3.3024 \(\int \frac{1}{a+b \left (c x^n\right )^{2/n}} \, dx\)

Optimal. Leaf size=44 \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

(x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(c*x^n)^n^(-1))

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Rubi [A]  time = 0.0307916, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(x*ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(c*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 3.99883, size = 39, normalized size = 0.89 \[ \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\sqrt{b} \left (c x^{n}\right )^{\frac{1}{n}}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c*x**n)**(-1/n)*atan(sqrt(b)*(c*x**n)**(1/n)/sqrt(a))/(sqrt(a)*sqrt(b))

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Mathematica [A]  time = 4.6918, size = 0, normalized size = 0. \[ \int \frac{1}{a+b \left (c x^n\right )^{2/n}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.185, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{-1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(c*x^n)^(2/n)),x)

[Out]

int(1/(a+b*(c*x^n)^(2/n)),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(1/((c*x^n)^(2/n)*b + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242305, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{2 \, a b c^{\frac{2}{n}} x +{\left (b c^{\frac{2}{n}} x^{2} - a\right )} \sqrt{-a b c^{\frac{2}{n}}}}{b c^{\frac{2}{n}} x^{2} + a}\right )}{2 \, \sqrt{-a b c^{\frac{2}{n}}}}, \frac{\arctan \left (\frac{\sqrt{a b c^{\frac{2}{n}}} x}{a}\right )}{\sqrt{a b c^{\frac{2}{n}}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log((2*a*b*c^(2/n)*x + (b*c^(2/n)*x^2 - a)*sqrt(-a*b*c^(2/n)))/(b*c^(2/n)*x
^2 + a))/sqrt(-a*b*c^(2/n)), arctan(sqrt(a*b*c^(2/n))*x/a)/sqrt(a*b*c^(2/n))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*(c*x**n)**(2/n)),x)

[Out]

Integral(1/(a + b*(c*x**n)**(2/n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^n)^(2/n)*b + a), x)