3.4.6 \(\int \frac {1}{(7+5 x^2)^3 \sqrt {2+3 x^2+x^4}} \, dx\) [306]

3.4.6.1 Optimal result

Integrand size = 24, antiderivative size = 237 \[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx=\frac {65 x \left (2+x^2\right )}{4704 \sqrt {2+3 x^2+x^4}}-\frac {25 x \sqrt {2+3 x^2+x^4}}{168 \left (7+5 x^2\right )^2}-\frac {325 x \sqrt {2+3 x^2+x^4}}{4704 \left (7+5 x^2\right )}-\frac {65 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{2352 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {631 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{9408 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {2525 \left (2+x^2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{65856 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}} \]

output

65/4704*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-2525/131712*(x^2+2)*(1/(x^2+1))^(1/2 
)*(x^2+1)^(1/2)*EllipticPi(x/(x^2+1)^(1/2),2/7,1/2*2^(1/2))*2^(1/2)/((x^2+ 
2)/(x^2+1))^(1/2)/(x^4+3*x^2+2)^(1/2)-65/4704*(x^2+1)^(3/2)*(1/(x^2+1))^(1 
/2)*EllipticE(x/(x^2+1)^(1/2),1/2*2^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2) 
/(x^4+3*x^2+2)^(1/2)+631/18816*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticF(x 
/(x^2+1)^(1/2),1/2*2^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2)/(x^4+3*x^2+2)^ 
(1/2)-25/168*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)^2-325/4704*x*(x^4+3*x^2+2)^(1 
/2)/(5*x^2+7)
 

3.4.6.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.38 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx=\frac {-175 x \left (238+487 x^2+314 x^4+65 x^6\right )-455 i \sqrt {1+x^2} \sqrt {2+x^2} \left (7+5 x^2\right )^2 E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+14 i \sqrt {1+x^2} \sqrt {2+x^2} \left (7+5 x^2\right )^2 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )-505 i \sqrt {1+x^2} \sqrt {2+x^2} \left (7+5 x^2\right )^2 \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{32928 \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \]

input

Integrate[1/((7 + 5*x^2)^3*Sqrt[2 + 3*x^2 + x^4]),x]
 

output

(-175*x*(238 + 487*x^2 + 314*x^4 + 65*x^6) - (455*I)*Sqrt[1 + x^2]*Sqrt[2 
+ x^2]*(7 + 5*x^2)^2*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + (14*I)*Sqrt[1 + 
x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)^2*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - (505 
*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)^2*EllipticPi[10/7, I*ArcSinh[x 
/Sqrt[2]], 2])/(32928*(7 + 5*x^2)^2*Sqrt[2 + 3*x^2 + x^4])
 

3.4.6.3 Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {1551, 2210, 27, 2234, 27, 1503, 1412, 1455, 1538, 27, 1412, 1786, 414}

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.

\(\displaystyle \int \frac {1}{\left (5 x^2+7\right )^3 \sqrt {x^4+3 x^2+2}} \, dx\)

\(\Big \downarrow \) 1551

\(\displaystyle \frac {1}{168} \int \frac {-25 x^4-10 x^2+74}{\left (5 x^2+7\right )^2 \sqrt {x^4+3 x^2+2}}dx-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 2210

\(\displaystyle \frac {1}{168} \left (\frac {1}{84} \int \frac {3 \left (325 x^4+770 x^2+946\right )}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \int \frac {325 x^4+770 x^2+946}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 2234

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (505 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx-\frac {1}{25} \int -\frac {25 \left (65 x^2+63\right )}{\sqrt {x^4+3 x^2+2}}dx\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (505 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx+\int \frac {65 x^2+63}{\sqrt {x^4+3 x^2+2}}dx\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (63 \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx+65 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx+505 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (65 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx+505 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx+\frac {63 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (505 \int \frac {1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx+\frac {63 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+65 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 1538

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (505 \left (\frac {1}{2} \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx-\frac {5}{4} \int \frac {2 \left (x^2+1\right )}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx\right )+\frac {63 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+65 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (505 \left (\frac {1}{2} \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx-\frac {5}{2} \int \frac {x^2+1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx\right )+\frac {63 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+65 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (505 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {5}{2} \int \frac {x^2+1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx\right )+\frac {63 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+65 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 1786

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (505 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {5 \sqrt {x^2+1} \sqrt {x^2+2} \int \frac {\sqrt {x^2+1}}{\sqrt {x^2+2} \left (5 x^2+7\right )}dx}{2 \sqrt {x^4+3 x^2+2}}\right )+\frac {63 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+65 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 414

\(\displaystyle \frac {1}{168} \left (\frac {1}{28} \left (\frac {63 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+65 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )+505 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {5 \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{14 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}\right )\right )-\frac {325 x \sqrt {x^4+3 x^2+2}}{28 \left (5 x^2+7\right )}\right )-\frac {25 x \sqrt {x^4+3 x^2+2}}{168 \left (5 x^2+7\right )^2}\)

input

Int[1/((7 + 5*x^2)^3*Sqrt[2 + 3*x^2 + x^4]),x]
 

output

(-25*x*Sqrt[2 + 3*x^2 + x^4])/(168*(7 + 5*x^2)^2) + ((-325*x*Sqrt[2 + 3*x^ 
2 + x^4])/(28*(7 + 5*x^2)) + (65*((x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] - (S 
qrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/Sqrt 
[2 + 3*x^2 + x^4]) + (63*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[Arc 
Tan[x], 1/2])/(Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + 505*(((1 + x^2)*Sqrt[(2 + 
x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(2*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4 
]) - (5*(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1/2])/(14*Sqrt[2]*Sqrt[(2 + x 
^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])))/28)/168
 

