Integrand size = 16, antiderivative size = 233 \[ \int \frac {1}{\left (-2+7 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {x \left (61+21 x^2\right )}{146 \sqrt {-2+7 x^2+3 x^4}}+\frac {7 \sqrt {7+\sqrt {73}} \sqrt {4-\left (7-\sqrt {73}\right ) x^2} \sqrt {4-\left (7+\sqrt {73}\right ) x^2} E\left (\arcsin \left (\frac {1}{2} \sqrt {7+\sqrt {73}} x\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )}{584 \sqrt {-2+7 x^2+3 x^4}}-\frac {\sqrt {\frac {1}{73} \left (7+\sqrt {73}\right )} \sqrt {4-\left (7-\sqrt {73}\right ) x^2} \sqrt {4-\left (7+\sqrt {73}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {7+\sqrt {73}} x\right ),\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )}{8 \sqrt {-2+7 x^2+3 x^4}} \] Output:
-1/146*x*(21*x^2+61)/(3*x^4+7*x^2-2)^(1/2)+7/584*(7+73^(1/2))^(1/2)*(4-(-7 3^(1/2)+7)*x^2)^(1/2)*(4-(7+73^(1/2))*x^2)^(1/2)*EllipticE(1/2*(7+73^(1/2) )^(1/2)*x,1/12*I*438^(1/2)-7/12*I*6^(1/2))/(3*x^4+7*x^2-2)^(1/2)-1/584*(51 1+73*73^(1/2))^(1/2)*(4-(-73^(1/2)+7)*x^2)^(1/2)*(4-(7+73^(1/2))*x^2)^(1/2 )*EllipticF(1/2*(7+73^(1/2))^(1/2)*x,1/12*I*438^(1/2)-7/12*I*6^(1/2))/(3*x ^4+7*x^2-2)^(1/2)
Result contains complex when optimal does not.
Time = 9.41 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (-2+7 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {2 x \left (61+21 x^2\right )-7 i \sqrt {2 \left (-7+\sqrt {73}\right )} \sqrt {2-7 x^2-3 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {6}{7+\sqrt {73}}} x\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )+i \sqrt {\frac {2}{-7+\sqrt {73}}} \left (-73+7 \sqrt {73}\right ) \sqrt {2-7 x^2-3 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{7+\sqrt {73}}} x\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{292 \sqrt {-2+7 x^2+3 x^4}} \] Input:
Integrate[(-2 + 7*x^2 + 3*x^4)^(-3/2),x]
Output:
-1/292*(2*x*(61 + 21*x^2) - (7*I)*Sqrt[2*(-7 + Sqrt[73])]*Sqrt[2 - 7*x^2 - 3*x^4]*EllipticE[I*ArcSinh[Sqrt[6/(7 + Sqrt[73])]*x], (-61 - 7*Sqrt[73])/ 12] + I*Sqrt[2/(-7 + Sqrt[73])]*(-73 + 7*Sqrt[73])*Sqrt[2 - 7*x^2 - 3*x^4] *EllipticF[I*ArcSinh[Sqrt[6/(7 + Sqrt[73])]*x], (-61 - 7*Sqrt[73])/12])/Sq rt[-2 + 7*x^2 + 3*x^4]
Time = 0.77 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.60, 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.
| \(\displaystyle \int \frac {1}{\left (3 x^4+7 x^2-2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {1}{146} \int -\frac {3 \left (4-7 x^2\right )}{\sqrt {3 x^4+7 x^2-2}}dx-\frac {x \left (21 x^2+61\right )}{146 \sqrt {3 x^4+7 x^2-2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{146} \int \frac {4-7 x^2}{\sqrt {3 x^4+7 x^2-2}}dx-\frac {x \left (21 x^2+61\right )}{146 \sqrt {3 x^4+7 x^2-2}}\) |
| \(\Big \downarrow \) 1501 |
| \(\displaystyle -\frac {3}{146} \left (\frac {1}{6} \left (73-7 \sqrt {73}\right ) \int \frac {1}{\sqrt {3 x^4+7 x^2-2}}dx-\frac {7}{6} \int \frac {6 x^2-\sqrt {73}+7}{\sqrt {3 x^4+7 x^2-2}}dx\right )-\frac {x \left (21 x^2+61\right )}{146 \sqrt {3 x^4+7 x^2-2}}\) |
| \(\Big \downarrow \) 1411 |
