3.1337 \(\int \frac{x^2}{(a+b x^6)^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 a^{3/2} \sqrt{b}}+\frac{x^3}{6 a \left (a+b x^6\right )} \]

[Out]

x^3/(6*a*(a + b*x^6)) + ArcTan[(Sqrt[b]*x^3)/Sqrt[a]]/(6*a^(3/2)*Sqrt[b])

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Rubi [A]  time = 0.0222814, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 a^{3/2} \sqrt{b}}+\frac{x^3}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^6)^2,x]

[Out]

x^3/(6*a*(a + b*x^6)) + ArcTan[(Sqrt[b]*x^3)/Sqrt[a]]/(6*a^(3/2)*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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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

Mathematica [A]  time = 0.0258621, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 a^{3/2} \sqrt{b}}+\frac{x^3}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^6)^2,x]

[Out]

x^3/(6*a*(a + b*x^6)) + ArcTan[(Sqrt[b]*x^3)/Sqrt[a]]/(6*a^(3/2)*Sqrt[b])

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Maple [A]  time = 0.006, size = 40, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{6\,a \left ( b{x}^{6}+a \right ) }}+{\frac{1}{6\,a}\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)^2,x)

[Out]

1/6*x^3/a/(b*x^6+a)+1/6/a/(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4801, size = 273, normalized size = 5.57 \begin{align*} \left [\frac{2 \, a b x^{3} -{\left (b x^{6} + a\right )} \sqrt{-a b} \log \left (\frac{b x^{6} - 2 \, \sqrt{-a b} x^{3} - a}{b x^{6} + a}\right )}{12 \,{\left (a^{2} b^{2} x^{6} + a^{3} b\right )}}, \frac{a b x^{3} +{\left (b x^{6} + a\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x^{3}}{a}\right )}{6 \,{\left (a^{2} b^{2} x^{6} + a^{3} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(2*a*b*x^3 - (b*x^6 + a)*sqrt(-a*b)*log((b*x^6 - 2*sqrt(-a*b)*x^3 - a)/(b*x^6 + a)))/(a^2*b^2*x^6 + a^3*
b), 1/6*(a*b*x^3 + (b*x^6 + a)*sqrt(a*b)*arctan(sqrt(a*b)*x^3/a))/(a^2*b^2*x^6 + a^3*b)]

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Sympy [B]  time = 0.950025, size = 83, normalized size = 1.69 \begin{align*} \frac{x^{3}}{6 a^{2} + 6 a b x^{6}} - \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + x^{3} \right )}}{12} + \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + x^{3} \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x**6+a)**2,x)

[Out]

x**3/(6*a**2 + 6*a*b*x**6) - sqrt(-1/(a**3*b))*log(-a**2*sqrt(-1/(a**3*b)) + x**3)/12 + sqrt(-1/(a**3*b))*log(
a**2*sqrt(-1/(a**3*b)) + x**3)/12

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Giac [A]  time = 1.14281, size = 53, normalized size = 1.08 \begin{align*} \frac{x^{3}}{6 \,{\left (b x^{6} + a\right )} a} + \frac{\arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{6 \, \sqrt{a b} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^3/((b*x^6 + a)*a) + 1/6*arctan(b*x^3/sqrt(a*b))/(sqrt(a*b)*a)