3.490 \(\int \frac{1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx\)

Optimal. Leaf size=57 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left (a+b x^2\right )} \]

[Out]

-3/(2*a^2*x) + 1/(2*a*x*(a + b*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
2*a^(5/2))

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Rubi [A]  time = 0.0719584, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]

[Out]

-3/(2*a^2*x) + 1/(2*a*x*(a + b*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
2*a^(5/2))

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Rubi in Sympy [A]  time = 19.6246, size = 48, normalized size = 0.84 \[ \frac{1}{2 a x \left (a + b x^{2}\right )} - \frac{3}{2 a^{2} x} - \frac{3 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

1/(2*a*x*(a + b*x**2)) - 3/(2*a**2*x) - 3*sqrt(b)*atan(sqrt(b)*x/sqrt(a))/(2*a**
(5/2))

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Mathematica [A]  time = 0.0648362, size = 54, normalized size = 0.95 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{b x}{2 a^2 \left (a+b x^2\right )}-\frac{1}{a^2 x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]

[Out]

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

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Maple [A]  time = 0.014, size = 46, normalized size = 0.8 \[ -{\frac{bx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,b}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{{a}^{2}x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

-1/2*b/a^2*x/(b*x^2+a)-3/2*b/a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))-1/a^2/x

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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(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263742, size = 1, normalized size = 0.02 \[ \left [-\frac{6 \, b x^{2} - 3 \,{\left (b x^{3} + a x\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 4 \, a}{4 \,{\left (a^{2} b x^{3} + a^{3} x\right )}}, -\frac{3 \, b x^{2} + 3 \,{\left (b x^{3} + a x\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 2 \, a}{2 \,{\left (a^{2} b x^{3} + a^{3} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^2),x, algorithm="fricas")

[Out]

[-1/4*(6*b*x^2 - 3*(b*x^3 + a*x)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(
b*x^2 + a)) + 4*a)/(a^2*b*x^3 + a^3*x), -1/2*(3*b*x^2 + 3*(b*x^3 + a*x)*sqrt(b/a
)*arctan(b*x/(a*sqrt(b/a))) + 2*a)/(a^2*b*x^3 + a^3*x)]

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Sympy [A]  time = 1.65214, size = 90, normalized size = 1.58 \[ \frac{3 \sqrt{- \frac{b}{a^{5}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b}{a^{5}}}}{b} + x \right )}}{4} - \frac{3 \sqrt{- \frac{b}{a^{5}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b}{a^{5}}}}{b} + x \right )}}{4} - \frac{2 a + 3 b x^{2}}{2 a^{3} x + 2 a^{2} b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

3*sqrt(-b/a**5)*log(-a**3*sqrt(-b/a**5)/b + x)/4 - 3*sqrt(-b/a**5)*log(a**3*sqrt
(-b/a**5)/b + x)/4 - (2*a + 3*b*x**2)/(2*a**3*x + 2*a**2*b*x**3)

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GIAC/XCAS [A]  time = 0.270102, size = 63, normalized size = 1.11 \[ -\frac{3 \, b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} - \frac{3 \, b x^{2} + 2 \, a}{2 \,{\left (b x^{3} + a x\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^2),x, algorithm="giac")

[Out]

-3/2*b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) - 1/2*(3*b*x^2 + 2*a)/((b*x^3 + a*x
)*a^2)