3.3069 \(\int \frac{\left (c x^n\right )^{\frac{1}{n}}}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \, dx\)

Optimal. Leaf size=32 \[ \frac{x \left (c x^n\right )^{\frac{1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0355623, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x \left (c x^n\right )^{\frac{1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.4883, size = 26, normalized size = 0.81 \[ \frac{x \left (c x^{n}\right )^{\frac{1}{n}}}{2 a \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0875845, size = 32, normalized size = 1. \[ \frac{x \left (c x^n\right )^{\frac{1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.042, size = 203, normalized size = 6.3 \[{\frac{x}{2\,a}{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \left ( a+b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*exp(1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*csgn(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)/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)*c
sgn(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 = 1.44334, size = 81, normalized size = 2.53 \[ \frac{c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{2 \,{\left (a b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 2 \, a^{2} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218446, size = 76, normalized size = 2.38 \[ -\frac{2 \, b c^{\left (\frac{1}{n}\right )} x + a}{2 \,{\left (b^{4} c^{\frac{3}{n}} x^{2} + 2 \, a b^{3} c^{\frac{2}{n}} x + a^{2} b^{2} c^{\left (\frac{1}{n}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*b*c^(1/n)*x + a)/(b^4*c^(3/n)*x^2 + 2*a*b^3*c^(2/n)*x + a^2*b^2*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((c*x**n)**(1/n)/(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. \[ \int \frac{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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