\(\int \frac {-34-65 x^2-25 x^4}{(2+3 x^2+x^4)^{3/2}} \, dx\) [16]

Optimal result

Integrand size = 27, antiderivative size = 116 \[ \int \frac {-34-65 x^2-25 x^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\frac {2 x}{\sqrt {2+3 x^2+x^4}}+\frac {4 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {2+3 x^2+x^4}}-\frac {27 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+3 x^2+x^4}} \] Output:

2*x/(x^4+3*x^2+2)^(1/2)+4*2^(1/2)*(x^2+1)*((x^2+2)/(x^2+1))^(1/2)*Elliptic 
E(x/(x^2+1)^(1/2),1/2*2^(1/2))/(x^4+3*x^2+2)^(1/2)-27/2*(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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {-34-65 x^2-25 x^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\frac {10 x+4 x^3+4 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )+23 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{\sqrt {2+3 x^2+x^4}} \] Input:

Integrate[(-34 - 65*x^2 - 25*x^4)/(2 + 3*x^2 + x^4)^(3/2),x]
 

Output:

(10*x + 4*x^3 + (4*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sq 
rt[2]], 2] + (23*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt 
[2]], 2])/Sqrt[2 + 3*x^2 + x^4]
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.27, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2206, 27, 1503, 1412, 1455}

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.

\(\displaystyle \int \frac {-25 x^4-65 x^2-34}{\left (x^4+3 x^2+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {2 x \left (2 x^2+5\right )}{\sqrt {x^4+3 x^2+2}}-\frac {1}{2} \int \frac {2 \left (4 x^2+27\right )}{\sqrt {x^4+3 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x \left (2 x^2+5\right )}{\sqrt {x^4+3 x^2+2}}-\int \frac {4 x^2+27}{\sqrt {x^4+3 x^2+2}}dx\)

\(\Big \downarrow \) 1503

\(\displaystyle -27 \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx-4 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx+\frac {2 x \left (2 x^2+5\right )}{\sqrt {x^4+3 x^2+2}}\)

\(\Big \downarrow \) 1412

\(\displaystyle -4 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx-\frac {27 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+\frac {2 x \left (2 x^2+5\right )}{\sqrt {x^4+3 x^2+2}}\)

\(\Big \downarrow \) 1455

\(\displaystyle -\frac {27 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}-4 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )+\frac {2 x \left (2 x^2+5\right )}{\sqrt {x^4+3 x^2+2}}\)

Input:

Int[(-34 - 65*x^2 - 25*x^4)/(2 + 3*x^2 + x^4)^(3/2),x]
 

Output:

(2*x*(5 + 2*x^2))/Sqrt[2 + 3*x^2 + x^4] - 4*((x*(2 + x^2))/Sqrt[2 + 3*x^2 
+ x^4] - (Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 
 1/2])/Sqrt[2 + 3*x^2 + x^4]) - (27*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*El 
lipticF[ArcTan[x], 1/2])/(Sqrt[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 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 1455

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

rule 1503

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d   Int[1/Sqrt[a + b*x^2 + c*x^4] 
, x], x] + Simp[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 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.10

method	result	size
risch	\(\frac {2 x \left (2 x^{2}+5\right )}{\sqrt {x^{4}+3 x^{2}+2}}+\frac {27 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{4}+3 x^{2}+2}}\)	\(128\)
elliptic	\(-\frac {2 \left (-2 x^{3}-5 x \right )}{\sqrt {x^{4}+3 x^{2}+2}}+\frac {27 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{4}+3 x^{2}+2}}\)	\(129\)
default	\(\frac {-51 x^{3}-85 x}{\sqrt {x^{4}+3 x^{2}+2}}+\frac {27 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}-\frac {2 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, \sqrt {2}\right )\right )}{\sqrt {x^{4}+3 x^{2}+2}}+\frac {130 x^{3}+195 x}{\sqrt {x^{4}+3 x^{2}+2}}+\frac {-75 x^{3}-100 x}{\sqrt {x^{4}+3 x^{2}+2}}\)	\(173\)

Input:

int((-25*x^4-65*x^2-34)/(x^4+3*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2*x*(2*x^2+5)/(x^4+3*x^2+2)^(1/2)+27/2*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*x*2^(1/2),2^(1/2))-2*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*x*2^(1/2) 
,2^(1/2))-EllipticE(1/2*I*x*2^(1/2),2^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.78 \[ \int \frac {-34-65 x^2-25 x^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {2} \sqrt {-\frac {1}{2}} {\left (x^{4} + 3 \, x^{2} + 2\right )} E(\arcsin \left (\sqrt {-\frac {1}{2}} x\right )\,|\,2) - 29 \, \sqrt {2} \sqrt {-\frac {1}{2}} {\left (x^{4} + 3 \, x^{2} + 2\right )} F(\arcsin \left (\sqrt {-\frac {1}{2}} x\right )\,|\,2) - 2 \, \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (2 \, x^{3} + 5 \, x\right )}}{x^{4} + 3 \, x^{2} + 2} \] Input:

integrate((-25*x^4-65*x^2-34)/(x^4+3*x^2+2)^(3/2),x, algorithm="fricas")
 

Output:

-(2*sqrt(2)*sqrt(-1/2)*(x^4 + 3*x^2 + 2)*elliptic_e(arcsin(sqrt(-1/2)*x), 
2) - 29*sqrt(2)*sqrt(-1/2)*(x^4 + 3*x^2 + 2)*elliptic_f(arcsin(sqrt(-1/2)* 
x), 2) - 2*sqrt(x^4 + 3*x^2 + 2)*(2*x^3 + 5*x))/(x^4 + 3*x^2 + 2)
 