| \(\displaystyle -\frac {3}{146} \left (\frac {\left (73-7 \sqrt {73}\right ) \sqrt {\frac {4-\left (7-\sqrt {73}\right ) x^2}{4-\left (7+\sqrt {73}\right ) x^2}} \sqrt {\left (7+\sqrt {73}\right ) x^2-4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {\left (7+\sqrt {73}\right ) x^2-4}}\right ),\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{12 \sqrt {2} \sqrt [4]{73} \sqrt {\frac {1}{4-\left (7+\sqrt {73}\right ) x^2}} \sqrt {3 x^4+7 x^2-2}}-\frac {7}{6} \int \frac {6 x^2-\sqrt {73}+7}{\sqrt {3 x^4+7 x^2-2}}dx\right )-\frac {x \left (21 x^2+61\right )}{146 \sqrt {3 x^4+7 x^2-2}}\) |
| \(\Big \downarrow \) 1498 |
| \(\displaystyle -\frac {3}{146} \left (\frac {\left (73-7 \sqrt {73}\right ) \sqrt {\frac {4-\left (7-\sqrt {73}\right ) x^2}{4-\left (7+\sqrt {73}\right ) x^2}} \sqrt {\left (7+\sqrt {73}\right ) x^2-4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {\left (7+\sqrt {73}\right ) x^2-4}}\right ),\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{12 \sqrt {2} \sqrt [4]{73} \sqrt {\frac {1}{4-\left (7+\sqrt {73}\right ) x^2}} \sqrt {3 x^4+7 x^2-2}}-\frac {7}{6} \left (\frac {x \left (6 x^2+\sqrt {73}+7\right )}{\sqrt {3 x^4+7 x^2-2}}-\frac {\sqrt [4]{73} \sqrt {\frac {4-\left (7-\sqrt {73}\right ) x^2}{4-\left (7+\sqrt {73}\right ) x^2}} \sqrt {\left (7+\sqrt {73}\right ) x^2-4} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {\left (7+\sqrt {73}\right ) x^2-4}}\right )|\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{\sqrt {2} \sqrt {\frac {1}{4-\left (7+\sqrt {73}\right ) x^2}} \sqrt {3 x^4+7 x^2-2}}\right )\right )-\frac {x \left (21 x^2+61\right )}{146 \sqrt {3 x^4+7 x^2-2}}\) |
Input:
Int[(-2 + 7*x^2 + 3*x^4)^(-3/2),x]
Output:
-1/146*(x*(61 + 21*x^2))/Sqrt[-2 + 7*x^2 + 3*x^4] - (3*((-7*((x*(7 + Sqrt[ 73] + 6*x^2))/Sqrt[-2 + 7*x^2 + 3*x^4] - (73^(1/4)*Sqrt[(4 - (7 - Sqrt[73] )*x^2)/(4 - (7 + Sqrt[73])*x^2)]*Sqrt[-4 + (7 + Sqrt[73])*x^2]*EllipticE[A rcSin[(Sqrt[2]*73^(1/4)*x)/Sqrt[-4 + (7 + Sqrt[73])*x^2]], (73 + 7*Sqrt[73 ])/146])/(Sqrt[2]*Sqrt[(4 - (7 + Sqrt[73])*x^2)^(-1)]*Sqrt[-2 + 7*x^2 + 3* x^4])))/6 + ((73 - 7*Sqrt[73])*Sqrt[(4 - (7 - Sqrt[73])*x^2)/(4 - (7 + Sqr t[73])*x^2)]*Sqrt[-4 + (7 + Sqrt[73])*x^2]*EllipticF[ArcSin[(Sqrt[2]*73^(1 /4)*x)/Sqrt[-4 + (7 + Sqrt[73])*x^2]], (73 + 7*Sqrt[73])/146])/(12*Sqrt[2] *73^(1/4)*Sqrt[(4 - (7 + Sqrt[73])*x^2)^(-1)]*Sqrt[-2 + 7*x^2 + 3*x^4])))/ 146
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 = 2.51 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(-\frac {x \left (21 x^{2}+61\right )}{146 \sqrt {3 x^{4}+7 x^{2}-2}}-\frac {12 \sqrt {1-\left (\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {-\sqrt {73}+7}\, \sqrt {3 x^{4}+7 x^{2}-2}}+\frac {84 \sqrt {1-\left (\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {-\sqrt {73}+7}\, \sqrt {3 x^{4}+7 x^{2}-2}\, \left (7+\sqrt {73}\right )}\) | \(228\) |
| default | \(-\frac {6 \left (\frac {61}{876} x +\frac {7}{292} x^{3}\right )}{\sqrt {3 x^{4}+7 x^{2}-2}}-\frac {12 \sqrt {1-\left (\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {-\sqrt {73}+7}\, \sqrt {3 x^{4}+7 x^{2}-2}}+\frac {84 \sqrt {1-\left (\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {-\sqrt {73}+7}\, \sqrt {3 x^{4}+7 x^{2}-2}\, \left (7+\sqrt {73}\right )}\) | \(229\) |
| elliptic | \(-\frac {6 \left (\frac {61}{876} x +\frac {7}{292} x^{3}\right )}{\sqrt {3 x^{4}+7 x^{2}-2}}-\frac {12 \sqrt {1-\left (\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {-\sqrt {73}+7}\, \sqrt {3 x^{4}+7 x^{2}-2}}+\frac {84 \sqrt {1-\left (\frac {7}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-\sqrt {73}+7}}{2}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {-\sqrt {73}+7}\, \sqrt {3 x^{4}+7 x^{2}-2}\, \left (7+\sqrt {73}\right )}\) | \(229\) |
