\(\int \frac {1}{x^3 (-2+3 x^2) (-1+3 x^2)^{3/4}} \, dx\) [1514]

Optimal result

Integrand size = 24, antiderivative size = 152 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {\sqrt [4]{-1+3 x^2}}{4 x^2}-\frac {3}{4} \arctan \left (\sqrt [4]{-1+3 x^2}\right )+\frac {15 \arctan \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {15 \arctan \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {3}{4} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {15 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+3 x^2}}{1+\sqrt {-1+3 x^2}}\right )}{8 \sqrt {2}} \] Output:

-1/4*(3*x^2-1)^(1/4)/x^2-3/4*arctan((3*x^2-1)^(1/4))-15/16*arctan(-1+2^(1/ 
2)*(3*x^2-1)^(1/4))*2^(1/2)-15/16*arctan(1+2^(1/2)*(3*x^2-1)^(1/4))*2^(1/2 
)-3/4*arctanh((3*x^2-1)^(1/4))-15/16*arctanh(2^(1/2)*(3*x^2-1)^(1/4)/(1+(3 
*x^2-1)^(1/2)))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {1}{16} \left (-\frac {4 \sqrt [4]{-1+3 x^2}}{x^2}-12 \arctan \left (\sqrt [4]{-1+3 x^2}\right )-15 \sqrt {2} \arctan \left (\frac {-1+\sqrt {-1+3 x^2}}{\sqrt {2} \sqrt [4]{-1+3 x^2}}\right )-12 \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )-15 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+3 x^2}}{1+\sqrt {-1+3 x^2}}\right )\right ) \] Input:

Integrate[1/(x^3*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]
 

Output:

((-4*(-1 + 3*x^2)^(1/4))/x^2 - 12*ArcTan[(-1 + 3*x^2)^(1/4)] - 15*Sqrt[2]* 
ArcTan[(-1 + Sqrt[-1 + 3*x^2])/(Sqrt[2]*(-1 + 3*x^2)^(1/4))] - 12*ArcTanh[ 
(-1 + 3*x^2)^(1/4)] - 15*Sqrt[2]*ArcTanh[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(1 + 
 Sqrt[-1 + 3*x^2])])/16
 

Rubi [A] (warning: unable to verify)

Time = 0.39 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.30, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {354, 25, 114, 27, 174, 73, 755, 27, 756, 216, 219, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {1}{x^3 \left (3 x^2-2\right ) \left (3 x^2-1\right )^{3/4}} \, dx\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} \int -\frac {1}{x^4 \left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{2} \int \frac {1}{x^4 \left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {3 \left (10-9 x^2\right )}{4 x^2 \left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \int \frac {10-9 x^2}{x^2 \left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (5 \int \frac {1}{x^2 \left (3 x^2-1\right )^{3/4}}dx^2+6 \int \frac {1}{\left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (8 \int \frac {1}{1-x^8}d\sqrt [4]{3 x^2-1}+\frac {20}{3} \int \frac {1}{\frac {x^8}{3}+\frac {1}{3}}d\sqrt [4]{3 x^2-1}\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (8 \int \frac {1}{1-x^8}d\sqrt [4]{3 x^2-1}+\frac {20}{3} \left (\frac {1}{2} \int \frac {3 \left (1-x^4\right )}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {1}{2} \int \frac {3 \left (x^4+1\right )}{x^8+1}d\sqrt [4]{3 x^2-1}\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (8 \int \frac {1}{1-x^8}d\sqrt [4]{3 x^2-1}+\frac {20}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{3 x^2-1}\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (8 \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{3 x^2-1}+\frac {1}{2} \int \frac {1}{x^4+1}d\sqrt [4]{3 x^2-1}\right )+\frac {20}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{3 x^2-1}\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (8 \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{3 x^2-1}+\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )\right )+\frac {20}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{3 x^2-1}\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {20}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{3 x^2-1}\right )+8 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {20}{3} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1}{x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}+\frac {1}{2} \int \frac {1}{x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}\right )+\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}\right )+8 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {20}{3} \left (\frac {3}{2} \left (\frac {\int \frac {1}{-x^4-1}d\left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x^4-1}d\left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}\right )+\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}\right )+8 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {20}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )\right )+8 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {20}{3} \left (\frac {3}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt [4]{3 x^2-1}}{x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}\right )+\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )\right )+8 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {20}{3} \left (\frac {3}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{3 x^2-1}}{x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}\right )+\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )\right )+8 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {20}{3} \left (\frac {3}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{3 x^2-1}}{x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt [4]{3 x^2-1}+1}{x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}\right )+\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )\right )+8 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (8 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )+\frac {20}{3} \left (\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )+\frac {3}{2} \left (\frac {\log \left (x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{2 \sqrt {2}}\right )\right )\right )-\frac {\sqrt [4]{3 x^2-1}}{2 x^2}\right )\)

