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

Optimal. Leaf size=55 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{5/2}}-\frac{x^3}{2 b \left (a+b x^2\right )}+\frac{3 x}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0708196, antiderivative size = 55, 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 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{5/2}}-\frac{x^3}{2 b \left (a+b x^2\right )}+\frac{3 x}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.4335, size = 48, normalized size = 0.87 \[ - \frac{3 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{5}{2}}} - \frac{x^{3}}{2 b \left (a + b x^{2}\right )} + \frac{3 x}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0588881, size = 51, normalized size = 0.93 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{5/2}}+\frac{a x}{2 b^2 \left (a+b x^2\right )}+\frac{x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 43, normalized size = 0.8 \[{\frac{x}{{b}^{2}}}+{\frac{ax}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,a}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26314, size = 1, normalized size = 0.02 \[ \left [\frac{4 \, b x^{3} + 3 \,{\left (b x^{2} + a\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 6 \, a x}{4 \,{\left (b^{3} x^{2} + a b^{2}\right )}}, \frac{2 \, b x^{3} - 3 \,{\left (b x^{2} + a\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 3 \, a x}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.45644, size = 83, normalized size = 1.51 \[ \frac{a x}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (- b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{4} - \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{4} + \frac{x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271422, size = 57, normalized size = 1.04 \[ -\frac{3 \, a \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} + \frac{a x}{2 \,{\left (b x^{2} + a\right )} b^{2}} + \frac{x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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