\(\int \frac {1}{\sqrt {x} \sqrt {a+b \sqrt {x^3}}} \, dx\) [55]

Optimal result

Integrand size = 21, antiderivative size = 269 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b \sqrt {x^3}}} \, dx=\frac {4 \sqrt {2+\sqrt {3}} \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{x^3}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [6]{x^3}+b^{2/3} \sqrt [3]{x^3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{x^3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{x^3}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt [6]{x^3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{x^3}\right )^2}} \sqrt {a+b \sqrt {x^3}}} \] Output:

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

Mathematica [C] (verified)

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

Time = 3.88 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.23 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b \sqrt {x^3}}} \, dx=\frac {2 \sqrt {x} \sqrt {1+\frac {b \sqrt {x^3}}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b \sqrt {x^3}}{a}\right )}{\sqrt {a+b \sqrt {x^3}}} \] Input:

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

Output:

(2*Sqrt[x]*Sqrt[1 + (b*Sqrt[x^3])/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b 
*Sqrt[x^3])/a)])/Sqrt[a + b*Sqrt[x^3]]
 

Rubi [A] (warning: unable to verify)

Time = 0.47 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {893, 864, 759}

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.

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

\(\Big \downarrow \) 893

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

\(\Big \downarrow \) 864

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

\(\Big \downarrow \) 759

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

Input:

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

Output:

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

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 864

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi 
nator[n]}, Simp[k   Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x 
^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
 

rule 893

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x 
], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, 
d, m, p, q}, x] && FractionQ[n]
 

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {2 i \sqrt {3}\, x^{\frac {3}{2}} \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-b^{2} a \right )^{\frac {1}{3}}+2 b \sqrt {x^{3}}+x \left (-b^{2} a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \sqrt {x^{3}}-x \left (-b^{2} a \right )^{\frac {1}{3}}}{x \left (-b^{2} a \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-b^{2} a \right )^{\frac {1}{3}}+x \left (-b^{2} a \right )^{\frac {1}{3}}+2 b \sqrt {x^{3}}\right ) \sqrt {3}}{x \left (-b^{2} a \right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-b^{2} a \right )^{\frac {1}{3}}+2 b \sqrt {x^{3}}+x \left (-b^{2} a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-b^{2} a \right )^{\frac {1}{3}}}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right )}{3 \sqrt {x^{3}}\, b \sqrt {a +b \sqrt {x^{3}}}}\)	\(273\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*sqrt(x^3)*sqrt(x) - a*sqrt(x))*sqrt(b*sqrt(x^3) + a)/(b^2*x^4 
- a^2*x), x)
 

Sympy [F]

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

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

Output:

Integral(1/(sqrt(x)*sqrt(a + b*sqrt(x**3))), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {a+b \sqrt {x^3}}} \, dx=-\left (\int \frac {\sqrt {\sqrt {x}\, b x +a}\, x}{-b^{2} x^{3}+a^{2}}d x \right ) b +\left (\int \frac {\sqrt {x}\, \sqrt {\sqrt {x}\, b x +a}}{-b^{2} x^{4}+a^{2} x}d x \right ) a \] Input:

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

Output:

 - int((sqrt(sqrt(x)*b*x + a)*x)/(a**2 - b**2*x**3),x)*b + int((sqrt(x)*sq 
rt(sqrt(x)*b*x + a))/(a**2*x - b**2*x**4),x)*a