3.3013 \(\int \frac{1}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \, dx\)

Optimal. Leaf size=34 \[ -\frac{x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0212443, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.28924, size = 27, normalized size = 0.79 \[ - \frac{x \left (c x^{n}\right )^{- \frac{1}{n}}}{2 b \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0212443, size = 34, normalized size = 1. \[ -\frac{x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.041, size = 209, normalized size = 6.2 \[{\frac{x}{2\,{a}^{2}} \left ( b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}+2\,a \right ) \left ( a+b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(c*x^n)^(1/n))^3,x)

[Out]

1/2*x*(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)+2*a)/a^2/(a+b*exp(1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*c
sgn(I*c)*csgn(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))^2

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Maxima [A]  time = 1.41803, size = 93, normalized size = 2.74 \[ \frac{b c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + 2 \, a x}{2 \,{\left (a^{2} b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 2 \, a^{3} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*c^(1/n)*x*(x^n)^(1/n) + 2*a*x)/(a^2*b^2*c^(2/n)*(x^n)^(2/n) + 2*a^3*b*c^(
1/n)*(x^n)^(1/n) + a^4)

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Fricas [A]  time = 0.232197, size = 58, normalized size = 1.71 \[ -\frac{1}{2 \,{\left (b^{3} c^{\frac{3}{n}} x^{2} + 2 \, a b^{2} c^{\frac{2}{n}} x + a^{2} b c^{\left (\frac{1}{n}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: RecursionError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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