3.586 \(\int \frac{1}{x \sqrt{a+b x^n+c x^{2 n}}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{\sqrt{a} n} \]

[Out]

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

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Rubi [A]  time = 0.0334797, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1357, 724, 206} \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a + b*x^n + c*x^(2*n)]),x]

[Out]

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

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{a+b x^n+c x^{2 n}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^n\right )}{n}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^n}{\sqrt{a+b x^n+c x^{2 n}}}\right )}{n}\\ &=-\frac{\tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{\sqrt{a} n}\\ \end{align*}

Mathematica [A]  time = 0.0656383, size = 47, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a + b*x^n + c*x^(2*n)]),x]

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt{a+b{x}^{n}+c{x}^{2\,n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2 \, n} + b x^{n} + a} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.86126, size = 347, normalized size = 7.38 \begin{align*} \left [\frac{\log \left (-\frac{8 \, a b x^{n} + 8 \, a^{2} +{\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \,{\left (\sqrt{a} b x^{n} + 2 \, a^{\frac{3}{2}}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right )}{2 \, \sqrt{a} n}, \frac{\sqrt{-a} \arctan \left (\frac{{\left (\sqrt{-a} b x^{n} + 2 \, \sqrt{-a} a\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{2 \,{\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right )}{a n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log(-(8*a*b*x^n + 8*a^2 + (b^2 + 4*a*c)*x^(2*n) - 4*(sqrt(a)*b*x^n + 2*a^(3/2))*sqrt(c*x^(2*n) + b*x^n +
a))/x^(2*n))/(sqrt(a)*n), sqrt(-a)*arctan(1/2*(sqrt(-a)*b*x^n + 2*sqrt(-a)*a)*sqrt(c*x^(2*n) + b*x^n + a)/(a*c
*x^(2*n) + a*b*x^n + a^2))/(a*n)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{a + b x^{n} + c x^{2 n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2 \, n} + b x^{n} + a} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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