Integrand size = 16, antiderivative size = 235 \[ \int \frac {1}{\left (-3+3 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {x \left (7+2 x^2\right )}{33 \sqrt {-3+3 x^2+2 x^4}}+\frac {\sqrt {\frac {1}{6} \left (3+\sqrt {33}\right )} \sqrt {6-\left (3-\sqrt {33}\right ) x^2} \sqrt {6-\left (3+\sqrt {33}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{6} \left (3+\sqrt {33}\right )} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{66 \sqrt {-3+3 x^2+2 x^4}}-\frac {\sqrt {\frac {1}{22} \left (3+\sqrt {33}\right )} \sqrt {6-\left (3-\sqrt {33}\right ) x^2} \sqrt {6-\left (3+\sqrt {33}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{6} \left (3+\sqrt {33}\right )} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{18 \sqrt {-3+3 x^2+2 x^4}} \] Output:
-1/33*x*(2*x^2+7)/(2*x^4+3*x^2-3)^(1/2)+1/396*(18+6*33^(1/2))^(1/2)*(6-(3- 33^(1/2))*x^2)^(1/2)*(6-(3+33^(1/2))*x^2)^(1/2)*EllipticE(1/6*(18+6*33^(1/ 2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))/(2*x^4+3*x^2-3)^(1/2)-1/396*(66+ 22*33^(1/2))^(1/2)*(6-(3-33^(1/2))*x^2)^(1/2)*(6-(3+33^(1/2))*x^2)^(1/2)*E llipticF(1/6*(18+6*33^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))/(2*x^4+ 3*x^2-3)^(1/2)
Result contains complex when optimal does not.
Time = 7.01 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (-3+3 x^2+2 x^4\right )^{3/2}} \, dx=\frac {-4 x \left (7+2 x^2\right )+2 i \sqrt {-3+\sqrt {33}} \sqrt {6-6 x^2-4 x^4} E\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {3+\sqrt {33}}}\right )|-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )-\frac {2 i \left (-11+\sqrt {33}\right ) \sqrt {6-6 x^2-4 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {3+\sqrt {33}}}\right ),-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )}{\sqrt {-3+\sqrt {33}}}}{132 \sqrt {-3+3 x^2+2 x^4}} \] Input:
Integrate[(-3 + 3*x^2 + 2*x^4)^(-3/2),x]
Output:
(-4*x*(7 + 2*x^2) + (2*I)*Sqrt[-3 + Sqrt[33]]*Sqrt[6 - 6*x^2 - 4*x^4]*Elli pticE[I*ArcSinh[(2*x)/Sqrt[3 + Sqrt[33]]], -7/4 - Sqrt[33]/4] - ((2*I)*(-1 1 + Sqrt[33])*Sqrt[6 - 6*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x)/Sqrt[3 + S qrt[33]]], -7/4 - Sqrt[33]/4])/Sqrt[-3 + Sqrt[33]])/(132*Sqrt[-3 + 3*x^2 + 2*x^4])
Time = 0.76 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.55, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1405, 27, 1501, 1411, 1498}
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.
| \(\displaystyle \int \frac {1}{\left (2 x^4+3 x^2-3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {1}{99} \int -\frac {6 \left (2-x^2\right )}{\sqrt {2 x^4+3 x^2-3}}dx-\frac {x \left (2 x^2+7\right )}{33 \sqrt {2 x^4+3 x^2-3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{33} \int \frac {2-x^2}{\sqrt {2 x^4+3 x^2-3}}dx-\frac {x \left (2 x^2+7\right )}{33 \sqrt {2 x^4+3 x^2-3}}\) |
| \(\Big \downarrow \) 1501 |
| \(\displaystyle -\frac {2}{33} \left (\frac {1}{4} \left (11-\sqrt {33}\right ) \int \frac {1}{\sqrt {2 x^4+3 x^2-3}}dx-\frac {1}{4} \int \frac {4 x^2-\sqrt {33}+3}{\sqrt {2 x^4+3 x^2-3}}dx\right )-\frac {x \left (2 x^2+7\right )}{33 \sqrt {2 x^4+3 x^2-3}}\) |
| \(\Big \downarrow \) 1411 |
| \(\displaystyle -\frac {2}{33} \left (\frac {\left (11-\sqrt {33}\right ) \sqrt {\frac {6-\left (3-\sqrt {33}\right ) x^2}{6-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-6} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-6}}\right ),\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{8\ 3^{3/4} \sqrt [4]{11} \sqrt {\frac {1}{6-\left (3+\sqrt {33}\right ) x^2}} \sqrt {2 x^4+3 x^2-3}}-\frac {1}{4} \int \frac {4 x^2-\sqrt {33}+3}{\sqrt {2 x^4+3 x^2-3}}dx\right )-\frac {x \left (2 x^2+7\right )}{33 \sqrt {2 x^4+3 x^2-3}}\) |
