\(\int \frac {1}{x \sqrt {a+b (c (d (e x)^m)^n)^p}} \, dx\) [180]

Optimal result

Integrand size = 25, antiderivative size = 44 \[ \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p} \] Output:

-2*arctanh((a+b*(c*(d*(e*x)^m)^n)^p)^(1/2)/a^(1/2))/a^(1/2)/m/n/p
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p} \] Input:

Integrate[1/(x*Sqrt[a + b*(c*(d*(e*x)^m)^n)^p]),x]
 

Output:

(-2*ArcTanh[Sqrt[a + b*(c*(d*(e*x)^m)^n)^p]/Sqrt[a]])/(Sqrt[a]*m*n*p)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {7282, 7282, 891, 27, 798, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx\)

\(\Big \downarrow \) 7282

\(\displaystyle \frac {\int \frac {(e x)^{-m}}{\sqrt {b \left (c \left (d (e x)^m\right )^n\right )^p+a}}d(e x)^m}{m}\)

\(\Big \downarrow \) 7282

\(\displaystyle \frac {\int \frac {(e x)^{-m}}{\sqrt {b \left (c \left (d (e x)^m\right )^n\right )^p+a}}d\left (d (e x)^m\right )^n}{m n}\)

\(\Big \downarrow \) 891

\(\displaystyle \frac {\int \frac {c (e x)^{-m}}{\sqrt {b \left (c \left (d (e x)^m\right )^n\right )^p+a}}d\left (c \left (d (e x)^m\right )^n\right )}{c m n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(e x)^{-m}}{\sqrt {b \left (c \left (d (e x)^m\right )^n\right )^p+a}}d\left (c \left (d (e x)^m\right )^n\right )}{m n}\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\int \frac {(e x)^{-m}}{\sqrt {b \left (c \left (d (e x)^m\right )^n\right )^p+a}}d\left (c \left (d (e x)^m\right )^n\right )^p}{m n p}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \int \frac {1}{\frac {(e x)^{2 m}}{b}-\frac {a}{b}}d\sqrt {b \left (c \left (d (e x)^m\right )^n\right )^p+a}}{b m n p}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p}\)

Input:

Int[1/(x*Sqrt[a + b*(c*(d*(e*x)^m)^n)^p]),x]
 

Output:

(-2*ArcTanh[Sqrt[a + b*(c*(d*(e*x)^m)^n)^p]/Sqrt[a]])/(Sqrt[a]*m*n*p)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 891

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

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b {\left (c \left (d \left (e x \right )^{m}\right )^{n}\right )}^{p}}}{\sqrt {a}}\right )}{\sqrt {a}\, m n p}\)	\(39\)
default	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b {\left (c \left (d \left (e x \right )^{m}\right )^{n}\right )}^{p}}}{\sqrt {a}}\right )}{\sqrt {a}\, m n p}\)	\(39\)

Input:

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

Output:

-2*arctanh((a+b*(c*(d*(e*x)^m)^n)^p)^(1/2)/a^(1/2))/a^(1/2)/m/n/p
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 3.27 \[ \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx=\left [\frac {\log \left ({\left (b e^{\left (m n p \log \left (e x\right ) + n p \log \left (d\right ) + p \log \left (c\right )\right )} - 2 \, \sqrt {b e^{\left (m n p \log \left (e x\right ) + n p \log \left (d\right ) + p \log \left (c\right )\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n p \log \left (e x\right ) - n p \log \left (d\right ) - p \log \left (c\right )\right )}\right )}{\sqrt {a} m n p}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b e^{\left (m n p \log \left (e x\right ) + n p \log \left (d\right ) + p \log \left (c\right )\right )} + a}}\right )}{a m n p}\right ] \] Input:

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

Output:

[log((b*e^(m*n*p*log(e*x) + n*p*log(d) + p*log(c)) - 2*sqrt(b*e^(m*n*p*log 
(e*x) + n*p*log(d) + p*log(c)) + a)*sqrt(a) + 2*a)*e^(-m*n*p*log(e*x) - n* 
p*log(d) - p*log(c)))/(sqrt(a)*m*n*p), 2*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*e 
^(m*n*p*log(e*x) + n*p*log(d) + p*log(c)) + a))/(a*m*n*p)]
 

Sympy [F]

\[ \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx=\int \frac {1}{x \sqrt {a + b \left (c \left (d \left (e x\right )^{m}\right )^{n}\right )^{p}}}\, dx \] Input:

integrate(1/x/(a+b*(c*(d*(e*x)**m)**n)**p)**(1/2),x)
 

Output:

Integral(1/(x*sqrt(a + b*(c*(d*(e*x)**m)**n)**p)), x)
 

Maxima [F]

\[ \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx=\int { \frac {1}{\sqrt {\left (\left (\left (e x\right )^{m} d\right )^{n} c\right )^{p} b + a} x} \,d x } \] Input:

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

Output:

integrate(1/(sqrt((((e*x)^m*d)^n*c)^p*b + a)*x), x)
 

Giac [F]

\[ \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx=\int { \frac {1}{\sqrt {\left (\left (\left (e x\right )^{m} d\right )^{n} c\right )^{p} b + a} x} \,d x } \] Input:

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

Output:

integrate(1/(sqrt((((e*x)^m*d)^n*c)^p*b + a)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx=\int \frac {1}{x\,\sqrt {a+b\,{\left (c\,{\left (d\,{\left (e\,x\right )}^m\right )}^n\right )}^p}} \,d x \] Input:

int(1/(x*(a + b*(c*(d*(e*x)^m)^n)^p)^(1/2)),x)
 

Output:

int(1/(x*(a + b*(c*(d*(e*x)^m)^n)^p)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx=\int \frac {\sqrt {x^{m n p} e^{m n p} d^{n p} c^{p} b +a}}{x^{m n p} e^{m n p} d^{n p} c^{p} b x +a x}d x \] Input:

int(1/x/(a+b*(c*(d*(e*x)^m)^n)^p)^(1/2),x)
 

Output:

int(sqrt(x**(m*n*p)*e**(m*n*p)*d**(n*p)*c**p*b + a)/(x**(m*n*p)*e**(m*n*p) 
*d**(n*p)*c**p*b*x + a*x),x)