Integrand size = 16, antiderivative size = 239 \[ \int \frac {1}{\left (-3+7 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {x \left (61+14 x^2\right )}{219 \sqrt {-3+7 x^2+2 x^4}}+\frac {7 \sqrt {\frac {1}{6} \left (7+\sqrt {73}\right )} \sqrt {6-\left (7-\sqrt {73}\right ) x^2} \sqrt {6-\left (7+\sqrt {73}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{6} \left (7+\sqrt {73}\right )} x\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )}{438 \sqrt {-3+7 x^2+2 x^4}}-\frac {\sqrt {\frac {1}{438} \left (7+\sqrt {73}\right )} \sqrt {6-\left (7-\sqrt {73}\right ) x^2} \sqrt {6-\left (7+\sqrt {73}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{6} \left (7+\sqrt {73}\right )} x\right ),\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )}{6 \sqrt {-3+7 x^2+2 x^4}} \] Output:
-1/219*x*(14*x^2+61)/(2*x^4+7*x^2-3)^(1/2)+7/2628*(42+6*73^(1/2))^(1/2)*(6 -(-73^(1/2)+7)*x^2)^(1/2)*(6-(7+73^(1/2))*x^2)^(1/2)*EllipticE(1/6*(42+6*7 3^(1/2))^(1/2)*x,1/12*I*438^(1/2)-7/12*I*6^(1/2))/(2*x^4+7*x^2-3)^(1/2)-1/ 2628*(3066+438*73^(1/2))^(1/2)*(6-(-73^(1/2)+7)*x^2)^(1/2)*(6-(7+73^(1/2)) *x^2)^(1/2)*EllipticF(1/6*(42+6*73^(1/2))^(1/2)*x,1/12*I*438^(1/2)-7/12*I* 6^(1/2))/(2*x^4+7*x^2-3)^(1/2)
Result contains complex when optimal does not.
Time = 7.47 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (-3+7 x^2+2 x^4\right )^{3/2}} \, dx=\frac {-4 x \left (61+14 x^2\right )+14 i \sqrt {-7+\sqrt {73}} \sqrt {6-14 x^2-4 x^4} E\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )-\frac {2 i \left (-73+7 \sqrt {73}\right ) \sqrt {6-14 x^2-4 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{\sqrt {-7+\sqrt {73}}}}{876 \sqrt {-3+7 x^2+2 x^4}} \] Input:
Integrate[(-3 + 7*x^2 + 2*x^4)^(-3/2),x]
Output:
(-4*x*(61 + 14*x^2) + (14*I)*Sqrt[-7 + Sqrt[73]]*Sqrt[6 - 14*x^2 - 4*x^4]* EllipticE[I*ArcSinh[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/12] - (( 2*I)*(-73 + 7*Sqrt[73])*Sqrt[6 - 14*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x) /Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/12])/Sqrt[-7 + Sqrt[73]])/(876*Sq rt[-3 + 7*x^2 + 2*x^4])
Time = 0.75 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.56, 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.
| \(\displaystyle \int \frac {1}{\left (2 x^4+7 x^2-3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {1}{219} \int -\frac {2 \left (6-7 x^2\right )}{\sqrt {2 x^4+7 x^2-3}}dx-\frac {x \left (14 x^2+61\right )}{219 \sqrt {2 x^4+7 x^2-3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{219} \int \frac {6-7 x^2}{\sqrt {2 x^4+7 x^2-3}}dx-\frac {x \left (14 x^2+61\right )}{219 \sqrt {2 x^4+7 x^2-3}}\) |
| \(\Big \downarrow \) 1501 |
| \(\displaystyle -\frac {2}{219} \left (\frac {1}{4} \left (73-7 \sqrt {73}\right ) \int \frac {1}{\sqrt {2 x^4+7 x^2-3}}dx-\frac {7}{4} \int \frac {4 x^2-\sqrt {73}+7}{\sqrt {2 x^4+7 x^2-3}}dx\right )-\frac {x \left (14 x^2+61\right )}{219 \sqrt {2 x^4+7 x^2-3}}\) |
| \(\Big \downarrow \) 1411 |
| \(\displaystyle -\frac {2}{219} \left (\frac {\left (73-7 \sqrt {73}\right ) \sqrt {\frac {6-\left (7-\sqrt {73}\right ) x^2}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {\left (7+\sqrt {73}\right ) x^2-6} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {\left (7+\sqrt {73}\right ) x^2-6}}\right ),\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{8 \sqrt {3} \sqrt [4]{73} \sqrt {\frac {1}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {2 x^4+7 x^2-3}}-\frac {7}{4} \int \frac {4 x^2-\sqrt {73}+7}{\sqrt {2 x^4+7 x^2-3}}dx\right )-\frac {x \left (14 x^2+61\right )}{219 \sqrt {2 x^4+7 x^2-3}}\) |
