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

Optimal. Leaf size=37 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]

[Out]

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

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Rubi [A]  time = 0.166927, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {367, 12, 266, 63, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0658406, size = 37, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 32, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{nm\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( c \left ( dx \right ) ^{m} \right ) ^{n}}}{\sqrt{a}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (\left (d x\right )^{m} c\right )^{n} b + a} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.58939, size = 300, normalized size = 8.11 \begin{align*} \left [\frac{\log \left ({\left (b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} - 2 \, \sqrt{b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} \sqrt{a} + 2 \, a\right )} e^{\left (-m n \log \left (d x\right ) - n \log \left (c\right )\right )}\right )}{\sqrt{a} m n}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} \sqrt{-a}}{a}\right )}{a m n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{a + b \left (c \left (d x\right )^{m}\right )^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (\left (d x\right )^{m} c\right )^{n} b + a} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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