\frac {143 a x}{65536 b^{2}}-\frac {1859 x^{3}}{98304 b}+\frac {20899 x^{5}}{163840 a}+\frac {241657 b \,x^{7}}{1146880 a^{2}}+\frac {143 b^{2} x^{9}}{630 a^{3}}+\frac {184327 b^{3} x^{11}}{1146880 a^{4}}+\frac {11869 b^{4} x^{13}}{163840 a^{5}}+\frac {1859 b^{5} x^{15}}{98304 a^{6}}+\frac {143 b^{6} x^{17}}{65536 a^{7}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {143 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 a^{7} b^{2} \sqrt {a b}}\)	\(122\)
risch	\(\frac {-\frac {143 a x}{65536 b^{2}}-\frac {1859 x^{3}}{98304 b}+\frac {20899 x^{5}}{163840 a}+\frac {241657 b \,x^{7}}{1146880 a^{2}}+\frac {143 b^{2} x^{9}}{630 a^{3}}+\frac {184327 b^{3} x^{11}}{1146880 a^{4}}+\frac {11869 b^{4} x^{13}}{163840 a^{5}}+\frac {1859 b^{5} x^{15}}{98304 a^{6}}+\frac {143 b^{6} x^{17}}{65536 a^{7}}}{\left (b \,x^{2}+a \right )^{9}}-\frac {143 \ln \left (b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{2} a^{7}}+\frac {143 \ln \left (-b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{2} a^{7}}\)	\(151\)

Input:

int(x^4/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
 

Output:

(-143/65536*a*x/b^2-1859/98304*x^3/b+20899/163840/a*x^5+241657/1146880*b/a 
^2*x^7+143/630*b^2/a^3*x^9+184327/1146880*b^3/a^4*x^11+11869/163840*b^4/a^ 
5*x^13+1859/98304*b^5/a^6*x^15+143/65536*b^6/a^7*x^17)/(b*x^2+a)^9+143/655 
36/a^7/b^2/(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 = 3.21 \[ \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {90090 \, a b^{9} x^{17} + 780780 \, a^{2} b^{8} x^{15} + 2990988 \, a^{3} b^{7} x^{13} + 6635772 \, a^{4} b^{6} x^{11} + 9371648 \, a^{5} b^{5} x^{9} + 8699652 \, a^{6} b^{4} x^{7} + 5266548 \, a^{7} b^{3} x^{5} - 780780 \, a^{8} b^{2} x^{3} - 90090 \, a^{9} b x - 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{41287680 \, {\left (a^{8} b^{12} x^{18} + 9 \, a^{9} b^{11} x^{16} + 36 \, a^{10} b^{10} x^{14} + 84 \, a^{11} b^{9} x^{12} + 126 \, a^{12} b^{8} x^{10} + 126 \, a^{13} b^{7} x^{8} + 84 \, a^{14} b^{6} x^{6} + 36 \, a^{15} b^{5} x^{4} + 9 \, a^{16} b^{4} x^{2} + a^{17} b^{3}\right )}}, \frac {45045 \, a b^{9} x^{17} + 390390 \, a^{2} b^{8} x^{15} + 1495494 \, a^{3} b^{7} x^{13} + 3317886 \, a^{4} b^{6} x^{11} + 4685824 \, a^{5} b^{5} x^{9} + 4349826 \, a^{6} b^{4} x^{7} + 2633274 \, a^{7} b^{3} x^{5} - 390390 \, a^{8} b^{2} x^{3} - 45045 \, a^{9} b x + 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{20643840 \, {\left (a^{8} b^{12} x^{18} + 9 \, a^{9} b^{11} x^{16} + 36 \, a^{10} b^{10} x^{14} + 84 \, a^{11} b^{9} x^{12} + 126 \, a^{12} b^{8} x^{10} + 126 \, a^{13} b^{7} x^{8} + 84 \, a^{14} b^{6} x^{6} + 36 \, a^{15} b^{5} x^{4} + 9 \, a^{16} b^{4} x^{2} + a^{17} b^{3}\right )}}\right ] \] Input:

integrate(x^4/(b*x^2+a)^10,x, algorithm="fricas")
 

Output:

