Integrand size = 24, antiderivative size = 55 \[ \int \frac {\left (7+5 x^2\right )^3}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\frac {x \left (4945+4897 x^2\right )}{18 \sqrt {2+x^2-x^4}}-\frac {7147}{18} E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {1763}{6} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right ) \]
-7147/18*EllipticE(1/2*x*2^(1/2),I*2^(1/2))+1763/6*EllipticF(1/2*x*2^(1/2) ,I*2^(1/2))+1/18*x*(4897*x^2+4945)/(-x^4+x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.44 \[ \int \frac {\left (7+5 x^2\right )^3}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\frac {1}{18} \left (\frac {4945 x}{\sqrt {2+x^2-x^4}}+\frac {4897 x^3}{\sqrt {2+x^2-x^4}}-7147 i \sqrt {2} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )+8076 i \sqrt {2} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )\right ) \]
((4945*x)/Sqrt[2 + x^2 - x^4] + (4897*x^3)/Sqrt[2 + x^2 - x^4] - (7147*I)* Sqrt[2]*EllipticE[I*ArcSinh[x], -1/2] + (8076*I)*Sqrt[2]*EllipticF[I*ArcSi nh[x], -1/2])/18
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1517, 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.
\(\displaystyle \int \frac {\left (5 x^2+7\right )^3}{\left (-x^4+x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1517 |
\(\displaystyle \frac {x \left (4897 x^2+4945\right )}{18 \sqrt {-x^4+x^2+2}}-\frac {1}{18} \int \frac {7147 x^2+1858}{\sqrt {-x^4+x^2+2}}dx\) |
\(\Big \downarrow \) 1494 |
\(\displaystyle \frac {x \left (4897 x^2+4945\right )}{18 \sqrt {-x^4+x^2+2}}-\frac {1}{9} \int \frac {7147 x^2+1858}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x \left (4897 x^2+4945\right )}{18 \sqrt {-x^4+x^2+2}}-\frac {1}{18} \int \frac {7147 x^2+1858}{\sqrt {2-x^2} \sqrt {x^2+1}}dx\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {1}{18} \left (5289 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx-7147 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx\right )+\frac {x \left (4897 x^2+4945\right )}{18 \sqrt {-x^4+x^2+2}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {1}{18} \left (5289 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-7147 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx\right )+\frac {x \left (4897 x^2+4945\right )}{18 \sqrt {-x^4+x^2+2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {1}{18} \left (5289 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-7147 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )+\frac {x \left (4897 x^2+4945\right )}{18 \sqrt {-x^4+x^2+2}}\) |
(x*(4945 + 4897*x^2))/(18*Sqrt[2 + x^2 - x^4]) + (-7147*EllipticE[ArcSin[x /Sqrt[2]], -2] + 5289*EllipticF[ArcSin[x/Sqrt[2]], -2])/18
3.4.41.3.1 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)/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]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* (p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], 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] && IGtQ[q, 1] && LtQ[p, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (53 ) = 106\).
Time = 2.52 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.40
method | result | size |
risch | \(\frac {x \left (4897 x^{2}+4945\right )}{18 \sqrt {-x^{4}+x^{2}+2}}-\frac {929 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{18 \sqrt {-x^{4}+x^{2}+2}}+\frac {7147 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}\) | \(132\) |
elliptic | \(\frac {\frac {4897}{18} x^{3}+\frac {4945}{18} x}{\sqrt {-x^{4}+x^{2}+2}}-\frac {929 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{18 \sqrt {-x^{4}+x^{2}+2}}+\frac {7147 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}\) | \(133\) |
default | \(\frac {\frac {1715}{18} x -\frac {343}{18} x^{3}}{\sqrt {-x^{4}+x^{2}+2}}-\frac {929 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{18 \sqrt {-x^{4}+x^{2}+2}}+\frac {7147 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}+\frac {\frac {625}{9} x^{3}+\frac {250}{9} x}{\sqrt {-x^{4}+x^{2}+2}}+\frac {\frac {175}{3} x^{3}+\frac {700}{3} x}{\sqrt {-x^{4}+x^{2}+2}}+\frac {\frac {490}{3} x^{3}-\frac {245}{3} x}{\sqrt {-x^{4}+x^{2}+2}}\) | \(202\) |
1/18*x*(4897*x^2+4945)/(-x^4+x^2+2)^(1/2)-929/18*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))+7147/3 6*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 = 101, normalized size of antiderivative = 1.84 \[ \int \frac {\left (7+5 x^2\right )^3}{\left (2+x^2-x^4\right )^{3/2}} \, dx=-\frac {14294 \, \sqrt {2} {\left (-i \, x^{5} + i \, x^{3} + 2 i \, x\right )} E(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) + 15223 \, \sqrt {2} {\left (i \, x^{5} - i \, x^{3} - 2 i \, x\right )} F(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) - 2 \, {\left (1125 \, x^{4} - 6046 \, x^{2} - 7147\right )} \sqrt {-x^{4} + x^{2} + 2}}{18 \, {\left (x^{5} - x^{3} - 2 \, x\right )}} \]
-1/18*(14294*sqrt(2)*(-I*x^5 + I*x^3 + 2*I*x)*elliptic_e(arcsin(sqrt(2)/x) , -1/2) + 15223*sqrt(2)*(I*x^5 - I*x^3 - 2*I*x)*elliptic_f(arcsin(sqrt(2)/ x), -1/2) - 2*(1125*x^4 - 6046*x^2 - 7147)*sqrt(-x^4 + x^2 + 2))/(x^5 - x^ 3 - 2*x)
\[ \int \frac {\left (7+5 x^2\right )^3}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int \frac {\left (5 x^{2} + 7\right )^{3}}{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (7+5 x^2\right )^3}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{3}}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (7+5 x^2\right )^3}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{3}}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (7+5 x^2\right )^3}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int \frac {{\left (5\,x^2+7\right )}^3}{{\left (-x^4+x^2+2\right )}^{3/2}} \,d x \]