| \(\Big \downarrow \) 1498 |
| \(\displaystyle -\frac {2}{219} \left (\frac {\left (73-7 \sqrt {73}\right ) \sqrt {\frac {6-\left (7-\sqrt {73}\right ) x^2}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {\left (7+\sqrt {73}\right ) x^2-6} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {\left (7+\sqrt {73}\right ) x^2-6}}\right ),\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{8 \sqrt {3} \sqrt [4]{73} \sqrt {\frac {1}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {2 x^4+7 x^2-3}}-\frac {7}{4} \left (\frac {x \left (4 x^2+\sqrt {73}+7\right )}{\sqrt {2 x^4+7 x^2-3}}-\frac {\sqrt [4]{73} \sqrt {\frac {6-\left (7-\sqrt {73}\right ) x^2}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {\left (7+\sqrt {73}\right ) x^2-6} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {\left (7+\sqrt {73}\right ) x^2-6}}\right )|\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{\sqrt {3} \sqrt {\frac {1}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {2 x^4+7 x^2-3}}\right )\right )-\frac {x \left (14 x^2+61\right )}{219 \sqrt {2 x^4+7 x^2-3}}\) |
Input:
Int[(-3 + 7*x^2 + 2*x^4)^(-3/2),x]
Output:
-1/219*(x*(61 + 14*x^2))/Sqrt[-3 + 7*x^2 + 2*x^4] - (2*((-7*((x*(7 + Sqrt[ 73] + 4*x^2))/Sqrt[-3 + 7*x^2 + 2*x^4] - (73^(1/4)*Sqrt[(6 - (7 - Sqrt[73] )*x^2)/(6 - (7 + Sqrt[73])*x^2)]*Sqrt[-6 + (7 + Sqrt[73])*x^2]*EllipticE[A rcSin[(Sqrt[2]*73^(1/4)*x)/Sqrt[-6 + (7 + Sqrt[73])*x^2]], (73 + 7*Sqrt[73 ])/146])/(Sqrt[3]*Sqrt[(6 - (7 + Sqrt[73])*x^2)^(-1)]*Sqrt[-3 + 7*x^2 + 2* x^4])))/4 + ((73 - 7*Sqrt[73])*Sqrt[(6 - (7 - Sqrt[73])*x^2)/(6 - (7 + Sqr t[73])*x^2)]*Sqrt[-6 + (7 + Sqrt[73])*x^2]*EllipticF[ArcSin[(Sqrt[2]*73^(1 /4)*x)/Sqrt[-6 + (7 + Sqrt[73])*x^2]], (73 + 7*Sqrt[73])/146])/(8*Sqrt[3]* 73^(1/4)*Sqrt[(6 - (7 + Sqrt[73])*x^2)^(-1)]*Sqrt[-3 + 7*x^2 + 2*x^4])))/2 19
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 = 2.47 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(-\frac {x \left (14 x^{2}+61\right )}{219 \sqrt {2 x^{4}+7 x^{2}-3}}-\frac {24 \sqrt {1-\left (-\frac {\sqrt {73}}{6}+\frac {7}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {42-6 \sqrt {73}}\, \sqrt {2 x^{4}+7 x^{2}-3}}+\frac {168 \sqrt {1-\left (-\frac {\sqrt {73}}{6}+\frac {7}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {42-6 \sqrt {73}}\, \sqrt {2 x^{4}+7 x^{2}-3}\, \left (7+\sqrt {73}\right )}\) | \(228\) |
| default | \(-\frac {4 \left (\frac {61}{876} x +\frac {7}{438} x^{3}\right )}{\sqrt {2 x^{4}+7 x^{2}-3}}-\frac {24 \sqrt {1-\left (-\frac {\sqrt {73}}{6}+\frac {7}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {42-6 \sqrt {73}}\, \sqrt {2 x^{4}+7 x^{2}-3}}+\frac {168 \sqrt {1-\left (-\frac {\sqrt {73}}{6}+\frac {7}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {42-6 \sqrt {73}}\, \sqrt {2 x^{4}+7 x^{2}-3}\, \left (7+\sqrt {73}\right )}\) | \(229\) |
| elliptic | \(-\frac {4 \left (\frac {61}{876} x +\frac {7}{438} x^{3}\right )}{\sqrt {2 x^{4}+7 x^{2}-3}}-\frac {24 \sqrt {1-\left (-\frac {\sqrt {73}}{6}+\frac {7}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {42-6 \sqrt {73}}\, \sqrt {2 x^{4}+7 x^{2}-3}}+\frac {168 \sqrt {1-\left (-\frac {\sqrt {73}}{6}+\frac {7}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {42-6 \sqrt {73}}\, \sqrt {2 x^{4}+7 x^{2}-3}\, \left (7+\sqrt {73}\right )}\) | \(229\) |
