Integrand size = 22, antiderivative size = 145 \[ \int \frac {7+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=-\frac {x \left (2+x^2\right )}{2 \sqrt {2+3 x^2+x^4}}+\frac {x \left (5+x^2\right )}{2 \sqrt {2+3 x^2+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+3 x^2+x^4}} \]
-1/2*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)+1/2*x*(x^2+5)/(x^4+3*x^2+2)^(1/2)+1/2*( 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)+1/2*(x^2+1)^(3/2)*(1/(x^2+ 1))^(1/2)*EllipticF(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)
Result contains complex when optimal does not.
Time = 10.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.67 \[ \int \frac {7+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\frac {5 x+x^3+i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-3 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{2 \sqrt {2+3 x^2+x^4}} \]
(5*x + x^3 + I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (3*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] )/(2*Sqrt[2 + 3*x^2 + x^4])
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1492, 25, 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.
\(\displaystyle \int \frac {5 x^2+7}{\left (x^4+3 x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1492 |
\(\displaystyle \frac {x \left (x^2+5\right )}{2 \sqrt {x^4+3 x^2+2}}-\frac {1}{2} \int -\frac {2-x^2}{\sqrt {x^4+3 x^2+2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \int \frac {2-x^2}{\sqrt {x^4+3 x^2+2}}dx+\frac {x \left (x^2+5\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
\(\Big \downarrow \) 1503 |
\(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx-\int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx\right )+\frac {x \left (x^2+5\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
\(\Big \downarrow \) 1412 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}}-\int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx\right )+\frac {x \left (x^2+5\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
\(\Big \downarrow \) 1455 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}}+\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}-\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}\right )+\frac {x \left (x^2+5\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
(x*(5 + x^2))/(2*Sqrt[2 + 3*x^2 + x^4]) + (-((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] + (Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2 )]*EllipticF[ArcTan[x], 1/2])/Sqrt[2 + 3*x^2 + x^4])/2
3.4.11.3.1 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)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], 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] && 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[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]
Result contains complex when optimal does not.
Time = 1.42 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {x \left (x^{2}+5\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{4 \sqrt {x^{4}+3 x^{2}+2}}\) | \(126\) |
elliptic | \(-\frac {2 \left (-\frac {1}{4} x^{3}-\frac {5}{4} x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{4 \sqrt {x^{4}+3 x^{2}+2}}\) | \(129\) |
default | \(-\frac {14 \left (-\frac {3}{4} x^{3}-\frac {5}{4} x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{4 \sqrt {x^{4}+3 x^{2}+2}}-\frac {10 \left (x^{3}+\frac {3}{2} x \right )}{\sqrt {x^{4}+3 x^{2}+2}}\) | \(150\) |
1/2*x*(x^2+5)/(x^4+3*x^2+2)^(1/2)-1/2*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))-1/4*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 = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.56 \[ \int \frac {7+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\frac {{\left (-i \, x^{4} - 3 i \, x^{2} - 2 i\right )} E(\arcsin \left (\frac {1}{2} i \, \sqrt {2} x\right )\,|\,2) - 3 \, {\left (i \, x^{4} + 3 i \, x^{2} + 2 i\right )} F(\arcsin \left (\frac {1}{2} i \, \sqrt {2} x\right )\,|\,2) + 2 \, \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (x^{3} + 5 \, x\right )}}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \]
1/4*((-I*x^4 - 3*I*x^2 - 2*I)*elliptic_e(arcsin(1/2*I*sqrt(2)*x), 2) - 3*( I*x^4 + 3*I*x^2 + 2*I)*elliptic_f(arcsin(1/2*I*sqrt(2)*x), 2) + 2*sqrt(x^4 + 3*x^2 + 2)*(x^3 + 5*x))/(x^4 + 3*x^2 + 2)
\[ \int \frac {7+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {5 x^{2} + 7}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {7+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {5 \, x^{2} + 7}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {7+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {5 \, x^{2} + 7}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {7+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {5\,x^2+7}{{\left (x^4+3\,x^2+2\right )}^{3/2}} \,d x \]