\(\int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx\) [411]

Optimal result

Integrand size = 16, antiderivative size = 33 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}-b x^2}}\right )}{3 \sqrt {b}} \]

[Out]

2/3*arctan(x*b^(1/2)/(a/x-b*x^2)^(1/2))/b^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2004, 2033, 209} \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}-b x^2}}\right )}{3 \sqrt {b}} \]

[In]

Int[1/Sqrt[(a - b*x^3)/x],x]

[Out]

(2*ArcTan[(Sqrt[b]*x)/Sqrt[a/x - b*x^2]])/(3*Sqrt[b])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}-b x^2}}\right )}{3 \sqrt {b}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=-\frac {2 i \sqrt {a-b x^3} \log \left (i \sqrt {b} x^{3/2}+\sqrt {a-b x^3}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {\frac {a-b x^3}{x}}} \]

[In]

Integrate[1/Sqrt[(a - b*x^3)/x],x]

[Out]

(((-2*I)/3)*Sqrt[a - b*x^3]*Log[I*Sqrt[b]*x^(3/2) + Sqrt[a - b*x^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. \(59\) vs. \(2(25)=50\).

Time = 2.44 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82

method	result	size
default	\(-\frac {2 \left (-b \,x^{3}+a \right ) \arctan \left (\frac {\sqrt {x \left (-b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right )}{3 \sqrt {\frac {-b \,x^{3}+a}{x}}\, \sqrt {x \left (-b \,x^{3}+a \right )}\, \sqrt {b}}\)	\(60\)

[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)*arctan((x*(-b*x^3+a))^(1/2)/x^2/b^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.57 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.36 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\left [-\frac {\sqrt {-b} \log \left (-8 \, b^{2} x^{6} + 8 \, a b x^{3} - a^{2} + 4 \, {\left (2 \, b x^{5} - a x^{2}\right )} \sqrt {-b} \sqrt {-\frac {b x^{3} - a}{x}}\right )}{6 \, b}, -\frac {\arctan \left (\frac {2 \, \sqrt {b} x^{2} \sqrt {-\frac {b x^{3} - a}{x}}}{2 \, b x^{3} - a}\right )}{3 \, \sqrt {b}}\right ] \]

[In]

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

[Out]

[-1/6*sqrt(-b)*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))/b, -1/3*a
rctan(2*sqrt(b)*x^2*sqrt(-(b*x^3 - a)/x)/(2*b*x^3 - a))/sqrt(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. \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\int \frac {1}{\sqrt {\frac {a-b\,x^3}{x}}} \,d x \]

[In]

int(1/((a - b*x^3)/x)^(1/2),x)

[Out]

int(1/((a - b*x^3)/x)^(1/2), x)