Input:
int(1/(2*x^4+7*x^2-3)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/219*x*(14*x^2+61)/(2*x^4+7*x^2-3)^(1/2)-24/73/(42-6*73^(1/2))^(1/2)*(1- (-1/6*73^(1/2)+7/6)*x^2)^(1/2)*(1-(7/6+1/6*73^(1/2))*x^2)^(1/2)/(2*x^4+7*x ^2-3)^(1/2)*EllipticF(1/6*(42-6*73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*43 8^(1/2))+168/73/(42-6*73^(1/2))^(1/2)*(1-(-1/6*73^(1/2)+7/6)*x^2)^(1/2)*(1 -(7/6+1/6*73^(1/2))*x^2)^(1/2)/(2*x^4+7*x^2-3)^(1/2)/(7+73^(1/2))*(Ellipti cF(1/6*(42-6*73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*438^(1/2))-EllipticE( 1/6*(42-6*73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*438^(1/2)))
Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-3+7 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {7 \, {\left (\sqrt {73} \sqrt {-3} {\left (2 \, x^{4} + 7 \, x^{2} - 3\right )} + 7 \, \sqrt {-3} {\left (2 \, x^{4} + 7 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{6} \, \sqrt {73} + \frac {7}{6}} E(\arcsin \left (x \sqrt {\frac {1}{6} \, \sqrt {73} + \frac {7}{6}}\right )\,|\,\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) - {\left (13 \, \sqrt {73} \sqrt {-3} {\left (2 \, x^{4} + 7 \, x^{2} - 3\right )} + 7 \, \sqrt {-3} {\left (2 \, x^{4} + 7 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{6} \, \sqrt {73} + \frac {7}{6}} F(\arcsin \left (x \sqrt {\frac {1}{6} \, \sqrt {73} + \frac {7}{6}}\right )\,|\,\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) + 6 \, \sqrt {2 \, x^{4} + 7 \, x^{2} - 3} {\left (14 \, x^{3} + 61 \, x\right )}}{1314 \, {\left (2 \, x^{4} + 7 \, x^{2} - 3\right )}} \] Input:
integrate(1/(2*x^4+7*x^2-3)^(3/2),x, algorithm="fricas")
Output:
-1/1314*(7*(sqrt(73)*sqrt(-3)*(2*x^4 + 7*x^2 - 3) + 7*sqrt(-3)*(2*x^4 + 7* x^2 - 3))*sqrt(1/6*sqrt(73) + 7/6)*elliptic_e(arcsin(x*sqrt(1/6*sqrt(73) + 7/6)), 7/12*sqrt(73) - 61/12) - (13*sqrt(73)*sqrt(-3)*(2*x^4 + 7*x^2 - 3) + 7*sqrt(-3)*(2*x^4 + 7*x^2 - 3))*sqrt(1/6*sqrt(73) + 7/6)*elliptic_f(arc sin(x*sqrt(1/6*sqrt(73) + 7/6)), 7/12*sqrt(73) - 61/12) + 6*sqrt(2*x^4 + 7 *x^2 - 3)*(14*x^3 + 61*x))/(2*x^4 + 7*x^2 - 3)
\[ \int \frac {1}{\left (-3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (2 x^{4} + 7 x^{2} - 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(2*x**4+7*x**2-3)**(3/2),x)
Output:
Integral((2*x**4 + 7*x**2 - 3)**(-3/2), x)
\[ \int \frac {1}{\left (-3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 7 \, x^{2} - 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(2*x^4+7*x^2-3)^(3/2),x, algorithm="maxima")
Output:
integrate((2*x^4 + 7*x^2 - 3)^(-3/2), x)
\[ \int \frac {1}{\left (-3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, x^{4} + 7 \, x^{2} - 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(2*x^4+7*x^2-3)^(3/2),x, algorithm="giac")
Output:
integrate((2*x^4 + 7*x^2 - 3)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (-3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (2\,x^4+7\,x^2-3\right )}^{3/2}} \,d x \] Input:
int(1/(7*x^2 + 2*x^4 - 3)^(3/2),x)
Output:
int(1/(7*x^2 + 2*x^4 - 3)^(3/2), x)
\[ \int \frac {1}{\left (-3+7 x^2+2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {2 x^{4}+7 x^{2}-3}}{4 x^{8}+28 x^{6}+37 x^{4}-42 x^{2}+9}d x \] Input:
int(1/(2*x^4+7*x^2-3)^(3/2),x)
Output:
int(sqrt(2*x**4 + 7*x**2 - 3)/(4*x**8 + 28*x**6 + 37*x**4 - 42*x**2 + 9),x )