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

Optimal. Leaf size=128 \[ \frac{60409 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )}{23110752}+\frac{645625 \sqrt{-x^4+x^2+2} x}{15407168 \left (5 x^2+7\right )}+\frac{625 \sqrt{-x^4+x^2+2} x}{32368 \left (5 x^2+7\right )^2}+\frac{\left (9830-4909 x^2\right ) x}{353736 \sqrt{-x^4+x^2+2}}+\frac{3086453 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{138664512}-\frac{6898575 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{107850176} \]

[Out]

(x*(9830 - 4909*x^2))/(353736*Sqrt[2 + x^2 - x^4]) + (625*x*Sqrt[2 + x^2 - x^4])/(32368*(7 + 5*x^2)^2) + (6456
25*x*Sqrt[2 + x^2 - x^4])/(15407168*(7 + 5*x^2)) + (3086453*EllipticE[ArcSin[x/Sqrt[2]], -2])/138664512 + (604
09*EllipticF[ArcSin[x/Sqrt[2]], -2])/23110752 - (6898575*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2])/107850176

________________________________________________________________________________________

Rubi [A]  time = 0.570137, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {1228, 1178, 1180, 524, 424, 419, 1223, 1696, 1716, 1212, 537} \[ \frac{645625 \sqrt{-x^4+x^2+2} x}{15407168 \left (5 x^2+7\right )}+\frac{625 \sqrt{-x^4+x^2+2} x}{32368 \left (5 x^2+7\right )^2}+\frac{\left (9830-4909 x^2\right ) x}{353736 \sqrt{-x^4+x^2+2}}+\frac{60409 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{23110752}+\frac{3086453 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{138664512}-\frac{6898575 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{107850176} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(9830 - 4909*x^2))/(353736*Sqrt[2 + x^2 - x^4]) + (625*x*Sqrt[2 + x^2 - x^4])/(32368*(7 + 5*x^2)^2) + (6456
25*x*Sqrt[2 + x^2 - x^4])/(15407168*(7 + 5*x^2)) + (3086453*EllipticE[ArcSin[x/Sqrt[2]], -2])/138664512 + (604
09*EllipticF[ArcSin[x/Sqrt[2]], -2])/23110752 - (6898575*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2])/107850176

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 1180

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[2*Sqrt[-c], Int[(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] && LtQ[c, 0]

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), 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 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 1212

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*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] && LtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{1}{\left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2}} \, dx &=\int \left (-\frac{-3278+1635 x^2}{39304 \left (2+x^2-x^4\right )^{3/2}}-\frac{25}{34 \left (7+5 x^2\right )^3 \sqrt{2+x^2-x^4}}-\frac{475}{1156 \left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4}}-\frac{8175}{39304 \left (7+5 x^2\right ) \sqrt{2+x^2-x^4}}\right ) \, dx\\ &=-\frac{\int \frac{-3278+1635 x^2}{\left (2+x^2-x^4\right )^{3/2}} \, dx}{39304}-\frac{8175 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{39304}-\frac{475 \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4}} \, dx}{1156}-\frac{25}{34} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{2+x^2-x^4}} \, dx\\ &=\frac{x \left (9830-4909 x^2\right )}{353736 \sqrt{2+x^2-x^4}}+\frac{625 x \sqrt{2+x^2-x^4}}{32368 \left (7+5 x^2\right )^2}+\frac{11875 x \sqrt{2+x^2-x^4}}{550256 \left (7+5 x^2\right )}+\frac{\int \frac{9842+9818 x^2}{\sqrt{2+x^2-x^4}} \, dx}{707472}-\frac{25 \int \frac{186-190 x^2+25 x^4}{\left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4}} \, dx}{32368}-\frac{475 \int \frac{118-70 x^2-25 x^4}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{550256}-\frac{8175 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2} \left (7+5 x^2\right )} \, dx}{19652}\\ &=\frac{x \left (9830-4909 x^2\right )}{353736 \sqrt{2+x^2-x^4}}+\frac{625 x \sqrt{2+x^2-x^4}}{32368 \left (7+5 x^2\right )^2}+\frac{645625 x \sqrt{2+x^2-x^4}}{15407168 \left (7+5 x^2\right )}-\frac{8175 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{275128}-\frac{25 \int \frac{37698-32690 x^2-12525 x^4}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{15407168}+\frac{\int \frac{9842+9818 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{353736}+\frac{19 \int \frac{175+125 x^2}{\sqrt{2+x^2-x^4}} \, dx}{550256}-\frac{79325 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{550256}\\ &=\frac{x \left (9830-4909 x^2\right )}{353736 \sqrt{2+x^2-x^4}}+\frac{625 x \sqrt{2+x^2-x^4}}{32368 \left (7+5 x^2\right )^2}+\frac{645625 x \sqrt{2+x^2-x^4}}{15407168 \left (7+5 x^2\right )}-\frac{8175 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{275128}+\frac{\int \frac{75775+62625 x^2}{\sqrt{2+x^2-x^4}} \, dx}{15407168}+\frac{\int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{14739}+\frac{19 \int \frac{175+125 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{275128}+\frac{4909 \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx}{353736}-\frac{1472875 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{15407168}-\frac{79325 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2} \left (7+5 x^2\right )} \, dx}{275128}\\ &=\frac{x \left (9830-4909 x^2\right )}{353736 \sqrt{2+x^2-x^4}}+\frac{625 x \sqrt{2+x^2-x^4}}{32368 \left (7+5 x^2\right )^2}+\frac{645625 x \sqrt{2+x^2-x^4}}{15407168 \left (7+5 x^2\right )}+\frac{4909 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{353736}+\frac{F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{29478}-\frac{193775 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3851792}+\frac{\int \frac{75775+62625 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{7703584}+\frac{475 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{137564}+\frac{2375 \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx}{550256}-\frac{1472875 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2} \left (7+5 x^2\right )} \, dx}{7703584}\\ &=\frac{x \left (9830-4909 x^2\right )}{353736 \sqrt{2+x^2-x^4}}+\frac{625 x \sqrt{2+x^2-x^4}}{32368 \left (7+5 x^2\right )^2}+\frac{645625 x \sqrt{2+x^2-x^4}}{15407168 \left (7+5 x^2\right )}+\frac{90101 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{4952304}+\frac{1453 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{825384}-\frac{6898575 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{107850176}+\frac{6575 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{3851792}+\frac{62625 \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx}{15407168}\\ &=\frac{x \left (9830-4909 x^2\right )}{353736 \sqrt{2+x^2-x^4}}+\frac{625 x \sqrt{2+x^2-x^4}}{32368 \left (7+5 x^2\right )^2}+\frac{645625 x \sqrt{2+x^2-x^4}}{15407168 \left (7+5 x^2\right )}+\frac{3086453 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{138664512}+\frac{60409 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{23110752}-\frac{6898575 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{107850176}\\ \end{align*}

