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

Optimal. Leaf size=200 \[ \frac{55 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{7/2} b^{13/2}}+\frac{55 x}{65536 a^3 b^6 \left (a+b x^2\right )}+\frac{55 x}{98304 a^2 b^6 \left (a+b x^2\right )^2}+\frac{11 x}{24576 a b^6 \left (a+b x^2\right )^3}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{x^{11}}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x^11/(18*b*(a + b*x^2)^9) - (11*x^9)/(288*b^2*(a + b*x^2)^8) - (11*x^7)/(448*b^
3*(a + b*x^2)^7) - (11*x^5)/(768*b^4*(a + b*x^2)^6) - (11*x^3)/(1536*b^5*(a + b*
x^2)^5) - (11*x)/(4096*b^6*(a + b*x^2)^4) + (11*x)/(24576*a*b^6*(a + b*x^2)^3) +
(55*x)/(98304*a^2*b^6*(a + b*x^2)^2) + (55*x)/(65536*a^3*b^6*(a + b*x^2)) + (55
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(7/2)*b^(13/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.305134, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{55 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{7/2} b^{13/2}}+\frac{55 x}{65536 a^3 b^6 \left (a+b x^2\right )}+\frac{55 x}{98304 a^2 b^6 \left (a+b x^2\right )^2}+\frac{11 x}{24576 a b^6 \left (a+b x^2\right )^3}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{x^{11}}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Int[x^12/(a + b*x^2)^10,x]

[Out]

-x^11/(18*b*(a + b*x^2)^9) - (11*x^9)/(288*b^2*(a + b*x^2)^8) - (11*x^7)/(448*b^
3*(a + b*x^2)^7) - (11*x^5)/(768*b^4*(a + b*x^2)^6) - (11*x^3)/(1536*b^5*(a + b*
x^2)^5) - (11*x)/(4096*b^6*(a + b*x^2)^4) + (11*x)/(24576*a*b^6*(a + b*x^2)^3) +
(55*x)/(98304*a^2*b^6*(a + b*x^2)^2) + (55*x)/(65536*a^3*b^6*(a + b*x^2)) + (55
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(7/2)*b^(13/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 48.5058, size = 190, normalized size = 0.95 \[ - \frac{x^{11}}{18 b \left (a + b x^{2}\right )^{9}} - \frac{11 x^{9}}{288 b^{2} \left (a + b x^{2}\right )^{8}} - \frac{11 x^{7}}{448 b^{3} \left (a + b x^{2}\right )^{7}} - \frac{11 x^{5}}{768 b^{4} \left (a + b x^{2}\right )^{6}} - \frac{11 x^{3}}{1536 b^{5} \left (a + b x^{2}\right )^{5}} - \frac{11 x}{4096 b^{6} \left (a + b x^{2}\right )^{4}} + \frac{11 x}{24576 a b^{6} \left (a + b x^{2}\right )^{3}} + \frac{55 x}{98304 a^{2} b^{6} \left (a + b x^{2}\right )^{2}} + \frac{55 x}{65536 a^{3} b^{6} \left (a + b x^{2}\right )} + \frac{55 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{65536 a^{\frac{7}{2}} b^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12/(b*x**2+a)**10,x)

[Out]

-x**11/(18*b*(a + b*x**2)**9) - 11*x**9/(288*b**2*(a + b*x**2)**8) - 11*x**7/(44
8*b**3*(a + b*x**2)**7) - 11*x**5/(768*b**4*(a + b*x**2)**6) - 11*x**3/(1536*b**
5*(a + b*x**2)**5) - 11*x/(4096*b**6*(a + b*x**2)**4) + 11*x/(24576*a*b**6*(a +
b*x**2)**3) + 55*x/(98304*a**2*b**6*(a + b*x**2)**2) + 55*x/(65536*a**3*b**6*(a
+ b*x**2)) + 55*atan(sqrt(b)*x/sqrt(a))/(65536*a**(7/2)*b**(13/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.142483, size = 138, normalized size = 0.69 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (-3465 a^8-30030 a^7 b x^2-115038 a^6 b^2 x^4-255222 a^5 b^3 x^6-360448 a^4 b^4 x^8-334602 a^3 b^5 x^{10}+115038 a^2 b^6 x^{12}+30030 a b^7 x^{14}+3465 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+3465 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{4128768 a^{7/2} b^{13/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*Sqrt[b]*x*(-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 + 30
030*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))

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 122, normalized size = 0.6 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{55\,{a}^{5}x}{65536\,{b}^{6}}}-{\frac{715\,{a}^{4}{x}^{3}}{98304\,{b}^{5}}}-{\frac{913\,{a}^{3}{x}^{5}}{32768\,{b}^{4}}}-{\frac{14179\,{a}^{2}{x}^{7}}{229376\,{b}^{3}}}-{\frac{11\,a{x}^{9}}{126\,{b}^{2}}}-{\frac{18589\,{x}^{11}}{229376\,b}}+{\frac{913\,{x}^{13}}{32768\,a}}+{\frac{715\,b{x}^{15}}{98304\,{a}^{2}}}+{\frac{55\,{b}^{2}{x}^{17}}{65536\,{a}^{3}}} \right ) }+{\frac{55}{65536\,{a}^{3}{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12/(b*x^2+a)^10,x)

