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]

-3/2*ln(b^(1/3)*x^(1/3*n)-(a+b*x^n)^(1/3))/b^(1/3)/n+arctan(1/3*(1+2*b^(1/3)*x^(1/3*n)/(a+b*x^n)^(1/3))*3^(1/2
))*3^(1/2)/b^(1/3)/n

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Rubi [A]  time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [C]  time = 0.02, 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))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {1}{3} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac {1}{3}}}\,{d x} \]

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)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {n}{3}-1}}{\left (b \,x^{n}+a \right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {1}{3} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac {1}{3}}}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{\frac {n}{3}-1}}{{\left (a+b\,x^n\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n/3 - 1)/(a + b*x^n)^(1/3),x)

[Out]

int(x^(n/3 - 1)/(a + b*x^n)^(1/3), x)

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sympy [C]  time = 7.81, size = 39, normalized size = 0.44 \[ \frac {x^{\frac {n}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} n \Gamma \left (\frac {4}{3}\right )} \]

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))

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