Mathematica [C]  time = 0.402093, size = 244, normalized size = 1.91 \[ \frac{-67352691 i \sqrt{2} \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )-1080258550 x^7-737347940 x^5+3876617542 x^3+43210342 i \sqrt{2} \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )[phone] i \sqrt{2} \sqrt{-x^4+x^2+2} x^4 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )[phone] i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )[phone] i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )[phone] x}{1941303168 \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3857257460*x + 3876617542*x^3 - 737347940*x^5 - 1080258550*x^7 + (43210342*I)*Sqrt[2]*(7 + 5*x^2)^2*Sqrt[2 +
x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] - (67352691*I)*Sqrt[2]*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticF[I*
ArcSinh[x], -1/2] + (3042271575*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] + (43461022
50*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] + (1552179375*I)*Sqrt[2]*x^4*Sqrt[2
+ x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(1941303168*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4])

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Maple [A]  time = 0.023, size = 212, normalized size = 1.7 \begin{align*}{\frac{625\,x}{32368\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{645625\,x}{77035840\,{x}^{2}[phone]}\sqrt{-{x}^{4}+{x}^{2}+2}}+2\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ( -{\frac{4909\,{x}^{3}}{707472}}+{\frac{4915\,x}{353736}} \right ) }+{\frac{60409\,\sqrt{2}}{46221504}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{3086453\,\sqrt{2}}{277329024}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{6898575\,\sqrt{2}}{107850176}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{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+x^2+2)^(3/2),x)

[Out]

625/32368*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)^2+645625/15407168*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)+2*(-4909/707472*x^3+
4915/353736*x)/(-x^4+x^2+2)^(1/2)+60409/46221504*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*Ell
ipticF(1/2*x*2^(1/2),I*2^(1/2))+3086453/277329024*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*El
lipticE(1/2*x*2^(1/2),I*2^(1/2))-6898575/107850176*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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x^{4} + 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+x^2+2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-x^4 + 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} + x^{2} + 2}}{125 \, x^{14} + 275 \, x^{12} - 690 \, x^{10} - 2202 \, x^{8} - 291 \, x^{6} + 4011 \, x^{4} + 4312 \, 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+x^2+2)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-x^4 + x^2 + 2)/(125*x^14 + 275*x^12 - 690*x^10 - 2202*x^8 - 291*x^6 + 4011*x^4 + 4312*x^2 + 137
2), x)

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

[Out]

Integral(1/((-(x**2 - 2)*(x**2 + 1))**(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} + 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+x^2+2)^(3/2),x, algorithm="giac")

[Out]

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