\(\int \frac {\sqrt {a x^2}}{\sqrt {1+x^3}} \, dx\) [499]

Optimal result

Integrand size = 19, antiderivative size = 260 \[ \int \frac {\sqrt {a x^2}}{\sqrt {1+x^3}} \, dx=\frac {2 \sqrt {a x^2} \sqrt {1+x^3}}{x \left (1+\sqrt {3}+x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {a x^2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{x \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {2 \sqrt {2} \sqrt {a x^2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} x \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \] Output:

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

Mathematica [C] (verified)

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

Time = 10.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt {a x^2}}{\sqrt {1+x^3}} \, dx=\frac {1}{2} x \sqrt {a x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-x^3\right ) \] Input:

Integrate[Sqrt[a*x^2]/Sqrt[1 + x^3],x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {34, 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.

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

\(\Big \downarrow \) 34

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

\(\Big \downarrow \) 832

\(\displaystyle \frac {\sqrt {a x^2} \left (\int \frac {x-\sqrt {3}+1}{\sqrt {x^3+1}}dx-\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^3+1}}dx\right )}{x}\)

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2416

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

Input:

Int[Sqrt[a*x^2]/Sqrt[1 + x^3],x]
 

Output:

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

Defintions of rubi rules used

rule 34

Int[(u_.)*((a_.)*(x_)^(m_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*x^m)^F 
racPart[p]/x^(m*FracPart[p]))   Int[u*x^(m*p), x], x] /; FreeQ[{a, m, p}, x 
] &&  !IntegerQ[p]
 

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 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 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 [C] (verified)

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

Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.08

method	result	size
meijerg	\(\frac {\sqrt {a \,x^{2}}\, x \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}\)	\(22\)
default	\(\frac {\sqrt {a \,x^{2}}\, \left (i \sqrt {3}-3\right ) \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \left (i \operatorname {EllipticE}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) \sqrt {3}-i \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) \sqrt {3}+3 \operatorname {EllipticE}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-\operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )\right )}{2 x \sqrt {x^{3}+1}}\)	\(270\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt {a x^2}}{\sqrt {1+x^3}} \, dx=-\frac {2 \, \sqrt {a x^{2}} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}{x} \] Input:

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

Output:

-2*sqrt(a*x^2)*weierstrassZeta(0, -4, weierstrassPInverse(0, -4, x))/x
 

Sympy [F]

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

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

Output:

Integral(sqrt(a*x**2)/sqrt((x + 1)*(x**2 - x + 1)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a x^2}}{\sqrt {1+x^3}} \, dx=\int { \frac {\sqrt {a x^{2}}}{\sqrt {x^{3} + 1}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {a x^2}}{\sqrt {1+x^3}} \, dx=\int { \frac {\sqrt {a x^{2}}}{\sqrt {x^{3} + 1}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^2}}{\sqrt {1+x^3}} \, dx=\int \frac {\sqrt {a\,x^2}}{\sqrt {x^3+1}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*int((sqrt(x**3 + 1)*x)/(x**3 + 1),x)