3.4.6.3.1 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 414

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) 
^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* 
Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ 
Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ 
[d/c]
 

rule 1412

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q)*x^2)/(2*a + 
(b + q)*x^2)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF 
[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && 
!(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[ 
{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1455

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[b^2 - 4*a*c, 2]}, Simp[x*((b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 
])), x] - Simp[Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q 
)*x^2)/(2*a + (b + q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan 
[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] &&  !(PosQ[ 
(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, 
c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1503

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[d   Int[1/Sqrt[a + b*x^2 + c*x^4] 
, x], x] + Simp[e   Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) 
/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1538

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*(c/(2*c*d - e*(b - q)))   I 
nt[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[e/(2*c*d - e*(b - q))   Int[(b 
- q + 2*c*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), 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 1786

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + ( 
b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(a + b*x^n + c*x 
^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]) 
   Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c 
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p]
 

rule 2210

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 
]
 

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]
 

3.4.6.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.48 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {25 \sqrt {x^{4}+3 x^{2}+2}\, x \left (65 x^{2}+119\right )}{4704 \left (5 x^{2}+7\right )^{2}}-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{448 \sqrt {x^{4}+3 x^{2}+2}}+\frac {65 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{9408 \sqrt {x^{4}+3 x^{2}+2}}-\frac {505 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{32928 \sqrt {x^{4}+3 x^{2}+2}}\)	\(183\)
default	\(-\frac {25 x \sqrt {x^{4}+3 x^{2}+2}}{168 \left (5 x^{2}+7\right )^{2}}-\frac {325 x \sqrt {x^{4}+3 x^{2}+2}}{4704 \left (5 x^{2}+7\right )}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{4704 \sqrt {x^{4}+3 x^{2}+2}}-\frac {65 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{9408 \sqrt {x^{4}+3 x^{2}+2}}-\frac {505 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{32928 \sqrt {x^{4}+3 x^{2}+2}}\)	\(186\)
elliptic	\(-\frac {25 x \sqrt {x^{4}+3 x^{2}+2}}{168 \left (5 x^{2}+7\right )^{2}}-\frac {325 x \sqrt {x^{4}+3 x^{2}+2}}{4704 \left (5 x^{2}+7\right )}+\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{4704 \sqrt {x^{4}+3 x^{2}+2}}-\frac {65 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{9408 \sqrt {x^{4}+3 x^{2}+2}}-\frac {505 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{32928 \sqrt {x^{4}+3 x^{2}+2}}\)	\(186\)

input

int(1/(5*x^2+7)^3/(x^4+3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
 

output

-25/4704*(x^4+3*x^2+2)^(1/2)*x*(65*x^2+119)/(5*x^2+7)^2-3/448*I*2^(1/2)*(2 
*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticF(1/2*I*2^(1/2)*x, 
2^(1/2))+65/9408*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/ 
2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(1/2*I*2^(1/2)*x,2^(1/2))) 
-505/32928*I*2^(1/2)*(1+1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*E 
llipticPi(1/2*I*2^(1/2)*x,10/7,2^(1/2))
 

3.4.6.5 Fricas [F]

\[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]

input

integrate(1/(5*x^2+7)^3/(x^4+3*x^2+2)^(1/2),x, algorithm="fricas")
 

output

integral(sqrt(x^4 + 3*x^2 + 2)/(125*x^10 + 900*x^8 + 2560*x^6 + 3598*x^4 + 
 2499*x^2 + 686), x)
 

3.4.6.6 Sympy [F]

\[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx=\int \frac {1}{\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )^{3}}\, dx \]

input

integrate(1/(5*x**2+7)**3/(x**4+3*x**2+2)**(1/2),x)
 

output

Integral(1/(sqrt((x**2 + 1)*(x**2 + 2))*(5*x**2 + 7)**3), x)
 

3.4.6.7 Maxima [F]

\[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]

input

integrate(1/(5*x^2+7)^3/(x^4+3*x^2+2)^(1/2),x, algorithm="maxima")
 

output

integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7)^3), x)
 

3.4.6.8 Giac [F]

\[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]

input

integrate(1/(5*x^2+7)^3/(x^4+3*x^2+2)^(1/2),x, algorithm="giac")
 

output

integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7)^3), x)
 

3.4.6.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx=\int \frac {1}{{\left (5\,x^2+7\right )}^3\,\sqrt {x^4+3\,x^2+2}} \,d x \]

input

int(1/((5*x^2 + 7)^3*(3*x^2 + x^4 + 2)^(1/2)),x)
 

output

int(1/((5*x^2 + 7)^3*(3*x^2 + x^4 + 2)^(1/2)), x)