3.2702 \(\int \frac{x^{-1+\frac{n}{3}}}{\sqrt [3]{a+b x^n}} \, dx\)

Optimal. Leaf size=89 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{\sqrt [3]{b} n}-\frac{3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0401447, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {345, 239} \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{\sqrt [3]{b} n}-\frac{3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n/3)/(a + b*x^n)^(1/3),x]

[Out]

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

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

Mathematica [C]  time = 0.0162502, size = 56, normalized size = 0.63 \[ \frac{3 x^{n/3} \sqrt [3]{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^n}{a}\right )}{n \sqrt [3]{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^(n/3)*(1 + (b*x^n)/a)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -((b*x^n)/a)])/(n*(a + b*x^n)^(1/3))

________________________________________________________________________________________

Maple [F]  time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+{\frac{n}{3}}}{\frac{1}{\sqrt [3]{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [C]  time = 16.9825, size = 39, normalized size = 0.44 \begin{align*} \frac{x^{\frac{n}{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} n \Gamma \left (\frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(n/3)*gamma(1/3)*hyper((1/3, 1/3), (4/3,), b*x**n*exp_polar(I*pi)/a)/(a**(1/3)*n*gamma(4/3))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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