\(\int \frac {\sqrt {2+3 x^2+x^4}}{(7+5 x^2)^3} \, dx\) [292]

Optimal result

Integrand size = 24, antiderivative size = 237 \[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (7+5 x^2\right )^3} \, dx=-\frac {11 x \left (2+x^2\right )}{11760 \sqrt {2+3 x^2+x^4}}+\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {11 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{5880 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {81 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{7840 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {1201 \left (2+x^2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{164640 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}} \]

[Out]

-11/11760*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-1201/329280*(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)+11/11760*(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)+81/15680*(
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)+1/28*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)^2+11/2352*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1242, 1237, 1710, 1730, 1203, 1113, 1149, 1228, 1470, 553} \[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (7+5 x^2\right )^3} \, dx=\frac {81 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{7840 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {11 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{5880 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {1201 \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{164640 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}+\frac {11 \sqrt {x^4+3 x^2+2} x}{2352 \left (5 x^2+7\right )}+\frac {\sqrt {x^4+3 x^2+2} x}{28 \left (5 x^2+7\right )^2}-\frac {11 \left (x^2+2\right ) x}{11760 \sqrt {x^4+3 x^2+2}} \]

[In]

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

[Out]

(-11*x*(2 + x^2))/(11760*Sqrt[2 + 3*x^2 + x^4]) + (x*Sqrt[2 + 3*x^2 + x^4])/(28*(7 + 5*x^2)^2) + (11*x*Sqrt[2
+ 3*x^2 + x^4])/(2352*(7 + 5*x^2)) + (11*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(5880*
Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + (81*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(7840*Sqrt
[2]*Sqrt[2 + 3*x^2 + x^4]) - (1201*(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1/2])/(164640*Sqrt[2]*Sqrt[(2 + x^2)/(
1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

Rule 553

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)), Ar
cTan[Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 1113

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]))*Elli
pticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSq
rtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1149

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)])] /; Fre
eQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1203

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[d, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[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 1228

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[2*(c/(2*c*d - e*(b - q))), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[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] &&  !LtQ[c, 0]

Rule 1237

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] + Dist[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] && ILtQ[q, -1]

Rule 1242

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{aa, bb, cc}, In
t[ExpandIntegrand[1/Sqrt[aa + bb*x^2 + cc*x^4], (d + e*x^2)^q*(aa + bb*x^2 + cc*x^4)^(p + 1/2), x] /. {aa -> a
, bb -> b, cc -> c}, 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, 0] && IntegerQ[p + 1/2]

Rule 1470

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^
(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPar
t[p]), Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{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 1710

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 = Coeff[P4x, x, 4]}, Simp[(-(C*d^2 - B*d*e + A*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] + Dist[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]
&& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, -1]

