\(\int \frac {x^3}{(a+\frac {b}{x^3})^{3/2}} \, dx\) [2051]

Optimal result

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

[Out]

-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)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {342, 296, 331, 309, 224, 1891} \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {55 b^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{12 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {55 \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{16\ 3^{3/4} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {55 b^{4/3} \sqrt {a+\frac {b}{x^3}}}{24 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {55 b x \sqrt {a+\frac {b}{x^3}}}{24 a^3}+\frac {11 x^4 \sqrt {a+\frac {b}{x^3}}}{12 a^2}-\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}} \]

[In]

Int[x^3/(a + b/x^3)^(3/2),x]

[Out]

(55*b^(4/3)*Sqrt[a + b/x^3])/(24*a^3*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) - (55*b*Sqrt[a + b/x^3]*x)/(24*a^3)
- (2*x^4)/(3*a*Sqrt[a + b/x^3]) + (11*Sqrt[a + b/x^3]*x^4)/(12*a^2) - (55*Sqrt[2 - Sqrt[3]]*b^(4/3)*(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]*Elliptic
E[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]])/(16*3^(3/4
)*a^(8/3)*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]) + (55*b
^(4/3)*(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[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqr
t[3]])/(12*Sqrt[2]*3^(1/4)*a^(8/3)*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])

Rule 224

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]*Sq
rt[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 296

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] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 309

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 - Sqrt[3]))*(s/r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]
, x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

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] && IntegerQ[m]

Rule 1891

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[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
] - Simp[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] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x^5 \left (a+b x^3\right )^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 x^4}{3 a \sqrt {a+\frac {b}{x^3}}}-\frac {11 \text {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{3 a} \\ & = -\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 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{24 a^2} \\ & = -\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 {\left (55 b^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{48 a^3} \\ & = -\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 {\left (55 b^{5/3}\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{48 a^3}-\frac {\left (55 \left (1-\sqrt {3}\right ) b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{48 a^{8/3}} \\ & = \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 (\sin ^{-1}\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}} F\left (\sin ^{-1}\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}}} \\ \end{align*}

Mathematica [C] (verified)

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 \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=\frac {x \left (-11 b+2 a x^3+11 b \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {3}{2},\frac {11}{6},-\frac {a x^3}{b}\right )\right )}{8 a^2 \sqrt {a+\frac {b}{x^3}}} \]

[In]

Integrate[x^3/(a + b/x^3)^(3/2),x]

[Out]

(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])

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 = 1.86 (sec) , antiderivative size = 2199, normalized size of antiderivative = 3.91

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

[In]

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

[Out]

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)*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))+(1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*Ellip
ticE(((-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))*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)-8*b*(2/3*x^
3/b/((x^3+b/a)*a*x)^(1/2)-2/3/b*(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)*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))+(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*(-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))*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)))/x^2/((a*x^3+b)/x^3)^(1/2)*(x*(a*x
^3+b))^(1/2)

Fricas [F]

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

[In]

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

[Out]

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 = 0.58 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.08 \[ \int \frac {x^3}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx=- \frac {x^{4} \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {3}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (- \frac {1}{3}\right )} \]

[In]

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

[Out]

-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]

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

[In]

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

[Out]

integrate(x^3/(a + b/x^3)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^3/(a + b/x^3)^(3/2), x)

Mupad [F(-1)]

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

[In]

int(x^3/(a + b/x^3)^(3/2),x)

[Out]

int(x^3/(a + b/x^3)^(3/2), x)