3.3025 \(\int \frac{1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx\)

Optimal. Leaf size=73 \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )} \]

[Out]

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

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Rubi [A]  time = 0.0548553, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.00033, size = 58, normalized size = 0.79 \[ \frac{x}{2 a \left (a + b \left (c x^{n}\right )^{\frac{2}{n}}\right )} + \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\sqrt{b} \left (c x^{n}\right )^{\frac{1}{n}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.2524, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.58, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{-2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(c*x^n)^(2/n))^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258476, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c^{\frac{2}{n}} x^{2} + a\right )} \log \left (\frac{2 \, a b c^{\frac{2}{n}} x +{\left (b c^{\frac{2}{n}} x^{2} - a\right )} \sqrt{-a b c^{\frac{2}{n}}}}{b c^{\frac{2}{n}} x^{2} + a}\right ) + 2 \, \sqrt{-a b c^{\frac{2}{n}}} x}{4 \,{\left (a b c^{\frac{2}{n}} x^{2} + a^{2}\right )} \sqrt{-a b c^{\frac{2}{n}}}}, \frac{{\left (b c^{\frac{2}{n}} x^{2} + a\right )} \arctan \left (\frac{\sqrt{a b c^{\frac{2}{n}}} x}{a}\right ) + \sqrt{a b c^{\frac{2}{n}}} x}{2 \,{\left (a b c^{\frac{2}{n}} x^{2} + a^{2}\right )} \sqrt{a b c^{\frac{2}{n}}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((b*c^(2/n)*x^2 + a)*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)) + 2*sqrt(-a*b*c^(2/n))*x)/((a*b*c^(2/n)*x^2 + a^2)
*sqrt(-a*b*c^(2/n))), 1/2*((b*c^(2/n)*x^2 + a)*arctan(sqrt(a*b*c^(2/n))*x/a) + s
qrt(a*b*c^(2/n))*x)/((a*b*c^(2/n)*x^2 + a^2)*sqrt(a*b*c^(2/n)))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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