\(\int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx\) [2035]

Optimal result

Integrand size = 15, antiderivative size = 544 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx=-\frac {80 a^2 \sqrt {a+\frac {b}{x^3}}}{91 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}+\frac {20 a \sqrt {a+\frac {b}{x^3}}}{91 b^2 x^2}+\frac {40 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{91 b^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {80 \sqrt {2} a^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{91 \sqrt [4]{3} b^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

-2/13*(a+b/x^3)^(1/2)/b/x^5+20/91*a*(a+b/x^3)^(1/2)/b^2/x^2-80/91*a^2*(a+b/x^3)^(1/2)/b^(8/3)/(b^(1/3)/x+a^(1/
3)*(1+3^(1/2)))-80/273*a^(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)*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^(3/4)/b^(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
)+40/91*3^(1/4)*a^(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)/b^(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 = 0.25 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {342, 327, 309, 224, 1891} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx=-\frac {80 \sqrt {2} a^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{91 \sqrt [4]{3} b^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {40 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{91 b^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {80 a^2 \sqrt {a+\frac {b}{x^3}}}{91 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}+\frac {20 a \sqrt {a+\frac {b}{x^3}}}{91 b^2 x^2}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5} \]

[In]

Int[1/(Sqrt[a + b/x^3]*x^9),x]

[Out]

(-80*a^2*Sqrt[a + b/x^3])/(91*b^(8/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) - (2*Sqrt[a + b/x^3])/(13*b*x^5) +
(20*a*Sqrt[a + b/x^3])/(91*b^2*x^2) + (40*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(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]])/(91*b^(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]) - (80*Sqrt[2]*a^(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[ArcSi
n[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(91*3^(1/4)*b^(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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 {x^7}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}+\frac {(10 a) \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{13 b} \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}+\frac {20 a \sqrt {a+\frac {b}{x^3}}}{91 b^2 x^2}-\frac {\left (40 a^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{91 b^2} \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}+\frac {20 a \sqrt {a+\frac {b}{x^3}}}{91 b^2 x^2}-\frac {\left (40 a^2\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 )}{91 b^{7/3}}+\frac {\left (40 \left (1-\sqrt {3}\right ) a^{7/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{91 b^{7/3}} \\ & = -\frac {80 a^2 \sqrt {a+\frac {b}{x^3}}}{91 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}+\frac {20 a \sqrt {a+\frac {b}{x^3}}}{91 b^2 x^2}+\frac {40 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{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 )}{91 b^{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 {80 \sqrt {2} a^{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 )}{91 \sqrt [4]{3} b^{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 = 10.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx=-\frac {2 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {13}{6},\frac {1}{2},-\frac {7}{6},-\frac {a x^3}{b}\right )}{13 \sqrt {a+\frac {b}{x^3}} x^8} \]

[In]

Integrate[1/(Sqrt[a + b/x^3]*x^9),x]

[Out]

(-2*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[-13/6, 1/2, -7/6, -((a*x^3)/b)])/(13*Sqrt[a + b/x^3]*x^8)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1137 vs. \(2 (406 ) = 812\).

Time = 1.80 (sec) , antiderivative size = 1138, normalized size of antiderivative = 2.09

method	result	size
risch	\(\text {Expression too large to display}\)	\(1138\)
default	\(\text {Expression too large to display}\)	\(3547\)

[In]

int(1/x^9/(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/91*(a*x^3+b)*(40*a^2*x^6-10*a*b*x^3+7*b^2)/b^3/x^8/((a*x^3+b)/x^3)^(1/2)+80/91*a^3/b^3*(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 [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx=\frac {2 \, {\left (40 \, a^{2} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right )\right ) + {\left (10 \, a b x^{3} - 7 \, b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{91 \, b^{3} x^{5}} \]

[In]

integrate(1/x^9/(a+b/x^3)^(1/2),x, algorithm="fricas")

[Out]

2/91*(40*a^2*sqrt(b)*x^5*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, 1/x)) + (10*a*b*x^3 - 7*b^2
)*sqrt((a*x^3 + b)/x^3))/(b^3*x^5)

Sympy [A] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.07 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx=- \frac {\Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} x^{8} \Gamma \left (\frac {11}{3}\right )} \]

[In]

integrate(1/x**9/(a+b/x**3)**(1/2),x)

[Out]

-gamma(8/3)*hyper((1/2, 8/3), (11/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt(a)*x**8*gamma(11/3))

Maxima [F]

\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{9}} \,d x } \]

[In]

integrate(1/x^9/(a+b/x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^9), x)

Giac [F]

\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{9}} \,d x } \]

[In]

integrate(1/x^9/(a+b/x^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^9), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^9} \, dx=\int \frac {1}{x^9\,\sqrt {a+\frac {b}{x^3}}} \,d x \]

[In]

int(1/(x^9*(a + b/x^3)^(1/2)),x)

[Out]

int(1/(x^9*(a + b/x^3)^(1/2)), x)