\(\int \frac {1}{a+b (c x^n)^{\frac {1}{n}}} \, dx\) [3009]

Optimal result

Integrand size = 15, antiderivative size = 30 \[ \int \frac {1}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {260, 31} \[ \int \frac {1}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b} \]

[In]

Int[(a + b*(c*x^n)^n^(-1))^(-1),x]

[Out]

(x*Log[a + b*(c*x^n)^n^(-1)])/(b*(c*x^n)^n^(-1))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b} \]

[In]

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

[Out]

(x*Log[a + b*(c*x^n)^n^(-1)])/(b*(c*x^n)^n^(-1))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.63

method	result	size
risch	\(\frac {\ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} x \,{\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{b}\)	\(139\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {1}{a+b \left (c x^n\right )^{\frac {1}{n}}} \, dx=\frac {\log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{b c^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

log(b*c^(1/n)*x + a)/(b*c^(1/n))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(1/((c*x^n)^(1/n)*b + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/((c*x^n)^(1/n)*b + a), x)

Mupad [F(-1)]

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

[In]

int(1/(a + b*(c*x^n)^(1/n)),x)

[Out]

int(1/(a + b*(c*x^n)^(1/n)), x)