[Out]

(-55/65536*a^5/b^6*x-715/98304*a^4/b^5*x^3-913/32768*a^3/b^4*x^5-14179/229376*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(x*b/
(a*b)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.213414, size = 1, normalized size = 0. \[ \left [\frac{3465 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3465 \, b^{8} x^{17} + 30030 \, a b^{7} x^{15} + 115038 \, a^{2} b^{6} x^{13} - 334602 \, a^{3} b^{5} x^{11} - 360448 \, a^{4} b^{4} x^{9} - 255222 \, a^{5} b^{3} x^{7} - 115038 \, a^{6} b^{2} x^{5} - 30030 \, a^{7} b x^{3} - 3465 \, a^{8} x\right )} \sqrt{-a b}}{8257536 \,{\left (a^{3} b^{15} x^{18} + 9 \, a^{4} b^{14} x^{16} + 36 \, a^{5} b^{13} x^{14} + 84 \, a^{6} b^{12} x^{12} + 126 \, a^{7} b^{11} x^{10} + 126 \, a^{8} b^{10} x^{8} + 84 \, a^{9} b^{9} x^{6} + 36 \, a^{10} b^{8} x^{4} + 9 \, a^{11} b^{7} x^{2} + a^{12} b^{6}\right )} \sqrt{-a b}}, \frac{3465 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3465 \, b^{8} x^{17} + 30030 \, a b^{7} x^{15} + 115038 \, a^{2} b^{6} x^{13} - 334602 \, a^{3} b^{5} x^{11} - 360448 \, a^{4} b^{4} x^{9} - 255222 \, a^{5} b^{3} x^{7} - 115038 \, a^{6} b^{2} x^{5} - 30030 \, a^{7} b x^{3} - 3465 \, a^{8} x\right )} \sqrt{a b}}{4128768 \,{\left (a^{3} b^{15} x^{18} + 9 \, a^{4} b^{14} x^{16} + 36 \, a^{5} b^{13} x^{14} + 84 \, a^{6} b^{12} x^{12} + 126 \, a^{7} b^{11} x^{10} + 126 \, a^{8} b^{10} x^{8} + 84 \, a^{9} b^{9} x^{6} + 36 \, a^{10} b^{8} x^{4} + 9 \, a^{11} b^{7} x^{2} + a^{12} b^{6}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8257536*(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)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(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)*s
qrt(-a*b))/((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)*sqrt(-a*b)), 1/4128768*(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)*arctan(sqrt(a*b)*x/a) + (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)*sqrt(a*b))/((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)*sqrt(a*b))]

_______________________________________________________________________________________

Sympy [A]  time = 34.9877, size = 291, normalized size = 1.46 \[ - \frac{55 \sqrt{- \frac{1}{a^{7} b^{13}}} \log{\left (- a^{4} b^{6} \sqrt{- \frac{1}{a^{7} b^{13}}} + x \right )}}{131072} + \frac{55 \sqrt{- \frac{1}{a^{7} b^{13}}} \log{\left (a^{4} b^{6} \sqrt{- \frac{1}{a^{7} b^{13}}} + x \right )}}{131072} + \frac{- 3465 a^{8} x - 30030 a^{7} b x^{3} - 115038 a^{6} b^{2} x^{5} - 255222 a^{5} b^{3} x^{7} - 360448 a^{4} b^{4} x^{9} - 334602 a^{3} b^{5} x^{11} + 115038 a^{2} b^{6} x^{13} + 30030 a b^{7} x^{15} + 3465 b^{8} x^{17}}{4128768 a^{12} b^{6} + 37158912 a^{11} b^{7} x^{2} + 148635648 a^{10} b^{8} x^{4} + 346816512 a^{9} b^{9} x^{6} + 520224768 a^{8} b^{10} x^{8} + 520224768 a^{7} b^{11} x^{10} + 346816512 a^{6} b^{12} x^{12} + 148635648 a^{5} b^{13} x^{14} + 37158912 a^{4} b^{14} x^{16} + 4128768 a^{3} b^{15} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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)/131072 + (-3465*a
**8*x - 30030*a**7*b*x**3 - 115038*a**6*b**2*x**5 - 255222*a**5*b**3*x**7 - 3604
48*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 + 148
635648*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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239501, size = 173, normalized size = 0.86 \[ \frac{55 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{3} b^{6}} + \frac{3465 \, b^{8} x^{17} + 30030 \, a b^{7} x^{15} + 115038 \, a^{2} b^{6} x^{13} - 334602 \, a^{3} b^{5} x^{11} - 360448 \, a^{4} b^{4} x^{9} - 255222 \, a^{5} b^{3} x^{7} - 115038 \, a^{6} b^{2} x^{5} - 30030 \, a^{7} b x^{3} - 3465 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} a^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 - 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)/((b*
x^2 + a)^9*a^3*b^6)