\(\int \frac {946+315 x^2}{(7+5 x^2) \sqrt {2+3 x^2+x^4}} \, dx\) [185]

Optimal result

Integrand size = 31, antiderivative size = 106 \[ \int \frac {946+315 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx=\frac {631 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {2+3 x^2+x^4}}-\frac {2525 \left (2+x^2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{14 \sqrt {2} \sqrt {\frac {2+x^2}{1+x^2}} \sqrt {2+3 x^2+x^4}} \] Output:

631/4*(x^2+1)*((x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(arctan(x),1/2*2^(1/2 
))*2^(1/2)/(x^4+3*x^2+2)^(1/2)-2525/28*(x^2+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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.70 \[ \int \frac {946+315 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx=-\frac {i \sqrt {1+x^2} \sqrt {2+x^2} \left (441 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )+505 \operatorname {EllipticPi}\left (\frac {10}{7},i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )\right )}{7 \sqrt {2+3 x^2+x^4}} \] Input:

Integrate[(946 + 315*x^2)/((7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4]),x]
 

Output:

((-1/7*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(441*EllipticF[I*ArcSinh[x/Sqrt[2]], 
 2] + 505*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2]))/Sqrt[2 + 3*x^2 + x^4 
]
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2218, 27, 1412, 1786, 414}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {315 x^2+946}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}} \, dx\)

\(\Big \downarrow \) 2218

\(\displaystyle \frac {631}{2} \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx-\frac {2525}{8} \int \frac {4 \left (x^2+1\right )}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {631}{2} \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx-\frac {2525}{2} \int \frac {x^2+1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {631 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {2525}{2} \int \frac {x^2+1}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+2}}dx\)

\(\Big \downarrow \) 1786

\(\displaystyle \frac {631 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {2525 \sqrt {x^2+1} \sqrt {x^2+2} \int \frac {\sqrt {x^2+1}}{\sqrt {x^2+2} \left (5 x^2+7\right )}dx}{2 \sqrt {x^4+3 x^2+2}}\)

\(\Big \downarrow \) 414

\(\displaystyle \frac {631 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+2}}-\frac {2525 \left (x^2+2\right ) \operatorname {EllipticPi}\left (\frac {2}{7},\arctan (x),\frac {1}{2}\right )}{14 \sqrt {2} \sqrt {\frac {x^2+2}{x^2+1}} \sqrt {x^4+3 x^2+2}}\)

Input:

Int[(946 + 315*x^2)/((7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4]),x]
 

Output:

(631*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(2*Sqr 
t[2]*Sqrt[2 + 3*x^2 + x^4]) - (2525*(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1 
/2])/(14*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 414

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)), ArcTan[ 
Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ 
[d/c]
 

rule 1412

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]))*EllipticF 
[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)])] /; FreeQ[ 
{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1786

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + ( 
b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(a + b*x^n + c*x 
^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]) 
   Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; Fre 
eQ[{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 2218

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Sqrt[b^2 - 4*a*c]}, Simp[(2*a*B - A 
*(b + q))/(2*a*e - d*(b + q))   Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Sim 
p[(B*d - A*e)/(2*a*e - d*(b + q))   Int[(2*a + (b + q)*x^2)/((d + e*x^2)*Sq 
rt[a + b*x^2 + c*x^4]), x], x] /; RationalQ[q]] /; FreeQ[{a, b, c, d, e, A, 
 B}, x] && GtQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*A^ 
2 - b*A*B + a*B^2, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {63 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {505 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \operatorname {EllipticPi}\left (\frac {i x \sqrt {2}}{2}, \frac {10}{7}, \sqrt {2}\right )}{7 \sqrt {x^{4}+3 x^{2}+2}}\)	\(93\)
elliptic	\(-\frac {63 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {505 i \sqrt {2}\, \sqrt {1+\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \operatorname {EllipticPi}\left (\frac {i x \sqrt {2}}{2}, \frac {10}{7}, \sqrt {2}\right )}{7 \sqrt {x^{4}+3 x^{2}+2}}\)	\(93\)

Input:

int((315*x^2+946)/(5*x^2+7)/(x^4+3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-63/2*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*Elliptic 
F(1/2*I*x*2^(1/2),2^(1/2))-505/7*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))
 

Fricas [F]

\[ \int \frac {946+315 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx=\int { \frac {315 \, x^{2} + 946}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}} \,d x } \] Input:

integrate((315*x^2+946)/(5*x^2+7)/(x^4+3*x^2+2)^(1/2),x, algorithm="fricas 
")
 

Output:

integral(sqrt(x^4 + 3*x^2 + 2)*(315*x^2 + 946)/(5*x^6 + 22*x^4 + 31*x^2 + 
14), x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {946+315 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx=\int \frac {315 x^{2} + 946}{\sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )}\, dx \] Input:

integrate((315*x**2+946)/(5*x**2+7)/(x**4+3*x**2+2)**(1/2),x)
 

Output:

Integral((315*x**2 + 946)/(sqrt((x**2 + 1)*(x**2 + 2))*(5*x**2 + 7)), x)
 

Maxima [F]

\[ \int \frac {946+315 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx=\int { \frac {315 \, x^{2} + 946}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}} \,d x } \] Input:

integrate((315*x^2+946)/(5*x^2+7)/(x^4+3*x^2+2)^(1/2),x, algorithm="maxima 
")
 

Output:

integrate((315*x^2 + 946)/(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7)), x)
 

Giac [F]

\[ \int \frac {946+315 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx=\int { \frac {315 \, x^{2} + 946}{\sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}} \,d x } \] Input:

integrate((315*x^2+946)/(5*x^2+7)/(x^4+3*x^2+2)^(1/2),x, algorithm="giac")
 

Output:

integrate((315*x^2 + 946)/(sqrt(x^4 + 3*x^2 + 2)*(5*x^2 + 7)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {946+315 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx=\int \frac {315\,x^2+946}{\left (5\,x^2+7\right )\,\sqrt {x^4+3\,x^2+2}} \,d x \] Input:

int((315*x^2 + 946)/((5*x^2 + 7)*(3*x^2 + x^4 + 2)^(1/2)),x)
 

Output:

int((315*x^2 + 946)/((5*x^2 + 7)*(3*x^2 + x^4 + 2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {946+315 x^2}{\left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4}} \, dx=946 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+2}}{5 x^{6}+22 x^{4}+31 x^{2}+14}d x \right )+315 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+2}\, x^{2}}{5 x^{6}+22 x^{4}+31 x^{2}+14}d x \right ) \] Input:

int((315*x^2+946)/(5*x^2+7)/(x^4+3*x^2+2)^(1/2),x)
 

Output:

946*int(sqrt(x**4 + 3*x**2 + 2)/(5*x**6 + 22*x**4 + 31*x**2 + 14),x) + 315 
*int((sqrt(x**4 + 3*x**2 + 2)*x**2)/(5*x**6 + 22*x**4 + 31*x**2 + 14),x)