\(\int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx\) [2030]

Optimal result

Integrand size = 15, antiderivative size = 566 \[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx=-\frac {55 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}+\frac {55 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^3}-\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}+\frac {55 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{224 a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {55 b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{56 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

55/112*b^2*x*(a+b/x^3)^(1/2)/a^3-11/56*b*x^4*(a+b/x^3)^(1/2)/a^2+1/7*x^7*(a+b/x^3)^(1/2)/a-55/112*b^(7/3)*(a+b
/x^3)^(1/2)/a^3/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))-55/336*b^(7/3)*(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)+55/224*3^(1/4)*b^(7/3)*(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)/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 = 566, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {342, 331, 309, 224, 1891} \[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx=-\frac {55 b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{56 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {55 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{224 a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {55 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}+\frac {55 b^2 x \sqrt {a+\frac {b}{x^3}}}{112 a^3}-\frac {11 b x^4 \sqrt {a+\frac {b}{x^3}}}{56 a^2}+\frac {x^7 \sqrt {a+\frac {b}{x^3}}}{7 a} \]

[In]

Int[x^6/Sqrt[a + b/x^3],x]

[Out]

(-55*b^(7/3)*Sqrt[a + b/x^3])/(112*a^3*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) + (55*b^2*Sqrt[a + b/x^3]*x)/(112*
a^3) - (11*b*Sqrt[a + b/x^3]*x^4)/(56*a^2) + (Sqrt[a + b/x^3]*x^7)/(7*a) + (55*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(7/
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]
])/(224*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^(7/3)*(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*Sqrt[3]])/(56*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 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^8 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}+\frac {(11 b) \text {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{14 a} \\ & = -\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}-\frac {\left (55 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{112 a^2} \\ & = \frac {55 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^3}-\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}-\frac {\left (55 b^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{224 a^3} \\ & = \frac {55 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^3}-\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}-\frac {\left (55 b^{8/3}\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 )}{224 a^3}+\frac {\left (55 \left (1-\sqrt {3}\right ) b^{8/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{224 a^{8/3}} \\ & = -\frac {55 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}+\frac {55 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^3}-\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}+\frac {55 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \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 )}{224 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^{7/3} \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 )}{56 \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 = 79, normalized size of antiderivative = 0.14 \[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {-11 b^2 x-3 a b x^4+8 a^2 x^7+11 b^2 x \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},-\frac {a x^3}{b}\right )}{56 a^2 \sqrt {a+\frac {b}{x^3}}} \]

[In]

Integrate[x^6/Sqrt[a + b/x^3],x]

[Out]

(-11*b^2*x - 3*a*b*x^4 + 8*a^2*x^7 + 11*b^2*x*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[1/2, 5/6, 11/6, -((a*x^3)/
b)])/(56*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. 1124 vs. \(2 (424 ) = 848\).

Time = 1.18 (sec) , antiderivative size = 1125, normalized size of antiderivative = 1.99

method	result	size
risch	\(\text {Expression too large to display}\)	\(1125\)
default	\(\text {Expression too large to display}\)	\(2806\)

[In]

int(x^6/(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/56*x*(8*a*x^3-11*b)/a^2*(a*x^3+b)/((a*x^3+b)/x^3)^(1/2)+55/112*b^2/a^2*(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]

\[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{6}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]

[In]

integrate(x^6/(a+b/x^3)^(1/2),x, algorithm="fricas")

[Out]

integral(x^9*sqrt((a*x^3 + b)/x^3)/(a*x^3 + b), x)

Sympy [A] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.08 \[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx=- \frac {x^{7} \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} \Gamma \left (- \frac {4}{3}\right )} \]

[In]

integrate(x**6/(a+b/x**3)**(1/2),x)

[Out]

-x**7*gamma(-7/3)*hyper((-7/3, 1/2), (-4/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt(a)*gamma(-4/3))

Maxima [F]

\[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{6}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]

[In]

integrate(x^6/(a+b/x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^6/sqrt(a + b/x^3), x)

Giac [F]

\[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{6}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]

[In]

integrate(x^6/(a+b/x^3)^(1/2),x, algorithm="giac")

[Out]

integrate(x^6/sqrt(a + b/x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \,d x \]

[In]

int(x^6/(a + b/x^3)^(1/2),x)

[Out]

int(x^6/(a + b/x^3)^(1/2), x)