| \(\Big \downarrow \) 1498 |
| \(\displaystyle -\frac {2}{33} \left (\frac {\left (11-\sqrt {33}\right ) \sqrt {\frac {6-\left (3-\sqrt {33}\right ) x^2}{6-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-6} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-6}}\right ),\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{8\ 3^{3/4} \sqrt [4]{11} \sqrt {\frac {1}{6-\left (3+\sqrt {33}\right ) x^2}} \sqrt {2 x^4+3 x^2-3}}+\frac {1}{4} \left (\frac {\sqrt [4]{\frac {11}{3}} \sqrt {\frac {6-\left (3-\sqrt {33}\right ) x^2}{6-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-6} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-6}}\right )|\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{\sqrt {\frac {1}{6-\left (3+\sqrt {33}\right ) x^2}} \sqrt {2 x^4+3 x^2-3}}-\frac {x \left (4 x^2+\sqrt {33}+3\right )}{\sqrt {2 x^4+3 x^2-3}}\right )\right )-\frac {x \left (2 x^2+7\right )}{33 \sqrt {2 x^4+3 x^2-3}}\) |
Input:
Int[(-3 + 3*x^2 + 2*x^4)^(-3/2),x]
Output:
-1/33*(x*(7 + 2*x^2))/Sqrt[-3 + 3*x^2 + 2*x^4] - (2*((-((x*(3 + Sqrt[33] + 4*x^2))/Sqrt[-3 + 3*x^2 + 2*x^4]) + ((11/3)^(1/4)*Sqrt[(6 - (3 - Sqrt[33] )*x^2)/(6 - (3 + Sqrt[33])*x^2)]*Sqrt[-6 + (3 + Sqrt[33])*x^2]*EllipticE[A rcSin[(Sqrt[2]*33^(1/4)*x)/Sqrt[-6 + (3 + Sqrt[33])*x^2]], (11 + Sqrt[33]) /22])/(Sqrt[(6 - (3 + Sqrt[33])*x^2)^(-1)]*Sqrt[-3 + 3*x^2 + 2*x^4]))/4 + ((11 - Sqrt[33])*Sqrt[(6 - (3 - Sqrt[33])*x^2)/(6 - (3 + Sqrt[33])*x^2)]*S qrt[-6 + (3 + Sqrt[33])*x^2]*EllipticF[ArcSin[(Sqrt[2]*33^(1/4)*x)/Sqrt[-6 + (3 + Sqrt[33])*x^2]], (11 + Sqrt[33])/22])/(8*3^(3/4)*11^(1/4)*Sqrt[(6 - (3 + Sqrt[33])*x^2)^(-1)]*Sqrt[-3 + 3*x^2 + 2*x^4])))/33
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*(Sqrt[( 2*a + (b + q)*x^2)/q]/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2) ]))*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] ] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[e*x*((b + q + 2*c*x^2)/(2*c*Sqrt[ a + b*x^2 + c*x^4])), x] - Simp[e*q*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q) *x^2)]*(Sqrt[(2*a + (b + q)*x^2)/q]/(2*c*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2* a + (b + q)*x^2)]))*EllipticE[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] /; EqQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(2*c*d - e*(b - q))/(2*c) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[e/(2*c) Int[(b - q + 2*c*x^2)/Sqr t[a + b*x^2 + c*x^4], x], x] /; NeQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
Time = 1.95 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(-\frac {x \left (2 x^{2}+7\right )}{33 \sqrt {2 x^{4}+3 x^{2}-3}}-\frac {8 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{11 \sqrt {18-6 \sqrt {33}}\, \sqrt {2 x^{4}+3 x^{2}-3}}+\frac {24 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {18-6 \sqrt {33}}\, \sqrt {2 x^{4}+3 x^{2}-3}\, \left (3+\sqrt {33}\right )}\) | \(228\) |
| default | \(-\frac {4 \left (\frac {7}{132} x +\frac {1}{66} x^{3}\right )}{\sqrt {2 x^{4}+3 x^{2}-3}}-\frac {8 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{11 \sqrt {18-6 \sqrt {33}}\, \sqrt {2 x^{4}+3 x^{2}-3}}+\frac {24 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {18-6 \sqrt {33}}\, \sqrt {2 x^{4}+3 x^{2}-3}\, \left (3+\sqrt {33}\right )}\) | \(229\) |
