\(\int \frac {x^{12}}{(a+b x^2)^{10}} \, dx\) [215]

Optimal result

Integrand size = 13, antiderivative size = 200 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {x^{11}}{18 b \left (a+b x^2\right )^9}-\frac {11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac {11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac {11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac {11 x^3}{1536 b^5 \left (a+b x^2\right )^5}-\frac {11 x}{4096 b^6 \left (a+b x^2\right )^4}+\frac {11 x}{24576 a b^6 \left (a+b x^2\right )^3}+\frac {55 x}{98304 a^2 b^6 \left (a+b x^2\right )^2}+\frac {55 x}{65536 a^3 b^6 \left (a+b x^2\right )}+\frac {55 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{7/2} b^{13/2}} \] Output:

-1/18*x^11/b/(b*x^2+a)^9-11/288*x^9/b^2/(b*x^2+a)^8-11/448*x^7/b^3/(b*x^2+ 
a)^7-11/768*x^5/b^4/(b*x^2+a)^6-11/1536*x^3/b^5/(b*x^2+a)^5-11/4096*x/b^6/ 
(b*x^2+a)^4+11/24576*x/a/b^6/(b*x^2+a)^3+55/98304*x/a^2/b^6/(b*x^2+a)^2+55 
/65536*x/a^3/b^6/(b*x^2+a)+55/65536*arctan(b^(1/2)*x/a^(1/2))/a^(7/2)/b^(1 
3/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {\sqrt {a} \sqrt {b} x \left (-3465 a^8-30030 a^7 b x^2-115038 a^6 b^2 x^4-255222 a^5 b^3 x^6-360448 a^4 b^4 x^8-334602 a^3 b^5 x^{10}+115038 a^2 b^6 x^{12}+30030 a b^7 x^{14}+3465 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+3465 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4128768 a^{7/2} b^{13/2}} \] Input:

Integrate[x^12/(a + b*x^2)^10,x]
 

Output:

((Sqrt[a]*Sqrt[b]*x*(-3465*a^8 - 30030*a^7*b*x^2 - 115038*a^6*b^2*x^4 - 25 
5222*a^5*b^3*x^6 - 360448*a^4*b^4*x^8 - 334602*a^3*b^5*x^10 + 115038*a^2*b 
^6*x^12 + 30030*a*b^7*x^14 + 3465*b^8*x^16))/(a + b*x^2)^9 + 3465*ArcTan[( 
Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(7/2)*b^(13/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.28, 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, 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.

\(\displaystyle \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {11 \int \frac {x^{10}}{\left (b x^2+a\right )^9}dx}{18 b}-\frac {x^{11}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {11 \left (\frac {9 \int \frac {x^8}{\left (b x^2+a\right )^8}dx}{16 b}-\frac {x^9}{16 b \left (a+b x^2\right )^8}\right )}{18 b}-\frac {x^{11}}{18 b \left (a+b x^2\right )^9}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 218

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

Input:

Int[x^12/(a + b*x^2)^10,x]
 

Output:

-1/18*x^11/(b*(a + b*x^2)^9) + (11*(-1/16*x^9/(b*(a + b*x^2)^8) + (9*(-1/1 
4*x^7/(b*(a + b*x^2)^7) + (-1/12*x^5/(b*(a + b*x^2)^6) + (5*(-1/10*x^3/(b* 
(a + b*x^2)^5) + (3*(-1/8*x/(b*(a + b*x^2)^4) + (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)/Sq 
rt[a]]/(2*a^(3/2)*Sqrt[b])))/(4*a)))/(6*a))/(8*b)))/(10*b)))/(12*b))/(2*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 = 0.38 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {-\frac {55 a^{5} x}{65536 b^{6}}-\frac {715 a^{4} x^{3}}{98304 b^{5}}-\frac {913 a^{3} x^{5}}{32768 b^{4}}-\frac {14179 a^{2} x^{7}}{229376 b^{3}}-\frac {11 a \,x^{9}}{126 b^{2}}-\frac {18589 x^{11}}{229376 b}+\frac {913 x^{13}}{32768 a}+\frac {715 b \,x^{15}}{98304 a^{2}}+\frac {55 b^{2} x^{17}}{65536 a^{3}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {55 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 a^{3} b^{6} \sqrt {a b}}\)	\(122\)
risch	\(\frac {-\frac {55 a^{5} x}{65536 b^{6}}-\frac {715 a^{4} x^{3}}{98304 b^{5}}-\frac {913 a^{3} x^{5}}{32768 b^{4}}-\frac {14179 a^{2} x^{7}}{229376 b^{3}}-\frac {11 a \,x^{9}}{126 b^{2}}-\frac {18589 x^{11}}{229376 b}+\frac {913 x^{13}}{32768 a}+\frac {715 b \,x^{15}}{98304 a^{2}}+\frac {55 b^{2} x^{17}}{65536 a^{3}}}{\left (b \,x^{2}+a \right )^{9}}-\frac {55 \ln \left (b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{6} a^{3}}+\frac {55 \ln \left (-b x +\sqrt {-a b}\right )}{131072 \sqrt {-a b}\, b^{6} a^{3}}\)	\(151\)

