Integrand size = 24, antiderivative size = 102 \[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}} \, dx=-\frac {25 x \sqrt {2+x^2-x^4}}{952 \left (7+5 x^2\right )^2}-\frac {12525 x \sqrt {2+x^2-x^4}}{453152 \left (7+5 x^2\right )}-\frac {2505 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{453152}-\frac {263 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{226576}+\frac {58915 \operatorname {EllipticPi}\left (-\frac {10}{7},\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{3172064} \]
-2505/453152*EllipticE(1/2*x*2^(1/2),I*2^(1/2))-263/226576*EllipticF(1/2*x *2^(1/2),I*2^(1/2))+58915/3172064*EllipticPi(1/2*x*2^(1/2),-10/7,I*2^(1/2) )-25/952*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)^2-12525/453152*x*(-x^4+x^2+2)^(1/2 )/(5*x^2+7)
Result contains complex when optimal does not.
Time = 10.43 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}} \, dx=\frac {\frac {350 x \left (-7966-8993 x^2+1478 x^4+2505 x^6\right )}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}}-35070 i \sqrt {2} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )+56287 i \sqrt {2} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )-58915 i \sqrt {2} \operatorname {EllipticPi}\left (\frac {5}{7},i \text {arcsinh}(x),-\frac {1}{2}\right )}{6344128} \]
((350*x*(-7966 - 8993*x^2 + 1478*x^4 + 2505*x^6))/((7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]) - (35070*I)*Sqrt[2]*EllipticE[I*ArcSinh[x], -1/2] + (56287*I)* Sqrt[2]*EllipticF[I*ArcSinh[x], -1/2] - (58915*I)*Sqrt[2]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/6344128
Time = 0.46 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1551, 2210, 2234, 27, 1494, 27, 399, 321, 327, 1536, 27, 412}
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.
| \(\displaystyle \int \frac {1}{\left (5 x^2+7\right )^3 \sqrt {-x^4+x^2+2}} \, dx\) |
| \(\Big \downarrow \) 1551 |
| \(\displaystyle \frac {1}{952} \int \frac {25 x^4-190 x^2+186}{\left (5 x^2+7\right )^2 \sqrt {-x^4+x^2+2}}dx-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 2210 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \int \frac {-12525 x^4-32690 x^2+37698}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 2234 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (58915 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-\frac {1}{25} \int \frac {25 \left (2505 x^2+3031\right )}{\sqrt {-x^4+x^2+2}}dx\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (58915 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-\int \frac {2505 x^2+3031}{\sqrt {-x^4+x^2+2}}dx\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (58915 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-2 \int \frac {2505 x^2+3031}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (58915 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-\int \frac {2505 x^2+3031}{\sqrt {2-x^2} \sqrt {x^2+1}}dx\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (-526 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx-2505 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx+58915 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (-2505 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx+58915 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-526 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (58915 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-526 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-2505 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 1536 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (117830 \int \frac {1}{2 \sqrt {2-x^2} \sqrt {x^2+1} \left (5 x^2+7\right )}dx-526 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-2505 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (58915 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1} \left (5 x^2+7\right )}dx-526 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-2505 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {1}{952} \left (\frac {1}{476} \left (-526 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-2505 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {58915}{7} \operatorname {EllipticPi}\left (-\frac {10}{7},\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )-\frac {12525 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{952 \left (5 x^2+7\right )^2}\) |
