\(\int \frac {\sqrt {\frac {a}{x^3}}}{\sqrt {1+x^3}} \, dx\) [503]

Optimal result

Integrand size = 19, antiderivative size = 312 \[ \int \frac {\sqrt {\frac {a}{x^3}}}{\sqrt {1+x^3}} \, dx=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^3}+\frac {2 \left (1+\sqrt {3}\right ) \sqrt {\frac {a}{x^3}} x^2 \sqrt {1+x^3}}{1+\left (1+\sqrt {3}\right ) x}-\frac {2 \sqrt [4]{3} \sqrt {\frac {a}{x^3}} x^2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} E\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {1+x^3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {\frac {a}{x^3}} x^2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {1+x^3}} \] Output:

-2*(a/x^3)^(1/2)*x*(x^3+1)^(1/2)+2*(1+3^(1/2))*(a/x^3)^(1/2)*x^2*(x^3+1)^( 
1/2)/(1+(1+3^(1/2))*x)-2*3^(1/4)*(a/x^3)^(1/2)*x^2*(1+x)*((x^2-x+1)/(1+(1+ 
3^(1/2))*x)^2)^(1/2)*EllipticE((1-(1+(1-3^(1/2))*x)^2/(1+(1+3^(1/2))*x)^2) 
^(1/2),1/4*6^(1/2)+1/4*2^(1/2))/(x*(1+x)/(1+(1+3^(1/2))*x)^2)^(1/2)/(x^3+1 
)^(1/2)-1/3*(1-3^(1/2))*(a/x^3)^(1/2)*x^2*(1+x)*((x^2-x+1)/(1+(1+3^(1/2))* 
x)^2)^(1/2)*InverseJacobiAM(arccos((1+(1-3^(1/2))*x)/(1+(1+3^(1/2))*x)),1/ 
4*6^(1/2)+1/4*2^(1/2))*3^(3/4)/(x*(1+x)/(1+(1+3^(1/2))*x)^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 = 10.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {\frac {a}{x^3}}}{\sqrt {1+x^3}} \, dx=-2 \sqrt {\frac {a}{x^3}} x \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},-x^3\right ) \] Input:

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

Output:

-2*Sqrt[a/x^3]*x*Hypergeometric2F1[-1/6, 1/2, 5/6, -x^3]
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {34, 847, 851, 837, 25, 766, 2420}

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.

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

\(\Big \downarrow \) 34

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

\(\Big \downarrow \) 847

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

\(\Big \downarrow \) 851

\(\displaystyle x^{3/2} \sqrt {\frac {a}{x^3}} \left (4 \int \frac {x^2}{\sqrt {x^3+1}}d\sqrt {x}-\frac {2 \sqrt {x^3+1}}{\sqrt {x}}\right )\)

\(\Big \downarrow \) 837

\(\displaystyle x^{3/2} \sqrt {\frac {a}{x^3}} \left (4 \left (-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^3+1}}d\sqrt {x}-\frac {1}{2} \int -\frac {2 x^2-\sqrt {3}+1}{\sqrt {x^3+1}}d\sqrt {x}\right )-\frac {2 \sqrt {x^3+1}}{\sqrt {x}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle x^{3/2} \sqrt {\frac {a}{x^3}} \left (4 \left (\frac {1}{2} \int \frac {2 x^2-\sqrt {3}+1}{\sqrt {x^3+1}}d\sqrt {x}-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^3+1}}d\sqrt {x}\right )-\frac {2 \sqrt {x^3+1}}{\sqrt {x}}\right )\)

\(\Big \downarrow \) 766

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

\(\Big \downarrow \) 2420

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

rule 837

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 
3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2))   Int[1/Sqrt[ 
a + b*x^6], x], x] - Simp[1/(2*r^2)   Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S 
qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
 

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 851

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

rule 2420

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = 
 Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr 
t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* 
(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 
*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) 
)*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 
 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 
- Sqrt[3])*d, 0]
 

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.07

method	result	size
meijerg	\(-2 \sqrt {\frac {a}{x^{3}}}\, x \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {1}{2}\right ], \left [\frac {5}{6}\right ], -x^{3}\right )\)	\(22\)
risch	\(-2 \sqrt {\frac {a}{x^{3}}}\, x \sqrt {x^{3}+1}+\frac {2 \left (x \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )+\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +1\right )}}\, \left (x +1\right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +1\right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +1\right )}}\, \left (-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +1\right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +1\right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )\right ) \sqrt {\frac {a}{x^{3}}}\, x \sqrt {x \left (x^{3}+1\right ) a}}{\sqrt {a x \left (x +1\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}\, \sqrt {x^{3}+1}}\)	\(355\)
default	\(\text {Expression too large to display}\)	\(1784\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.04 \[ \int \frac {\sqrt {\frac {a}{x^3}}}{\sqrt {1+x^3}} \, dx=2 \, \sqrt {a} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, \frac {1}{x}\right )\right ) \] Input:

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

Output:

2*sqrt(a)*weierstrassZeta(0, -4, weierstrassPInverse(0, -4, 1/x))
 

Sympy [F]

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

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

Output:

Integral(sqrt(a/x**3)/sqrt((x + 1)*(x**2 - x + 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {\frac {a}{x^3}}}{\sqrt {1+x^3}} \, dx=\sqrt {a}\, \left (\int \frac {\sqrt {x}\, \sqrt {x^{3}+1}}{x^{5}+x^{2}}d x \right ) \] Input:

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

Output:

sqrt(a)*int((sqrt(x)*sqrt(x**3 + 1))/(x**5 + x**2),x)