Integrand size = 24, antiderivative size = 142 \[ \int \frac {\left (7+5 x^2\right )^2}{\sqrt {2+3 x^2+x^4}} \, dx=\frac {20 x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {25}{3} x \sqrt {2+3 x^2+x^4}-\frac {20 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {2+3 x^2+x^4}}+\frac {97 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {2+3 x^2+x^4}} \]
20*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)+97/6*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*Elli pticF(x/(x^2+1)^(1/2),1/2*2^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2)/(x^4+3* x^2+2)^(1/2)-20*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticE(x/(x^2+1)^(1/2), 1/2*2^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2)/(x^4+3*x^2+2)^(1/2)+25/3*x*(x ^4+3*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.73 \[ \int \frac {\left (7+5 x^2\right )^2}{\sqrt {2+3 x^2+x^4}} \, dx=\frac {25 x \left (2+3 x^2+x^4\right )-60 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-37 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{3 \sqrt {2+3 x^2+x^4}} \]
(25*x*(2 + 3*x^2 + x^4) - (60*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*A rcSinh[x/Sqrt[2]], 2] - (37*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*Arc Sinh[x/Sqrt[2]], 2])/(3*Sqrt[2 + 3*x^2 + x^4])
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1518, 1503, 1412, 1455}
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.
| \(\displaystyle \int \frac {\left (5 x^2+7\right )^2}{\sqrt {x^4+3 x^2+2}} \, dx\) |
| \(\Big \downarrow \) 1518 |
| \(\displaystyle \frac {1}{3} \int \frac {60 x^2+97}{\sqrt {x^4+3 x^2+2}}dx+\frac {25}{3} \sqrt {x^4+3 x^2+2} x\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{3} \left (97 \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx+60 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx\right )+\frac {25}{3} \sqrt {x^4+3 x^2+2} x\) |
| \(\Big \downarrow \) 1412 |
| \(\displaystyle \frac {1}{3} \left (60 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx+\frac {97 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}\right )+\frac {25}{3} \sqrt {x^4+3 x^2+2} x\) |
| \(\Big \downarrow \) 1455 |
| \(\displaystyle \frac {1}{3} \left (\frac {97 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+60 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )+\frac {25}{3} \sqrt {x^4+3 x^2+2} x\) |
(25*x*Sqrt[2 + 3*x^2 + x^4])/3 + (60*((x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] - (Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/ Sqrt[2 + 3*x^2 + x^4]) + (97*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF [ArcTan[x], 1/2])/(Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]))/3
3.4.1.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q )*x^2)/(2*a + (b + q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && !(PosQ[ (b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*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[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Simp[e^q*x^(2*q - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c*(4*p + 2*q + 1)) Int[(a + b*x^2 + c*x^4)^p*Expand ToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2* p + 2*q - 1)*e^q*x^(2*q - 2) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1]
Result contains complex when optimal does not.
Time = 1.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(-\frac {97 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x \sqrt {x^{4}+3 x^{2}+2}}{3}+\frac {10 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{4}+3 x^{2}+2}}\) | \(121\) |
| risch | \(-\frac {97 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x \sqrt {x^{4}+3 x^{2}+2}}{3}+\frac {10 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{4}+3 x^{2}+2}}\) | \(121\) |
| elliptic | \(-\frac {97 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x \sqrt {x^{4}+3 x^{2}+2}}{3}+\frac {10 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{4}+3 x^{2}+2}}\) | \(121\) |
-97/6*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*Elliptic F(1/2*I*2^(1/2)*x,2^(1/2))+25/3*x*(x^4+3*x^2+2)^(1/2)+10*I*2^(1/2)*(2*x^2+ 4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1 /2))-EllipticE(1/2*I*2^(1/2)*x,2^(1/2)))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.35 \[ \int \frac {\left (7+5 x^2\right )^2}{\sqrt {2+3 x^2+x^4}} \, dx=\frac {-60 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 157 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 5 \, \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 12\right )}}{3 \, x} \]
1/3*(-60*I*x*elliptic_e(arcsin(I/x), 2) + 157*I*x*elliptic_f(arcsin(I/x), 2) + 5*sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 12))/x
\[ \int \frac {\left (7+5 x^2\right )^2}{\sqrt {2+3 x^2+x^4}} \, dx=\int \frac {\left (5 x^{2} + 7\right )^{2}}{\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}\, dx \]
\[ \int \frac {\left (7+5 x^2\right )^2}{\sqrt {2+3 x^2+x^4}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{2}}{\sqrt {x^{4} + 3 \, x^{2} + 2}} \,d x } \]
\[ \int \frac {\left (7+5 x^2\right )^2}{\sqrt {2+3 x^2+x^4}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{2}}{\sqrt {x^{4} + 3 \, x^{2} + 2}} \,d x } \]
Timed out. \[ \int \frac {\left (7+5 x^2\right )^2}{\sqrt {2+3 x^2+x^4}} \, dx=\int \frac {{\left (5\,x^2+7\right )}^2}{\sqrt {x^4+3\,x^2+2}} \,d x \]