3.1318 \(\int \frac{x^8}{a+b x^6} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^3}{3 b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0208056, antiderivative size = 40, 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, 321, 205} \[ \frac{x^3}{3 b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^6),x]

[Out]

x^3/(3*b) - (Sqrt[a]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]])/(3*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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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}{a+b x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{a+b x^2} \, dx,x,x^3\right )\\ &=\frac{x^3}{3 b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^3\right )}{3 b}\\ &=\frac{x^3}{3 b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0139895, size = 40, normalized size = 1. \[ \frac{x^3}{3 b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^6),x]

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{3\,b}}-{\frac{a}{3\,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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67614, size = 180, normalized size = 4.5 \begin{align*} \left [\frac{2 \, x^{3} + \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{6} - 2 \, b x^{3} \sqrt{-\frac{a}{b}} - a}{b x^{6} + a}\right )}{6 \, b}, \frac{x^{3} - \sqrt{\frac{a}{b}} \arctan \left (\frac{b x^{3} \sqrt{\frac{a}{b}}}{a}\right )}{3 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.379238, size = 63, normalized size = 1.58 \begin{align*} \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (- b \sqrt{- \frac{a}{b^{3}}} + x^{3} \right )}}{6} - \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (b \sqrt{- \frac{a}{b^{3}}} + x^{3} \right )}}{6} + \frac{x^{3}}{3 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19006, size = 42, normalized size = 1.05 \begin{align*} \frac{x^{3}}{3 \, b} - \frac{a \arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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