\(\int \frac {(2+3 x^2+x^4)^{3/2}}{(7+5 x^2)^3} \, dx\) [299]

Optimal result

Integrand size = 24, antiderivative size = 231 \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\frac {3 x \left (2+x^2\right )}{392 \sqrt {2+3 x^2+x^4}}-\frac {3 x \sqrt {2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac {17 x \sqrt {2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{2+2 x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{196 \sqrt {2+3 x^2+x^4}}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{2+2 x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{784 \sqrt {2+3 x^2+x^4}}+\frac {141 \left (2+x^2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{27440 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}} \]

[Out]

3/392*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)+141/54880*(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)-3/196*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*
EllipticE(x/(x^2+1)^(1/2),1/2*2^(1/2))*((x^2+2)/(2*x^2+2))^(1/2)/(x^4+3*x^2+2)^(1/2)+5/784*(x^2+1)^(3/2)*(1/(x
^2+1))^(1/2)*EllipticF(x/(x^2+1)^(1/2),1/2*2^(1/2))*((x^2+2)/(2*x^2+2))^(1/2)/(x^4+3*x^2+2)^(1/2)-3/350*x*(x^4
+3*x^2+2)^(1/2)/(5*x^2+7)^2+17/9800*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.25, number of steps used = 27, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1242, 1113, 1149, 1237, 1710, 1730, 1203, 1228, 1470, 553} \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{784 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {6 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{875 \sqrt {x^4+3 x^2+2}}-\frac {39 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{24500 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {141 \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{27440 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}+\frac {17 \sqrt {x^4+3 x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac {3 \sqrt {x^4+3 x^2+2} x}{350 \left (5 x^2+7\right )^2}+\frac {3 \left (x^2+2\right ) x}{392 \sqrt {x^4+3 x^2+2}} \]

[In]

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

[Out]

(3*x*(2 + x^2))/(392*Sqrt[2 + 3*x^2 + x^4]) - (3*x*Sqrt[2 + 3*x^2 + x^4])/(350*(7 + 5*x^2)^2) + (17*x*Sqrt[2 +
3*x^2 + x^4])/(9800*(7 + 5*x^2)) - (39*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(24500*
Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) - (6*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(87
5*Sqrt[2 + 3*x^2 + x^4]) + (5*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(784*Sqrt[2]*Sqrt
[2 + 3*x^2 + x^4]) + (141*(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1/2])/(27440*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 {9}{625 \sqrt {2+3 x^2+x^4}}+\frac {x^2}{125 \sqrt {2+3 x^2+x^4}}+\frac {36}{625 \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}}-\frac {12}{625 \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}}-\frac {11}{625 \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}}\right ) \, dx \\ & = \frac {1}{125} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {9}{625} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx-\frac {11}{625} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx-\frac {12}{625} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx+\frac {36}{625} \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}-\frac {3 x \sqrt {2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac {x \sqrt {2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {9 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{625 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {\int \frac {62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{4375}+\frac {3 \int \frac {74-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4}} \, dx}{8750}-\frac {11 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{1250}+\frac {11}{500} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}-\frac {3 x \sqrt {2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac {17 x \sqrt {2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {7 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1250 \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}{245000}+\frac {\int \frac {-175-125 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{109375}-\frac {13 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{4375}+\frac {\left (11 \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}{500 \sqrt {2+3 x^2+x^4}} \\ & = \frac {x \left (2+x^2\right )}{125 \sqrt {2+3 x^2+x^4}}-\frac {3 x \sqrt {2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac {17 x \sqrt {2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{125 \sqrt {2+3 x^2+x^4}}+\frac {7 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{1250 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {11 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{1750 \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}{6125000}-\frac {1}{875} \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}{8750}-\frac {1}{625} \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}{3500}+\frac {303 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{49000} \\ & = \frac {6 x \left (2+x^2\right )}{875 \sqrt {2+3 x^2+x^4}}-\frac {3 x \sqrt {2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac {17 x \sqrt {2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac {6 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {2+3 x^2+x^4}}+\frac {11 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{4375 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {11 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{1750 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}+\frac {27 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{35000}+\frac {39 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{49000}+\frac {303 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{98000}-\frac {303 \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx}{39200}+\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}{3500 \sqrt {2+3 x^2+x^4}} \\ & = \frac {3 x \left (2+x^2\right )}{392 \sqrt {2+3 x^2+x^4}}-\frac {3 x \sqrt {2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac {17 x \sqrt {2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac {39 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{24500 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {6 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {2+3 x^2+x^4}}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{784 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {9 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{1225 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}}-\frac {\left (303 \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}{39200 \sqrt {2+3 x^2+x^4}} \\ & = \frac {3 x \left (2+x^2\right )}{392 \sqrt {2+3 x^2+x^4}}-\frac {3 x \sqrt {2+3 x^2+x^4}}{350 \left (7+5 x^2\right )^2}+\frac {17 x \sqrt {2+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}-\frac {39 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{24500 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {6 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {2+3 x^2+x^4}}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{784 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {141 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{27440 \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 = 10.39 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.75 \[ \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\frac {-\frac {588 x \left (2+3 x^2+x^4\right )}{\left (7+5 x^2\right )^2}+\frac {119 x \left (2+3 x^2+x^4\right )}{7+5 x^2}-525 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-406 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )+141 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{68600 \sqrt {2+3 x^2+x^4}} \]

[In]

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

[Out]

((-588*x*(2 + 3*x^2 + x^4))/(7 + 5*x^2)^2 + (119*x*(2 + 3*x^2 + x^4))/(7 + 5*x^2) - (525*I)*Sqrt[1 + x^2]*Sqrt
[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (406*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[
2]], 2] + (141*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(68600*Sqrt[2 + 3*x^2
+ x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.53 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {\sqrt {x^{4}+3 x^{2}+2}\, x \left (17 x^{2}+7\right )}{1960 \left (5 x^{2}+7\right )^{2}}-\frac {19 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{2800 \sqrt {x^{4}+3 x^{2}+2}}+\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{784 \sqrt {x^{4}+3 x^{2}+2}}+\frac {141 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{68600 \sqrt {x^{4}+3 x^{2}+2}}\)	\(183\)
default	\(-\frac {3 x \sqrt {x^{4}+3 x^{2}+2}}{350 \left (5 x^{2}+7\right )^{2}}+\frac {17 x \sqrt {x^{4}+3 x^{2}+2}}{9800 \left (5 x^{2}+7\right )}-\frac {29 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{9800 \sqrt {x^{4}+3 x^{2}+2}}-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{784 \sqrt {x^{4}+3 x^{2}+2}}+\frac {141 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{68600 \sqrt {x^{4}+3 x^{2}+2}}\)	\(186\)
elliptic	\(-\frac {3 x \sqrt {x^{4}+3 x^{2}+2}}{350 \left (5 x^{2}+7\right )^{2}}+\frac {17 x \sqrt {x^{4}+3 x^{2}+2}}{9800 \left (5 x^{2}+7\right )}-\frac {29 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{9800 \sqrt {x^{4}+3 x^{2}+2}}-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{784 \sqrt {x^{4}+3 x^{2}+2}}+\frac {141 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{68600 \sqrt {x^{4}+3 x^{2}+2}}\)	\(186\)

[In]

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

[Out]

1/1960*(x^4+3*x^2+2)^(1/2)*x*(17*x^2+7)/(5*x^2+7)^2-19/2800*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))+3/784*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)))+141/68600*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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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