3.3076 \(\int \frac{(c x^n)^{\frac{1}{n}}}{(a+b (c x^n)^{\frac{1}{n}})^3} \, dx\)

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 = 0.0129269, 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, Rules used = {15, 368, 37} \[ \frac{x \left (c x^n\right )^{\frac{1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

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)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 368

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0078196, 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 = 0.026, size = 203, normalized size = 6.3 \begin{align*}{\frac{x}{2\,a}{{\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}}}} \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((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*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*csgn
(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-2*ln(c)-2*ln(x^n))/n)/(a+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*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-2*ln(c)-2*ln(x^n))/n))^2/
a

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Maxima [A]  time = 1.03809, size = 81, normalized size = 2.53 \begin{align*} \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 )}} \end{align*}

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, 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 = 1.63759, size = 111, normalized size = 3.47 \begin{align*} -\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 )}} \end{align*}

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, 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(-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((c*x**n)**(1/n)/(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{\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} \end{align*}

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, algorithm="giac")

[Out]

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