3.3009 \(\int \frac{1}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

Optimal. Leaf size=20 \[ \frac{x}{a^2+a b \left (c x^n\right )^{\frac{1}{n}}} \]

[Out]

x/(a^2 + a*b*(c*x^n)^n^(-1))

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Rubi [A]  time = 0.0213441, antiderivative size = 32, normalized size of antiderivative = 1.6, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{x \left (c x^n\right )^{-1/n}}{b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(x/(b*(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))))

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Rubi in Sympy [A]  time = 2.32961, size = 24, normalized size = 1.2 \[ - \frac{x \left (c x^{n}\right )^{- \frac{1}{n}}}{b \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(c*x**n)**(-1/n)/(b*(a + b*(c*x**n)**(1/n)))

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Mathematica [A]  time = 0.0214296, size = 33, normalized size = 1.65 \[ -\frac{x \left (c x^n\right )^{-1/n}}{a b+b^2 \left (c x^n\right )^{\frac{1}{n}}} \]

Antiderivative was successfully verified.

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

[Out]

-(x/((c*x^n)^n^(-1)*(a*b + b^2*(c*x^n)^n^(-1))))

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Maple [C]  time = 0.037, size = 107, normalized size = 5.4 \[{\frac{x}{a} \left ( a+b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(c*x^n)^(1/n))^2,x)

[Out]

x/a/(a+b*exp(1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*csgn(I*c)*cs
gn(I*c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*ln(c)+2*ln(x^n
))/n))

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Maxima [A]  time = 1.36733, size = 31, normalized size = 1.55 \[ \frac{x}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(a*b*c^(1/n)*(x^n)^(1/n) + a^2)

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Fricas [A]  time = 0.232289, size = 34, normalized size = 1.7 \[ -\frac{1}{b^{2} c^{\frac{2}{n}} x + a b c^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(b^2*c^(2/n)*x + a*b*c^(1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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