| elliptic | \(-\frac {4 \left (\frac {7}{132} x +\frac {1}{66} x^{3}\right )}{\sqrt {2 x^{4}+3 x^{2}-3}}-\frac {8 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{11 \sqrt {18-6 \sqrt {33}}\, \sqrt {2 x^{4}+3 x^{2}-3}}+\frac {24 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {18-6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {18-6 \sqrt {33}}\, \sqrt {2 x^{4}+3 x^{2}-3}\, \left (3+\sqrt {33}\right )}\) | \(229\) |
Input:
int(1/(2*x^4+3*x^2-3)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/33*x*(2*x^2+7)/(2*x^4+3*x^2-3)^(1/2)-8/11/(18-6*33^(1/2))^(1/2)*(1-(1/2 -1/6*33^(1/2))*x^2)^(1/2)*(1-(1/2+1/6*33^(1/2))*x^2)^(1/2)/(2*x^4+3*x^2-3) ^(1/2)*EllipticF(1/6*(18-6*33^(1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2)) +24/11/(18-6*33^(1/2))^(1/2)*(1-(1/2-1/6*33^(1/2))*x^2)^(1/2)*(1-(1/2+1/6* 33^(1/2))*x^2)^(1/2)/(2*x^4+3*x^2-3)^(1/2)/(3+33^(1/2))*(EllipticF(1/6*(18 -6*33^(1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2))-EllipticE(1/6*(18-6*33^ (1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2)))
Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-3+3 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {{\left (\sqrt {\frac {11}{3}} \sqrt {-3} {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )} + \sqrt {-3} {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {\frac {11}{3}} + \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {\frac {11}{3}} + \frac {1}{2}}\right )\,|\,\frac {3}{4} \, \sqrt {\frac {11}{3}} - \frac {7}{4}) - {\left (3 \, \sqrt {\frac {11}{3}} \sqrt {-3} {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )} - \sqrt {-3} {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {\frac {11}{3}} + \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {\frac {11}{3}} + \frac {1}{2}}\right )\,|\,\frac {3}{4} \, \sqrt {\frac {11}{3}} - \frac {7}{4}) + 2 \, \sqrt {2 \, x^{4} + 3 \, x^{2} - 3} {\left (2 \, x^{3} + 7 \, x\right )}}{66 \, {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )}} \] Input:
integrate(1/(2*x^4+3*x^2-3)^(3/2),x, algorithm="fricas")
Output:
-1/66*((sqrt(11/3)*sqrt(-3)*(2*x^4 + 3*x^2 - 3) + sqrt(-3)*(2*x^4 + 3*x^2 - 3))*sqrt(1/2*sqrt(11/3) + 1/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(11/3) + 1/2)), 3/4*sqrt(11/3) - 7/4) - (3*sqrt(11/3)*sqrt(-3)*(2*x^4 + 3*x^2 - 3) - sqrt(-3)*(2*x^4 + 3*x^2 - 3))*sqrt(1/2*sqrt(11/3) + 1/2)*elliptic_f(arc sin(x*sqrt(1/2*sqrt(11/3) + 1/2)), 3/4*sqrt(11/3) - 7/4) + 2*sqrt(2*x^4 + 3*x^2 - 3)*(2*x^3 + 7*x))/(2*x^4 + 3*x^2 - 3)
\[ \int \frac {1}{\left (-3+3 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (2 x^{4} + 3 x^{2} - 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(2*x**4+3*x**2-3)**(3/2),x)
Output:
Integral((2*x**4 + 3*x**2 - 3)**(-3/2), x)
\[ \int \frac {1}{\left (-3+3 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 3 \, x^{2} - 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(2*x^4+3*x^2-3)^(3/2),x, algorithm="maxima")
Output:
integrate((2*x^4 + 3*x^2 - 3)^(-3/2), x)
\[ \int \frac {1}{\left (-3+3 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 3 \, x^{2} - 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(2*x^4+3*x^2-3)^(3/2),x, algorithm="giac")
Output:
integrate((2*x^4 + 3*x^2 - 3)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (-3+3 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (2\,x^4+3\,x^2-3\right )}^{3/2}} \,d x \] Input:
int(1/(3*x^2 + 2*x^4 - 3)^(3/2),x)
Output:
int(1/(3*x^2 + 2*x^4 - 3)^(3/2), x)
\[ \int \frac {1}{\left (-3+3 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {2 x^{4}+3 x^{2}-3}}{4 x^{8}+12 x^{6}-3 x^{4}-18 x^{2}+9}d x \] Input:
int(1/(2*x^4+3*x^2-3)^(3/2),x)
Output:
int(sqrt(2*x**4 + 3*x**2 - 3)/(4*x**8 + 12*x**6 - 3*x**4 - 18*x**2 + 9),x)