\(\int \sqrt {a+\frac {b}{x^3}} x^6 \, dx\) [2004]

Optimal result

Integrand size = 15, antiderivative size = 563 \[ \int \sqrt {a+\frac {b}{x^3}} x^6 \, dx=\frac {15 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {15 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^2}+\frac {3 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a}+\frac {1}{7} \sqrt {a+\frac {b}{x^3}} x^7-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {5\ 3^{3/4} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{56 \sqrt {2} a^{5/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

-15/112*b^2*x*(a+b/x^3)^(1/2)/a^2+3/56*b*x^4*(a+b/x^3)^(1/2)/a+1/7*x^7*(a+b/x^3)^(1/2)+15/112*b^(7/3)*(a+b/x^3
)^(1/2)/a^2/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))+5/112*3^(3/4)*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)/a^(5/3)*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)-15/224*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^(5/3)/(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 = 0.33 (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, 283, 331, 309, 224, 1891} \[ \int \sqrt {a+\frac {b}{x^3}} x^6 \, dx=\frac {5\ 3^{3/4} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{56 \sqrt {2} a^{5/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {15 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {15 b^2 x \sqrt {a+\frac {b}{x^3}}}{112 a^2}+\frac {1}{7} x^7 \sqrt {a+\frac {b}{x^3}}+\frac {3 b x^4 \sqrt {a+\frac {b}{x^3}}}{56 a} \]

[In]

Int[Sqrt[a + b/x^3]*x^6,x]

[Out]

(15*b^(7/3)*Sqrt[a + b/x^3])/(112*a^2*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) - (15*b^2*Sqrt[a + b/x^3]*x)/(112*a
^2) + (3*b*Sqrt[a + b/x^3]*x^4)/(56*a) + (Sqrt[a + b/x^3]*x^7)/7 - (15*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]*Ell
ipticE[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^(5/3)*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]) + (5*3^(3
/4)*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]*a^(5/3)*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 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && 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 {\sqrt {a+b x^3}}{x^8} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{7} \sqrt {a+\frac {b}{x^3}} x^7-\frac {1}{14} (3 b) \text {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a}+\frac {1}{7} \sqrt {a+\frac {b}{x^3}} x^7+\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{112 a} \\ & = -\frac {15 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^2}+\frac {3 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a}+\frac {1}{7} \sqrt {a+\frac {b}{x^3}} x^7+\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{224 a^2} \\ & = -\frac {15 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^2}+\frac {3 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a}+\frac {1}{7} \sqrt {a+\frac {b}{x^3}} x^7+\frac {\left (15 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^2}-\frac {\left (15 \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^{5/3}} \\ & = \frac {15 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {15 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^2}+\frac {3 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a}+\frac {1}{7} \sqrt {a+\frac {b}{x^3}} x^7-\frac {15 \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^{5/3} \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 {5\ 3^{3/4} 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} a^{5/3} \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 = 80, normalized size of antiderivative = 0.14 \[ \int \sqrt {a+\frac {b}{x^3}} x^6 \, dx=\frac {\sqrt {a+\frac {b}{x^3}} x^4 \left (\left (b+a x^3\right ) \sqrt {1+\frac {a x^3}{b}}-b \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{6},\frac {11}{6},-\frac {a x^3}{b}\right )\right )}{7 a \sqrt {1+\frac {a x^3}{b}}} \]

[In]

Integrate[Sqrt[a + b/x^3]*x^6,x]

[Out]

(Sqrt[a + b/x^3]*x^4*((b + a*x^3)*Sqrt[1 + (a*x^3)/b] - b*Hypergeometric2F1[-1/2, 5/6, 11/6, -((a*x^3)/b)]))/(
7*a*Sqrt[1 + (a*x^3)/b])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1126 vs. \(2 (421 ) = 842\).

Time = 0.44 (sec) , antiderivative size = 1127, normalized size of antiderivative = 2.00

method	result	size
risch	\(\text {Expression too large to display}\)	\(1127\)
default	\(\text {Expression too large to display}\)	\(2799\)

[In]

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

[Out]

1/56*x^4*(8*a*x^3+3*b)/a*((a*x^3+b)/x^3)^(1/2)-15/112/a*b^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))*E
llipticE(((-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*((a*x
^3+b)/x^3)^(1/2)*(x*(a*x^3+b))^(1/2)/(a*x^3+b)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.09 \[ \int \sqrt {a+\frac {b}{x^3}} x^6 \, dx=- \frac {\sqrt {a} x^{7} \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \Gamma \left (- \frac {4}{3}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(a + b/x^3)*x^6, x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(a + b/x^3)*x^6, x)

Mupad [F(-1)]

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

[In]

int(x^6*(a + b/x^3)^(1/2),x)

[Out]

int(x^6*(a + b/x^3)^(1/2), x)