Input:

int(x^12/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
 

Output:

(-55/65536*a^5/b^6*x-715/98304*a^4/b^5*x^3-913/32768*a^3/b^4*x^5-14179/229 
376*a^2/b^3*x^7-11/126*a/b^2*x^9-18589/229376/b*x^11+913/32768/a*x^13+715/ 
98304*b/a^2*x^15+55/65536*b^2/a^3*x^17)/(b*x^2+a)^9+55/65536/a^3/b^6/(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 = 3.27 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx=\left [\frac {6930 \, a b^{9} x^{17} + 60060 \, a^{2} b^{8} x^{15} + 230076 \, a^{3} b^{7} x^{13} - 669204 \, a^{4} b^{6} x^{11} - 720896 \, a^{5} b^{5} x^{9} - 510444 \, a^{6} b^{4} x^{7} - 230076 \, a^{7} b^{3} x^{5} - 60060 \, a^{8} b^{2} x^{3} - 6930 \, a^{9} b x - 3465 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{8257536 \, {\left (a^{4} b^{16} x^{18} + 9 \, a^{5} b^{15} x^{16} + 36 \, a^{6} b^{14} x^{14} + 84 \, a^{7} b^{13} x^{12} + 126 \, a^{8} b^{12} x^{10} + 126 \, a^{9} b^{11} x^{8} + 84 \, a^{10} b^{10} x^{6} + 36 \, a^{11} b^{9} x^{4} + 9 \, a^{12} b^{8} x^{2} + a^{13} b^{7}\right )}}, \frac {3465 \, a b^{9} x^{17} + 30030 \, a^{2} b^{8} x^{15} + 115038 \, a^{3} b^{7} x^{13} - 334602 \, a^{4} b^{6} x^{11} - 360448 \, a^{5} b^{5} x^{9} - 255222 \, a^{6} b^{4} x^{7} - 115038 \, a^{7} b^{3} x^{5} - 30030 \, a^{8} b^{2} x^{3} - 3465 \, a^{9} b x + 3465 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{4128768 \, {\left (a^{4} b^{16} x^{18} + 9 \, a^{5} b^{15} x^{16} + 36 \, a^{6} b^{14} x^{14} + 84 \, a^{7} b^{13} x^{12} + 126 \, a^{8} b^{12} x^{10} + 126 \, a^{9} b^{11} x^{8} + 84 \, a^{10} b^{10} x^{6} + 36 \, a^{11} b^{9} x^{4} + 9 \, a^{12} b^{8} x^{2} + a^{13} b^{7}\right )}}\right ] \] Input:

integrate(x^12/(b*x^2+a)^10,x, algorithm="fricas")
 

Output:

[1/8257536*(6930*a*b^9*x^17 + 60060*a^2*b^8*x^15 + 230076*a^3*b^7*x^13 - 6 
69204*a^4*b^6*x^11 - 720896*a^5*b^5*x^9 - 510444*a^6*b^4*x^7 - 230076*a^7* 
b^3*x^5 - 60060*a^8*b^2*x^3 - 6930*a^9*b*x - 3465*(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^4*b^16*x^18 + 9*a^5*b^15*x^16 + 
36*a^6*b^14*x^14 + 84*a^7*b^13*x^12 + 126*a^8*b^12*x^10 + 126*a^9*b^11*x^8 
 + 84*a^10*b^10*x^6 + 36*a^11*b^9*x^4 + 9*a^12*b^8*x^2 + a^13*b^7), 1/4128 
768*(3465*a*b^9*x^17 + 30030*a^2*b^8*x^15 + 115038*a^3*b^7*x^13 - 334602*a 
^4*b^6*x^11 - 360448*a^5*b^5*x^9 - 255222*a^6*b^4*x^7 - 115038*a^7*b^3*x^5 
 - 30030*a^8*b^2*x^3 - 3465*a^9*b*x + 3465*(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^4*b^16*x^18 + 9*a^5*b^15*x^16 + 36*a^6*b^14*x^14 + 84*a^7*b^13*x 
^12 + 126*a^8*b^12*x^10 + 126*a^9*b^11*x^8 + 84*a^10*b^10*x^6 + 36*a^11*b^ 
9*x^4 + 9*a^12*b^8*x^2 + a^13*b^7)]
 

