\(\int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx\) [2019]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

[Out]

2/3*arctanh((a+b/x^3)^(1/2)/a^(1/2))/a^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, 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} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

[In]

Int[1/(Sqrt[a + b/x^3]*x),x]

[Out]

(2*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/(3*Sqrt[a])

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}& = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^3}}\right )}{3 b} \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(27)=54\).

Time = 0.36 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx=\frac {2 \sqrt {b+a x^3} \log \left (\sqrt {a} x^{3/2}+\sqrt {b+a x^3}\right )}{3 \sqrt {a} \sqrt {a+\frac {b}{x^3}} x^{3/2}} \]

[In]

Integrate[1/(Sqrt[a + b/x^3]*x),x]

[Out]

(2*Sqrt[b + a*x^3]*Log[Sqrt[a]*x^(3/2) + Sqrt[b + a*x^3]])/(3*Sqrt[a]*Sqrt[a + b/x^3]*x^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(19)=38\).

Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.19

method	result	size
default	\(\frac {2 \left (a \,x^{3}+b \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right )}{3 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, x \sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a}}\)	\(59\)

[In]

int(1/x/(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3/((a*x^3+b)/x^3)^(1/2)/x*(a*x^3+b)/(x*(a*x^3+b))^(1/2)/a^(1/2)*arctanh((x*(a*x^3+b))^(1/2)/x^2/a^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).

Time = 0.37 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.78 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx=\left [\frac {\log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}{6 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{3 \, a}\right ] \]

[In]

integrate(1/x/(a+b/x^3)^(1/2),x, algorithm="fricas")

[Out]

[1/6*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3)*sqrt(a)*sqrt((a*x^3 + b)/x^3))/sqrt(a), -1/3*sqrt(
-a)*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b))/a]

Sympy [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx=\frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {3}{2}}}{\sqrt {b}} \right )}}{3 \sqrt {a}} \]

[In]

integrate(1/x/(a+b/x**3)**(1/2),x)

[Out]

2*asinh(sqrt(a)*x**(3/2)/sqrt(b))/(3*sqrt(a))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx=-\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right )}{3 \, \sqrt {a}} \]

[In]

integrate(1/x/(a+b/x^3)^(1/2),x, algorithm="maxima")

[Out]

-1/3*log((sqrt(a + b/x^3) - sqrt(a))/(sqrt(a + b/x^3) + sqrt(a)))/sqrt(a)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x/(a+b/x^3)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 6.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3\,\sqrt {a}} \]

[In]

int(1/(x*(a + b/x^3)^(1/2)),x)

[Out]

(2*atanh((a + b/x^3)^(1/2)/a^(1/2)))/(3*a^(1/2))