Integrand size = 22, antiderivative size = 81 \[ \int \left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx=\frac {1}{315} x \left (1087+669 x^2\right ) \sqrt {2+x^2-x^4}+\frac {1}{63} x \left (48+35 x^2\right ) \left (2+x^2-x^4\right )^{3/2}+\frac {4432}{315} E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {418}{105} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right ) \]
1/63*x*(35*x^2+48)*(-x^4+x^2+2)^(3/2)+4432/315*EllipticE(1/2*x*2^(1/2),I*2 ^(1/2))+418/105*EllipticF(1/2*x*2^(1/2),I*2^(1/2))+1/315*x*(669*x^2+1087)* (-x^4+x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 7.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.32 \[ \int \left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx=\frac {3134 x+4085 x^3-438 x^5-1674 x^7-110 x^9+175 x^{11}+4432 i \sqrt {4+2 x^2-2 x^4} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )-7275 i \sqrt {4+2 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )}{315 \sqrt {2+x^2-x^4}} \]
(3134*x + 4085*x^3 - 438*x^5 - 1674*x^7 - 110*x^9 + 175*x^11 + (4432*I)*Sq rt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (7275*I)*Sqrt[4 + 2* x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(315*Sqrt[2 + x^2 - x^4])
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1490, 25, 1490, 27, 1494, 27, 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.
| \(\displaystyle \int \left (5 x^2+7\right ) \left (-x^4+x^2+2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}-\frac {1}{21} \int -\left (\left (223 x^2+262\right ) \sqrt {-x^4+x^2+2}\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{21} \int \left (223 x^2+262\right ) \sqrt {-x^4+x^2+2}dx+\frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{15} x \left (669 x^2+1087\right ) \sqrt {-x^4+x^2+2}-\frac {1}{15} \int -\frac {2 \left (2216 x^2+2843\right )}{\sqrt {-x^4+x^2+2}}dx\right )+\frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{15} \int \frac {2216 x^2+2843}{\sqrt {-x^4+x^2+2}}dx+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (669 x^2+1087\right )\right )+\frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {1}{21} \left (\frac {4}{15} \int \frac {2216 x^2+2843}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (669 x^2+1087\right )\right )+\frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{15} \int \frac {2216 x^2+2843}{\sqrt {2-x^2} \sqrt {x^2+1}}dx+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (669 x^2+1087\right )\right )+\frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{15} \left (627 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx+2216 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx\right )+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (669 x^2+1087\right )\right )+\frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{15} \left (2216 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx+627 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (669 x^2+1087\right )\right )+\frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{15} \left (627 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )+2216 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (669 x^2+1087\right )\right )+\frac {1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}\) |
(x*(48 + 35*x^2)*(2 + x^2 - x^4)^(3/2))/63 + ((x*(1087 + 669*x^2)*Sqrt[2 + x^2 - x^4])/15 + (2*(2216*EllipticE[ArcSin[x/Sqrt[2]], -2] + 627*Elliptic F[ArcSin[x/Sqrt[2]], -2]))/15)/21
3.4.27.3.1 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]*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[((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[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. 151 vs. \(2 (75 ) = 150\).
Time = 1.71 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.88
| method | result | size |
| risch | \(\frac {x \left (175 x^{6}+65 x^{4}-1259 x^{2}-1567\right ) \left (x^{4}-x^{2}-2\right )}{315 \sqrt {-x^{4}+x^{2}+2}}+\frac {2843 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{315 \sqrt {-x^{4}+x^{2}+2}}-\frac {2216 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{315 \sqrt {-x^{4}+x^{2}+2}}\) | \(152\) |
| default | \(-\frac {13 x^{5} \sqrt {-x^{4}+x^{2}+2}}{63}+\frac {1259 x^{3} \sqrt {-x^{4}+x^{2}+2}}{315}+\frac {1567 x \sqrt {-x^{4}+x^{2}+2}}{315}+\frac {2843 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{315 \sqrt {-x^{4}+x^{2}+2}}-\frac {2216 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{315 \sqrt {-x^{4}+x^{2}+2}}-\frac {5 x^{7} \sqrt {-x^{4}+x^{2}+2}}{9}\) | \(176\) |
| elliptic | \(-\frac {13 x^{5} \sqrt {-x^{4}+x^{2}+2}}{63}+\frac {1259 x^{3} \sqrt {-x^{4}+x^{2}+2}}{315}+\frac {1567 x \sqrt {-x^{4}+x^{2}+2}}{315}+\frac {2843 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{315 \sqrt {-x^{4}+x^{2}+2}}-\frac {2216 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{315 \sqrt {-x^{4}+x^{2}+2}}-\frac {5 x^{7} \sqrt {-x^{4}+x^{2}+2}}{9}\) | \(176\) |
1/315*x*(175*x^6+65*x^4-1259*x^2-1567)*(x^4-x^2-2)/(-x^4+x^2+2)^(1/2)+2843 /315*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticF(1 /2*x*2^(1/2),I*2^(1/2))-2216/315*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(- x^4+x^2+2)^(1/2)*(EllipticF(1/2*x*2^(1/2),I*2^(1/2))-EllipticE(1/2*x*2^(1/ 2),I*2^(1/2)))
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx=\frac {-8864 i \, \sqrt {2} x E(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) + 11707 i \, \sqrt {2} x F(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) - {\left (175 \, x^{8} + 65 \, x^{6} - 1259 \, x^{4} - 1567 \, x^{2} + 4432\right )} \sqrt {-x^{4} + x^{2} + 2}}{315 \, x} \]
1/315*(-8864*I*sqrt(2)*x*elliptic_e(arcsin(sqrt(2)/x), -1/2) + 11707*I*sqr t(2)*x*elliptic_f(arcsin(sqrt(2)/x), -1/2) - (175*x^8 + 65*x^6 - 1259*x^4 - 1567*x^2 + 4432)*sqrt(-x^4 + x^2 + 2))/x
\[ \int \left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx=\int \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}} \cdot \left (5 x^{2} + 7\right )\, dx \]
\[ \int \left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )} \,d x } \]
\[ \int \left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )} \,d x } \]
Timed out. \[ \int \left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx=\int \left (5\,x^2+7\right )\,{\left (-x^4+x^2+2\right )}^{3/2} \,d x \]