\(\int \frac {\sqrt {2+3 x^2+x^4}}{7+5 x^2} \, dx\) [290]

Optimal result

Integrand size = 24, antiderivative size = 178 \[ \int \frac {\sqrt {2+3 x^2+x^4}}{7+5 x^2} \, dx=\frac {x \left (2+x^2\right )}{5 \sqrt {2+3 x^2+x^4}}-\frac {\sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{5 \sqrt {2+3 x^2+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {2+x^2}{2+2 x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{5 \sqrt {2+3 x^2+x^4}}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{35 \sqrt {2} \sqrt {2+3 x^2+x^4}} \]

[Out]

1/5*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)+3/70*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticPi(x/(x^2+1)^(1/2),2/7,1/2*2^(1
/2))*((x^2+2)/(x^2+1))^(1/2)*2^(1/2)/(x^4+3*x^2+2)^(1/2)-1/5*(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)+1/5*(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)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.30, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1222, 1203, 1113, 1149, 1228, 1470, 553} \[ \int \frac {\sqrt {2+3 x^2+x^4}}{7+5 x^2} \, dx=\frac {4 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{25 \sqrt {x^4+3 x^2+2}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{25 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{5 \sqrt {x^4+3 x^2+2}}+\frac {3 \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{35 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}+\frac {x \left (x^2+2\right )}{5 \sqrt {x^4+3 x^2+2}} \]

[In]

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

[Out]

(x*(2 + x^2))/(5*Sqrt[2 + 3*x^2 + x^4]) - (Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/
2])/(5*Sqrt[2 + 3*x^2 + x^4]) - (3*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(25*Sqrt[2]*
Sqrt[2 + 3*x^2 + x^4]) + (4*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(25*Sqrt[2
+ 3*x^2 + x^4]) + (3*(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1/2])/(35*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 1222

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

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{25} \int \frac {-8-5 x^2}{\sqrt {2+3 x^2+x^4}} \, dx\right )-\frac {6}{25} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx \\ & = -\left (\frac {3}{25} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx\right )+\frac {1}{5} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {3}{10} \int \frac {2+2 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx+\frac {8}{25} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {x \left (2+x^2\right )}{5 \sqrt {2+3 x^2+x^4}}-\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 )}{5 \sqrt {2+3 x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {4 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2+3 x^2+x^4}}+\frac {\left (3 \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}{10 \sqrt {2+3 x^2+x^4}} \\ & = \frac {x \left (2+x^2\right )}{5 \sqrt {2+3 x^2+x^4}}-\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 )}{5 \sqrt {2+3 x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {4 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{25 \sqrt {2+3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \Pi \left (\frac {2}{7};\tan ^{-1}(x)|\frac {1}{2}\right )}{35 \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 = 9.53 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {2+3 x^2+x^4}}{7+5 x^2} \, dx=-\frac {i \sqrt {1+x^2} \sqrt {2+x^2} \left (35 E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+21 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )-6 \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )\right )}{175 \sqrt {2+3 x^2+x^4}} \]

[In]

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

[Out]

((-1/175*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(35*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + 21*EllipticF[I*ArcSinh[x/Sqrt
[2]], 2] - 6*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2]))/Sqrt[2 + 3*x^2 + x^4]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.78

method	result	size
default	\(-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{50 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{10 \sqrt {x^{4}+3 x^{2}+2}}+\frac {6 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{175 \sqrt {x^{4}+3 x^{2}+2}}\)	\(138\)
elliptic	\(-\frac {3 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{50 \sqrt {x^{4}+3 x^{2}+2}}-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{10 \sqrt {x^{4}+3 x^{2}+2}}+\frac {6 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {i \sqrt {2}\, x}{2}, \frac {10}{7}, \sqrt {2}\right )}{175 \sqrt {x^{4}+3 x^{2}+2}}\)	\(138\)

[In]

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

[Out]

-3/50*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))-1/10*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))+6/175*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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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