[1/41287680*(90090*a*b^9*x^17 + 780780*a^2*b^8*x^15 + 2990988*a^3*b^7*x^13 
 + 6635772*a^4*b^6*x^11 + 9371648*a^5*b^5*x^9 + 8699652*a^6*b^4*x^7 + 5266 
548*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^8*b^12*x^18 + 9*a^9*b 
^11*x^16 + 36*a^10*b^10*x^14 + 84*a^11*b^9*x^12 + 126*a^12*b^8*x^10 + 126* 
a^13*b^7*x^8 + 84*a^14*b^6*x^6 + 36*a^15*b^5*x^4 + 9*a^16*b^4*x^2 + a^17*b 
^3), 1/20643840*(45045*a*b^9*x^17 + 390390*a^2*b^8*x^15 + 1495494*a^3*b^7* 
x^13 + 3317886*a^4*b^6*x^11 + 4685824*a^5*b^5*x^9 + 4349826*a^6*b^4*x^7 + 
2633274*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 + 1 
26*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^8*b^12*x^18 + 9*a^9*b^11*x^16 + 36*a^10*b^ 
10*x^14 + 84*a^11*b^9*x^12 + 126*a^12*b^8*x^10 + 126*a^13*b^7*x^8 + 84*a^1 
4*b^6*x^6 + 36*a^15*b^5*x^4 + 9*a^16*b^4*x^2 + a^17*b^3)]
 

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.43 \[ \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx=- \frac {143 \sqrt {- \frac {1}{a^{15} b^{5}}} \log {\left (- a^{8} b^{2} \sqrt {- \frac {1}{a^{15} b^{5}}} + x \right )}}{131072} + \frac {143 \sqrt {- \frac {1}{a^{15} b^{5}}} \log {\left (a^{8} b^{2} \sqrt {- \frac {1}{a^{15} b^{5}}} + x \right )}}{131072} + \frac {- 45045 a^{8} x - 390390 a^{7} b x^{3} + 2633274 a^{6} b^{2} x^{5} + 4349826 a^{5} b^{3} x^{7} + 4685824 a^{4} b^{4} x^{9} + 3317886 a^{3} b^{5} x^{11} + 1495494 a^{2} b^{6} x^{13} + 390390 a b^{7} x^{15} + 45045 b^{8} x^{17}}{20643840 a^{16} b^{2} + 185794560 a^{15} b^{3} x^{2} + 743178240 a^{14} b^{4} x^{4} + 1734082560 a^{13} b^{5} x^{6} + 2601123840 a^{12} b^{6} x^{8} + 2601123840 a^{11} b^{7} x^{10} + 1734082560 a^{10} b^{8} x^{12} + 743178240 a^{9} b^{9} x^{14} + 185794560 a^{8} b^{10} x^{16} + 20643840 a^{7} b^{11} x^{18}} \] Input:

integrate(x**4/(b*x**2+a)**10,x)
 

Output:

-143*sqrt(-1/(a**15*b**5))*log(-a**8*b**2*sqrt(-1/(a**15*b**5)) + x)/13107 
2 + 143*sqrt(-1/(a**15*b**5))*log(a**8*b**2*sqrt(-1/(a**15*b**5)) + x)/131 
072 + (-45045*a**8*x - 390390*a**7*b*x**3 + 2633274*a**6*b**2*x**5 + 43498 
26*a**5*b**3*x**7 + 4685824*a**4*b**4*x**9 + 3317886*a**3*b**5*x**11 + 149 
5494*a**2*b**6*x**13 + 390390*a*b**7*x**15 + 45045*b**8*x**17)/(20643840*a 
**16*b**2 + 185794560*a**15*b**3*x**2 + 743178240*a**14*b**4*x**4 + 173408 
2560*a**13*b**5*x**6 + 2601123840*a**12*b**6*x**8 + 2601123840*a**11*b**7* 
x**10 + 1734082560*a**10*b**8*x**12 + 743178240*a**9*b**9*x**14 + 18579456 
0*a**8*b**10*x**16 + 20643840*a**7*b**11*x**18)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.08 \[ \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx=\frac {45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{20643840 \, {\left (a^{7} b^{11} x^{18} + 9 \, a^{8} b^{10} x^{16} + 36 \, a^{9} b^{9} x^{14} + 84 \, a^{10} b^{8} x^{12} + 126 \, a^{11} b^{7} x^{10} + 126 \, a^{12} b^{6} x^{8} + 84 \, a^{13} b^{5} x^{6} + 36 \, a^{14} b^{4} x^{4} + 9 \, a^{15} b^{3} x^{2} + a^{16} b^{2}\right )}} + \frac {143 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{7} b^{2}} \] Input:

integrate(x^4/(b*x^2+a)^10,x, algorithm="maxima")
 

