3.525 \(\int \frac{x^{12}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=131 \[ -\frac{693 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{13/2}}-\frac{231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac{231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}+\frac{693 x}{256 b^6} \]

[Out]

(693*x)/(256*b^6) - x^11/(10*b*(a + b*x^2)^5) - (11*x^9)/(80*b^2*(a + b*x^2)^4)
- (33*x^7)/(160*b^3*(a + b*x^2)^3) - (231*x^5)/(640*b^4*(a + b*x^2)^2) - (231*x^
3)/(256*b^5*(a + b*x^2)) - (693*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*b^(13/
2))

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Rubi [A]  time = 0.203562, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{693 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{13/2}}-\frac{231 x^3}{256 b^5 \left (a+b x^2\right )}-\frac{231 x^5}{640 b^4 \left (a+b x^2\right )^2}-\frac{33 x^7}{160 b^3 \left (a+b x^2\right )^3}-\frac{11 x^9}{80 b^2 \left (a+b x^2\right )^4}-\frac{x^{11}}{10 b \left (a+b x^2\right )^5}+\frac{693 x}{256 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(693*x)/(256*b^6) - x^11/(10*b*(a + b*x^2)^5) - (11*x^9)/(80*b^2*(a + b*x^2)^4)
- (33*x^7)/(160*b^3*(a + b*x^2)^3) - (231*x^5)/(640*b^4*(a + b*x^2)^2) - (231*x^
3)/(256*b^5*(a + b*x^2)) - (693*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*b^(13/
2))

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Rubi in Sympy [A]  time = 42.8991, size = 122, normalized size = 0.93 \[ - \frac{693 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{13}{2}}} - \frac{x^{11}}{10 b \left (a + b x^{2}\right )^{5}} - \frac{11 x^{9}}{80 b^{2} \left (a + b x^{2}\right )^{4}} - \frac{33 x^{7}}{160 b^{3} \left (a + b x^{2}\right )^{3}} - \frac{231 x^{5}}{640 b^{4} \left (a + b x^{2}\right )^{2}} - \frac{231 x^{3}}{256 b^{5} \left (a + b x^{2}\right )} + \frac{693 x}{256 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-693*sqrt(a)*atan(sqrt(b)*x/sqrt(a))/(256*b**(13/2)) - x**11/(10*b*(a + b*x**2)*
*5) - 11*x**9/(80*b**2*(a + b*x**2)**4) - 33*x**7/(160*b**3*(a + b*x**2)**3) - 2
31*x**5/(640*b**4*(a + b*x**2)**2) - 231*x**3/(256*b**5*(a + b*x**2)) + 693*x/(2
56*b**6)

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Mathematica [A]  time = 0.0970371, size = 100, normalized size = 0.76 \[ \frac{\frac{\sqrt{b} x \left (3465 a^5+16170 a^4 b x^2+29568 a^3 b^2 x^4+26070 a^2 b^3 x^6+10615 a b^4 x^8+1280 b^5 x^{10}\right )}{\left (a+b x^2\right )^5}-3465 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{1280 b^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((Sqrt[b]*x*(3465*a^5 + 16170*a^4*b*x^2 + 29568*a^3*b^2*x^4 + 26070*a^2*b^3*x^6
+ 10615*a*b^4*x^8 + 1280*b^5*x^10))/(a + b*x^2)^5 - 3465*Sqrt[a]*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(1280*b^(13/2))

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Maple [A]  time = 0.02, size = 123, normalized size = 0.9 \[{\frac{x}{{b}^{6}}}+{\frac{843\,a{x}^{9}}{256\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{1327\,{a}^{2}{x}^{7}}{128\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{131\,{a}^{3}{x}^{5}}{10\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{977\,{a}^{4}{x}^{3}}{128\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{437\,{a}^{5}x}{256\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{693\,a}{256\,{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

x/b^6+843/256/b^2*a/(b*x^2+a)^5*x^9+1327/128/b^3*a^2/(b*x^2+a)^5*x^7+131/10/b^4*
a^3/(b*x^2+a)^5*x^5+977/128/b^5*a^4/(b*x^2+a)^5*x^3+437/256/b^6*a^5/(b*x^2+a)^5*
x-693/256/b^6*a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))

