Integrand size = 15, antiderivative size = 563 \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {55 b^{4/3} \sqrt {a+\frac {b}{x^3}}}{24 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {55 b \sqrt {a+\frac {b}{x^3}} x}{24 a^3}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {11 \sqrt {a+\frac {b}{x^3}} x^4}{12 a^2}-\frac {55 \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{16\ 3^{3/4} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {55 b^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{12 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]
-2/3*x^4/a/(a+b/x^3)^(1/2)-55/24*b*x*(a+b/x^3)^(1/2)/a^3+11/12*x^4*(a+b/x^ 3)^(1/2)/a^2+55/24*b^(4/3)*(a+b/x^3)^(1/2)/a^3/(b^(1/3)/x+a^(1/3)*(1+3^(1/ 2)))+55/72*b^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-3^( 1/2)))/(b^(1/3)/x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*((a^(2/3)+b^(2/3)/x^ 2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/a^(8 /3)*2^(1/2)/(a+b/x^3)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3 )*(1+3^(1/2)))^2)^(1/2)-55/48*b^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/ 3)/x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)/x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*( 1/2*6^(1/2)-1/2*2^(1/2))*((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3) /x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(1/4)/a^(8/3)/(a+b/x^3)^(1/2)/(a^(1/3)* (a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.12 \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {x \left (-11 b+2 a x^3+11 b \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {3}{2},\frac {11}{6},-\frac {a x^3}{b}\right )\right )}{8 a^2 \sqrt {a+\frac {b}{x^3}}} \]
(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])
Time = 0.59 (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 |
\(\displaystyle -\frac {11 \left (-\frac {5 b \left (\frac {b \left (\frac {\frac {2 \sqrt {a+\frac {b}{x^3}}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{2 a}-\frac {x \sqrt {a+\frac {b}{x^3}}}{a}\right )}{8 a}-\frac {x^4 \sqrt {a+\frac {b}{x^3}}}{4 a}\right )}{3 a}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}\) |
(-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)
3.21.51.3.1 Defintions of rubi rules used
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]
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]
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]
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]
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]
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]
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 = 1.86 (sec) , antiderivative size = 2199, normalized size of antiderivative = 3.91
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2199\) |
default | \(\text {Expression too large to display}\) | \(2936\) |
1/4/a^2*x*(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*(...
\[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.58 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.08 \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=- \frac {x^{4} \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {3}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (- \frac {1}{3}\right )} \]
-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))
\[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{x^3}\right )}^{3/2}} \,d x \]