3.315 \(\int \frac{1}{(7+5 x^2)^3 (2+3 x^2+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=263 \[ -\frac{49907 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{56448 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{41875 \sqrt{x^4+3 x^2+2} x}{84672 \left (5 x^2+7\right )}+\frac{625 \sqrt{x^4+3 x^2+2} x}{1008 \left (5 x^2+7\right )^2}-\frac{5797 \left (x^2+2\right ) x}{28224 \sqrt{x^4+3 x^2+2}}+\frac{\left (23 x^2+50\right ) x}{216 \sqrt{x^4+3 x^2+2}}+\frac{5797 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{14112 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{192625 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{395136 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

[Out]

(-5797*x*(2 + x^2))/(28224*Sqrt[2 + 3*x^2 + x^4]) + (x*(50 + 23*x^2))/(216*Sqrt[2 + 3*x^2 + x^4]) + (625*x*Sqr
t[2 + 3*x^2 + x^4])/(1008*(7 + 5*x^2)^2) + (41875*x*Sqrt[2 + 3*x^2 + x^4])/(84672*(7 + 5*x^2)) + (5797*(1 + x^
2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(14112*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) - (49907*(1 + x^
2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(56448*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + (192625*(2 + x
^2)*EllipticPi[2/7, ArcTan[x], 1/2])/(395136*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.760469, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {1228, 1178, 1189, 1099, 1135, 1223, 1696, 1716, 1214, 1456, 539} \[ \frac{41875 \sqrt{x^4+3 x^2+2} x}{84672 \left (5 x^2+7\right )}+\frac{625 \sqrt{x^4+3 x^2+2} x}{1008 \left (5 x^2+7\right )^2}-\frac{5797 \left (x^2+2\right ) x}{28224 \sqrt{x^4+3 x^2+2}}+\frac{\left (23 x^2+50\right ) x}{216 \sqrt{x^4+3 x^2+2}}-\frac{49907 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{56448 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{5797 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{14112 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{192625 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{395136 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5797*x*(2 + x^2))/(28224*Sqrt[2 + 3*x^2 + x^4]) + (x*(50 + 23*x^2))/(216*Sqrt[2 + 3*x^2 + x^4]) + (625*x*Sqr
t[2 + 3*x^2 + x^4])/(1008*(7 + 5*x^2)^2) + (41875*x*Sqrt[2 + 3*x^2 + x^4])/(84672*(7 + 5*x^2)) + (5797*(1 + x^
2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(14112*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) - (49907*(1 + x^
2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(56448*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + (192625*(2 + x
^2)*EllipticPi[2/7, ArcTan[x], 1/2])/(395136*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

Rule 1228

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 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), 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] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1189

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 1099

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)]*EllipticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)]
)/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]), 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 1135

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)]*EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)])/(2*c*Sqrt[a + b*x^2
+ c*x^4]), 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 1223

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)*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])/Sqrt[a + b*x^2 + c*x^4], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b
^2 - 4*a*c, 0] && ILtQ[q, -1]

Rule 1696

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)*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
])/Sqrt[a + b*x^2 + c*x^4], 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 1716

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]

Rule 1214

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 1456

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*x^n)/e)^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 539