Input:

Int[1/(x^3*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]
 

Output:

(-1/2*(-1 + 3*x^2)^(1/4)/x^2 - (3*(8*(ArcTan[(-1 + 3*x^2)^(1/4)]/2 + ArcTa 
nh[(-1 + 3*x^2)^(1/4)]/2) + (20*((3*(-(ArcTan[1 - Sqrt[2]*(-1 + 3*x^2)^(1/ 
4)]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*(-1 + 3*x^2)^(1/4)]/Sqrt[2]))/2 + (3*(-1 
/2*Log[1 + x^4 - Sqrt[2]*(-1 + 3*x^2)^(1/4)]/Sqrt[2] + Log[1 + x^4 + Sqrt[ 
2]*(-1 + 3*x^2)^(1/4)]/(2*Sqrt[2])))/2))/3))/8)/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 12.48 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(\frac {-15 \ln \left (\frac {-\sqrt {2}\, \left (3 x^{2}-1\right )^{\frac {1}{4}}-\sqrt {3 x^{2}-1}-1}{\sqrt {2}\, \left (3 x^{2}-1\right )^{\frac {1}{4}}-\sqrt {3 x^{2}-1}-1}\right ) \sqrt {2}\, x^{2}-30 \arctan \left (1+\sqrt {2}\, \left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) \sqrt {2}\, x^{2}-30 \arctan \left (-1+\sqrt {2}\, \left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) \sqrt {2}\, x^{2}+12 \ln \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) x^{2}-24 \arctan \left (\left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) x^{2}-12 \ln \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) x^{2}-8 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{32 x^{2}}\)	\(180\)
trager	\(-\frac {\left (3 x^{2}-1\right )^{\frac {1}{4}}}{4 x^{2}}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (3 x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{2}}\right )}{16}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {3 x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \left (3 x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{16}-\frac {3 \ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}+3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {3 x^{2}-1}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}\)	\(315\)
risch	\(\text {Expression too large to display}\)	\(917\)

Input:

int(1/x^3/(3*x^2-2)/(3*x^2-1)^(3/4),x,method=_RETURNVERBOSE)
 

Output:

1/32*(-15*ln((-2^(1/2)*(3*x^2-1)^(1/4)-(3*x^2-1)^(1/2)-1)/(2^(1/2)*(3*x^2- 
1)^(1/4)-(3*x^2-1)^(1/2)-1))*2^(1/2)*x^2-30*arctan(1+2^(1/2)*(3*x^2-1)^(1/ 
4))*2^(1/2)*x^2-30*arctan(-1+2^(1/2)*(3*x^2-1)^(1/4))*2^(1/2)*x^2+12*ln(-1 
+(3*x^2-1)^(1/4))*x^2-24*arctan((3*x^2-1)^(1/4))*x^2-12*ln(1+(3*x^2-1)^(1/ 
4))*x^2-8*(3*x^2-1)^(1/4))/x^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {30 \, \sqrt {2} x^{2} \arctan \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + 30 \, \sqrt {2} x^{2} \arctan \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) + 15 \, \sqrt {2} x^{2} \log \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - 15 \, \sqrt {2} x^{2} \log \left (-\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) + 24 \, x^{2} \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) + 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) - 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) + 8 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{32 \, x^{2}} \] Input:

integrate(1/x^3/(3*x^2-2)/(3*x^2-1)^(3/4),x, algorithm="fricas")
 