Input:
int(1/(3*x^4+7*x^2-2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/146*x*(21*x^2+61)/(3*x^4+7*x^2-2)^(1/2)-12/73/(-73^(1/2)+7)^(1/2)*(1-(7 /4-1/4*73^(1/2))*x^2)^(1/2)*(1-(7/4+1/4*73^(1/2))*x^2)^(1/2)/(3*x^4+7*x^2- 2)^(1/2)*EllipticF(1/2*x*(-73^(1/2)+7)^(1/2),7/12*I*6^(1/2)+1/12*I*438^(1/ 2))+84/73/(-73^(1/2)+7)^(1/2)*(1-(7/4-1/4*73^(1/2))*x^2)^(1/2)*(1-(7/4+1/4 *73^(1/2))*x^2)^(1/2)/(3*x^4+7*x^2-2)^(1/2)/(7+73^(1/2))*(EllipticF(1/2*x* (-73^(1/2)+7)^(1/2),7/12*I*6^(1/2)+1/12*I*438^(1/2))-EllipticE(1/2*x*(-73^ (1/2)+7)^(1/2),7/12*I*6^(1/2)+1/12*I*438^(1/2)))
Time = 0.07 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (-2+7 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {7 \, {\left (\sqrt {73} \sqrt {-2} {\left (3 \, x^{4} + 7 \, x^{2} - 2\right )} + 7 \, \sqrt {-2} {\left (3 \, x^{4} + 7 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {73} + 7} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {73} + 7}\right )\,|\,\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) - {\left (11 \, \sqrt {73} \sqrt {-2} {\left (3 \, x^{4} + 7 \, x^{2} - 2\right )} + 21 \, \sqrt {-2} {\left (3 \, x^{4} + 7 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {73} + 7} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {73} + 7}\right )\,|\,\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) + 8 \, \sqrt {3 \, x^{4} + 7 \, x^{2} - 2} {\left (21 \, x^{3} + 61 \, x\right )}}{1168 \, {\left (3 \, x^{4} + 7 \, x^{2} - 2\right )}} \] Input:
integrate(1/(3*x^4+7*x^2-2)^(3/2),x, algorithm="fricas")
Output:
-1/1168*(7*(sqrt(73)*sqrt(-2)*(3*x^4 + 7*x^2 - 2) + 7*sqrt(-2)*(3*x^4 + 7* x^2 - 2))*sqrt(sqrt(73) + 7)*elliptic_e(arcsin(1/2*x*sqrt(sqrt(73) + 7)), 7/12*sqrt(73) - 61/12) - (11*sqrt(73)*sqrt(-2)*(3*x^4 + 7*x^2 - 2) + 21*sq rt(-2)*(3*x^4 + 7*x^2 - 2))*sqrt(sqrt(73) + 7)*elliptic_f(arcsin(1/2*x*sqr t(sqrt(73) + 7)), 7/12*sqrt(73) - 61/12) + 8*sqrt(3*x^4 + 7*x^2 - 2)*(21*x ^3 + 61*x))/(3*x^4 + 7*x^2 - 2)
\[ \int \frac {1}{\left (-2+7 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (3 x^{4} + 7 x^{2} - 2\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(3*x**4+7*x**2-2)**(3/2),x)
Output:
Integral((3*x**4 + 7*x**2 - 2)**(-3/2), x)
\[ \int \frac {1}{\left (-2+7 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 7 \, x^{2} - 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(3*x^4+7*x^2-2)^(3/2),x, algorithm="maxima")
Output:
integrate((3*x^4 + 7*x^2 - 2)^(-3/2), x)
\[ \int \frac {1}{\left (-2+7 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} + 7 \, x^{2} - 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(3*x^4+7*x^2-2)^(3/2),x, algorithm="giac")
Output:
integrate((3*x^4 + 7*x^2 - 2)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (-2+7 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (3\,x^4+7\,x^2-2\right )}^{3/2}} \,d x \] Input:
int(1/(7*x^2 + 3*x^4 - 2)^(3/2),x)
Output:
int(1/(7*x^2 + 3*x^4 - 2)^(3/2), x)
\[ \int \frac {1}{\left (-2+7 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {3 x^{4}+7 x^{2}-2}}{9 x^{8}+42 x^{6}+37 x^{4}-28 x^{2}+4}d x \] Input:
int(1/(3*x^4+7*x^2-2)^(3/2),x)
Output:
int(sqrt(3*x**4 + 7*x**2 - 2)/(9*x**8 + 42*x**6 + 37*x**4 - 28*x**2 + 4),x )