\(\int \frac {1}{\sqrt {a+\frac {b}{x^3}}} \, dx\) [495]

Optimal result

Integrand size = 11, antiderivative size = 513 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}}} \, dx=-\frac {\sqrt [3]{b} \sqrt {a+\frac {b}{x^3}}}{a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}+\frac {\sqrt {a+\frac {b}{x^3}} x}{a}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{2 a^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {\sqrt {2} \sqrt [3]{b} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} a^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \] Output:

-b^(1/3)*(a+b/x^3)^(1/2)/a/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)+(a+b/x^3)^(1/2) 
*x/a+1/2*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*b^(1/3)*(a^(1/3)+b^(1/3)/x)*((a 
^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)^2)^( 
1/2)*EllipticE(((1-3^(1/2))*a^(1/3)+b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3 
)/x),I*3^(1/2)+2*I)/a^(2/3)/(a+b/x^3)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)/x)/( 
(1+3^(1/2))*a^(1/3)+b^(1/3)/x)^2)^(1/2)-1/3*2^(1/2)*b^(1/3)*(a^(1/3)+b^(1/ 
3)/x)*((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3 
)/x)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)/x)/((1+3^(1/2))*a^(1/ 
3)+b^(1/3)/x),I*3^(1/2)+2*I)*3^(3/4)/a^(2/3)/(a+b/x^3)^(1/2)/(a^(1/3)*(a^( 
1/3)+b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)^2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {2 x \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},-\frac {a x^3}{b}\right )}{5 \sqrt {a+\frac {b}{x^3}}} \] Input:

Integrate[1/Sqrt[a + b/x^3],x]
 

Output:

(2*x*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[1/2, 5/6, 11/6, -((a*x^3)/b)])/ 
(5*Sqrt[a + b/x^3])
 

Rubi [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {773, 847, 832, 759, 2416}

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.

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

\(\Big \downarrow \) 773

\(\displaystyle -\int \frac {x^2}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x^3}}}{a}-\frac {b \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x}d\frac {1}{x}}{2 a}\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x^3}}}{a}-\frac {b \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}\right )}{2 a}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x^3}}}{a}-\frac {b \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{2 a}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {x \sqrt {a+\frac {b}{x^3}}}{a}-\frac {b \left (\frac {\frac {2 \sqrt {a+\frac {b}{x^3}}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{2 a}\)

Input:

Int[1/Sqrt[a + b/x^3],x]
 

Output:

(Sqrt[a + b/x^3]*x)/a - (b*(((2*Sqrt[a + b/x^3])/(b^(1/3)*((1 + Sqrt[3])*a 
^(1/3) + b^(1/3)/x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/ 
3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^ 
(1/3) + b^(1/3)/x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x) 
/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt[a + 
b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/ 
3)/x)^2]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) + 
 b^(1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[ 
3])*a^(1/3) + b^(1/3)/x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1 
/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(3^(1/4)*b^( 
2/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a 
^(1/3) + b^(1/3)/x)^2])))/(2*a)
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 773

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 
2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] &&  !IntegerQ[p]
 

rule 832

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && PosQ[a]
 

rule 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2373 vs. \(2 (385 ) = 770\).

Time = 0.35 (sec) , antiderivative size = 2374, normalized size of antiderivative = 4.63

method	result	size
default	\(\text {Expression too large to display}\)	\(2374\)

Input:

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

Output:

-2/((a*x^3+b)/x^3)^(1/2)/x*(a*x^3+b)/a^2*(-I*(-a^2*b)^(1/3)*3^(1/2)*(-(I*3 
^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2 
*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2) 
*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^ 
2*b)^(1/3)))^(1/2)*EllipticE((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2 
*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3) 
)^(1/2))*a*x^2+2*I*(-a^2*b)^(2/3)*3^(1/2)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1 
)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^( 
1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3 
)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*Ellipti 
cE((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1 
/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*x-2*(-a^2*b)^(1/3 
)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/ 
2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3) 
))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(- 
a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a 
*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^ 
(1/2)-3))^(1/2))*a*x^2+3*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/ 
(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/ 
3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/...
 

Fricas [F]

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

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

Output:

integral(x^3*sqrt((a*x^3 + b)/x^3)/(a*x^3 + b), x)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.08 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}}} \, dx=- \frac {x \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {2}{3}\right )} \] Input:

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

Output:

-x*gamma(-1/3)*hyper((-1/3, 1/2), (2/3,), b*exp_polar(I*pi)/(a*x**3))/(3*s 
qrt(a)*gamma(2/3))
 

Maxima [F]

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

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

Output:

integrate(1/sqrt(a + b/x^3), x)
 

Giac [F]

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

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

Output:

integrate(1/sqrt(a + b/x^3), x)
 

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.07 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {2\,x\,\sqrt {\frac {a\,x^3}{b}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{6};\ \frac {11}{6};\ -\frac {a\,x^3}{b}\right )}{5\,\sqrt {a+\frac {b}{x^3}}} \] Input:

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

Output:

(2*x*((a*x^3)/b + 1)^(1/2)*hypergeom([1/2, 5/6], 11/6, -(a*x^3)/b))/(5*(a 
+ b/x^3)^(1/2))
 

Reduce [F]

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

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

Output:

int((sqrt(x)*sqrt(a*x**3 + b)*x)/(a*x**3 + b),x)