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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^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267743, size = 1, normalized size = 0.01 \[ \left [\frac{2560 \, b^{5} x^{11} + 21230 \, a b^{4} x^{9} + 52140 \, a^{2} b^{3} x^{7} + 59136 \, a^{3} b^{2} x^{5} + 32340 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{2560 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}, \frac{1280 \, b^{5} x^{11} + 10615 \, a b^{4} x^{9} + 26070 \, a^{2} b^{3} x^{7} + 29568 \, a^{3} b^{2} x^{5} + 16170 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right )}{1280 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

[1/2560*(2560*b^5*x^11 + 21230*a*b^4*x^9 + 52140*a^2*b^3*x^7 + 59136*a^3*b^2*x^5
+ 32340*a^4*b*x^3 + 6930*a^5*x + 3465*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6
+ 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a^5)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) -
a)/(b*x^2 + a)))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 +
5*a^4*b^7*x^2 + a^5*b^6), 1/1280*(1280*b^5*x^11 + 10615*a*b^4*x^9 + 26070*a^2*b^
3*x^7 + 29568*a^3*b^2*x^5 + 16170*a^4*b*x^3 + 3465*a^5*x - 3465*(b^5*x^10 + 5*a*
b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a^5)*sqrt(a/b)*arctan(
x/sqrt(a/b)))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^
4*b^7*x^2 + a^5*b^6)]

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Sympy [A]  time = 5.02746, size = 178, normalized size = 1.36 \[ \frac{693 \sqrt{- \frac{a}{b^{13}}} \log{\left (- b^{6} \sqrt{- \frac{a}{b^{13}}} + x \right )}}{512} - \frac{693 \sqrt{- \frac{a}{b^{13}}} \log{\left (b^{6} \sqrt{- \frac{a}{b^{13}}} + x \right )}}{512} + \frac{2185 a^{5} x + 9770 a^{4} b x^{3} + 16768 a^{3} b^{2} x^{5} + 13270 a^{2} b^{3} x^{7} + 4215 a b^{4} x^{9}}{1280 a^{5} b^{6} + 6400 a^{4} b^{7} x^{2} + 12800 a^{3} b^{8} x^{4} + 12800 a^{2} b^{9} x^{6} + 6400 a b^{10} x^{8} + 1280 b^{11} x^{10}} + \frac{x}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

693*sqrt(-a/b**13)*log(-b**6*sqrt(-a/b**13) + x)/512 - 693*sqrt(-a/b**13)*log(b*
*6*sqrt(-a/b**13) + x)/512 + (2185*a**5*x + 9770*a**4*b*x**3 + 16768*a**3*b**2*x
**5 + 13270*a**2*b**3*x**7 + 4215*a*b**4*x**9)/(1280*a**5*b**6 + 6400*a**4*b**7*
x**2 + 12800*a**3*b**8*x**4 + 12800*a**2*b**9*x**6 + 6400*a*b**10*x**8 + 1280*b*
*11*x**10) + x/b**6

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GIAC/XCAS [A]  time = 0.271582, size = 117, normalized size = 0.89 \[ -\frac{693 \, a \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} b^{6}} + \frac{x}{b^{6}} + \frac{4215 \, a b^{4} x^{9} + 13270 \, a^{2} b^{3} x^{7} + 16768 \, a^{3} b^{2} x^{5} + 9770 \, a^{4} b x^{3} + 2185 \, a^{5} x}{1280 \,{\left (b x^{2} + a\right )}^{5} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

-693/256*a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + x/b^6 + 1/1280*(4215*a*b^4*x^
9 + 13270*a^2*b^3*x^7 + 16768*a^3*b^2*x^5 + 9770*a^4*b*x^3 + 2185*a^5*x)/((b*x^2
+ a)^5*b^6)