3.1322 \(\int \frac{x^2}{a+b x^6} \, dx\)

Optimal. Leaf size=29 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 \sqrt{a} \sqrt{b}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0372156, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^6),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.14479, size = 26, normalized size = 0.9 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{3}}{\sqrt{a}} \right )}}{3 \sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**6+a),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0111824, size = 29, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^6),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 19, normalized size = 0.7 \[{\frac{1}{3}\arctan \left ({b{x}^{3}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^6+a),x)

[Out]

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

_______________________________________________________________________________________

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^6 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.225422, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, a b x^{3} +{\left (b x^{6} - a\right )} \sqrt{-a b}}{b x^{6} + a}\right )}{6 \, \sqrt{-a b}}, \frac{\arctan \left (\frac{\sqrt{a b} x^{3}}{a}\right )}{3 \, \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^6 + a),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.545832, size = 56, normalized size = 1.93 \[ - \frac{\sqrt{- \frac{1}{a b}} \log{\left (- a \sqrt{- \frac{1}{a b}} + x^{3} \right )}}{6} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (a \sqrt{- \frac{1}{a b}} + x^{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**6+a),x)

[Out]

-sqrt(-1/(a*b))*log(-a*sqrt(-1/(a*b)) + x**3)/6 + sqrt(-1/(a*b))*log(a*sqrt(-1/(
a*b)) + x**3)/6

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224465, size = 24, normalized size = 0.83 \[ \frac{\arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^6 + a),x, algorithm="giac")

[Out]

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