\(\int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx\) [2036]

Optimal result

Integrand size = 15, antiderivative size = 568 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx=\frac {1280 a^3 \sqrt {a+\frac {b}{x^3}}}{1729 b^{11/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 b x^8}+\frac {32 a \sqrt {a+\frac {b}{x^3}}}{247 b^2 x^5}-\frac {320 a^2 \sqrt {a+\frac {b}{x^3}}}{1729 b^3 x^2}-\frac {640 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{11/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {1280 \sqrt {2} a^{10/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{1729 \sqrt [4]{3} b^{11/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

-2/19*(a+b/x^3)^(1/2)/b/x^8+32/247*a*(a+b/x^3)^(1/2)/b^2/x^5-320/1729*a^2*(a+b/x^3)^(1/2)/b^3/x^2+1280/1729*a^
3*(a+b/x^3)^(1/2)/b^(11/3)/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))+1280/5187*a^(10/3)*(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^(11/3)/(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)-640/1729*3^(1/4)*a^(10/3)*(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^(11/3)/(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 = 0.30 (sec) , antiderivative size = 568, 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, 327, 309, 224, 1891} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx=\frac {1280 \sqrt {2} a^{10/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{1729 \sqrt [4]{3} b^{11/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {640 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{11/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {1280 a^3 \sqrt {a+\frac {b}{x^3}}}{1729 b^{11/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {320 a^2 \sqrt {a+\frac {b}{x^3}}}{1729 b^3 x^2}+\frac {32 a \sqrt {a+\frac {b}{x^3}}}{247 b^2 x^5}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 b x^8} \]

[In]

Int[1/(Sqrt[a + b/x^3]*x^12),x]

[Out]

(1280*a^3*Sqrt[a + b/x^3])/(1729*b^(11/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) - (2*Sqrt[a + b/x^3])/(19*b*x^8
) + (32*a*Sqrt[a + b/x^3])/(247*b^2*x^5) - (320*a^2*Sqrt[a + b/x^3])/(1729*b^3*x^2) - (640*3^(1/4)*Sqrt[2 - Sq
rt[3]]*a^(10/3)*(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]])/(1729*b^(11/3)*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]) + (1280*Sqrt[2]*a^(10/3)*(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]])/(1729*3^(1/4)*b^(11/3)*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^{10}}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 b x^8}+\frac {(16 a) \text {Subst}\left (\int \frac {x^7}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{19 b} \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 b x^8}+\frac {32 a \sqrt {a+\frac {b}{x^3}}}{247 b^2 x^5}-\frac {\left (160 a^2\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{247 b^2} \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 b x^8}+\frac {32 a \sqrt {a+\frac {b}{x^3}}}{247 b^2 x^5}-\frac {320 a^2 \sqrt {a+\frac {b}{x^3}}}{1729 b^3 x^2}+\frac {\left (640 a^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{1729 b^3} \\ & = -\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 b x^8}+\frac {32 a \sqrt {a+\frac {b}{x^3}}}{247 b^2 x^5}-\frac {320 a^2 \sqrt {a+\frac {b}{x^3}}}{1729 b^3 x^2}+\frac {\left (640 a^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 )}{1729 b^{10/3}}-\frac {\left (640 \left (1-\sqrt {3}\right ) a^{10/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{1729 b^{10/3}} \\ & = \frac {1280 a^3 \sqrt {a+\frac {b}{x^3}}}{1729 b^{11/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 b x^8}+\frac {32 a \sqrt {a+\frac {b}{x^3}}}{247 b^2 x^5}-\frac {320 a^2 \sqrt {a+\frac {b}{x^3}}}{1729 b^3 x^2}-\frac {640 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} \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 )}{1729 b^{11/3} \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 {1280 \sqrt {2} a^{10/3} \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 )}{1729 \sqrt [4]{3} b^{11/3} \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 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx=-\frac {2 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {19}{6},\frac {1}{2},-\frac {13}{6},-\frac {a x^3}{b}\right )}{19 \sqrt {a+\frac {b}{x^3}} x^{11}} \]

[In]

Integrate[1/(Sqrt[a + b/x^3]*x^12),x]

[Out]

(-2*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[-19/6, 1/2, -13/6, -((a*x^3)/b)])/(19*Sqrt[a + b/x^3]*x^11)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (426 ) = 852\).

Time = 2.84 (sec) , antiderivative size = 1149, normalized size of antiderivative = 2.02

method	result	size
risch	\(\text {Expression too large to display}\)	\(1149\)
default	\(\text {Expression too large to display}\)	\(3779\)

[In]

int(1/x^12/(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/1729*(a*x^3+b)*(640*a^3*x^9-160*a^2*b*x^6+112*a*b^2*x^3-91*b^3)/b^4/x^11/((a*x^3+b)/x^3)^(1/2)-1280/1729*a^4
/b^4*(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 = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx=-\frac {2 \, {\left (640 \, a^{3} \sqrt {b} x^{8} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right )\right ) + {\left (160 \, a^{2} b x^{6} - 112 \, a b^{2} x^{3} + 91 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{1729 \, b^{4} x^{8}} \]

[In]

integrate(1/x^12/(a+b/x^3)^(1/2),x, algorithm="fricas")

[Out]

-2/1729*(640*a^3*sqrt(b)*x^8*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, 1/x)) + (160*a^2*b*x^6
- 112*a*b^2*x^3 + 91*b^3)*sqrt((a*x^3 + b)/x^3))/(b^4*x^8)

Sympy [A] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.07 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx=- \frac {\Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} x^{11} \Gamma \left (\frac {14}{3}\right )} \]

[In]

integrate(1/x**12/(a+b/x**3)**(1/2),x)

[Out]

-gamma(11/3)*hyper((1/2, 11/3), (14/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt(a)*x**11*gamma(14/3))

Maxima [F]

\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{12}} \,d x } \]

[In]

integrate(1/x^12/(a+b/x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^12), x)

Giac [F]

\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{12}} \,d x } \]

[In]

integrate(1/x^12/(a+b/x^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^12), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{12}} \, dx=\int \frac {1}{x^{12}\,\sqrt {a+\frac {b}{x^3}}} \,d x \]

[In]

int(1/(x^12*(a + b/x^3)^(1/2)),x)

[Out]

int(1/(x^12*(a + b/x^3)^(1/2)), x)