\(\int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx\) [407]

Optimal result

Integrand size = 15, antiderivative size = 32 \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, 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 = {2004, 2033, 212} \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}} \]

[In]

Int[1/Sqrt[(a + b*x^3)/x],x]

[Out]

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

Rule 212

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

Rule 2004

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2033

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\frac {2 \sqrt {a+b x^3} \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {\frac {a+b x^3}{x}}} \]

[In]

Integrate[1/Sqrt[(a + b*x^3)/x],x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(24)=48\).

Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {2 \left (b \,x^{3}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right )}{3 \sqrt {\frac {b \,x^{3}+a}{x}}\, \sqrt {x \left (b \,x^{3}+a \right )}\, \sqrt {b}}\)	\(56\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\left [\frac {\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - a^{2} - 4 \, {\left (2 \, b x^{5} + a x^{2}\right )} \sqrt {b} \sqrt {\frac {b x^{3} + a}{x}}\right )}{6 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {-b} x^{2} \sqrt {\frac {b x^{3} + a}{x}}}{2 \, b x^{3} + a}\right )}{3 \, b}\right ] \]

[In]

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

[Out]

[1/6*log(-8*b^2*x^6 - 8*a*b*x^3 - a^2 - 4*(2*b*x^5 + a*x^2)*sqrt(b)*sqrt((b*x^3 + a)/x))/sqrt(b), -1/3*sqrt(-b
)*arctan(2*sqrt(-b)*x^2*sqrt((b*x^3 + a)/x)/(2*b*x^3 + a))/b]

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt((a + b*x**3)/x), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\int { \frac {1}{\sqrt {\frac {b x^{3} + a}{x}}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt((b*x^3 + a)/x), x)

Giac [F(-2)]

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

[In]

integrate(1/((b*x^3+a)/x)^(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 [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\int \frac {1}{\sqrt {\frac {b\,x^3+a}{x}}} \,d x \]

[In]

int(1/((a + b*x^3)/x)^(1/2),x)

[Out]

int(1/((a + b*x^3)/x)^(1/2), x)