\(\int \frac {x}{(a+\frac {b}{x^3})^{3/2}} \, dx\) [509]

Optimal result

Integrand size = 13, antiderivative size = 269 \[ \int \frac {x}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=-\frac {2 x^2}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {7 \sqrt {a+\frac {b}{x^3}} x^2}{6 a^2}+\frac {7 \sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{6 \sqrt [4]{3} a^2 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \] Output:

-2/3*x^2/a/(a+b/x^3)^(1/2)+7/6*(a+b/x^3)^(1/2)*x^2/a^2+7/18*(1/2*6^(1/2)+1 
/2*2^(1/2))*b^(2/3)*(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 
/(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 = 67, normalized size of antiderivative = 0.25 \[ \int \frac {x}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {7 b+3 a x^3-7 b \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {a x^3}{b}\right )}{6 a^2 \sqrt {a+\frac {b}{x^3}} x} \] Input:

Integrate[x/(a + b/x^3)^(3/2),x]
 

Output:

(7*b + 3*a*x^3 - 7*b*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[1/6, 1/2, 7/6, 
-((a*x^3)/b)])/(6*a^2*Sqrt[a + b/x^3]*x)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {858, 819, 847, 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.

\(\displaystyle \int \frac {x}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 858

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

\(\Big \downarrow \) 819

\(\displaystyle -\frac {7 \int \frac {x^3}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{3 a}-\frac {2 x^2}{3 a \sqrt {a+\frac {b}{x^3}}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{4 a}-\frac {x^2 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )}{3 a}-\frac {2 x^2}{3 a \sqrt {a+\frac {b}{x^3}}}\)

\(\Big \downarrow \) 759

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

Input:

Int[x/(a + b/x^3)^(3/2),x]
 

Output:

(-2*x^2)/(3*a*Sqrt[a + b/x^3]) - (7*(-1/2*(Sqrt[a + b/x^3]*x^2)/a - (Sqrt[ 
2 + Sqrt[3]]*b^(2/3)*(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[ArcSi 
n[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)] 
, -7 - 4*Sqrt[3]])/(2*3^(1/4)*a*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])))/(3*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 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, 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 858

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1430 vs. \(2 (206 ) = 412\).

Time = 2.39 (sec) , antiderivative size = 1431, normalized size of antiderivative = 5.32

method	result	size
risch	\(\text {Expression too large to display}\)	\(1431\)
default	\(\text {Expression too large to display}\)	\(2052\)

Input:

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

Output:

1/2/a^2/x*(a*x^3+b)/((a*x^3+b)/x^3)^(1/2)-1/4/a^2*b*(10*(1/2/a*(-a^2*b)^(1 
/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*((-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/ 
a*(-a^2*b)^(1/3))*x/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)) 
/(x-1/a*(-a^2*b)^(1/3)))^(1/2)*(x-1/a*(-a^2*b)^(1/3))^2*(1/a*(-a^2*b)^(1/3 
)*(x+1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(-1/2/a*(-a^2*b) 
^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3)))^(1/2)*(1/a* 
(-a^2*b)^(1/3)*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(-1 
/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3)) 
)^(1/2)/(-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*a/(-a^2*b)^ 
(1/3)/(a*x*(x-1/a*(-a^2*b)^(1/3))*(x+1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a* 
(-a^2*b)^(1/3))*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)))^( 
1/2)*EllipticF(((-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*x/( 
-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3 
)))^(1/2),((3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*(1/2/a*(- 
a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(1/2/a*(-a^2*b)^(1/3)+1/2*I*3 
^(1/2)/a*(-a^2*b)^(1/3))/(3/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1 
/3)))^(1/2))-4*b*(2/3*x/b/((x^3+b/a)*a*x)^(1/2)+4/3/b*(1/2/a*(-a^2*b)^(1/3 
)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*((-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a* 
(-a^2*b)^(1/3))*x/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/( 
x-1/a*(-a^2*b)^(1/3)))^(1/2)*(x-1/a*(-a^2*b)^(1/3))^2*(1/a*(-a^2*b)^(1/...
 

Fricas [F]

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

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

Output:

integral(x^7*sqrt((a*x^3 + b)/x^3)/(a^2*x^6 + 2*a*b*x^3 + b^2), x)
 

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.16 \[ \int \frac {x}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=- \frac {x^{2} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {1}{3}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

integrate(x/(a + b/x^3)^(3/2), x)
 

Giac [F]

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

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

Output:

integrate(x/(a + b/x^3)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int \frac {x}{{\left (a+\frac {b}{x^3}\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {4 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, a \,x^{3}+14 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, b -7 \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}}{a^{2} x^{7}+2 a b \,x^{4}+b^{2} x}d x \right ) a \,b^{2} x^{3}-7 \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}}{a^{2} x^{7}+2 a b \,x^{4}+b^{2} x}d x \right ) b^{3}}{8 a^{2} \left (a \,x^{3}+b \right )} \] Input:

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

Output:

(4*sqrt(x)*sqrt(a*x**3 + b)*a*x**3 + 14*sqrt(x)*sqrt(a*x**3 + b)*b - 7*int 
((sqrt(x)*sqrt(a*x**3 + b))/(a**2*x**7 + 2*a*b*x**4 + b**2*x),x)*a*b**2*x* 
*3 - 7*int((sqrt(x)*sqrt(a*x**3 + b))/(a**2*x**7 + 2*a*b*x**4 + b**2*x),x) 
*b**3)/(8*a**2*(a*x**3 + b))