3.4.37 \(\int \frac {1}{(7+5 x^2)^2 \sqrt {2+x^2-x^4}} \, dx\) [337]

3.4.37.1 Optimal result

Integrand size = 24, antiderivative size = 74 \[ \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx=-\frac {25 x \sqrt {2+x^2-x^4}}{476 \left (7+5 x^2\right )}-\frac {5}{476} E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {1}{238} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )+\frac {167 \operatorname {EllipticPi}\left (-\frac {10}{7},\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{3332} \]

output

-5/476*EllipticE(1/2*x*2^(1/2),I*2^(1/2))-1/238*EllipticF(1/2*x*2^(1/2),I* 
2^(1/2))+167/3332*EllipticPi(1/2*x*2^(1/2),-10/7,I*2^(1/2))-25/476*x*(-x^4 
+x^2+2)^(1/2)/(5*x^2+7)
 

3.4.37.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.32 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.65 \[ \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx=\frac {-700 x-350 x^3+350 x^5-70 i \sqrt {2} \left (7+5 x^2\right ) \sqrt {2+x^2-x^4} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )+119 i \sqrt {2} \left (7+5 x^2\right ) \sqrt {2+x^2-x^4} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )-1169 i \sqrt {2} \sqrt {2+x^2-x^4} \operatorname {EllipticPi}\left (\frac {5}{7},i \text {arcsinh}(x),-\frac {1}{2}\right )-835 i \sqrt {2} x^2 \sqrt {2+x^2-x^4} \operatorname {EllipticPi}\left (\frac {5}{7},i \text {arcsinh}(x),-\frac {1}{2}\right )}{6664 \left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \]

input

Integrate[1/((7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]),x]
 

output

(-700*x - 350*x^3 + 350*x^5 - (70*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2 - x^ 
4]*EllipticE[I*ArcSinh[x], -1/2] + (119*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^ 
2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] - (1169*I)*Sqrt[2]*Sqrt[2 + x^2 - x 
^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (835*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 
 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(6664*(7 + 5*x^2)*Sqrt[2 + x^ 
2 - x^4])
 

3.4.37.3 Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1551, 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.

\(\displaystyle \int \frac {1}{\left (5 x^2+7\right )^2 \sqrt {-x^4+x^2+2}} \, dx\)

\(\Big \downarrow \) 1551

\(\displaystyle \frac {1}{476} \int \frac {-25 x^4-70 x^2+118}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 2234

\(\displaystyle \frac {1}{476} \left (167 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-\frac {1}{25} \int \frac {25 \left (5 x^2+7\right )}{\sqrt {-x^4+x^2+2}}dx\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{476} \left (167 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-\int \frac {5 x^2+7}{\sqrt {-x^4+x^2+2}}dx\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {1}{476} \left (167 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-2 \int \frac {5 x^2+7}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{476} \left (167 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-\int \frac {5 x^2+7}{\sqrt {2-x^2} \sqrt {x^2+1}}dx\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{476} \left (-2 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx-5 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx+167 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{476} \left (-5 \int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx+167 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-2 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{476} \left (167 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}dx-2 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-5 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 1536

\(\displaystyle \frac {1}{476} \left (334 \int \frac {1}{2 \sqrt {2-x^2} \sqrt {x^2+1} \left (5 x^2+7\right )}dx-2 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-5 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{476} \left (167 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1} \left (5 x^2+7\right )}dx-2 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-5 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {1}{476} \left (-2 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )-5 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {167}{7} \operatorname {EllipticPi}\left (-\frac {10}{7},\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )-\frac {25 x \sqrt {-x^4+x^2+2}}{476 \left (5 x^2+7\right )}\)

input

Int[1/((7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]),x]
 

output

(-25*x*Sqrt[2 + x^2 - x^4])/(476*(7 + 5*x^2)) + (-5*EllipticE[ArcSin[x/Sqr 
t[2]], -2] - 2*EllipticF[ArcSin[x/Sqrt[2]], -2] + (167*EllipticPi[-10/7, A 
rcSin[x/Sqrt[2]], -2])/7)/476
 

3.4.37.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

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])
 

rule 327

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]
 

rule 399

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])))))
 

rule 412

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])
 

rule 1494

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]
 

rule 1536

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]
 

rule 1551

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]
 

rule 2234

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]
 

3.4.37.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (71 ) = 142\).

Time = 2.95 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.23

method	result	size
default	\(-\frac {25 x \sqrt {-x^{4}+x^{2}+2}}{476 \left (5 x^{2}+7\right )}-\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{476 \sqrt {-x^{4}+x^{2}+2}}-\frac {5 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{952 \sqrt {-x^{4}+x^{2}+2}}+\frac {167 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{3332 \sqrt {-x^{4}+x^{2}+2}}\)	\(165\)
elliptic	\(-\frac {25 x \sqrt {-x^{4}+x^{2}+2}}{476 \left (5 x^{2}+7\right )}-\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{476 \sqrt {-x^{4}+x^{2}+2}}-\frac {5 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{952 \sqrt {-x^{4}+x^{2}+2}}+\frac {167 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{3332 \sqrt {-x^{4}+x^{2}+2}}\)	\(165\)
risch	\(\frac {25 \left (x^{4}-x^{2}-2\right ) x}{476 \left (5 x^{2}+7\right ) \sqrt {-x^{4}+x^{2}+2}}-\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{136 \sqrt {-x^{4}+x^{2}+2}}+\frac {5 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{952 \sqrt {-x^{4}+x^{2}+2}}+\frac {167 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{3332 \sqrt {-x^{4}+x^{2}+2}}\)	\(191\)

input

int(1/(5*x^2+7)^2/(-x^4+x^2+2)^(1/2),x,method=_RETURNVERBOSE)
 

output

-25/476*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)-1/476*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))-5/952*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))+167/3332*2^(1/2)*(1-1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+ 
2)^(1/2)*EllipticPi(1/2*x*2^(1/2),-10/7,I*2^(1/2))
 

3.4.37.5 Fricas [F]

\[ \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]

input

integrate(1/(5*x^2+7)^2/(-x^4+x^2+2)^(1/2),x, algorithm="fricas")
 

output

integral(-sqrt(-x^4 + x^2 + 2)/(25*x^8 + 45*x^6 - 71*x^4 - 189*x^2 - 98), 
x)
 

3.4.37.6 Sympy [F]

\[ \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx=\int \frac {1}{\sqrt {- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{2}}\, dx \]

input

integrate(1/(5*x**2+7)**2/(-x**4+x**2+2)**(1/2),x)
 

output

Integral(1/(sqrt(-(x**2 - 2)*(x**2 + 1))*(5*x**2 + 7)**2), x)
 

3.4.37.7 Maxima [F]

\[ \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]

input

integrate(1/(5*x^2+7)^2/(-x^4+x^2+2)^(1/2),x, algorithm="maxima")
 

output

integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2), x)
 

3.4.37.8 Giac [F]

\[ \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]

input

integrate(1/(5*x^2+7)^2/(-x^4+x^2+2)^(1/2),x, algorithm="giac")
 

output

integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)^2), x)
 

3.4.37.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx=\int \frac {1}{{\left (5\,x^2+7\right )}^2\,\sqrt {-x^4+x^2+2}} \,d x \]

input

int(1/((5*x^2 + 7)^2*(x^2 - x^4 + 2)^(1/2)),x)
 

output

int(1/((5*x^2 + 7)^2*(x^2 - x^4 + 2)^(1/2)), x)