3.4.70 \(\int \frac {1}{(7+5 x^2)^3 \sqrt {4+3 x^2+x^4}} \, dx\) [370]

3.4.70.1 Optimal result

Integrand size = 24, antiderivative size = 314 \[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {4+3 x^2+x^4}} \, dx=-\frac {555 x \sqrt {4+3 x^2+x^4}}{758912 \left (2+x^2\right )}+\frac {25 x \sqrt {4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac {2775 x \sqrt {4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}-\frac {3285 \sqrt {\frac {5}{77}} \arctan \left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{3035648}+\frac {555 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{379456 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {\left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{8624 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {18615 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{21249536 \sqrt {2} \sqrt {4+3 x^2+x^4}} \]

output

-3285/233744896*arctan(2/35*x*385^(1/2)/(x^4+3*x^2+4)^(1/2))*385^(1/2)-555 
/758912*x*(x^4+3*x^2+4)^(1/2)/(x^2+2)+25/1232*x*(x^4+3*x^2+4)^(1/2)/(5*x^2 
+7)^2+2775/758912*x*(x^4+3*x^2+4)^(1/2)/(5*x^2+7)+555/758912*(x^2+2)*(cos( 
2*arctan(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*EllipticE(s 
in(2*arctan(1/2*x*2^(1/2))),1/4*2^(1/2))*2^(1/2)*((x^4+3*x^2+4)/(x^2+2)^2) 
^(1/2)/(x^4+3*x^2+4)^(1/2)-1/17248*(x^2+2)*(cos(2*arctan(1/2*x*2^(1/2)))^2 
)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*EllipticF(sin(2*arctan(1/2*x*2^(1/2)) 
),1/4*2^(1/2))*((x^4+3*x^2+4)/(x^2+2)^2)^(1/2)*2^(1/2)/(x^4+3*x^2+4)^(1/2) 
-18615/42499072*(x^2+2)*(cos(2*arctan(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arcta 
n(1/2*x*2^(1/2)))*EllipticPi(sin(2*arctan(1/2*x*2^(1/2))),-9/280,1/4*2^(1/ 
2))*((x^4+3*x^2+4)/(x^2+2)^2)^(1/2)*2^(1/2)/(x^4+3*x^2+4)^(1/2)
 

3.4.70.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.72 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {4+3 x^2+x^4}} \, dx=\frac {\frac {700 x \left (1393+555 x^2\right ) \left (4+3 x^2+x^4\right )}{\left (7+5 x^2\right )^2}+i \sqrt {6+2 i \sqrt {7}} \sqrt {1-\frac {2 i x^2}{-3 i+\sqrt {7}}} \sqrt {1+\frac {2 i x^2}{3 i+\sqrt {7}}} \left (3885 \left (3-i \sqrt {7}\right ) E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+\left (-9401+3885 i \sqrt {7}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+6570 \operatorname {EllipticPi}\left (\frac {5}{14} \left (3+i \sqrt {7}\right ),i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )\right )}{21249536 \sqrt {4+3 x^2+x^4}} \]

input

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

output

((700*x*(1393 + 555*x^2)*(4 + 3*x^2 + x^4))/(7 + 5*x^2)^2 + I*Sqrt[6 + (2* 
I)*Sqrt[7]]*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3 
*I + Sqrt[7])]*(3885*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I 
 + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + (-9401 + (3885*I)*Sqrt 
[7])*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7]) 
/(3*I + Sqrt[7])] + 6570*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt 
[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])]))/(21249536 
*Sqrt[4 + 3*x^2 + x^4])
 

3.4.70.3 Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1551, 25, 2210, 2232, 27, 1509, 2226, 27, 1416, 2220}

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.

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

\(\Big \downarrow \) 1551

\(\displaystyle \frac {25 x \sqrt {x^4+3 x^2+4}}{1232 \left (5 x^2+7\right )^2}-\frac {\int -\frac {25 x^4+10 x^2+76}{\left (5 x^2+7\right )^2 \sqrt {x^4+3 x^2+4}}dx}{1232}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {25 x^4+10 x^2+76}{\left (5 x^2+7\right )^2 \sqrt {x^4+3 x^2+4}}dx}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 2210

\(\displaystyle \frac {\frac {2775 x \sqrt {x^4+3 x^2+4}}{616 \left (5 x^2+7\right )}-\frac {1}{616} \int \frac {2775 x^4+4690 x^2+4412}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 2232

\(\displaystyle \frac {\frac {1}{616} \left (1110 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx-\frac {1}{5} \int \frac {5 \left (6355 x^2+12182\right )}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\right )+\frac {2775 \sqrt {x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{616} \left (555 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx-\int \frac {6355 x^2+12182}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\right )+\frac {2775 \sqrt {x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {1}{616} \left (555 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )-\int \frac {6355 x^2+12182}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\right )+\frac {2775 \sqrt {x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 2226

\(\displaystyle \frac {\frac {1}{616} \left (-176 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-10950 \int \frac {x^2+2}{2 \left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+555 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+\frac {2775 \sqrt {x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{616} \left (-176 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-5475 \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+555 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+\frac {2775 \sqrt {x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {1}{616} \left (-5475 \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx-\frac {44 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}+555 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+\frac {2775 \sqrt {x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

