Integrand size = 24, antiderivative size = 121 \[ \int \left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2} \, dx=\frac {x \left (2512273+5712051 x^2\right ) \sqrt {2+x^2-x^4}}{15015}+\frac {x \left (33792+374045 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{3003}-\frac {7825}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac {125}{13} x^3 \left (2+x^2-x^4\right )^{5/2}+\frac {31072528 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{15015}-\frac {3199778 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{5005} \]
1/3003*x*(374045*x^2+33792)*(-x^4+x^2+2)^(3/2)-7825/143*x*(-x^4+x^2+2)^(5/ 2)-125/13*x^3*(-x^4+x^2+2)^(5/2)+31072528/15015*EllipticE(1/2*x*2^(1/2),I* 2^(1/2))-3199778/5005*EllipticF(1/2*x*2^(1/2),I*2^(1/2))+1/15015*x*(571205 1*x^2+2512273)*(-x^4+x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97 \[ \int \left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2} \, dx=\frac {-872614 x+11078615 x^3+13371048 x^5-1756521 x^7-4448240 x^9-1027775 x^{11}+388500 x^{13}+144375 x^{15}+31072528 i \sqrt {4+2 x^2-2 x^4} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )-41809125 i \sqrt {4+2 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )}{15015 \sqrt {2+x^2-x^4}} \]
(-872614*x + 11078615*x^3 + 13371048*x^5 - 1756521*x^7 - 4448240*x^9 - 102 7775*x^11 + 388500*x^13 + 144375*x^15 + (31072528*I)*Sqrt[4 + 2*x^2 - 2*x^ 4]*EllipticE[I*ArcSinh[x], -1/2] - (41809125*I)*Sqrt[4 + 2*x^2 - 2*x^4]*El lipticF[I*ArcSinh[x], -1/2])/(15015*Sqrt[2 + x^2 - x^4])
Time = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {1518, 25, 2207, 25, 1490, 27, 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.
| \(\displaystyle \int \left (5 x^2+7\right )^3 \left (-x^4+x^2+2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1518 |
| \(\displaystyle -\frac {1}{13} \int -\left (-x^4+x^2+2\right )^{3/2} \left (7825 x^4+10305 x^2+4459\right )dx-\frac {125}{13} \left (-x^4+x^2+2\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{13} \int \left (-x^4+x^2+2\right )^{3/2} \left (7825 x^4+10305 x^2+4459\right )dx-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{13} \left (-\frac {1}{11} \int -\left (\left (160305 x^2+64699\right ) \left (-x^4+x^2+2\right )^{3/2}\right )dx-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \left (160305 x^2+64699\right ) \left (-x^4+x^2+2\right )^{3/2}dx-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}-\frac {1}{21} \int -3 \left (1904017 x^2+883258\right ) \sqrt {-x^4+x^2+2}dx\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{7} \int \left (1904017 x^2+883258\right ) \sqrt {-x^4+x^2+2}dx+\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{7} \left (\frac {1}{15} x \left (5712051 x^2+2512273\right ) \sqrt {-x^4+x^2+2}-\frac {1}{15} \int -\frac {2 \left (15536264 x^2+10736597\right )}{\sqrt {-x^4+x^2+2}}dx\right )+\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{7} \left (\frac {2}{15} \int \frac {15536264 x^2+10736597}{\sqrt {-x^4+x^2+2}}dx+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (5712051 x^2+2512273\right )\right )+\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{7} \left (\frac {4}{15} \int \frac {15536264 x^2+10736597}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (5712051 x^2+2512273\right )\right )+\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{7} \left (\frac {2}{15} \int \frac {15536264 x^2+10736597}{\sqrt {2-x^2} \sqrt {x^2+1}}dx+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (5712051 x^2+2512273\right )\right )+\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{7} \left (\frac {2}{15} \left (15536264 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx-4799667 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx\right )+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (5712051 x^2+2512273\right )\right )+\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{7} \left (\frac {2}{15} \left (15536264 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx-4799667 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (5712051 x^2+2512273\right )\right )+\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{7} \left (\frac {2}{15} \left (15536264 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-4799667 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )+\frac {1}{15} x \sqrt {-x^4+x^2+2} \left (5712051 x^2+2512273\right )\right )+\frac {1}{21} x \left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2}\right )-\frac {7825}{11} x \left (-x^4+x^2+2\right )^{5/2}\right )-\frac {125}{13} x^3 \left (-x^4+x^2+2\right )^{5/2}\) |
