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

Optimal. Leaf size=85 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x}{16 a^2 b \left (a+b x^2\right )}+\frac{x}{24 a b \left (a+b x^2\right )^2}-\frac{x}{6 b \left (a+b x^2\right )^3} \]

[Out]

-x/(6*b*(a + b*x^2)^3) + x/(24*a*b*(a + b*x^2)^2) + x/(16*a^2*b*(a + b*x^2)) + A
rcTan[(Sqrt[b]*x)/Sqrt[a]]/(16*a^(5/2)*b^(3/2))

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Rubi [A]  time = 0.100726, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x}{16 a^2 b \left (a+b x^2\right )}+\frac{x}{24 a b \left (a+b x^2\right )^2}-\frac{x}{6 b \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-x/(6*b*(a + b*x^2)^3) + x/(24*a*b*(a + b*x^2)^2) + x/(16*a^2*b*(a + b*x^2)) + A
rcTan[(Sqrt[b]*x)/Sqrt[a]]/(16*a^(5/2)*b^(3/2))

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Rubi in Sympy [A]  time = 20.3255, size = 68, normalized size = 0.8 \[ - \frac{x}{6 b \left (a + b x^{2}\right )^{3}} + \frac{x}{24 a b \left (a + b x^{2}\right )^{2}} + \frac{x}{16 a^{2} b \left (a + b x^{2}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(6*b*(a + b*x**2)**3) + x/(24*a*b*(a + b*x**2)**2) + x/(16*a**2*b*(a + b*x**2
)) + atan(sqrt(b)*x/sqrt(a))/(16*a**(5/2)*b**(3/2))

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Mathematica [A]  time = 0.0755909, size = 69, normalized size = 0.81 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{-3 a^2 x+8 a b x^3+3 b^2 x^5}{48 a^2 b \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a^2*x + 8*a*b*x^3 + 3*b^2*x^5)/(48*a^2*b*(a + b*x^2)^3) + ArcTan[(Sqrt[b]*x)
/Sqrt[a]]/(16*a^(5/2)*b^(3/2))

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Maple [A]  time = 0.012, size = 58, normalized size = 0.7 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{3}} \left ({\frac{b{x}^{5}}{16\,{a}^{2}}}+{\frac{{x}^{3}}{6\,a}}-{\frac{x}{16\,b}} \right ) }+{\frac{1}{16\,{a}^{2}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/16*b/a^2*x^5+1/6/a*x^3-1/16/b*x)/(b*x^2+a)^3+1/16/a^2/b/(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^2/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275281, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \, b^{2} x^{5} + 8 \, a b x^{3} - 3 \, a^{2} x\right )} \sqrt{-a b}}{96 \,{\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \sqrt{-a b}}, \frac{3 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, b^{2} x^{5} + 8 \, a b x^{3} - 3 \, a^{2} x\right )} \sqrt{a b}}{48 \,{\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(3*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*log((2*a*b*x + (b*x^2 - a)*
sqrt(-a*b))/(b*x^2 + a)) + 2*(3*b^2*x^5 + 8*a*b*x^3 - 3*a^2*x)*sqrt(-a*b))/((a^2
*b^4*x^6 + 3*a^3*b^3*x^4 + 3*a^4*b^2*x^2 + a^5*b)*sqrt(-a*b)), 1/48*(3*(b^3*x^6
+ 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*arctan(sqrt(a*b)*x/a) + (3*b^2*x^5 + 8*a*b*x^
3 - 3*a^2*x)*sqrt(a*b))/((a^2*b^4*x^6 + 3*a^3*b^3*x^4 + 3*a^4*b^2*x^2 + a^5*b)*s
qrt(a*b))]

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Sympy [A]  time = 2.15994, size = 139, normalized size = 1.64 \[ - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \log{\left (- a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \log{\left (a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{32} + \frac{- 3 a^{2} x + 8 a b x^{3} + 3 b^{2} x^{5}}{48 a^{5} b + 144 a^{4} b^{2} x^{2} + 144 a^{3} b^{3} x^{4} + 48 a^{2} b^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**5*b**3))*log(-a**3*b*sqrt(-1/(a**5*b**3)) + x)/32 + sqrt(-1/(a**5*b
**3))*log(a**3*b*sqrt(-1/(a**5*b**3)) + x)/32 + (-3*a**2*x + 8*a*b*x**3 + 3*b**2
*x**5)/(48*a**5*b + 144*a**4*b**2*x**2 + 144*a**3*b**3*x**4 + 48*a**2*b**4*x**6)

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GIAC/XCAS [A]  time = 0.271589, size = 84, normalized size = 0.99 \[ \frac{\arctan \left (\frac{b x}{\sqrt{a b}}\right )}{16 \, \sqrt{a b} a^{2} b} + \frac{3 \, b^{2} x^{5} + 8 \, a b x^{3} - 3 \, a^{2} x}{48 \,{\left (b x^{2} + a\right )}^{3} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b) + 1/48*(3*b^2*x^5 + 8*a*b*x^3 - 3*a
^2*x)/((b*x^2 + a)^3*a^2*b)