Rule 1730

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]}, Dist[-(e^2)^(-1), Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^
2 + c*x^4], x], x] + Dist[(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[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Ne
Q[c*d^2 - a*e^2, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {6}{25 \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}}+\frac {1}{25 \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}}+\frac {1}{25 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}}\right ) \, dx \\ & = \frac {1}{25} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx+\frac {1}{25} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx-\frac {6}{25} \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}-\frac {x \sqrt {2+3 x^2+x^4}}{84 \left (7+5 x^2\right )}+\frac {\int \frac {62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{2100}-\frac {1}{700} \int \frac {74-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx+\frac {1}{50} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {1}{20} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{50 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {\int \frac {2838+2310 x^2+975 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{58800}-\frac {\int \frac {-175-125 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{52500}+\frac {13 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{2100}-\frac {\left (\sqrt {1+\frac {x^2}{2}} \sqrt {2+2 x^2}\right ) \int \frac {\sqrt {2+2 x^2}}{\sqrt {1+\frac {x^2}{2}} \left (7+5 x^2\right )} \, dx}{20 \sqrt {2+3 x^2+x^4}} \\ & = \frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{50 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {\left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{70 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {\int \frac {-4725-4875 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{1470000}+\frac {1}{420} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {13 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{4200}+\frac {1}{300} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {13 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{1680}-\frac {101 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{3920} \\ & = \frac {x \left (2+x^2\right )}{420 \sqrt {2+3 x^2+x^4}}+\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}-\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{210 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {37 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1400 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {\left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{70 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}-\frac {9 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{2800}-\frac {13 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{3920}-\frac {101 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{7840}+\frac {101 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{3136}-\frac {\left (13 \sqrt {1+\frac {x^2}{2}} \sqrt {2+2 x^2}\right ) \int \frac {\sqrt {2+2 x^2}}{\sqrt {1+\frac {x^2}{2}} \left (7+5 x^2\right )} \, dx}{1680 \sqrt {2+3 x^2+x^4}} \\ & = -\frac {11 x \left (2+x^2\right )}{11760 \sqrt {2+3 x^2+x^4}}+\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {11 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5880 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {81 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{7840 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {97 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{5880 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {\left (101 \sqrt {1+\frac {x^2}{2}} \sqrt {2+2 x^2}\right ) \int \frac {\sqrt {2+2 x^2}}{\sqrt {1+\frac {x^2}{2}} \left (7+5 x^2\right )} \, dx}{3136 \sqrt {2+3 x^2+x^4}} \\ & = -\frac {11 x \left (2+x^2\right )}{11760 \sqrt {2+3 x^2+x^4}}+\frac {x \sqrt {2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac {11 x \sqrt {2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac {11 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5880 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {81 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{7840 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {1201 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{164640 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.35 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {2+3 x^2+x^4}}{\left (7+5 x^2\right )^3} \, dx=\frac {\frac {14700 x \left (2+3 x^2+x^4\right )}{\left (7+5 x^2\right )^2}+\frac {1925 x \left (2+3 x^2+x^4\right )}{7+5 x^2}+385 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-434 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )-1201 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{411600 \sqrt {2+3 x^2+x^4}} \]

[In]

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

[Out]

((14700*x*(2 + 3*x^2 + x^4))/(7 + 5*x^2)^2 + (1925*x*(2 + 3*x^2 + x^4))/(7 + 5*x^2) + (385*I)*Sqrt[1 + x^2]*Sq
rt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (434*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqr
t[2]], 2] - (1201*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(411600*Sqrt[2 + 3
*x^2 + x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.41 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.77

method	result	size
risch	\(\frac {\sqrt {x^{4}+3 x^{2}+2}\, x \left (55 x^{2}+161\right )}{2352 \left (5 x^{2}+7\right )^{2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{16800 \sqrt {x^{4}+3 x^{2}+2}}-\frac {11 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{23520 \sqrt {x^{4}+3 x^{2}+2}}-\frac {1201 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{411600 \sqrt {x^{4}+3 x^{2}+2}}\)	\(183\)
default	\(\frac {x \sqrt {x^{4}+3 x^{2}+2}}{28 \left (5 x^{2}+7\right )^{2}}+\frac {11 x \sqrt {x^{4}+3 x^{2}+2}}{2352 \left (5 x^{2}+7\right )}-\frac {31 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{58800 \sqrt {x^{4}+3 x^{2}+2}}+\frac {11 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{23520 \sqrt {x^{4}+3 x^{2}+2}}-\frac {1201 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{411600 \sqrt {x^{4}+3 x^{2}+2}}\)	\(186\)
elliptic	\(\frac {x \sqrt {x^{4}+3 x^{2}+2}}{28 \left (5 x^{2}+7\right )^{2}}+\frac {11 x \sqrt {x^{4}+3 x^{2}+2}}{2352 \left (5 x^{2}+7\right )}-\frac {31 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{58800 \sqrt {x^{4}+3 x^{2}+2}}+\frac {11 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{23520 \sqrt {x^{4}+3 x^{2}+2}}-\frac {1201 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{411600 \sqrt {x^{4}+3 x^{2}+2}}\)	\(186\)

[In]

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

[Out]

1/2352*(x^4+3*x^2+2)^(1/2)*x*(55*x^2+161)/(5*x^2+7)^2-1/16800*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))-11/23520*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)))-1201/411600*I*2^(1/2)*(1+1/2*x^2
)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticPi(1/2*I*2^(1/2)*x,10/7,2^(1/2))

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(x^4 + 3*x^2 + 2)/(125*x^6 + 525*x^4 + 735*x^2 + 343), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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