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

Optimal. Leaf size=89 \[ \frac{x \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{a} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(x*(a + b*x^2))/(b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (Sqrt[a]*(a + b*x^2)*ArcTa
n[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.104396, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{x \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\sqrt{a} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^2))/(b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (Sqrt[a]*(a + b*x^2)*ArcTa
n[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*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. \[ \int \frac{x^{2}}{\sqrt{\left (a + b x^{2}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0327067, size = 54, normalized size = 0.61 \[ \frac{\left (a+b x^2\right ) \left (\sqrt{b} x-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{b^{3/2} \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 48, normalized size = 0.5 \[{\frac{b{x}^{2}+a}{b} \left ( x\sqrt{ab}-a\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/((b*x^2+a)^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268572, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 2 \, x}{2 \, b}, -\frac{\sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - x}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.19954, size = 56, normalized size = 0.63 \[ \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (- b \sqrt{- \frac{a}{b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (b \sqrt{- \frac{a}{b^{3}}} + x \right )}}{2} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271855, size = 57, normalized size = 0.64 \[ -\frac{a \arctan \left (\frac{b x}{\sqrt{a b}}\right ){\rm sign}\left (b x^{2} + a\right )}{\sqrt{a b} b} + \frac{x{\rm sign}\left (b x^{2} + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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