3.2695 \(\int \frac{x^{-1+\frac{n}{3}}}{\sqrt [3]{a+b x^n}} \, dx\)

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)

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Rubi [A]  time = 0.100837, 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 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{\sqrt [3]{b} n}-\frac{3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n} \]

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)

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Rubi in Sympy [A]  time = 19.1853, size = 129, normalized size = 1.45 \[ - \frac{\log{\left (- \frac{\sqrt [3]{b} x^{\frac{n}{3}}}{\sqrt [3]{a + b x^{n}}} + 1 \right )}}{\sqrt [3]{b} n} + \frac{\log{\left (\frac{b^{\frac{2}{3}} x^{\frac{2 n}{3}}}{\left (a + b x^{n}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x^{\frac{n}{3}}}{\sqrt [3]{a + b x^{n}}} + 1 \right )}}{2 \sqrt [3]{b} n} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x^{\frac{n}{3}}}{3 \sqrt [3]{a + b x^{n}}} + \frac{1}{3}\right ) \right )}}{\sqrt [3]{b} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+1/3*n)/(a+b*x**n)**(1/3),x)

[Out]

-log(-b**(1/3)*x**(n/3)/(a + b*x**n)**(1/3) + 1)/(b**(1/3)*n) + log(b**(2/3)*x**
(2*n/3)/(a + b*x**n)**(2/3) + b**(1/3)*x**(n/3)/(a + b*x**n)**(1/3) + 1)/(2*b**(
1/3)*n) + sqrt(3)*atan(sqrt(3)*(2*b**(1/3)*x**(n/3)/(3*(a + b*x**n)**(1/3)) + 1/
3))/(b**(1/3)*n)

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Mathematica [C]  time = 0.0555622, size = 57, normalized size = 0.64 \[ \frac{3 x^{n/3} \sqrt [3]{\frac{a+b x^n}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^n}{a}\right )}{n \sqrt [3]{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

(3*x^(n/3)*((a + 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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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{1{x}^{-1+{\frac{n}{3}}}{\frac{1}{\sqrt [3]{a+b{x}^{n}}}}}\, dx \]

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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