(-125*x^3*(2 + x^2 - x^4)^(5/2))/13 + ((-7825*x*(2 + x^2 - x^4)^(5/2))/11 + ((x*(33792 + 374045*x^2)*(2 + x^2 - x^4)^(3/2))/21 + ((x*(2512273 + 5712 051*x^2)*Sqrt[2 + x^2 - x^4])/15 + (2*(15536264*EllipticE[ArcSin[x/Sqrt[2] ], -2] - 4799667*EllipticF[ArcSin[x/Sqrt[2]], -2]))/15)/7)/11)/13
3.4.25.3.1 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]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Simp[e^q*x^(2*q - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c*(4*p + 2*q + 1)) Int[(a + b*x^2 + c*x^4)^p*Expand ToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2* p + 2*q - 1)*e^q*x^(2*q - 2) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 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 = 2.90 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(\frac {x \left (144375 x^{10}+532875 x^{8}-206150 x^{6}-3588640 x^{4}-5757461 x^{2}+436307\right ) \left (x^{4}-x^{2}-2\right )}{15015 \sqrt {-x^{4}+x^{2}+2}}+\frac {10736597 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{15015 \sqrt {-x^{4}+x^{2}+2}}-\frac {15536264 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{15015 \sqrt {-x^{4}+x^{2}+2}}\) | \(162\) |
| default | \(\frac {65248 x^{5} \sqrt {-x^{4}+x^{2}+2}}{273}+\frac {5757461 x^{3} \sqrt {-x^{4}+x^{2}+2}}{15015}-\frac {436307 x \sqrt {-x^{4}+x^{2}+2}}{15015}+\frac {10736597 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{15015 \sqrt {-x^{4}+x^{2}+2}}-\frac {15536264 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{15015 \sqrt {-x^{4}+x^{2}+2}}-\frac {125 x^{11} \sqrt {-x^{4}+x^{2}+2}}{13}-\frac {5075 x^{9} \sqrt {-x^{4}+x^{2}+2}}{143}+\frac {5890 x^{7} \sqrt {-x^{4}+x^{2}+2}}{429}\) | \(210\) |
| elliptic | \(\frac {65248 x^{5} \sqrt {-x^{4}+x^{2}+2}}{273}+\frac {5757461 x^{3} \sqrt {-x^{4}+x^{2}+2}}{15015}-\frac {436307 x \sqrt {-x^{4}+x^{2}+2}}{15015}+\frac {10736597 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{15015 \sqrt {-x^{4}+x^{2}+2}}-\frac {15536264 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{15015 \sqrt {-x^{4}+x^{2}+2}}-\frac {125 x^{11} \sqrt {-x^{4}+x^{2}+2}}{13}-\frac {5075 x^{9} \sqrt {-x^{4}+x^{2}+2}}{143}+\frac {5890 x^{7} \sqrt {-x^{4}+x^{2}+2}}{429}\) | \(210\) |
1/15015*x*(144375*x^10+532875*x^8-206150*x^6-3588640*x^4-5757461*x^2+43630 7)*(x^4-x^2-2)/(-x^4+x^2+2)^(1/2)+10736597/15015*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))-155362 64/15015*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*(Ellipt icF(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 = 84, normalized size of antiderivative = 0.69 \[ \int \left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2} \, dx=\frac {-62145056 i \, \sqrt {2} x E(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) + 72881653 i \, \sqrt {2} x F(\arcsin \left (\frac {\sqrt {2}}{x}\right )\,|\,-\frac {1}{2}) - {\left (144375 \, x^{12} + 532875 \, x^{10} - 206150 \, x^{8} - 3588640 \, x^{6} - 5757461 \, x^{4} + 436307 \, x^{2} + 31072528\right )} \sqrt {-x^{4} + x^{2} + 2}}{15015 \, x} \]
1/15015*(-62145056*I*sqrt(2)*x*elliptic_e(arcsin(sqrt(2)/x), -1/2) + 72881 653*I*sqrt(2)*x*elliptic_f(arcsin(sqrt(2)/x), -1/2) - (144375*x^12 + 53287 5*x^10 - 206150*x^8 - 3588640*x^6 - 5757461*x^4 + 436307*x^2 + 31072528)*s qrt(-x^4 + x^2 + 2))/x
\[ \int \left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2} \, dx=\int \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{3}\, dx \]
\[ \int \left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3} \,d x } \]
\[ \int \left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3} \,d x } \]
Timed out. \[ \int \left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2} \, dx=\int {\left (5\,x^2+7\right )}^3\,{\left (-x^4+x^2+2\right )}^{3/2} \,d x \]