Sympy [F]

\[ \int \frac {-34-65 x^2-25 x^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=- \int \frac {65 x^{2}}{x^{4} \sqrt {x^{4} + 3 x^{2} + 2} + 3 x^{2} \sqrt {x^{4} + 3 x^{2} + 2} + 2 \sqrt {x^{4} + 3 x^{2} + 2}}\, dx - \int \frac {25 x^{4}}{x^{4} \sqrt {x^{4} + 3 x^{2} + 2} + 3 x^{2} \sqrt {x^{4} + 3 x^{2} + 2} + 2 \sqrt {x^{4} + 3 x^{2} + 2}}\, dx - \int \frac {34}{x^{4} \sqrt {x^{4} + 3 x^{2} + 2} + 3 x^{2} \sqrt {x^{4} + 3 x^{2} + 2} + 2 \sqrt {x^{4} + 3 x^{2} + 2}}\, dx \] Input:

integrate((-25*x**4-65*x**2-34)/(x**4+3*x**2+2)**(3/2),x)
 

Output:

-Integral(65*x**2/(x**4*sqrt(x**4 + 3*x**2 + 2) + 3*x**2*sqrt(x**4 + 3*x** 
2 + 2) + 2*sqrt(x**4 + 3*x**2 + 2)), x) - Integral(25*x**4/(x**4*sqrt(x**4 
 + 3*x**2 + 2) + 3*x**2*sqrt(x**4 + 3*x**2 + 2) + 2*sqrt(x**4 + 3*x**2 + 2 
)), x) - Integral(34/(x**4*sqrt(x**4 + 3*x**2 + 2) + 3*x**2*sqrt(x**4 + 3* 
x**2 + 2) + 2*sqrt(x**4 + 3*x**2 + 2)), x)
 

Maxima [F]

\[ \int \frac {-34-65 x^2-25 x^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int { -\frac {25 \, x^{4} + 65 \, x^{2} + 34}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((-25*x^4-65*x^2-34)/(x^4+3*x^2+2)^(3/2),x, algorithm="maxima")
 

Output:

-integrate((25*x^4 + 65*x^2 + 34)/(x^4 + 3*x^2 + 2)^(3/2), x)
 

Giac [F]

\[ \int \frac {-34-65 x^2-25 x^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int { -\frac {25 \, x^{4} + 65 \, x^{2} + 34}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((-25*x^4-65*x^2-34)/(x^4+3*x^2+2)^(3/2),x, algorithm="giac")
 

Output:

integrate(-(25*x^4 + 65*x^2 + 34)/(x^4 + 3*x^2 + 2)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-34-65 x^2-25 x^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int -\frac {25\,x^4+65\,x^2+34}{{\left (x^4+3\,x^2+2\right )}^{3/2}} \,d x \] Input:

int(-(65*x^2 + 25*x^4 + 34)/(3*x^2 + x^4 + 2)^(3/2),x)
 

Output:

int(-(65*x^2 + 25*x^4 + 34)/(3*x^2 + x^4 + 2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {-34-65 x^2-25 x^4}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\frac {25 \sqrt {x^{4}+3 x^{2}+2}\, x -84 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+2}}{x^{8}+6 x^{6}+13 x^{4}+12 x^{2}+4}d x \right ) x^{4}-252 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+2}}{x^{8}+6 x^{6}+13 x^{4}+12 x^{2}+4}d x \right ) x^{2}-168 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+2}}{x^{8}+6 x^{6}+13 x^{4}+12 x^{2}+4}d x \right )-65 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+2}\, x^{2}}{x^{8}+6 x^{6}+13 x^{4}+12 x^{2}+4}d x \right ) x^{4}-195 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+2}\, x^{2}}{x^{8}+6 x^{6}+13 x^{4}+12 x^{2}+4}d x \right ) x^{2}-130 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+2}\, x^{2}}{x^{8}+6 x^{6}+13 x^{4}+12 x^{2}+4}d x \right )}{x^{4}+3 x^{2}+2} \] Input:

int((-25*x^4-65*x^2-34)/(x^4+3*x^2+2)^(3/2),x)
 

Output:

(25*sqrt(x**4 + 3*x**2 + 2)*x - 84*int(sqrt(x**4 + 3*x**2 + 2)/(x**8 + 6*x 
**6 + 13*x**4 + 12*x**2 + 4),x)*x**4 - 252*int(sqrt(x**4 + 3*x**2 + 2)/(x* 
*8 + 6*x**6 + 13*x**4 + 12*x**2 + 4),x)*x**2 - 168*int(sqrt(x**4 + 3*x**2 
+ 2)/(x**8 + 6*x**6 + 13*x**4 + 12*x**2 + 4),x) - 65*int((sqrt(x**4 + 3*x* 
*2 + 2)*x**2)/(x**8 + 6*x**6 + 13*x**4 + 12*x**2 + 4),x)*x**4 - 195*int((s 
qrt(x**4 + 3*x**2 + 2)*x**2)/(x**8 + 6*x**6 + 13*x**4 + 12*x**2 + 4),x)*x* 
*2 - 130*int((sqrt(x**4 + 3*x**2 + 2)*x**2)/(x**8 + 6*x**6 + 13*x**4 + 12* 
x**2 + 4),x))/(x**4 + 3*x**2 + 2)