Integrand size = 24, antiderivative size = 93 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\frac {x \left (1419985+1419793 x^2\right )}{18 \sqrt {2+x^2-x^4}}+\frac {27500}{3} x \sqrt {2+x^2-x^4}+625 x^3 \sqrt {2+x^2-x^4}-\frac {3482293}{18} E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {627857}{6} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right ) \]
-3482293/18*EllipticE(1/2*x*2^(1/2),I*2^(1/2))+627857/6*EllipticF(1/2*x*2^ (1/2),I*2^(1/2))+1/18*x*(1419793*x^2+1419985)/(-x^4+x^2+2)^(1/2)+27500/3*x *(-x^4+x^2+2)^(1/2)+625*x^3*(-x^4+x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\frac {1749985 x+1607293 x^3-153750 x^5-11250 x^7-3482293 i \sqrt {4+2 x^2-2 x^4} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )+4281654 i \sqrt {4+2 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )}{18 \sqrt {2+x^2-x^4}} \]
(1749985*x + 1607293*x^3 - 153750*x^5 - 11250*x^7 - (3482293*I)*Sqrt[4 + 2 *x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] + (4281654*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(18*Sqrt[2 + x^2 - x^4])
Time = 0.36 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1517, 2207, 27, 2207, 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.
| \(\displaystyle \int \frac {\left (5 x^2+7\right )^5}{\left (-x^4+x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}-\frac {1}{18} \int \frac {56250 x^6+450000 x^4+3084793 x^2+1268722}{\sqrt {-x^4+x^2+2}}dx\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{5} \int -\frac {5 \left (495000 x^4+3152293 x^2+1268722\right )}{\sqrt {-x^4+x^2+2}}dx+11250 \sqrt {-x^4+x^2+2} x^3\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (11250 x^3 \sqrt {-x^4+x^2+2}-\int \frac {495000 x^4+3152293 x^2+1268722}{\sqrt {-x^4+x^2+2}}dx\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{3} \int -\frac {3 \left (3482293 x^2+1598722\right )}{\sqrt {-x^4+x^2+2}}dx+165000 \sqrt {-x^4+x^2+2} x+11250 \sqrt {-x^4+x^2+2} x^3\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (-\int \frac {3482293 x^2+1598722}{\sqrt {-x^4+x^2+2}}dx+165000 \sqrt {-x^4+x^2+2} x+11250 \sqrt {-x^4+x^2+2} x^3\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {1}{18} \left (-2 \int \frac {3482293 x^2+1598722}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx+165000 \sqrt {-x^4+x^2+2} x+11250 \sqrt {-x^4+x^2+2} x^3\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (-\int \frac {3482293 x^2+1598722}{\sqrt {2-x^2} \sqrt {x^2+1}}dx+165000 \sqrt {-x^4+x^2+2} x+11250 \sqrt {-x^4+x^2+2} x^3\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {1}{18} \left (1883571 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx-3482293 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx+165000 \sqrt {-x^4+x^2+2} x+11250 \sqrt {-x^4+x^2+2} x^3\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{18} \left (-3482293 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx+1883571 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )+165000 \sqrt {-x^4+x^2+2} x+11250 \sqrt {-x^4+x^2+2} x^3\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{18} \left (1883571 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-3482293 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+165000 \sqrt {-x^4+x^2+2} x+11250 \sqrt {-x^4+x^2+2} x^3\right )+\frac {x \left (1419793 x^2+1419985\right )}{18 \sqrt {-x^4+x^2+2}}\) |
(x*(1419985 + 1419793*x^2))/(18*Sqrt[2 + x^2 - x^4]) + (165000*x*Sqrt[2 + x^2 - x^4] + 11250*x^3*Sqrt[2 + x^2 - x^4] - 3482293*EllipticE[ArcSin[x/Sq rt[2]], -2] + 1883571*EllipticF[ArcSin[x/Sqrt[2]], -2])/18
3.4.39.3.1 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]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p + 1)) Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 *n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) *x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && !LtQ[p, -1]
Time = 8.85 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.53
| method | result | size |
| risch | \(-\frac {x \left (11250 x^{6}+153750 x^{4}-1607293 x^{2}-1749985\right )}{18 \sqrt {-x^{4}+x^{2}+2}}-\frac {799361 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{18 \sqrt {-x^{4}+x^{2}+2}}+\frac {3482293 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}\) | \(142\) |
| elliptic | \(\frac {\frac {1419793}{18} x^{3}+\frac {1419985}{18} x}{\sqrt {-x^{4}+x^{2}+2}}+625 x^{3} \sqrt {-x^{4}+x^{2}+2}+\frac {27500 x \sqrt {-x^{4}+x^{2}+2}}{3}-\frac {799361 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{18 \sqrt {-x^{4}+x^{2}+2}}+\frac {3482293 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}\) | \(165\) |
| default | \(\frac {\frac {84035}{18} x -\frac {16807}{18} x^{3}}{\sqrt {-x^{4}+x^{2}+2}}-\frac {799361 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{18 \sqrt {-x^{4}+x^{2}+2}}+\frac {3482293 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}+\frac {\frac {53125}{9} x^{3}+\frac {43750}{9} x}{\sqrt {-x^{4}+x^{2}+2}}+625 x^{3} \sqrt {-x^{4}+x^{2}+2}+\frac {27500 x \sqrt {-x^{4}+x^{2}+2}}{3}+\frac {\frac {153125}{9} x^{3}+\frac {218750}{9} x}{\sqrt {-x^{4}+x^{2}+2}}+\frac {\frac {306250}{9} x^{3}+\frac {122500}{9} x}{\sqrt {-x^{4}+x^{2}+2}}+\frac {\frac {85750}{9} x^{3}+\frac {343000}{9} x}{\sqrt {-x^{4}+x^{2}+2}}+\frac {\frac {120050}{9} x^{3}-\frac {60025}{9} x}{\sqrt {-x^{4}+x^{2}+2}}\) | \(280\) |
-1/18*x*(11250*x^6+153750*x^4-1607293*x^2-1749985)/(-x^4+x^2+2)^(1/2)-7993 61/18*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))+3482293/36*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 = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+x^2-x^4\right )^{3/2}} \, dx=-\frac {6964586 \, \sqrt {2} {\left (-i \, x^{5} + i \, x^{3} + 2 i \, x\right )} E(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) + 7763947 \, \sqrt {2} {\left (i \, x^{5} - i \, x^{3} - 2 i \, x\right )} F(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) - 2 \, {\left (5625 \, x^{8} + 76875 \, x^{6} + 937500 \, x^{4} - 2616139 \, x^{2} - 3482293\right )} \sqrt {-x^{4} + x^{2} + 2}}{18 \, {\left (x^{5} - x^{3} - 2 \, x\right )}} \]
-1/18*(6964586*sqrt(2)*(-I*x^5 + I*x^3 + 2*I*x)*elliptic_e(arcsin(sqrt(2)/ x), -1/2) + 7763947*sqrt(2)*(I*x^5 - I*x^3 - 2*I*x)*elliptic_f(arcsin(sqrt (2)/x), -1/2) - 2*(5625*x^8 + 76875*x^6 + 937500*x^4 - 2616139*x^2 - 34822 93)*sqrt(-x^4 + x^2 + 2))/(x^5 - x^3 - 2*x)
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int \frac {\left (5 x^{2} + 7\right )^{5}}{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{5}}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{5}}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int \frac {{\left (5\,x^2+7\right )}^5}{{\left (-x^4+x^2+2\right )}^{3/2}} \,d x \]