Output:

-1/32*(30*sqrt(2)*x^2*arctan(sqrt(2)*(3*x^2 - 1)^(1/4) + 1) + 30*sqrt(2)*x 
^2*arctan(sqrt(2)*(3*x^2 - 1)^(1/4) - 1) + 15*sqrt(2)*x^2*log(sqrt(2)*(3*x 
^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) - 15*sqrt(2)*x^2*log(-sqrt(2)*(3*x^2 
- 1)^(1/4) + sqrt(3*x^2 - 1) + 1) + 24*x^2*arctan((3*x^2 - 1)^(1/4)) + 12* 
x^2*log((3*x^2 - 1)^(1/4) + 1) - 12*x^2*log((3*x^2 - 1)^(1/4) - 1) + 8*(3* 
x^2 - 1)^(1/4))/x^2
 

Sympy [F]

\[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int \frac {1}{x^{3} \cdot \left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(1/x**3/(3*x**2-2)/(3*x**2-1)**(3/4),x)
 

Output:

Integral(1/(x**3*(3*x**2 - 2)*(3*x**2 - 1)**(3/4)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )} x^{3}} \,d x } \] Input:

integrate(1/x^3/(3*x^2-2)/(3*x^2-1)^(3/4),x, algorithm="maxima")
 

Output:

integrate(1/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)*x^3), x)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {15}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {15}{16} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {15}{32} \, \sqrt {2} \log \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) + \frac {15}{32} \, \sqrt {2} \log \left (-\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - \frac {{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{4 \, x^{2}} - \frac {3}{4} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{8} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{8} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \] Input:

integrate(1/x^3/(3*x^2-2)/(3*x^2-1)^(3/4),x, algorithm="giac")
 

Output:

-15/16*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(3*x^2 - 1)^(1/4))) - 15/16 
*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(3*x^2 - 1)^(1/4))) - 15/32*sqrt 
(2)*log(sqrt(2)*(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) + 15/32*sqrt(2)*l 
og(-sqrt(2)*(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) - 1/4*(3*x^2 - 1)^(1/ 
4)/x^2 - 3/4*arctan((3*x^2 - 1)^(1/4)) - 3/8*log((3*x^2 - 1)^(1/4) + 1) + 
3/8*log(abs((3*x^2 - 1)^(1/4) - 1))
 

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {3\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{4}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{4}-\frac {{\left (3\,x^2-1\right )}^{1/4}}{4\,x^2}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (3\,x^2-1\right )}^{1/4}\right )\,15{}\mathrm {i}}{8}-\frac {{\left (-1\right )}^{3/4}\,\mathrm {atan}\left ({\left (-1\right )}^{3/4}\,{\left (3\,x^2-1\right )}^{1/4}\right )\,15{}\mathrm {i}}{8} \] Input:

int(1/(x^3*(3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x)
 

Output:

(atan((3*x^2 - 1)^(1/4)*1i)*3i)/4 - (3*atan((3*x^2 - 1)^(1/4)))/4 - (3*x^2 
 - 1)^(1/4)/(4*x^2) + ((-1)^(1/4)*atan((-1)^(1/4)*(3*x^2 - 1)^(1/4))*15i)/ 
8 - ((-1)^(3/4)*atan((-1)^(3/4)*(3*x^2 - 1)^(1/4))*15i)/8
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int \frac {1}{3 \left (3 x^{2}-1\right )^{\frac {3}{4}} x^{5}-2 \left (3 x^{2}-1\right )^{\frac {3}{4}} x^{3}}d x \] Input:

int(1/x^3/(3*x^2-2)/(3*x^2-1)^(3/4),x)
 

Output:

int(1/(3*(3*x**2 - 1)**(3/4)*x**5 - 2*(3*x**2 - 1)**(3/4)*x**3),x)