\(\int \frac {x^3}{(a+\frac {b}{x^3})^{3/2}} \, dx\) [514]

Optimal result

Integrand size = 15, antiderivative size = 563 \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {55 b^{4/3} \sqrt {a+\frac {b}{x^3}}}{24 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {55 b \sqrt {a+\frac {b}{x^3}} x}{24 a^3}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {11 \sqrt {a+\frac {b}{x^3}} x^4}{12 a^2}-\frac {55 \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{16\ 3^{3/4} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {55 b^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{12 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \] Output:

55/24*b^(4/3)*(a+b/x^3)^(1/2)/a^3/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)-55/24*b* 
(a+b/x^3)^(1/2)*x/a^3-2/3*x^4/a/(a+b/x^3)^(1/2)+11/12*(a+b/x^3)^(1/2)*x^4/ 
a^2-55/48*(1/2*6^(1/2)-1/2*2^(1/2))*b^(4/3)*(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)*El 
lipticE(((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^(1/4)/a^(8/3)/(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)+55/72*b^(4/3)*(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)*2^(1/2)*3^(3/4)/a^(8/3)/(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 = 65, normalized size of antiderivative = 0.12 \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {x \left (-11 b+2 a x^3+11 b \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {3}{2},\frac {11}{6},-\frac {a x^3}{b}\right )\right )}{8 a^2 \sqrt {a+\frac {b}{x^3}}} \] Input:

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

Output:

(x*(-11*b + 2*a*x^3 + 11*b*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[5/6, 3/2, 
 11/6, -((a*x^3)/b)]))/(8*a^2*Sqrt[a + b/x^3])
 

Rubi [A] (warning: unable to verify)

Time = 0.98 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {858, 819, 847, 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.

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

\(\Big \downarrow \) 858

\(\displaystyle -\int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^{3/2}}d\frac {1}{x}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {11 \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{3 a}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {11 \left (-\frac {5 b \int \frac {x^2}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{8 a}-\frac {x^4 \sqrt {a+\frac {b}{x^3}}}{4 a}\right )}{3 a}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {11 \left (-\frac {5 b \left (\frac {b \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x}d\frac {1}{x}}{2 a}-\frac {x \sqrt {a+\frac {b}{x^3}}}{a}\right )}{8 a}-\frac {x^4 \sqrt {a+\frac {b}{x^3}}}{4 a}\right )}{3 a}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}\)

\(\Big \downarrow \) 832

\(\displaystyle -\frac {11 \left (-\frac {5 b \left (\frac {b \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}\right )}{2 a}-\frac {x \sqrt {a+\frac {b}{x^3}}}{a}\right )}{8 a}-\frac {x^4 \sqrt {a+\frac {b}{x^3}}}{4 a}\right )}{3 a}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {11 \left (-\frac {5 b \left (\frac {b \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{2 a}-\frac {x \sqrt {a+\frac {b}{x^3}}}{a}\right )}{8 a}-\frac {x^4 \sqrt {a+\frac {b}{x^3}}}{4 a}\right )}{3 a}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}\)

\(\Big \downarrow \) 2416

\(\displaystyle -\frac {11 \left (-\frac {5 b \left (\frac {b \left (\frac {\frac {2 \sqrt {a+\frac {b}{x^3}}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{2 a}-\frac {x \sqrt {a+\frac {b}{x^3}}}{a}\right )}{8 a}-\frac {x^4 \sqrt {a+\frac {b}{x^3}}}{4 a}\right )}{3 a}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}\)

Input:

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

Output:

(-2*x^4)/(3*a*Sqrt[a + b/x^3]) - (11*(-1/4*(Sqrt[a + b/x^3]*x^4)/a - (5*b* 
(-((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 + S 
qrt[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)))/(8*a)))/(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 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 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]
 

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. 2198 vs. \(2 (421 ) = 842\).

Time = 3.14 (sec) , antiderivative size = 2199, normalized size of antiderivative = 3.91

method	result	size
risch	\(\text {Expression too large to display}\)	\(2199\)
default	\(\text {Expression too large to display}\)	\(2936\)

Input:

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

Output:

1/4*x/a^2*(a*x^3+b)/((a*x^3+b)/x^3)^(1/2)-1/8/a^2*b*(13*(x*(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/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)*(((-1/2/a*(-a^2*b)^(1/3)+ 
1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/a*(-a^2*b)^(1/3)+1/a^2*(-a^2*b)^(2/3))/(-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)*Ellip 
ticF(((-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 
))+(1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*EllipticE(((-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*(...
 

Fricas [F]

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

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

Output:

integral(x^9*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 = 0.65 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.08 \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=- \frac {x^{4} \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {3}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (- \frac {1}{3}\right )} \] Input:

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

Output:

-x**4*gamma(-4/3)*hyper((-4/3, 3/2), (-1/3,), b*exp_polar(I*pi)/(a*x**3))/ 
(3*a**(3/2)*gamma(-1/3))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {4 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, a \,x^{5}-22 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, b \,x^{2}+55 \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}\, x}{a^{2} x^{6}+2 a b \,x^{3}+b^{2}}d x \right ) a \,b^{2} x^{3}+55 \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}\, x}{a^{2} x^{6}+2 a b \,x^{3}+b^{2}}d x \right ) b^{3}}{16 a^{2} \left (a \,x^{3}+b \right )} \] Input:

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

Output:

(4*sqrt(x)*sqrt(a*x**3 + b)*a*x**5 - 22*sqrt(x)*sqrt(a*x**3 + b)*b*x**2 + 
55*int((sqrt(x)*sqrt(a*x**3 + b)*x)/(a**2*x**6 + 2*a*b*x**3 + b**2),x)*a*b 
**2*x**3 + 55*int((sqrt(x)*sqrt(a*x**3 + b)*x)/(a**2*x**6 + 2*a*b*x**3 + b 
**2),x)*b**3)/(16*a**2*(a*x**3 + b))