Integrand size = 16, antiderivative size = 33 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}-b x^2}}\right )}{3 \sqrt {b}} \] Output:
2/3*arctan(b^(1/2)*x/(a/x-b*x^2)^(1/2))/b^(1/2)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=-\frac {2 i \sqrt {a-b x^3} \log \left (i \sqrt {b} x^{3/2}+\sqrt {a-b x^3}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {\frac {a-b x^3}{x}}} \] Input:
Integrate[1/Sqrt[(a - b*x^3)/x],x]
Output:
(((-2*I)/3)*Sqrt[a - b*x^3]*Log[I*Sqrt[b]*x^(3/2) + Sqrt[a - b*x^3]])/(Sqr t[b]*Sqrt[x]*Sqrt[(a - b*x^3)/x])
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2078, 1914, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx\) |
\(\Big \downarrow \) 2078 |
\(\displaystyle \int \frac {1}{\sqrt {\frac {a}{x}-b x^2}}dx\) |
\(\Big \downarrow \) 1914 |
\(\displaystyle \frac {2}{3} \int \frac {1}{\frac {b x^2}{\frac {a}{x}-b x^2}+1}d\frac {x}{\sqrt {\frac {a}{x}-b x^2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}-b x^2}}\right )}{3 \sqrt {b}}\) |
Input:
Int[1/Sqrt[(a - b*x^3)/x],x]
Output:
(2*ArcTan[(Sqrt[b]*x)/Sqrt[a/x - b*x^2]])/(3*Sqrt[b])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[2/(2 - n) Subst[Int[1/(1 - a*x^2), x], x, x/Sqrt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]
Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && G eneralizedBinomialQ[u, x] && !GeneralizedBinomialMatchQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(25)=50\).
Time = 2.79 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82
method | result | size |
default | \(-\frac {2 \left (-b \,x^{3}+a \right ) \arctan \left (\frac {\sqrt {x \left (-b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right )}{3 \sqrt {\frac {-b \,x^{3}+a}{x}}\, \sqrt {x \left (-b \,x^{3}+a \right )}\, \sqrt {b}}\) | \(60\) |
Input:
int(1/((-b*x^3+a)/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-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))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).
Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.67 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\left [-\frac {\sqrt {-b} \log \left (-8 \, b^{2} x^{6} + 8 \, a b x^{3} - a^{2} + 4 \, {\left (2 \, b x^{5} - a x^{2}\right )} \sqrt {-b} \sqrt {-\frac {b x^{3} - a}{x}}\right )}{6 \, b}, -\frac {\arctan \left (\frac {{\left (2 \, b x^{3} - a\right )} \sqrt {b} \sqrt {-\frac {b x^{3} - a}{x}}}{2 \, {\left (b^{2} x^{4} - a b x\right )}}\right )}{3 \, \sqrt {b}}\right ] \] Input:
integrate(1/((-b*x^3+a)/x)^(1/2),x, algorithm="fricas")
Output:
[-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*arctan(1/2*(2*b*x^3 - a)*sqrt(b)*sqrt(- (b*x^3 - a)/x)/(b^2*x^4 - a*b*x))/sqrt(b)]
\[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\int \frac {1}{\sqrt {\frac {a - b x^{3}}{x}}}\, dx \] Input:
integrate(1/((-b*x**3+a)/x)**(1/2),x)
Output:
Integral(1/sqrt((a - b*x**3)/x), x)
\[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\int { \frac {1}{\sqrt {-\frac {b x^{3} - a}{x}}} \,d x } \] Input:
integrate(1/((-b*x^3+a)/x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(-(b*x^3 - a)/x), x)
Exception generated. \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((-b*x^3+a)/x)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=\int \frac {1}{\sqrt {\frac {a-b\,x^3}{x}}} \,d x \] Input:
int(1/((a - b*x^3)/x)^(1/2),x)
Output:
int(1/((a - b*x^3)/x)^(1/2), x)
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\sqrt {\frac {a-b x^3}{x}}} \, dx=-\frac {\sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {b}\, \sqrt {-b \,x^{3}+a}\, a -2 \sqrt {x}\, \sqrt {b}\, \sqrt {-b \,x^{3}+a}\, b \,x^{3}}{-2 b^{2} x^{5}+2 a b \,x^{2}}\right )}{3 b} \] Input:
int(1/((-b*x^3+a)/x)^(1/2),x)
Output:
( - sqrt(b)*atan((sqrt(x)*sqrt(b)*sqrt(a - b*x**3)*a - 2*sqrt(x)*sqrt(b)*s qrt(a - b*x**3)*b*x**3)/(2*a*b*x**2 - 2*b**2*x**5)))/(3*b)