3.2703 \(\int x^{-1-\frac{2 n}{3}} (a+b x^n)^{2/3} \, dx\)

Optimal. Leaf size=114 \[ -\frac{3 b^{2/3} \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 n}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{n}-\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3}}{2 n} \]

[Out]

(-3*(a + b*x^n)^(2/3))/(2*n*x^((2*n)/3)) + (Sqrt[3]*b^(2/3)*ArcTan[(1 + (2*b^(1/3)*x^(n/3))/(a + b*x^n)^(1/3))
/Sqrt[3]])/n - (3*b^(2/3)*Log[b^(1/3)*x^(n/3) - (a + b*x^n)^(1/3)])/(2*n)

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Rubi [A]  time = 0.0541963, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {349, 345, 239} \[ -\frac{3 b^{2/3} \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 n}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{n}-\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - (2*n)/3)*(a + b*x^n)^(2/3),x]

[Out]

(-3*(a + b*x^n)^(2/3))/(2*n*x^((2*n)/3)) + (Sqrt[3]*b^(2/3)*ArcTan[(1 + (2*b^(1/3)*x^(n/3))/(a + b*x^n)^(1/3))
/Sqrt[3]])/n - (3*b^(2/3)*Log[b^(1/3)*x^(n/3) - (a + b*x^n)^(1/3)])/(2*n)

Rule 349

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

Rule 345

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

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [C]  time = 0.0186392, size = 58, normalized size = 0.51 \[ -\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{b x^n}{a}\right )}{2 n \left (\frac{b x^n}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - (2*n)/3)*(a + b*x^n)^(2/3),x]

[Out]

(-3*(a + b*x^n)^(2/3)*Hypergeometric2F1[-2/3, -2/3, 1/3, -((b*x^n)/a)])/(2*n*x^((2*n)/3)*(1 + (b*x^n)/a)^(2/3)
)

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-{\frac{2\,n}{3}}} \left ( a+b{x}^{n} \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-2/3*n)*(a+b*x^n)^(2/3),x)

[Out]

int(x^(-1-2/3*n)*(a+b*x^n)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{\frac{2}{3}} x^{-\frac{2}{3} \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(2/3)*x^(-2/3*n - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2/3*n)*(a+b*x^n)^(2/3),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-2/3*n)*(a+b*x**n)**(2/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{\frac{2}{3}} x^{-\frac{2}{3} \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2/3*n)*(a+b*x^n)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(2/3)*x^(-2/3*n - 1), x)