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))

_______________________________________________________________________________________

Rubi [A]  time = 0.0856025, 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 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{\sqrt{a} n} \]

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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.4458, size = 41, normalized size = 0.87 \[ - \frac{\operatorname{atanh}{\left (\frac{2 a + b x^{n}}{2 \sqrt{a} \sqrt{a + b x^{n} + c x^{2 n}}} \right )}}{\sqrt{a} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a+b*x**n+c*x**(2*n))**(1/2),x)

[Out]

-atanh((2*a + b*x**n)/(2*sqrt(a)*sqrt(a + b*x**n + c*x**(2*n))))/(sqrt(a)*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.113137, size = 50, normalized size = 1.06 \[ \frac{n \log (x)-\log \left (2 \sqrt{a} \sqrt{a+x^n \left (b+c x^n\right )}+2 a+b x^n\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt{a+b{x}^{n}+c{x}^{2\,n}}}}}\, dx \]

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)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.285165, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{8 \, a^{\frac{3}{2}} b x^{n} + 8 \, a^{\frac{5}{2}} +{\left (b^{2} + 4 \, a c\right )} \sqrt{a} x^{2 \, n} - 4 \,{\left (a b x^{n} + 2 \, a^{2}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right )}{2 \, \sqrt{a} n}, -\frac{\arctan \left (\frac{\sqrt{-a} b x^{n} + 2 \, \sqrt{-a} a}{2 \, \sqrt{c x^{2 \, n} + b x^{n} + a} a}\right )}{\sqrt{-a} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log(-(8*a^(3/2)*b*x^n + 8*a^(5/2) + (b^2 + 4*a*c)*sqrt(a)*x^(2*n) - 4*(a*b*
x^n + 2*a^2)*sqrt(c*x^(2*n) + b*x^n + a))/x^(2*n))/(sqrt(a)*n), -arctan(1/2*(sqr
t(-a)*b*x^n + 2*sqrt(-a)*a)/(sqrt(c*x^(2*n) + b*x^n + a)*a))/(sqrt(-a)*n)]

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2 \, n} + b x^{n} + a} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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