\(\int \frac {1}{(a+b (c x^n)^{\frac {1}{n}})^2} \, dx\) [3016]

Optimal result

Integrand size = 15, antiderivative size = 20 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x}{a^2+a b \left (c x^n\right )^{\frac {1}{n}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {260, 32} \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {x \left (c x^n\right )^{-1/n}}{b \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \]

[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))))

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]

Rule 260

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)]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {x \left (c x^n\right )^{-1/n}}{a b+b^2 \left (c x^n\right )^{\frac {1}{n}}} \]

[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))))

Maple [A] (verified)

Time = 4.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05

method	result	size
parallelrisch	\(\frac {x}{a \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}\)	\(21\)
risch	\(\frac {x}{a \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}\)	\(74\)

[In]

int(1/(a+b*(c*x^n)^(1/n))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {1}{b^{2} c^{\frac {2}{n}} x + a b c^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

-1/(b^2*c^(2/n)*x + a*b*c^(1/n))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (15) = 30\).

Time = 2.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\begin {cases} \tilde {\infty } x \left (c x^{n}\right )^{- \frac {2}{n}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {x \left (c x^{n}\right )^{- \frac {2}{n}}}{b^{2}} & \text {for}\: a = 0 \\\tilde {\infty } x & \text {for}\: b = - a \left (c x^{n}\right )^{- \frac {1}{n}} \\\frac {x}{a^{2} + a b \left (c x^{n}\right )^{\frac {1}{n}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*x/(c*x**n)**(2/n), Eq(a, 0) & Eq(b, 0)), (-x/(b**2*(c*x**n)**(2/n)), Eq(a, 0)), (zoo*x, Eq(b, -
a/(c*x**n)**(1/n))), (x/(a**2 + a*b*(c*x**n)**(1/n)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}} \]

[In]

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

[Out]

x/(a*b*c^(1/n)*(x^n)^(1/n) + a^2)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x}{a\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )} \]

[In]

int(1/(a + b*(c*x^n)^(1/n))^2,x)

[Out]

x/(a*(a + b*(c*x^n)^(1/n)))