3.2696 \(\int x^{-1-\frac{2 n}{3}} \left (a+b x^n\right )^{2/3} \, dx\)

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 = 0.128699, 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 \[ -\frac{3 b^{2/3} \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 n}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{n}-\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3}}{2 n} \]

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)

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Rubi in Sympy [A]  time = 24.0984, size = 151, normalized size = 1.32 \[ - \frac{b^{\frac{2}{3}} \log{\left (- \frac{\sqrt [3]{b} x^{\frac{n}{3}}}{\sqrt [3]{a + b x^{n}}} + 1 \right )}}{n} + \frac{b^{\frac{2}{3}} \log{\left (\frac{b^{\frac{2}{3}} x^{\frac{2 n}{3}}}{\left (a + b x^{n}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x^{\frac{n}{3}}}{\sqrt [3]{a + b x^{n}}} + 1 \right )}}{2 n} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x^{\frac{n}{3}}}{3 \sqrt [3]{a + b x^{n}}} + \frac{1}{3}\right ) \right )}}{n} - \frac{3 x^{- \frac{2 n}{3}} \left (a + b x^{n}\right )^{\frac{2}{3}}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0972262, size = 71, normalized size = 0.62 \[ -\frac{3 x^{-2 n/3} \left (-2 b x^n \sqrt [3]{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^n}{a}\right )+a+b x^n\right )}{2 n \sqrt [3]{a+b x^n}} \]

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*b*x^n*(1 + (b*x^n)/a)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3,
-((b*x^n)/a)]))/(2*n*x^((2*n)/3)*(a + b*x^n)^(1/3))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{x}^{-1-{\frac{2\,n}{3}}} \left ( a+b{x}^{n} \right ) ^{{\frac{2}{3}}}\, dx \]

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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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