3.528 \(\int \frac{x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=123 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{5/2} b^{7/2}}+\frac{3 x}{256 a^2 b^3 \left (a+b x^2\right )}+\frac{x}{128 a b^3 \left (a+b x^2\right )^2}-\frac{x}{32 b^3 \left (a+b x^2\right )^3}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}-\frac{x^5}{10 b \left (a+b x^2\right )^5} \]

[Out]

-x^5/(10*b*(a + b*x^2)^5) - x^3/(16*b^2*(a + b*x^2)^4) - x/(32*b^3*(a + b*x^2)^3
) + x/(128*a*b^3*(a + b*x^2)^2) + (3*x)/(256*a^2*b^3*(a + b*x^2)) + (3*ArcTan[(S
qrt[b]*x)/Sqrt[a]])/(256*a^(5/2)*b^(7/2))

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Rubi [A]  time = 0.180164, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{5/2} b^{7/2}}+\frac{3 x}{256 a^2 b^3 \left (a+b x^2\right )}+\frac{x}{128 a b^3 \left (a+b x^2\right )^2}-\frac{x}{32 b^3 \left (a+b x^2\right )^3}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}-\frac{x^5}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-x^5/(10*b*(a + b*x^2)^5) - x^3/(16*b^2*(a + b*x^2)^4) - x/(32*b^3*(a + b*x^2)^3
) + x/(128*a*b^3*(a + b*x^2)^2) + (3*x)/(256*a^2*b^3*(a + b*x^2)) + (3*ArcTan[(S
qrt[b]*x)/Sqrt[a]])/(256*a^(5/2)*b^(7/2))

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Rubi in Sympy [A]  time = 33.9606, size = 109, normalized size = 0.89 \[ - \frac{x^{5}}{10 b \left (a + b x^{2}\right )^{5}} - \frac{x^{3}}{16 b^{2} \left (a + b x^{2}\right )^{4}} - \frac{x}{32 b^{3} \left (a + b x^{2}\right )^{3}} + \frac{x}{128 a b^{3} \left (a + b x^{2}\right )^{2}} + \frac{3 x}{256 a^{2} b^{3} \left (a + b x^{2}\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 a^{\frac{5}{2}} b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-x**5/(10*b*(a + b*x**2)**5) - x**3/(16*b**2*(a + b*x**2)**4) - x/(32*b**3*(a +
b*x**2)**3) + x/(128*a*b**3*(a + b*x**2)**2) + 3*x/(256*a**2*b**3*(a + b*x**2))
+ 3*atan(sqrt(b)*x/sqrt(a))/(256*a**(5/2)*b**(7/2))

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Mathematica [A]  time = 0.105152, size = 91, normalized size = 0.74 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{5/2} b^{7/2}}+\frac{-15 a^4 x-70 a^3 b x^3-128 a^2 b^2 x^5+70 a b^3 x^7+15 b^4 x^9}{1280 a^2 b^3 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(-15*a^4*x - 70*a^3*b*x^3 - 128*a^2*b^2*x^5 + 70*a*b^3*x^7 + 15*b^4*x^9)/(1280*a
^2*b^3*(a + b*x^2)^5) + (3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(5/2)*b^(7/2))

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Maple [A]  time = 0.015, size = 78, normalized size = 0.6 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{5}} \left ({\frac{3\,b{x}^{9}}{256\,{a}^{2}}}+{\frac{7\,{x}^{7}}{128\,a}}-{\frac{{x}^{5}}{10\,b}}-{\frac{7\,a{x}^{3}}{128\,{b}^{2}}}-{\frac{3\,{a}^{2}x}{256\,{b}^{3}}} \right ) }+{\frac{3}{256\,{a}^{2}{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

(3/256*b/a^2*x^9+7/128/a*x^7-1/10/b*x^5-7/128*a/b^2*x^3-3/256*a^2/b^3*x)/(b*x^2+
a)^5+3/256/a^2/b^3/(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^6/(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 = 0.272864, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (15 \, b^{4} x^{9} + 70 \, a b^{3} x^{7} - 128 \, a^{2} b^{2} x^{5} - 70 \, a^{3} b x^{3} - 15 \, a^{4} x\right )} \sqrt{-a b}}{2560 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \sqrt{-a b}}, \frac{15 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (15 \, b^{4} x^{9} + 70 \, a b^{3} x^{7} - 128 \, a^{2} b^{2} x^{5} - 70 \, a^{3} b x^{3} - 15 \, a^{4} x\right )} \sqrt{a b}}{1280 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

[1/2560*(15*(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)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(15*b^4*x^9 +
70*a*b^3*x^7 - 128*a^2*b^2*x^5 - 70*a^3*b*x^3 - 15*a^4*x)*sqrt(-a*b))/((a^2*b^8
*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^
3)*sqrt(-a*b)), 1/1280*(15*(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)*arctan(sqrt(a*b)*x/a) + (15*b^4*x^9 + 70*a*b^3*x^7 - 1
28*a^2*b^2*x^5 - 70*a^3*b*x^3 - 15*a^4*x)*sqrt(a*b))/((a^2*b^8*x^10 + 5*a^3*b^7*
x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)*sqrt(a*b))]

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Sympy [A]  time = 4.26422, size = 196, normalized size = 1.59 \[ - \frac{3 \sqrt{- \frac{1}{a^{5} b^{7}}} \log{\left (- a^{3} b^{3} \sqrt{- \frac{1}{a^{5} b^{7}}} + x \right )}}{512} + \frac{3 \sqrt{- \frac{1}{a^{5} b^{7}}} \log{\left (a^{3} b^{3} \sqrt{- \frac{1}{a^{5} b^{7}}} + x \right )}}{512} + \frac{- 15 a^{4} x - 70 a^{3} b x^{3} - 128 a^{2} b^{2} x^{5} + 70 a b^{3} x^{7} + 15 b^{4} x^{9}}{1280 a^{7} b^{3} + 6400 a^{6} b^{4} x^{2} + 12800 a^{5} b^{5} x^{4} + 12800 a^{4} b^{6} x^{6} + 6400 a^{3} b^{7} x^{8} + 1280 a^{2} b^{8} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-3*sqrt(-1/(a**5*b**7))*log(-a**3*b**3*sqrt(-1/(a**5*b**7)) + x)/512 + 3*sqrt(-1
/(a**5*b**7))*log(a**3*b**3*sqrt(-1/(a**5*b**7)) + x)/512 + (-15*a**4*x - 70*a**
3*b*x**3 - 128*a**2*b**2*x**5 + 70*a*b**3*x**7 + 15*b**4*x**9)/(1280*a**7*b**3 +
6400*a**6*b**4*x**2 + 12800*a**5*b**5*x**4 + 12800*a**4*b**6*x**6 + 6400*a**3*b
**7*x**8 + 1280*a**2*b**8*x**10)

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GIAC/XCAS [A]  time = 0.270989, size = 113, normalized size = 0.92 \[ \frac{3 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a^{2} b^{3}} + \frac{15 \, b^{4} x^{9} + 70 \, a b^{3} x^{7} - 128 \, a^{2} b^{2} x^{5} - 70 \, a^{3} b x^{3} - 15 \, a^{4} x}{1280 \,{\left (b x^{2} + a\right )}^{5} a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

3/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^3) + 1/1280*(15*b^4*x^9 + 70*a*b^3*
x^7 - 128*a^2*b^2*x^5 - 70*a^3*b*x^3 - 15*a^4*x)/((b*x^2 + a)^5*a^2*b^3)