\(\int \frac {1}{x \sqrt {a+b x^n}} \, dx\) [2504]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {1}{x \sqrt {a+b x^n}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 214} \[ \int \frac {1}{x \sqrt {a+b x^n}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \]

[In]

Int[1/(x*Sqrt[a + b*x^n]),x]

[Out]

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

Rule 65

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 214

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 272

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]]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+b x^n}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \]

[In]

Integrate[1/(x*Sqrt[a + b*x^n]),x]

[Out]

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

Maple [A] (verified)

Time = 4.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{n \sqrt {a}}\)	\(23\)
default	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{n \sqrt {a}}\)	\(23\)

[In]

int(1/x/(a+b*x^n)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {1}{x \sqrt {a+b x^n}} \, dx=\left [\frac {\log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right )}{\sqrt {a} n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right )}{a n}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \sqrt {a+b x^n}} \, dx=- \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{\sqrt {a} n} \]

[In]

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

[Out]

-2*asinh(sqrt(a)/(sqrt(b)*x**(n/2)))/(sqrt(a)*n)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x \sqrt {a+b x^n}} \, dx=\frac {\log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{\sqrt {a} n} \]

[In]

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

[Out]

log((sqrt(b*x^n + a) - sqrt(a))/(sqrt(b*x^n + a) + sqrt(a)))/(sqrt(a)*n)

Giac [F]

\[ \int \frac {1}{x \sqrt {a+b x^n}} \, dx=\int { \frac {1}{\sqrt {b x^{n} + a} x} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(b*x^n + a)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {a+b x^n}} \, dx=\int \frac {1}{x\,\sqrt {a+b\,x^n}} \,d x \]

[In]

int(1/(x*(a + b*x^n)^(1/2)),x)

[Out]

int(1/(x*(a + b*x^n)^(1/2)), x)