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.46 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx=- \frac {55 \sqrt {- \frac {1}{a^{7} b^{13}}} \log {\left (- a^{4} b^{6} \sqrt {- \frac {1}{a^{7} b^{13}}} + x \right )}}{131072} + \frac {55 \sqrt {- \frac {1}{a^{7} b^{13}}} \log {\left (a^{4} b^{6} \sqrt {- \frac {1}{a^{7} b^{13}}} + x \right )}}{131072} + \frac {- 3465 a^{8} x - 30030 a^{7} b x^{3} - 115038 a^{6} b^{2} x^{5} - 255222 a^{5} b^{3} x^{7} - 360448 a^{4} b^{4} x^{9} - 334602 a^{3} b^{5} x^{11} + 115038 a^{2} b^{6} x^{13} + 30030 a b^{7} x^{15} + 3465 b^{8} x^{17}}{4128768 a^{12} b^{6} + 37158912 a^{11} b^{7} x^{2} + 148635648 a^{10} b^{8} x^{4} + 346816512 a^{9} b^{9} x^{6} + 520224768 a^{8} b^{10} x^{8} + 520224768 a^{7} b^{11} x^{10} + 346816512 a^{6} b^{12} x^{12} + 148635648 a^{5} b^{13} x^{14} + 37158912 a^{4} b^{14} x^{16} + 4128768 a^{3} b^{15} x^{18}} \] Input:

integrate(x**12/(b*x**2+a)**10,x)
 

Output:

-55*sqrt(-1/(a**7*b**13))*log(-a**4*b**6*sqrt(-1/(a**7*b**13)) + x)/131072 
 + 55*sqrt(-1/(a**7*b**13))*log(a**4*b**6*sqrt(-1/(a**7*b**13)) + x)/13107 
2 + (-3465*a**8*x - 30030*a**7*b*x**3 - 115038*a**6*b**2*x**5 - 255222*a** 
5*b**3*x**7 - 360448*a**4*b**4*x**9 - 334602*a**3*b**5*x**11 + 115038*a**2 
*b**6*x**13 + 30030*a*b**7*x**15 + 3465*b**8*x**17)/(4128768*a**12*b**6 + 
37158912*a**11*b**7*x**2 + 148635648*a**10*b**8*x**4 + 346816512*a**9*b**9 
*x**6 + 520224768*a**8*b**10*x**8 + 520224768*a**7*b**11*x**10 + 346816512 
*a**6*b**12*x**12 + 148635648*a**5*b**13*x**14 + 37158912*a**4*b**14*x**16 
 + 4128768*a**3*b**15*x**18)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.10 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx=\frac {3465 \, b^{8} x^{17} + 30030 \, a b^{7} x^{15} + 115038 \, a^{2} b^{6} x^{13} - 334602 \, a^{3} b^{5} x^{11} - 360448 \, a^{4} b^{4} x^{9} - 255222 \, a^{5} b^{3} x^{7} - 115038 \, a^{6} b^{2} x^{5} - 30030 \, a^{7} b x^{3} - 3465 \, a^{8} x}{4128768 \, {\left (a^{3} b^{15} x^{18} + 9 \, a^{4} b^{14} x^{16} + 36 \, a^{5} b^{13} x^{14} + 84 \, a^{6} b^{12} x^{12} + 126 \, a^{7} b^{11} x^{10} + 126 \, a^{8} b^{10} x^{8} + 84 \, a^{9} b^{9} x^{6} + 36 \, a^{10} b^{8} x^{4} + 9 \, a^{11} b^{7} x^{2} + a^{12} b^{6}\right )}} + \frac {55 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{3} b^{6}} \] Input:

integrate(x^12/(b*x^2+a)^10,x, algorithm="maxima")
 

Output:

1/4128768*(3465*b^8*x^17 + 30030*a*b^7*x^15 + 115038*a^2*b^6*x^13 - 334602 
*a^3*b^5*x^11 - 360448*a^4*b^4*x^9 - 255222*a^5*b^3*x^7 - 115038*a^6*b^2*x 
^5 - 30030*a^7*b*x^3 - 3465*a^8*x)/(a^3*b^15*x^18 + 9*a^4*b^14*x^16 + 36*a 
^5*b^13*x^14 + 84*a^6*b^12*x^12 + 126*a^7*b^11*x^10 + 126*a^8*b^10*x^8 + 8 
4*a^9*b^9*x^6 + 36*a^10*b^8*x^4 + 9*a^11*b^7*x^2 + a^12*b^6) + 55/65536*ar 
ctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3*b^6)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.64 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx=\frac {55 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{3} b^{6}} + \frac {3465 \, b^{8} x^{17} + 30030 \, a b^{7} x^{15} + 115038 \, a^{2} b^{6} x^{13} - 334602 \, a^{3} b^{5} x^{11} - 360448 \, a^{4} b^{4} x^{9} - 255222 \, a^{5} b^{3} x^{7} - 115038 \, a^{6} b^{2} x^{5} - 30030 \, a^{7} b x^{3} - 3465 \, a^{8} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} a^{3} b^{6}} \] Input:

integrate(x^12/(b*x^2+a)^10,x, algorithm="giac")
 

