3.21 \(\int \frac{x^2}{\left (a x+b x^3\right )^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0328066, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a*x + b*x^3)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.94367, size = 36, normalized size = 0.8 \[ \frac{x}{2 a \left (a + b x^{2}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**3+a*x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0418397, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a*x + b*x^3)^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \[{\frac{x}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{1}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^3+a*x)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21109, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b x^{2} + a\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \, \sqrt{-a b} x}{4 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{-a b}}, \frac{{\left (b x^{2} + a\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + \sqrt{a b} x}{2 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^3 + a*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.43121, size = 78, normalized size = 1.73 \[ \frac{x}{2 a^{2} + 2 a b x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**3+a*x)**2,x)

[Out]

x/(2*a**2 + 2*a*b*x**2) - sqrt(-1/(a**3*b))*log(-a**2*sqrt(-1/(a**3*b)) + x)/4 +
sqrt(-1/(a**3*b))*log(a**2*sqrt(-1/(a**3*b)) + x)/4

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GIAC/XCAS [A]  time = 0.218348, size = 47, normalized size = 1.04 \[ \frac{\arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a} + \frac{x}{2 \,{\left (b x^{2} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^3 + a*x)^2,x, algorithm="giac")

[Out]

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