Output:

1/20643840*(45045*b^8*x^17 + 390390*a*b^7*x^15 + 1495494*a^2*b^6*x^13 + 33 
17886*a^3*b^5*x^11 + 4685824*a^4*b^4*x^9 + 4349826*a^5*b^3*x^7 + 2633274*a 
^6*b^2*x^5 - 390390*a^7*b*x^3 - 45045*a^8*x)/(a^7*b^11*x^18 + 9*a^8*b^10*x 
^16 + 36*a^9*b^9*x^14 + 84*a^10*b^8*x^12 + 126*a^11*b^7*x^10 + 126*a^12*b^ 
6*x^8 + 84*a^13*b^5*x^6 + 36*a^14*b^4*x^4 + 9*a^15*b^3*x^2 + a^16*b^2) + 1 
43/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7*b^2)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.63 \[ \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx=\frac {143 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{7} b^{2}} + \frac {45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{20643840 \, {\left (b x^{2} + a\right )}^{9} a^{7} b^{2}} \] Input:

integrate(x^4/(b*x^2+a)^10,x, algorithm="giac")
 

Output:

143/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^7*b^2) + 1/20643840*(45045*b^ 
8*x^17 + 390390*a*b^7*x^15 + 1495494*a^2*b^6*x^13 + 3317886*a^3*b^5*x^11 + 
 4685824*a^4*b^4*x^9 + 4349826*a^5*b^3*x^7 + 2633274*a^6*b^2*x^5 - 390390* 
a^7*b*x^3 - 45045*a^8*x)/((b*x^2 + a)^9*a^7*b^2)
 

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {20899\,x^5}{163840\,a}-\frac {1859\,x^3}{98304\,b}+\frac {241657\,b\,x^7}{1146880\,a^2}+\frac {143\,b^2\,x^9}{630\,a^3}+\frac {184327\,b^3\,x^{11}}{1146880\,a^4}+\frac {11869\,b^4\,x^{13}}{163840\,a^5}+\frac {1859\,b^5\,x^{15}}{98304\,a^6}+\frac {143\,b^6\,x^{17}}{65536\,a^7}-\frac {143\,a\,x}{65536\,b^2}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {143\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{15/2}\,b^{5/2}} \] Input:

int(x^4/(a + b*x^2)^10,x)
 

Output:

((20899*x^5)/(163840*a) - (1859*x^3)/(98304*b) + (241657*b*x^7)/(1146880*a 
^2) + (143*b^2*x^9)/(630*a^3) + (184327*b^3*x^11)/(1146880*a^4) + (11869*b 
^4*x^13)/(163840*a^5) + (1859*b^5*x^15)/(98304*a^6) + (143*b^6*x^17)/(6553 
6*a^7) - (143*a*x)/(65536*b^2))/(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) + (143*atan((b^(1/2)*x)/a^(1/2)))/(65 
536*a^(15/2)*b^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.24 \[ \int \frac {x^4}{\left (a+b x^2\right )^{10}} \, dx=\frac {45045 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{9}+405405 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{8} b \,x^{2}+1621620 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{7} b^{2} x^{4}+3783780 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6} b^{3} x^{6}+5675670 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{4} x^{8}+5675670 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{5} x^{10}+3783780 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{6} x^{12}+1621620 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{7} x^{14}+405405 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{8} x^{16}+45045 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{9} x^{18}-45045 a^{9} b x -390390 a^{8} b^{2} x^{3}+2633274 a^{7} b^{3} x^{5}+4349826 a^{6} b^{4} x^{7}+4685824 a^{5} b^{5} x^{9}+3317886 a^{4} b^{6} x^{11}+1495494 a^{3} b^{7} x^{13}+390390 a^{2} b^{8} x^{15}+45045 a \,b^{9} x^{17}}{20643840 a^{8} b^{3} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:

int(x^4/(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 + 2633274*a* 
*7*b**3*x**5 + 4349826*a**6*b**4*x**7 + 4685824*a**5*b**5*x**9 + 3317886*a 
**4*b**6*x**11 + 1495494*a**3*b**7*x**13 + 390390*a**2*b**8*x**15 + 45045* 
a*b**9*x**17)/(20643840*a**8*b**3*(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))