\(\Big \downarrow \) 2220

\(\displaystyle \frac {\frac {1}{616} \left (-\frac {44 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}+555 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )-5475 \left (\frac {3 \arctan \left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{4 \sqrt {385}}+\frac {17 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{140 \sqrt {2} \sqrt {x^4+3 x^2+4}}\right )\right )+\frac {2775 \sqrt {x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}}{1232}+\frac {25 \sqrt {x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}\)

input

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

output

(25*x*Sqrt[4 + 3*x^2 + x^4])/(1232*(7 + 5*x^2)^2) + ((2775*x*Sqrt[4 + 3*x^ 
2 + x^4])/(616*(7 + 5*x^2)) + (555*(-((x*Sqrt[4 + 3*x^2 + x^4])/(2 + x^2)) 
 + (Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcT 
an[x/Sqrt[2]], 1/8])/Sqrt[4 + 3*x^2 + x^4]) - (44*Sqrt[2]*(2 + x^2)*Sqrt[( 
4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/Sqrt[4 
+ 3*x^2 + x^4] - 5475*((3*ArcTan[(2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]]) 
/(4*Sqrt[385]) + (17*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*Ellipti 
cPi[-9/280, 2*ArcTan[x/Sqrt[2]], 1/8])/(140*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) 
))/616)/1232
 

3.4.70.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1416

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

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

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 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 2220

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a 
+ b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*El 
lipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
 /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] & 
& EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[-b + c*(d/e) + a*(e/d)]
 

rule 2226

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(A*(c*d + a*e*q) 
- a*B*(e + d*q))/(c*d^2 - a*e^2)   Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + 
Simp[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2))   Int[(1 + q*x^2)/((d + e*x^ 
2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && 
NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
 

rule 2232

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x 
, 2], C = Coeff[P4x, x, 4]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + b*x^2 
 + c*x^4], x], x] + Simp[1/(c*e)   Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - 
 a*e*q))*x^2)/((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] && PosQ[c/a] 
&&  !GtQ[b^2 - 4*a*c, 0]
 

3.4.70.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.10

method	result	size
risch	\(\frac {25 \sqrt {x^{4}+3 x^{2}+4}\, x \left (555 x^{2}+1393\right )}{758912 \left (5 x^{2}+7\right )^{2}}-\frac {23 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{27104 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}+\frac {555 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{23716 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}-\frac {3285 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{5312384 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\)	\(346\)
default	\(\frac {25 x \sqrt {x^{4}+3 x^{2}+4}}{1232 \left (5 x^{2}+7\right )^{2}}+\frac {2775 x \sqrt {x^{4}+3 x^{2}+4}}{758912 \left (5 x^{2}+7\right )}-\frac {23 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{27104 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}+\frac {555 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{23716 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}-\frac {555 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{23716 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}-\frac {3285 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{5312384 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\)	\(434\)
elliptic	\(\frac {25 x \sqrt {x^{4}+3 x^{2}+4}}{1232 \left (5 x^{2}+7\right )^{2}}+\frac {2775 x \sqrt {x^{4}+3 x^{2}+4}}{758912 \left (5 x^{2}+7\right )}-\frac {23 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{27104 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}+\frac {555 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{23716 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}-\frac {555 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{23716 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}-\frac {3285 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{5312384 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\)	\(434\)

input

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

output

25/758912*(x^4+3*x^2+4)^(1/2)*x*(555*x^2+1393)/(5*x^2+7)^2-23/27104/(-6+2* 
I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2) 
)*x^2)^(1/2)/(x^4+3*x^2+4)^(1/2)*EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/ 
4*(2+6*I*7^(1/2))^(1/2))+555/23716/(-6+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7 
^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2+4)^(1/2)/ 
(3+I*7^(1/2))*(EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^ 
(1/2))-EllipticE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2)))- 
3285/5312384/(-3/8+1/8*I*7^(1/2))^(1/2)*(1+3/8*x^2-1/8*I*x^2*7^(1/2))^(1/2 
)*(1+3/8*x^2+1/8*I*x^2*7^(1/2))^(1/2)/(x^4+3*x^2+4)^(1/2)*EllipticPi((-3/8 
+1/8*I*7^(1/2))^(1/2)*x,-5/7/(-3/8+1/8*I*7^(1/2)),(-3/8-1/8*I*7^(1/2))^(1/ 
2)/(-3/8+1/8*I*7^(1/2))^(1/2))
 

3.4.70.5 Fricas [F]

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

input

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

output

integral(sqrt(x^4 + 3*x^2 + 4)/(125*x^10 + 900*x^8 + 2810*x^6 + 4648*x^4 + 
 3969*x^2 + 1372), x)
 

3.4.70.6 Sympy [F]

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

input

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

output

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

3.4.70.7 Maxima [F]

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

input

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

output

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

3.4.70.8 Giac [F]

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

input

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

output

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

3.4.70.9 Mupad [F(-1)]

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

input

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

output

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