3.1324 \(\int \frac{x^2}{a+b x^6} \, dx\)

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])

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Rubi [A]  time = 0.0121066, 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, Rules used = {275, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 \sqrt{a} \sqrt{b}} \]

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])

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^2}{a+b x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^3\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 \sqrt{a} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0054046, 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])

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Maple [A]  time = 0.003, size = 19, normalized size = 0.7 \begin{align*}{\frac{1}{3}\arctan \left ({b{x}^{3}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 1.68494, size = 162, normalized size = 5.59 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{b x^{6} - 2 \, \sqrt{-a b} x^{3} - a}{b x^{6} + a}\right )}{6 \, a b}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} x^{3}}{a}\right )}{3 \, a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.188683, size = 56, normalized size = 1.93 \begin{align*} - \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} \end{align*}

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

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Giac [A]  time = 1.16937, size = 24, normalized size = 0.83 \begin{align*} \frac{\arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b}} \end{align*}

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)