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

Optimal. Leaf size=129 \[ -\frac{a x \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^3 \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

-((a*x*(a + b*x^2))/(b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])) + (x^3*(a + b*x^2))/(
3*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (a^(3/2)*(a + b*x^2)*ArcTan[(Sqrt[b]*x)/S
qrt[a]])/(b^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.139212, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{a x \left (a+b x^2\right )}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x^3 \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-((a*x*(a + b*x^2))/(b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])) + (x^3*(a + b*x^2))/(
3*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (a^(3/2)*(a + b*x^2)*ArcTan[(Sqrt[b]*x)/S
qrt[a]])/(b^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{\left (a + b x^{2}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4/sqrt((a + b*x**2)**2), x)

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Mathematica [A]  time = 0.0486806, size = 66, normalized size = 0.51 \[ \frac{\left (a+b x^2\right ) \left (3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x \left (b x^2-3 a\right )\right )}{3 b^{5/2} \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 63, normalized size = 0.5 \[{\frac{b{x}^{2}+a}{3\,{b}^{2}} \left ( \sqrt{ab}{x}^{3}b-3\,\sqrt{ab}xa+3\,{a}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/((b*x^2+a)^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264613, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, b x^{3} + 3 \, a \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 6 \, a x}{6 \, b^{2}}, \frac{b x^{3} + 3 \, a \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 3 \, a x}{3 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.27894, size = 80, normalized size = 0.62 \[ - \frac{a x}{b^{2}} - \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x - \frac{b^{2} \sqrt{- \frac{a^{3}}{b^{5}}}}{a} \right )}}{2} + \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x + \frac{b^{2} \sqrt{- \frac{a^{3}}{b^{5}}}}{a} \right )}}{2} + \frac{x^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273083, size = 86, normalized size = 0.67 \[ \frac{a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right ){\rm sign}\left (b x^{2} + a\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} x^{3}{\rm sign}\left (b x^{2} + a\right ) - 3 \, a b x{\rm sign}\left (b x^{2} + a\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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