\(\int (4-7 x^2+x^4) \sqrt {2+5 x^2+3 x^4} \, dx\) [193]

Optimal result

Integrand size = 27, antiderivative size = 188 \[ \int \left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4} \, dx=\frac {8488 x \left (2+3 x^2\right )}{2835 \sqrt {2+5 x^2+3 x^4}}+\frac {1}{945} x \left (395-1503 x^2\right ) \sqrt {2+5 x^2+3 x^4}+\frac {1}{21} x \left (2+5 x^2+3 x^4\right )^{3/2}-\frac {8488 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{2835 \sqrt {2+5 x^2+3 x^4}}+\frac {659 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{189 \sqrt {2+5 x^2+3 x^4}} \] Output:

8488/2835*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+1/945*x*(-1503*x^2+395)*(3*x^4 
+5*x^2+2)^(1/2)+1/21*x*(3*x^4+5*x^2+2)^(3/2)-8488/2835*2^(1/2)*(x^2+1)*((3 
*x^2+2)/(x^2+1))^(1/2)*EllipticE(x/(x^2+1)^(1/2),1/2*I*2^(1/2))/(3*x^4+5*x 
^2+2)^(1/2)+659/189*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacob 
iAM(arctan(x),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.56 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.74 \[ \int \left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4} \, dx=\frac {3 x \left (970-131 x^2-4665 x^4-3159 x^6+405 x^8\right )-8488 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+1898 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{2835 \sqrt {2+5 x^2+3 x^4}} \] Input:

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

Output:

(3*x*(970 - 131*x^2 - 4665*x^4 - 3159*x^6 + 405*x^8) - (8488*I)*Sqrt[3]*Sq 
rt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (1898 
*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x] 
, 2/3])/(2835*Sqrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 1490, 27, 1503, 1413, 1456}

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.

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

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{21} \int \left (82-167 x^2\right ) \sqrt {3 x^4+5 x^2+2}dx+\frac {1}{21} x \left (3 x^4+5 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{21} \left (\frac {1}{45} \int \frac {2 \left (4244 x^2+3295\right )}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (395-1503 x^2\right )\right )+\frac {1}{21} x \left (3 x^4+5 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \left (\frac {2}{45} \int \frac {4244 x^2+3295}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (395-1503 x^2\right )\right )+\frac {1}{21} x \left (3 x^4+5 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{21} \left (\frac {2}{45} \left (3295 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+4244 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (395-1503 x^2\right )\right )+\frac {1}{21} x \left (3 x^4+5 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{21} \left (\frac {2}{45} \left (4244 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {3295 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}}\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (395-1503 x^2\right )\right )+\frac {1}{21} x \left (3 x^4+5 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{21} \left (\frac {2}{45} \left (\frac {3295 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}}+4244 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (395-1503 x^2\right )\right )+\frac {1}{21} x \left (3 x^4+5 x^2+2\right )^{3/2}\)

Input:

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

Output:

(x*(2 + 5*x^2 + 3*x^4)^(3/2))/21 + ((x*(395 - 1503*x^2)*Sqrt[2 + 5*x^2 + 3 
*x^4])/45 + (2*(4244*((x*(2 + 3*x^2))/(3*Sqrt[2 + 5*x^2 + 3*x^4]) - (Sqrt[ 
2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(3*Sq 
rt[2 + 5*x^2 + 3*x^4])) + (3295*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*Elli 
pticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4])))/45)/21
 

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 1413

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

rule 1456

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

rule 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), 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] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

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 2207

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

Maple [A] (verified)

Time = 3.92 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {x \left (135 x^{4}-1278 x^{2}+485\right ) \sqrt {3 x^{4}+5 x^{2}+2}}{945}-\frac {659 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{189 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {8488 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{2835 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(130\)
default	\(\frac {x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{7}-\frac {142 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{105}+\frac {97 x \sqrt {3 x^{4}+5 x^{2}+2}}{189}-\frac {659 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{189 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {8488 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{2835 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(156\)
elliptic	\(\frac {x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{7}-\frac {142 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{105}+\frac {97 x \sqrt {3 x^{4}+5 x^{2}+2}}{189}-\frac {659 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{189 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {8488 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{2835 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(156\)

Input:

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

Output:

1/945*x*(135*x^4-1278*x^2+485)*(3*x^4+5*x^2+2)^(1/2)-659/189*I*(x^2+1)^(1/ 
2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticF(I*x,1/2*6^(1/2))+8488/2 
835*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*(EllipticF(I*x,1 
/2*6^(1/2))-EllipticE(I*x,1/2*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.41 \[ \int \left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4} \, dx=-\frac {16976 \, \sqrt {3} \sqrt {-\frac {2}{3}} x E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 46631 \, \sqrt {3} \sqrt {-\frac {2}{3}} x F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 3 \, {\left (405 \, x^{6} - 3834 \, x^{4} + 1455 \, x^{2} + 8488\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{8505 \, x} \] Input:

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

Output:

-1/8505*(16976*sqrt(3)*sqrt(-2/3)*x*elliptic_e(arcsin(sqrt(-2/3)/x), 3/2) 
- 46631*sqrt(3)*sqrt(-2/3)*x*elliptic_f(arcsin(sqrt(-2/3)/x), 3/2) - 3*(40 
5*x^6 - 3834*x^4 + 1455*x^2 + 8488)*sqrt(3*x^4 + 5*x^2 + 2))/x
 

Sympy [F]

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

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

Output:

Integral(sqrt((x**2 + 1)*(3*x**2 + 2))*(x**4 - 7*x**2 + 4), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4} \, dx=\frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}}{7}-\frac {142 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}}{105}+\frac {97 \sqrt {3 x^{4}+5 x^{2}+2}\, x}{189}+\frac {1318 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{3 x^{4}+5 x^{2}+2}d x \right )}{189}+\frac {8488 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{3 x^{4}+5 x^{2}+2}d x \right )}{945} \] Input:

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

Output:

(135*sqrt(3*x**4 + 5*x**2 + 2)*x**5 - 1278*sqrt(3*x**4 + 5*x**2 + 2)*x**3 
+ 485*sqrt(3*x**4 + 5*x**2 + 2)*x + 6590*int(sqrt(3*x**4 + 5*x**2 + 2)/(3* 
x**4 + 5*x**2 + 2),x) + 8488*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(3*x**4 
+ 5*x**2 + 2),x))/945