\(\int \frac {1}{(a+\frac {b}{x^3})^{3/2} x^{12}} \, dx\) [519]

Optimal result

Integrand size = 15, antiderivative size = 565 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{12}} \, dx=-\frac {1280 a^2 \sqrt {a+\frac {b}{x^3}}}{273 b^{11/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}+\frac {2}{3 b \sqrt {a+\frac {b}{x^3}} x^8}-\frac {32 \sqrt {a+\frac {b}{x^3}}}{39 b^2 x^5}+\frac {320 a \sqrt {a+\frac {b}{x^3}}}{273 b^3 x^2}+\frac {640 \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{91\ 3^{3/4} b^{11/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {1280 \sqrt {2} a^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{273 \sqrt [4]{3} b^{11/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \] Output:

-1280/273*a^2*(a+b/x^3)^(1/2)/b^(11/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)+2/3 
/b/(a+b/x^3)^(1/2)/x^8-32/39*(a+b/x^3)^(1/2)/b^2/x^5+320/273*a*(a+b/x^3)^( 
1/2)/b^3/x^2+640/273*(1/2*6^(1/2)-1/2*2^(1/2))*a^(7/3)*(a^(1/3)+b^(1/3)/x) 
*((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)^ 
2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)+b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^ 
(1/3)/x),I*3^(1/2)+2*I)*3^(1/4)/b^(11/3)/(a+b/x^3)^(1/2)/(a^(1/3)*(a^(1/3) 
+b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)^2)^(1/2)-1280/819*2^(1/2)*a^(7 
/3)*(a^(1/3)+b^(1/3)/x)*((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/((1+3^(1/ 
2))*a^(1/3)+b^(1/3)/x)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)/x)/ 
((1+3^(1/2))*a^(1/3)+b^(1/3)/x),I*3^(1/2)+2*I)*3^(3/4)/b^(11/3)/(a+b/x^3)^ 
(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)^2)^(1/2 
)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{12}} \, dx=-\frac {2 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {13}{6},\frac {3}{2},-\frac {7}{6},-\frac {a x^3}{b}\right )}{13 b \sqrt {a+\frac {b}{x^3}} x^8} \] Input:

Integrate[1/((a + b/x^3)^(3/2)*x^12),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.94 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {858, 817, 843, 843, 832, 759, 2416}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^{12} \left (a+\frac {b}{x^3}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 858

\(\displaystyle -\int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{10}}d\frac {1}{x}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {2}{3 b x^8 \sqrt {a+\frac {b}{x^3}}}-\frac {16 \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^7}d\frac {1}{x}}{3 b}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {2}{3 b x^8 \sqrt {a+\frac {b}{x^3}}}-\frac {16 \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^4}d\frac {1}{x}}{13 b}\right )}{3 b}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {2}{3 b x^8 \sqrt {a+\frac {b}{x^3}}}-\frac {16 \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{7 b x^2}-\frac {4 a \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x}d\frac {1}{x}}{7 b}\right )}{13 b}\right )}{3 b}\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {2}{3 b x^8 \sqrt {a+\frac {b}{x^3}}}-\frac {16 \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{7 b x^2}-\frac {4 a \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}\right )}{7 b}\right )}{13 b}\right )}{3 b}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {2}{3 b x^8 \sqrt {a+\frac {b}{x^3}}}-\frac {16 \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{7 b x^2}-\frac {4 a \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{7 b}\right )}{13 b}\right )}{3 b}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {2}{3 b x^8 \sqrt {a+\frac {b}{x^3}}}-\frac {16 \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{7 b x^2}-\frac {4 a \left (\frac {\frac {2 \sqrt {a+\frac {b}{x^3}}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{7 b}\right )}{13 b}\right )}{3 b}\)

Input:

Int[1/((a + b/x^3)^(3/2)*x^12),x]
 

Output:

2/(3*b*Sqrt[a + b/x^3]*x^8) - (16*((2*Sqrt[a + b/x^3])/(13*b*x^5) - (10*a* 
((2*Sqrt[a + b/x^3])/(7*b*x^2) - (4*a*(((2*Sqrt[a + b/x^3])/(b^(1/3)*((1 + 
 Sqrt[3])*a^(1/3) + b^(1/3)/x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(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]])/(b^(1/3 
)*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]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3) 
*(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]])/( 
3^(1/4)*b^(2/3)*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])))/(7*b)))/(13*b)))/(3*b)
 

Defintions of rubi rules used

rule 759

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]*Sqrt[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 817

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*n*(p + 1))), x] - Simp[c^n 
*((m - n + 1)/(b*n*(p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x 
] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  ! 
ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 832

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

rule 843

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] - Simp[ 
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 858

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] && Int 
egerQ[m]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[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] - S 
imp[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] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2223 vs. \(2 (423 ) = 846\).

Time = 5.18 (sec) , antiderivative size = 2224, normalized size of antiderivative = 3.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(2224\)
default	\(\text {Expression too large to display}\)	\(3554\)

Input:

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

Output:

-2/91*(a*x^3+b)*(183*a^2*x^6-23*a*b*x^3+7*b^2)/b^4/x^8/((a*x^3+b)/x^3)^(1/ 
2)+1/91*a^3/b^4*(366*(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))/...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.18 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{12}} \, dx=\frac {2 \, {\left (640 \, {\left (a^{3} x^{8} + a^{2} b x^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right )\right ) + {\left (160 \, a^{2} b x^{6} + 48 \, a b^{2} x^{3} - 21 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{273 \, {\left (a b^{4} x^{8} + b^{5} x^{5}\right )}} \] Input:

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

Output:

2/273*(640*(a^3*x^8 + a^2*b*x^5)*sqrt(b)*weierstrassZeta(0, -4*a/b, weiers 
trassPInverse(0, -4*a/b, 1/x)) + (160*a^2*b*x^6 + 48*a*b^2*x^3 - 21*b^3)*s 
qrt((a*x^3 + b)/x^3))/(a*b^4*x^8 + b^5*x^5)
 

Sympy [A] (verification not implemented)

Time = 1.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.07 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{12}} \, dx=- \frac {\Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{2}} x^{11} \Gamma \left (\frac {14}{3}\right )} \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{12}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} x^{12}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{12}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} x^{12}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{12}} \, dx=\int \frac {1}{x^{12}\,{\left (a+\frac {b}{x^3}\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x^{12}} \, dx=\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}}{a^{2} x^{14}+2 a b \,x^{11}+b^{2} x^{8}}d x \] Input:

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

Output:

int((sqrt(x)*sqrt(a*x**3 + b))/(a**2*x**14 + 2*a*b*x**11 + b**2*x**8),x)