3.1331 \(\int \frac{x^8}{(a+b x^6)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0236848, 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, 288, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 \sqrt{a} b^{3/2}}-\frac{x^3}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^6)^2,x]

[Out]

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

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 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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^8}{\left (a+b x^6\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right )^2} \, dx,x,x^3\right )\\ &=-\frac{x^3}{6 b \left (a+b x^6\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^3\right )}{6 b}\\ &=-\frac{x^3}{6 b \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 \sqrt{a} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.024416, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 \sqrt{a} b^{3/2}}-\frac{x^3}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^6)^2,x]

[Out]

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

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Maple [A]  time = 0.009, size = 40, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{6\,b \left ( b{x}^{6}+a \right ) }}+{\frac{1}{6\,b}\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^8/(b*x^6+a)^2,x)

[Out]

-1/6*x^3/b/(b*x^6+a)+1/6/b/(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^8/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.47044, size = 275, normalized size = 5.61 \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 b^{3} x^{6} + a^{2} b^{2}\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 b^{3} x^{6} + a^{2} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(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*b^3*x^6 + a^2*b
^2), -1/6*(a*b*x^3 - (b*x^6 + a)*sqrt(a*b)*arctan(sqrt(a*b)*x^3/a))/(a*b^3*x^6 + a^2*b^2)]

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Sympy [B]  time = 0.97011, size = 83, normalized size = 1.69 \begin{align*} - \frac{x^{3}}{6 a b + 6 b^{2} x^{6}} - \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x^{3} \right )}}{12} + \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x^{3} \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**6+a)**2,x)

[Out]

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

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Giac [A]  time = 1.17586, size = 53, normalized size = 1.08 \begin{align*} -\frac{x^{3}}{6 \,{\left (b x^{6} + a\right )} b} + \frac{\arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{6 \, \sqrt{a b} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^6+a)^2,x, algorithm="giac")

[Out]

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