Integrand size = 16, antiderivative size = 125 \[ \int \frac {1}{\left (3+7 x^2-2 x^4\right )^{3/2}} \, dx=\frac {x \left (61-14 x^2\right )}{219 \sqrt {3+7 x^2-2 x^4}}+\frac {7}{219} \sqrt {\frac {1}{2} \left (-7+\sqrt {73}\right )} E\left (\arcsin \left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )+\frac {1}{3} \sqrt {\frac {1}{146} \left (-7+\sqrt {73}\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right ) \] Output:
1/219*x*(-14*x^2+61)/(-2*x^4+7*x^2+3)^(1/2)+7/438*(-14+2*73^(1/2))^(1/2)*E llipticE(2*x/(7+73^(1/2))^(1/2),7/12*I*6^(1/2)+1/12*I*438^(1/2))+1/438*(-1 022+146*73^(1/2))^(1/2)*EllipticF(2*x/(7+73^(1/2))^(1/2),7/12*I*6^(1/2)+1/ 12*I*438^(1/2))
Result contains complex when optimal does not.
Time = 9.48 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\left (3+7 x^2-2 x^4\right )^{3/2}} \, dx=\frac {244 x-56 x^3+14 i \sqrt {2 \left (7+\sqrt {73}\right )} \sqrt {3+7 x^2-2 x^4} E\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )-\frac {2 i \left (73+7 \sqrt {73}\right ) \sqrt {6+14 x^2-4 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )}{\sqrt {7+\sqrt {73}}}}{876 \sqrt {3+7 x^2-2 x^4}} \] Input:
Integrate[(3 + 7*x^2 - 2*x^4)^(-3/2),x]
Output:
(244*x - 56*x^3 + (14*I)*Sqrt[2*(7 + Sqrt[73])]*Sqrt[3 + 7*x^2 - 2*x^4]*El lipticE[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)/S qrt[-7 + Sqrt[73]]], (-61 + 7*Sqrt[73])/12])/Sqrt[7 + Sqrt[73]])/(876*Sqrt [3 + 7*x^2 - 2*x^4])
Time = 0.49 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1494, 399, 321, 327}
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 |
\(\displaystyle \frac {x \left (61-14 x^2\right )}{219 \sqrt {-2 x^4+7 x^2+3}}-\frac {1}{219} \int -\frac {2 \left (7 x^2+6\right )}{\sqrt {-2 x^4+7 x^2+3}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{219} \int \frac {7 x^2+6}{\sqrt {-2 x^4+7 x^2+3}}dx+\frac {x \left (61-14 x^2\right )}{219 \sqrt {-2 x^4+7 x^2+3}}\) |
\(\Big \downarrow \) 1494 |
\(\displaystyle \frac {4}{219} \sqrt {2} \int \frac {7 x^2+6}{\sqrt {-4 x^2+\sqrt {73}+7} \sqrt {4 x^2+\sqrt {73}-7}}dx+\frac {x \left (61-14 x^2\right )}{219 \sqrt {-2 x^4+7 x^2+3}}\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {4}{219} \sqrt {2} \left (\frac {1}{4} \left (73-7 \sqrt {73}\right ) \int \frac {1}{\sqrt {-4 x^2+\sqrt {73}+7} \sqrt {4 x^2+\sqrt {73}-7}}dx+\frac {7}{4} \int \frac {\sqrt {4 x^2+\sqrt {73}-7}}{\sqrt {-4 x^2+\sqrt {73}+7}}dx\right )+\frac {x \left (61-14 x^2\right )}{219 \sqrt {-2 x^4+7 x^2+3}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {4}{219} \sqrt {2} \left (\frac {7}{4} \int \frac {\sqrt {4 x^2+\sqrt {73}-7}}{\sqrt {-4 x^2+\sqrt {73}+7}}dx+\frac {\left (73-7 \sqrt {73}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{8 \sqrt {\sqrt {73}-7}}\right )+\frac {x \left (61-14 x^2\right )}{219 \sqrt {-2 x^4+7 x^2+3}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {4}{219} \sqrt {2} \left (\frac {\left (73-7 \sqrt {73}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{8 \sqrt {\sqrt {73}-7}}+\frac {7}{8} \sqrt {\sqrt {73}-7} E\left (\arcsin \left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )\right )+\frac {x \left (61-14 x^2\right )}{219 \sqrt {-2 x^4+7 x^2+3}}\) |
Input:
Int[(3 + 7*x^2 - 2*x^4)^(-3/2),x]
Output:
(x*(61 - 14*x^2))/(219*Sqrt[3 + 7*x^2 - 2*x^4]) + (4*Sqrt[2]*((7*Sqrt[-7 + Sqrt[73]]*EllipticE[ArcSin[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/ 12])/8 + ((73 - 7*Sqrt[73])*EllipticF[ArcSin[(2*x)/Sqrt[7 + Sqrt[73]]], (- 61 - 7*Sqrt[73])/12])/(8*Sqrt[-7 + Sqrt[73]])))/219
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
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[((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*Sqrt[-c] Int[(d + e*x^2)/(Sqr t[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e }, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (95 ) = 190\).
Time = 2.51 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.82
method | result | size |
risch | \(-\frac {x \left (14 x^{2}-61\right )}{219 \sqrt {-2 x^{4}+7 x^{2}+3}}+\frac {24 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}}-\frac {168 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}\, \left (7+\sqrt {73}\right )}\) | \(228\) |
default | \(\frac {\frac {61}{219} x -\frac {14}{219} x^{3}}{\sqrt {-2 x^{4}+7 x^{2}+3}}+\frac {24 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}}-\frac {168 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}\, \left (7+\sqrt {73}\right )}\) | \(229\) |
elliptic | \(\frac {\frac {61}{219} x -\frac {14}{219} x^{3}}{\sqrt {-2 x^{4}+7 x^{2}+3}}+\frac {24 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{73 \sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}}-\frac {168 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-42+6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )\right )}{73 \sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}\, \left (7+\sqrt {73}\right )}\) | \(229\) |
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-(-7/6+1/6*73^(1/2))*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*438^(1/2))-168/73/(-42+6*73^(1/2))^(1/2)*(1-(-7/6+1/6*73^(1/2))*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))* (EllipticF(1/6*(-42+6*73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*438^(1/2))-E llipticE(1/6*(-42+6*73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*438^(1/2)))
Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\left (3+7 x^2-2 x^4\right )^{3/2}} \, dx=\frac {7 \, {\left (\sqrt {73} \sqrt {3} {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )} - 7 \, \sqrt {3} {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{6} \, \sqrt {73} - \frac {7}{6}} E(\arcsin \left (x \sqrt {\frac {1}{6} \, \sqrt {73} - \frac {7}{6}}\right )\,|\,-\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) - {\left (\sqrt {73} \sqrt {3} {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )} - 91 \, \sqrt {3} {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{6} \, \sqrt {73} - \frac {7}{6}} F(\arcsin \left (x \sqrt {\frac {1}{6} \, \sqrt {73} - \frac {7}{6}}\right )\,|\,-\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) + 6 \, \sqrt {-2 \, x^{4} + 7 \, x^{2} + 3} {\left (14 \, x^{3} - 61 \, x\right )}}{1314 \, {\left (2 \, x^{4} - 7 \, x^{2} - 3\right )}} \] 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) - (sqrt(73)*sqrt(3)*(2*x^4 - 7*x^2 - 3) - 91* sqrt(3)*(2*x^4 - 7*x^2 - 3))*sqrt(1/6*sqrt(73) - 7/6)*elliptic_f(arcsin(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)