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

Rubi steps

\begin{align*} \int \frac{1}{\left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2}} \, dx &=\int \left (-\frac{-62-35 x^2}{216 \left (2+3 x^2+x^4\right )^{3/2}}-\frac{25}{6 \left (7+5 x^2\right )^3 \sqrt{2+3 x^2+x^4}}-\frac{25}{36 \left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}}-\frac{175}{216 \left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}}\right ) \, dx\\ &=-\left (\frac{1}{216} \int \frac{-62-35 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx\right )-\frac{25}{36} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx-\frac{175}{216} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx-\frac{25}{6} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{x \left (50+23 x^2\right )}{216 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac{625 x \sqrt{2+3 x^2+x^4}}{3024 \left (7+5 x^2\right )}+\frac{1}{432} \int \frac{-38-46 x^2}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{25 \int \frac{62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{3024}-\frac{25 \int \frac{74-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx}{1008}-\frac{175}{432} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{875}{864} \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{x \left (50+23 x^2\right )}{216 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac{41875 x \sqrt{2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}-\frac{175 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{432 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{25 \int \frac{2838+2310 x^2+975 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{84672}+\frac{\int \frac{-175-125 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{3024}-\frac{19}{216} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{23}{216} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{325 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{3024}+\frac{\left (875 \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}{864 \sqrt{2+3 x^2+x^4}}\\ &=-\frac{23 x \left (2+x^2\right )}{216 \sqrt{2+3 x^2+x^4}}+\frac{x \left (50+23 x^2\right )}{216 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac{41875 x \sqrt{2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}+\frac{23 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{108 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{71 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{144 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{125 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{432 \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}{84672}-\frac{125 \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{3024}-\frac{325 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{6048}-\frac{25}{432} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{1625 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{12096}-\frac{12625 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{28224}\\ &=-\frac{149 x \left (2+x^2\right )}{1008 \sqrt{2+3 x^2+x^4}}+\frac{x \left (50+23 x^2\right )}{216 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac{41875 x \sqrt{2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}+\frac{149 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{504 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{1219 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2016 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{125 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{432 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}-\frac{25}{448} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{1625 \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{28224}-\frac{12625 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{56448}+\frac{63125 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{112896}+\frac{\left (1625 \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}{12096 \sqrt{2+3 x^2+x^4}}\\ &=-\frac{5797 x \left (2+x^2\right )}{28224 \sqrt{2+3 x^2+x^4}}+\frac{x \left (50+23 x^2\right )}{216 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac{41875 x \sqrt{2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}+\frac{5797 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{14112 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{49907 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{56448 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{4625 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{14112 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}+\frac{\left (63125 \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}{112896 \sqrt{2+3 x^2+x^4}}\\ &=-\frac{5797 x \left (2+x^2\right )}{28224 \sqrt{2+3 x^2+x^4}}+\frac{x \left (50+23 x^2\right )}{216 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{1008 \left (7+5 x^2\right )^2}+\frac{41875 x \sqrt{2+3 x^2+x^4}}{84672 \left (7+5 x^2\right )}+\frac{5797 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{14112 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{49907 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{56448 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{192625 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{395136 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.488673, size = 159, normalized size = 0.6 \[ \frac{-742 i \sqrt{x^2+1} \sqrt{x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+\frac{7 x \left (144925 x^6+698290 x^4+1089803 x^2+550550\right )}{\left (5 x^2+7\right )^2}+40579 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+38525 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )}{197568 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*x*(550550 + 1089803*x^2 + 698290*x^4 + 144925*x^6))/(7 + 5*x^2)^2 + (40579*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*
EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (742*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] +
(38525*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(197568*Sqrt[2 + 3*x^2 + x^4]
)

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Maple [C]  time = 0.023, size = 209, normalized size = 0.8 \begin{align*}{\frac{625\,x}{1008\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{41875\,x}{423360\,{x}^{2}+592704}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-2\,{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}} \left ( -{\frac{23\,{x}^{3}}{432}}-{\frac{25\,x}{216}} \right ) }-{{\frac{53\,i}{28224}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{5797\,i}{56448}}\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{38525\,i}{197568}}\sqrt{2}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},{\frac{10}{7}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/1008*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)^2+41875/84672*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)-2*(-23/432*x^3-25/216*x
)/(x^4+3*x^2+2)^(1/2)-53/28224*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*x*2
^(1/2),2^(1/2))+5797/56448*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticE(1/2*I*x*2^(1/
2),2^(1/2))+38525/197568*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*x*2^(1
/2),10/7,2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{4} + 3 \, x^{2} + 2}}{125 \, x^{14} + 1275 \, x^{12} + 5510 \, x^{10} + 13078 \, x^{8} + 18413 \, x^{6} + 15379 \, x^{4} + 7056 \, x^{2} + 1372}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^4 + 3*x^2 + 2)/(125*x^14 + 1275*x^12 + 5510*x^10 + 13078*x^8 + 18413*x^6 + 15379*x^4 + 7056*x^
2 + 1372), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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