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

Optimal. Leaf size=44 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.37, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {367, 12, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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 367

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b \left (c (d x)^n\right )^p}} \, dx,x,(e x)^m\right )}{m}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b (c x)^p}} \, dx,x,\left (d (e x)^m\right )^n\right )}{m n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {c}{x \sqrt {a+b x^p}} \, dx,x,c \left (d (e x)^m\right )^n\right )}{c m n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^p}} \, dx,x,c \left (d (e x)^m\right )^n\right )}{m n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c \left (d (e x)^m\right )^n\right )^p\right )}{m n p}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}\right )}{b m n p}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 44, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 147, normalized size = 3.34 \[ \left [\frac {\log \left ({\left (b e^{\left (m n p \log \left (e x\right ) + n p \log \relax (d) + p \log \relax (c)\right )} - 2 \, \sqrt {b e^{\left (m n p \log \left (e x\right ) + n p \log \relax (d) + p \log \relax (c)\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n p \log \left (e x\right ) - n p \log \relax (d) - p \log \relax (c)\right )}\right )}{\sqrt {a} m n p}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (m n p \log \left (e x\right ) + n p \log \relax (d) + p \log \relax (c)\right )} + a} \sqrt {-a}}{a}\right )}{a m n p}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (\left (\left (e x\right )^{m} d\right )^{n} c\right )^{p} b + a} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 39, normalized size = 0.89 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b \left (c \left (d \left (e x \right )^{m}\right )^{n}\right )^{p}+a}}{\sqrt {a}}\right )}{\sqrt {a}\, m n p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (\left (\left (e x\right )^{m} d\right )^{n} c\right )^{p} b + a} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,\sqrt {a+b\,{\left (c\,{\left (d\,{\left (e\,x\right )}^m\right )}^n\right )}^p}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {a + b \left (c \left (d \left (e x\right )^{m}\right )^{n}\right )^{p}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________