3.3020 \(\int \frac{1}{(a+b (c x^n)^{\frac{1}{n}})^3} \, dx\)

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 = 0.0081762, 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, Rules used = {254, 32} \[ -\frac{x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

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)

Rule 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^3} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=-\frac{x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0132898, 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 = 0.027, size = 209, normalized size = 6.2 \begin{align*}{\frac{x}{2\,{a}^{2}} \left ( b{{\rm e}^{-{\frac{i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}+2\,a \right ) \left ( a+b{{\rm e}^{-{\frac{i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-2}} \end{align*}

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

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Maxima [B]  time = 1.00105, size = 93, normalized size = 2.74 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(c*x^n)^(1/n))^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 = 1.51808, size = 81, normalized size = 2.38 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(c*x^n)^(1/n))^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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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