Integrand size = 22, antiderivative size = 179 \[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {116 x \left (2+x^2\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {1}{105} x \left (519+149 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}-\frac {116 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {197 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{35 \sqrt {2+3 x^2+x^4}} \]
1/63*x*(35*x^2+108)*(x^4+3*x^2+2)^(3/2)+116/15*x*(x^2+2)/(x^4+3*x^2+2)^(1/ 2)-116/15*(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)+197/35*(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)+1/105*x*(149*x^2+519)*(x^4+3*x^2+ 2)^(1/2)
Result contains complex when optimal does not.
Time = 6.82 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {5274 x+12745 x^3+12018 x^5+5962 x^7+1590 x^9+175 x^{11}-2436 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-1110 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{315 \sqrt {2+3 x^2+x^4}} \]
(5274*x + 12745*x^3 + 12018*x^5 + 5962*x^7 + 1590*x^9 + 175*x^11 - (2436*I )*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (1110*I )*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(315*Sqr t[2 + 3*x^2 + x^4])
Time = 0.32 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1490, 1490, 27, 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.
| \(\displaystyle \int \left (5 x^2+7\right ) \left (x^4+3 x^2+2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{21} \int \left (149 x^2+222\right ) \sqrt {x^4+3 x^2+2}dx+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{15} \int \frac {6 \left (406 x^2+591\right )}{\sqrt {x^4+3 x^2+2}}dx+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (149 x^2+519\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \int \frac {406 x^2+591}{\sqrt {x^4+3 x^2+2}}dx+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (149 x^2+519\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \left (591 \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx+406 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (149 x^2+519\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1412 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \left (406 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx+\frac {591 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (149 x^2+519\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1455 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \left (\frac {591 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+406 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (149 x^2+519\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}\) |
(x*(108 + 35*x^2)*(2 + 3*x^2 + x^4)^(3/2))/63 + ((x*(519 + 149*x^2)*Sqrt[2 + 3*x^2 + x^4])/5 + (2*(406*((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]) + (591*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan [x], 1/2])/(Sqrt[2]*Sqrt[2 + 3*x^2 + x^4])))/5)/21
3.3.95.3.1 Defintions of rubi rules used
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 + (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*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*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] && GtQ[p, 0] && FractionQ[p] && 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 = 1.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(\frac {x \left (175 x^{6}+1065 x^{4}+2417 x^{2}+2637\right ) \sqrt {x^{4}+3 x^{2}+2}}{315}-\frac {197 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{35 \sqrt {x^{4}+3 x^{2}+2}}+\frac {58 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}\) | \(138\) |
| default | \(\frac {71 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{21}+\frac {2417 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{315}+\frac {293 x \sqrt {x^{4}+3 x^{2}+2}}{35}-\frac {197 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{35 \sqrt {x^{4}+3 x^{2}+2}}+\frac {58 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{9}\) | \(172\) |
| elliptic | \(\frac {71 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{21}+\frac {2417 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{315}+\frac {293 x \sqrt {x^{4}+3 x^{2}+2}}{35}-\frac {197 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{35 \sqrt {x^{4}+3 x^{2}+2}}+\frac {58 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{9}\) | \(172\) |
1/315*x*(175*x^6+1065*x^4+2417*x^2+2637)*(x^4+3*x^2+2)^(1/2)-197/35*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))+58/15*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 = 63, normalized size of antiderivative = 0.35 \[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {-2436 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 5982 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + {\left (175 \, x^{8} + 1065 \, x^{6} + 2417 \, x^{4} + 2637 \, x^{2} + 2436\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{315 \, x} \]
1/315*(-2436*I*x*elliptic_e(arcsin(I/x), 2) + 5982*I*x*elliptic_f(arcsin(I /x), 2) + (175*x^8 + 1065*x^6 + 2417*x^4 + 2637*x^2 + 2436)*sqrt(x^4 + 3*x ^2 + 2))/x
\[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \cdot \left (5 x^{2} + 7\right )\, dx \]
\[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )} \,d x } \]
\[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )} \,d x } \]
Timed out. \[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int \left (5\,x^2+7\right )\,{\left (x^4+3\,x^2+2\right )}^{3/2} \,d x \]