Integrand size = 24, antiderivative size = 222 \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\frac {9 x \left (2+x^2\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {1}{75} x \sqrt {2+3 x^2+x^4}-\frac {3 x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {9 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{175 \sqrt {2+3 x^2+x^4}}+\frac {59 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{2+2 x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{1050 \sqrt {2+3 x^2+x^4}}+\frac {9 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{2+2 x^2}} \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{2450 \sqrt {2+3 x^2+x^4}} \]
9/175*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-9/175*(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)+59/1050*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticF(x/(x^2+ 1)^(1/2),1/2*2^(1/2))*((x^2+2)/(2*x^2+2))^(1/2)/(x^4+3*x^2+2)^(1/2)+9/2450 *(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticPi(x/(x^2+1)^(1/2),2/7,1/2*2^(1/2 ))*((x^2+2)/(2*x^2+2))^(1/2)/(x^4+3*x^2+2)^(1/2)+1/75*x*(x^4+3*x^2+2)^(1/2 )-3/175*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)
Result contains complex when optimal does not.
Time = 10.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.96 \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\frac {2800 x+6650 x^3+5075 x^5+1225 x^7-945 i \sqrt {1+x^2} \sqrt {2+x^2} \left (7+5 x^2\right ) E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-182 i \sqrt {1+x^2} \sqrt {2+x^2} \left (7+5 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )+189 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )+135 i x^2 \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{18375 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \]
(2800*x + 6650*x^3 + 5075*x^5 + 1225*x^7 - (945*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (182*I)*Sqrt[1 + x^2 ]*Sqrt[2 + x^2]*(7 + 5*x^2)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (189*I)*S qrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2] + (13 5*I)*x^2*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]] , 2])/(18375*(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4])
Time = 0.60 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1556, 2009}
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.
| \(\displaystyle \int \frac {\left (x^4+3 x^2+2\right )^{3/2}}{\left (5 x^2+7\right )^2} \, dx\) |
| \(\Big \downarrow \) 1556 |
| \(\displaystyle \int \left (\frac {x^4}{25 \sqrt {x^4+3 x^2+2}}+\frac {16 x^2}{125 \sqrt {x^4+3 x^2+2}}-\frac {12}{625 \left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}+\frac {36}{625 \left (5 x^2+7\right )^2 \sqrt {x^4+3 x^2+2}}+\frac {52}{625 \sqrt {x^4+3 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {44 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{1875 \sqrt {x^4+3 x^2+2}}+\frac {81 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{8750 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {9 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{175 \sqrt {x^4+3 x^2+2}}+\frac {3 \sqrt {2} \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{875 \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}-\frac {39 \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{12250 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}-\frac {3 \sqrt {x^4+3 x^2+2} x}{175 \left (5 x^2+7\right )}+\frac {1}{75} \sqrt {x^4+3 x^2+2} x+\frac {9 \left (x^2+2\right ) x}{175 \sqrt {x^4+3 x^2+2}}\) |
(9*x*(2 + x^2))/(175*Sqrt[2 + 3*x^2 + x^4]) + (x*Sqrt[2 + 3*x^2 + x^4])/75 - (3*x*Sqrt[2 + 3*x^2 + x^4])/(175*(7 + 5*x^2)) - (9*Sqrt[2]*(1 + x^2)*Sq rt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(175*Sqrt[2 + 3*x^2 + x ^4]) + (81*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/ (8750*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + (44*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2 )/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(1875*Sqrt[2 + 3*x^2 + x^4]) - (39 *(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1/2])/(12250*Sqrt[2]*Sqrt[(2 + x^2)/ (1 + x^2)]*Sqrt[2 + 3*x^2 + x^4]) + (3*Sqrt[2]*(2 + x^2)*EllipticPi[2/7, A rcTan[x], 1/2])/(875*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])
3.3.98.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Module[{aa, bb, cc}, Int[ExpandIntegrand[1/Sqrt[aa + bb*x^2 + c c*x^4], (d + e*x^2)^q*(aa + bb*x^2 + cc*x^4)^(p + 1/2), x] /. {aa -> a, bb -> b, cc -> c}, x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, 0] && IntegerQ[p + 1/2]
Result contains complex when optimal does not.
Time = 2.64 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(-\frac {3 x \sqrt {x^{4}+3 x^{2}+2}}{175 \left (5 x^{2}+7\right )}+\frac {x \sqrt {x^{4}+3 x^{2}+2}}{75}-\frac {13 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2625 \sqrt {x^{4}+3 x^{2}+2}}-\frac {9 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{350 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{6125 \sqrt {x^{4}+3 x^{2}+2}}\) | \(177\) |
| elliptic | \(-\frac {3 x \sqrt {x^{4}+3 x^{2}+2}}{175 \left (5 x^{2}+7\right )}+\frac {x \sqrt {x^{4}+3 x^{2}+2}}{75}-\frac {13 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2625 \sqrt {x^{4}+3 x^{2}+2}}-\frac {9 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{350 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{6125 \sqrt {x^{4}+3 x^{2}+2}}\) | \(177\) |
| risch | \(\frac {\sqrt {x^{4}+3 x^{2}+2}\, x \left (7 x^{2}+8\right )}{525 x^{2}+735}-\frac {23 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{750 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{350 \sqrt {x^{4}+3 x^{2}+2}}+\frac {9 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{6125 \sqrt {x^{4}+3 x^{2}+2}}\) | \(183\) |
-3/175*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)+1/75*x*(x^4+3*x^2+2)^(1/2)-13/2625* 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))-9/350*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3* x^2+2)^(1/2)*EllipticE(1/2*I*2^(1/2)*x,2^(1/2))+9/6125*I*2^(1/2)*(1+1/2*x^ 2)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticPi(1/2*I*2^(1/2)*x,10/7 ,2^(1/2))
\[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
\[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int \frac {\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{2}}\, dx \]
\[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
\[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int \frac {{\left (x^4+3\,x^2+2\right )}^{3/2}}{{\left (5\,x^2+7\right )}^2} \,d x \]