Output:

55/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3*b^6) + 1/4128768*(3465*b^8*x 
^17 + 30030*a*b^7*x^15 + 115038*a^2*b^6*x^13 - 334602*a^3*b^5*x^11 - 36044 
8*a^4*b^4*x^9 - 255222*a^5*b^3*x^7 - 115038*a^6*b^2*x^5 - 30030*a^7*b*x^3 
- 3465*a^8*x)/((b*x^2 + a)^9*a^3*b^6)
 

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.02 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx=\frac {55\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{7/2}\,b^{13/2}}-\frac {\frac {18589\,x^{11}}{229376\,b}-\frac {913\,x^{13}}{32768\,a}+\frac {11\,a\,x^9}{126\,b^2}+\frac {55\,a^5\,x}{65536\,b^6}-\frac {715\,b\,x^{15}}{98304\,a^2}+\frac {14179\,a^2\,x^7}{229376\,b^3}+\frac {913\,a^3\,x^5}{32768\,b^4}+\frac {715\,a^4\,x^3}{98304\,b^5}-\frac {55\,b^2\,x^{17}}{65536\,a^3}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \] Input:

int(x^12/(a + b*x^2)^10,x)
 

Output:

(55*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^(7/2)*b^(13/2)) - ((18589*x^11)/(2 
29376*b) - (913*x^13)/(32768*a) + (11*a*x^9)/(126*b^2) + (55*a^5*x)/(65536 
*b^6) - (715*b*x^15)/(98304*a^2) + (14179*a^2*x^7)/(229376*b^3) + (913*a^3 
*x^5)/(32768*b^4) + (715*a^4*x^3)/(98304*b^5) - (55*b^2*x^17)/(65536*a^3)) 
/(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)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.28 \[ \int \frac {x^{12}}{\left (a+b x^2\right )^{10}} \, dx=\frac {3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{9}+31185 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{8} b \,x^{2}+124740 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{7} b^{2} x^{4}+291060 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{6} b^{3} x^{6}+436590 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} b^{4} x^{8}+436590 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b^{5} x^{10}+291060 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{6} x^{12}+124740 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{7} x^{14}+31185 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{8} x^{16}+3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{9} x^{18}-3465 a^{9} b x -30030 a^{8} b^{2} x^{3}-115038 a^{7} b^{3} x^{5}-255222 a^{6} b^{4} x^{7}-360448 a^{5} b^{5} x^{9}-334602 a^{4} b^{6} x^{11}+115038 a^{3} b^{7} x^{13}+30030 a^{2} b^{8} x^{15}+3465 a \,b^{9} x^{17}}{4128768 a^{4} b^{7} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:

int(x^12/(b*x^2+a)^10,x)
 

Output:

(3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**9 + 31185*sqrt(b)*s 
qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**8*b*x**2 + 124740*sqrt(b)*sqrt(a)* 
atan((b*x)/(sqrt(b)*sqrt(a)))*a**7*b**2*x**4 + 291060*sqrt(b)*sqrt(a)*atan 
((b*x)/(sqrt(b)*sqrt(a)))*a**6*b**3*x**6 + 436590*sqrt(b)*sqrt(a)*atan((b* 
x)/(sqrt(b)*sqrt(a)))*a**5*b**4*x**8 + 436590*sqrt(b)*sqrt(a)*atan((b*x)/( 
sqrt(b)*sqrt(a)))*a**4*b**5*x**10 + 291060*sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*a**3*b**6*x**12 + 124740*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b 
)*sqrt(a)))*a**2*b**7*x**14 + 31185*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a*b**8*x**16 + 3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))* 
b**9*x**18 - 3465*a**9*b*x - 30030*a**8*b**2*x**3 - 115038*a**7*b**3*x**5 
- 255222*a**6*b**4*x**7 - 360448*a**5*b**5*x**9 - 334602*a**4*b**6*x**11 + 
 115038*a**3*b**7*x**13 + 30030*a**2*b**8*x**15 + 3465*a*b**9*x**17)/(4128 
768*a**4*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))