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

Optimal. Leaf size=231 \[ \frac {17 \sqrt {x^4+3 x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac {3 \sqrt {x^4+3 x^2+2} x}{350 \left (5 x^2+7\right )^2}+\frac {3 \left (x^2+2\right ) x}{392 \sqrt {x^4+3 x^2+2}}+\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{2 x^2+2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{784 \sqrt {x^4+3 x^2+2}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{2 x^2+2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{196 \sqrt {x^4+3 x^2+2}}+\frac {141 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{27440 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]

[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]  time = 0.67, antiderivative size = 288, normalized size of antiderivative = 1.25, number of steps used = 27, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1228, 1099, 1135, 1223, 1696, 1716, 1189, 1214, 1456, 539} \[ \frac {17 \sqrt {x^4+3 x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac {3 \sqrt {x^4+3 x^2+2} x}{350 \left (5 x^2+7\right )^2}+\frac {3 \left (x^2+2\right ) x}{392 \sqrt {x^4+3 x^2+2}}+\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{784 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {6 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{875 \sqrt {x^4+3 x^2+2}}-\frac {39 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{24500 \sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {141 \left (x^2+2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{27440 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[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 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]

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 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 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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx &=\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]  time = 0.38, size = 174, normalized size = 0.75 \[ \frac {-406 i \sqrt {x^2+1} \sqrt {x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-525 i \sqrt {x^2+1} \sqrt {x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+141 i \sqrt {x^2+1} \sqrt {x^2+2} \Pi \left (\frac {10}{7};\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+\frac {119 x \left (x^4+3 x^2+2\right )}{5 x^2+7}-\frac {588 x \left (x^4+3 x^2+2\right )}{\left (5 x^2+7\right )^2}}{68600 \sqrt {x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[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])

________________________________________________________________________________________

fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [C]  time = 0.02, size = 186, normalized size = 0.81 \[ -\frac {3 \sqrt {x^{4}+3 x^{2}+2}\, x}{350 \left (5 x^{2}+7\right )^{2}}+\frac {17 \sqrt {x^{4}+3 x^{2}+2}\, x}{9800 \left (5 x^{2}+7\right )}-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{784 \sqrt {x^{4}+3 x^{2}+2}}-\frac {29 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{9800 \sqrt {x^{4}+3 x^{2}+2}}+\frac {141 i \sqrt {2}\, \sqrt {\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{68600 \sqrt {x^{4}+3 x^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/350*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)^2*x+17/9800*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)*x-29/9800*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)*EllipticE(1/2*I*2^(1/2)*x,2^(1/2))+141/68600*I*2^(1/2)*(1/2*x^2+1)^(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))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (x^4+3\,x^2+2\right )}^{3/2}}{{\left (5\,x^2+7\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________