(-25*x*Sqrt[2 + x^2 - x^4])/(952*(7 + 5*x^2)^2) + ((-12525*x*Sqrt[2 + x^2 - x^4])/(476*(7 + 5*x^2)) + (-2505*EllipticE[ArcSin[x/Sqrt[2]], -2] - 526* EllipticF[ArcSin[x/Sqrt[2]], -2] + (58915*EllipticPi[-10/7, ArcSin[x/Sqrt[ 2]], -2])/7)/476)/952
3.4.38.3.1 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[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -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[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c] Int[1/((d + e*x^ 2)*Sqrt[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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[ c, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_ Symbol] :> Simp[(-e^2)*x*(d + e*x^2)^(q + 1)*(Sqrt[a + b*x^2 + c*x^4]/(2*d* (q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)) Int[((d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[a*e^2*(2 *q + 3) + 2*d*(c*d - b*e)*(q + 1) - 2*e*(c*d*(q + 1) - b*e*(q + 2))*x^2 + c *e^2*(2*q + 5)*x^4, 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] && ILtQ[q, -1]
Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x _)^4], x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = C oeff[P4x, x, 4]}, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1)*(Sq rt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/( 2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)) Int[((d + e*x^2)^(q + 1)/Sqrt[a + b* x^2 + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*( q + 1)) - 2*((B*d - A*e)*(b*e*(q + 2) - c*d*(q + 1)) - C*d*(b*d + a*e*(q + 1)))*x^2 + c*(C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a , b, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && ILtQ[q, -1 ]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) , x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[ P4x, x, 4]}, Simp[-(e^2)^(-1) Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[(C*d^2 - B*d*e + A*e^2)/e^2 Int[1/((d + e*x^2)*Sqrt [a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^ 2, 2] && NeQ[c*d^2 - a*e^2, 0]
Time = 3.54 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.85
| method | result | size |
| default | \(-\frac {25 x \sqrt {-x^{4}+x^{2}+2}}{952 \left (5 x^{2}+7\right )^{2}}-\frac {12525 x \sqrt {-x^{4}+x^{2}+2}}{453152 \left (5 x^{2}+7\right )}-\frac {263 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{453152 \sqrt {-x^{4}+x^{2}+2}}-\frac {2505 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{906304 \sqrt {-x^{4}+x^{2}+2}}+\frac {58915 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{3172064 \sqrt {-x^{4}+x^{2}+2}}\) | \(189\) |
| elliptic | \(-\frac {25 x \sqrt {-x^{4}+x^{2}+2}}{952 \left (5 x^{2}+7\right )^{2}}-\frac {12525 x \sqrt {-x^{4}+x^{2}+2}}{453152 \left (5 x^{2}+7\right )}-\frac {263 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{453152 \sqrt {-x^{4}+x^{2}+2}}-\frac {2505 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{906304 \sqrt {-x^{4}+x^{2}+2}}+\frac {58915 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{3172064 \sqrt {-x^{4}+x^{2}+2}}\) | \(189\) |
| risch | \(\frac {25 \left (x^{4}-x^{2}-2\right ) x \left (2505 x^{2}+3983\right )}{453152 \left (5 x^{2}+7\right )^{2} \sqrt {-x^{4}+x^{2}+2}}-\frac {433 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{129472 \sqrt {-x^{4}+x^{2}+2}}+\frac {2505 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{906304 \sqrt {-x^{4}+x^{2}+2}}+\frac {58915 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{3172064 \sqrt {-x^{4}+x^{2}+2}}\) | \(198\) |
-25/952*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)^2-12525/453152*x*(-x^4+x^2+2)^(1/2) /(5*x^2+7)-263/453152*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))-2505/906304*2^(1/2)*(-2*x^2+4)^(1 /2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticE(1/2*x*2^(1/2),I*2^(1/2))+58 915/3172064*2^(1/2)*(1-1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*Ell ipticPi(1/2*x*2^(1/2),-10/7,I*2^(1/2))
\[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]
integral(-sqrt(-x^4 + x^2 + 2)/(125*x^10 + 400*x^8 - 40*x^6 - 1442*x^4 - 1 813*x^2 - 686), x)
\[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}} \, dx=\int \frac {1}{\sqrt {- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{3}}\, dx \]
\[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]
\[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}} \, dx=\int \frac {1}{{\left (5\,x^2+7\right )}^3\,\sqrt {-x^4+x^2+2}} \,d x \]