\(\int (4-7 x^2+x^4) (2+5 x^2+3 x^4)^{3/2} \, dx\) [200]

Optimal result

Integrand size = 27, antiderivative size = 216 \[ \int \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {104954 x \left (2+3 x^2\right )}{31185 \sqrt {2+5 x^2+3 x^4}}+\frac {2 x \left (19910+10863 x^2\right ) \sqrt {2+5 x^2+3 x^4}}{10395}-\frac {1}{231} x \left (15+203 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}+\frac {1}{33} x \left (2+5 x^2+3 x^4\right )^{5/2}-\frac {104954 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{31185 \sqrt {2+5 x^2+3 x^4}}+\frac {8686 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{2079 \sqrt {2+5 x^2+3 x^4}} \] Output:

104954/31185*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+2/10395*x*(10863*x^2+19910) 
*(3*x^4+5*x^2+2)^(1/2)-1/231*x*(203*x^2+15)*(3*x^4+5*x^2+2)^(3/2)+1/33*x*( 
3*x^4+5*x^2+2)^(5/2)-104954/31185*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)+8686/2079 
*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(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 = 10.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.69 \[ \int \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {3 x \left (79460+211412 x^2+79005 x^4-196992 x^6-192240 x^8-39690 x^{10}+8505 x^{12}\right )-104954 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+18094 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{31185 \sqrt {2+5 x^2+3 x^4}} \] Input:

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

Output:

(3*x*(79460 + 211412*x^2 + 79005*x^4 - 196992*x^6 - 192240*x^8 - 39690*x^1 
0 + 8505*x^12) - (104954*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*Elliptic 
E[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (18094*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 
3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/(31185*Sqrt[2 + 5*x^2 + 3*x 
^4])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2207, 1490, 27, 1490, 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.

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

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{33} \int \left (130-261 x^2\right ) \left (3 x^4+5 x^2+2\right )^{3/2}dx+\frac {1}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{33} \left (\frac {1}{63} \int 18 \left (1207 x^2+925\right ) \sqrt {3 x^4+5 x^2+2}dx-\frac {1}{7} x \left (203 x^2+15\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {1}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{33} \left (\frac {2}{7} \int \left (1207 x^2+925\right ) \sqrt {3 x^4+5 x^2+2}dx-\frac {1}{7} x \left (203 x^2+15\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {1}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {1}{33} \left (\frac {2}{7} \left (\frac {1}{45} \int \frac {52477 x^2+43430}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (10863 x^2+19910\right )\right )-\frac {1}{7} x \left (203 x^2+15\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {1}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{33} \left (\frac {2}{7} \left (\frac {1}{45} \left (43430 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+52477 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (10863 x^2+19910\right )\right )-\frac {1}{7} x \left (203 x^2+15\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {1}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{33} \left (\frac {2}{7} \left (\frac {1}{45} \left (52477 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {21715 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (10863 x^2+19910\right )\right )-\frac {1}{7} x \left (203 x^2+15\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {1}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{33} \left (\frac {2}{7} \left (\frac {1}{45} \left (\frac {21715 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}+52477 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )+\frac {1}{45} x \sqrt {3 x^4+5 x^2+2} \left (10863 x^2+19910\right )\right )-\frac {1}{7} x \left (203 x^2+15\right ) \left (3 x^4+5 x^2+2\right )^{3/2}\right )+\frac {1}{33} x \left (3 x^4+5 x^2+2\right )^{5/2}\)

Input:

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

Output:

(x*(2 + 5*x^2 + 3*x^4)^(5/2))/33 + (-1/7*(x*(15 + 203*x^2)*(2 + 5*x^2 + 3* 
x^4)^(3/2)) + (2*((x*(19910 + 10863*x^2)*Sqrt[2 + 5*x^2 + 3*x^4])/45 + (52 
477*((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*Sqrt[2 + 5*x^2 + 3* 
x^4])) + (21715*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[Ar 
cTan[x], -1/2])/Sqrt[2 + 5*x^2 + 3*x^4])/45))/7)/33
 

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 = 3.95 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {x \left (2835 x^{8}-17955 x^{6}-36045 x^{4}+6381 x^{2}+39730\right ) \sqrt {3 x^{4}+5 x^{2}+2}}{10395}-\frac {8686 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2079 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {104954 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{31185 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(140\)
default	\(\frac {3 x^{9} \sqrt {3 x^{4}+5 x^{2}+2}}{11}-\frac {19 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{11}-\frac {267 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{77}+\frac {709 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{1155}+\frac {7946 x \sqrt {3 x^{4}+5 x^{2}+2}}{2079}-\frac {8686 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2079 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {104954 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{31185 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(194\)
elliptic	\(\frac {3 x^{9} \sqrt {3 x^{4}+5 x^{2}+2}}{11}-\frac {19 x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{11}-\frac {267 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{77}+\frac {709 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{1155}+\frac {7946 x \sqrt {3 x^{4}+5 x^{2}+2}}{2079}-\frac {8686 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{2079 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {104954 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{31185 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(194\)

Input:

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

Output:

1/10395*x*(2835*x^8-17955*x^6-36045*x^4+6381*x^2+39730)*(3*x^4+5*x^2+2)^(1 
/2)-8686/2079*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*Ellipt 
icF(I*x,1/2*6^(1/2))+104954/31185*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 = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.40 \[ \int \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=-\frac {209908 \, \sqrt {3} \sqrt {-\frac {2}{3}} x E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 600778 \, \sqrt {3} \sqrt {-\frac {2}{3}} x F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 3 \, {\left (8505 \, x^{10} - 53865 \, x^{8} - 108135 \, x^{6} + 19143 \, x^{4} + 119190 \, x^{2} + 104954\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{93555 \, x} \] Input:

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

Output:

-1/93555*(209908*sqrt(3)*sqrt(-2/3)*x*elliptic_e(arcsin(sqrt(-2/3)/x), 3/2 
) - 600778*sqrt(3)*sqrt(-2/3)*x*elliptic_f(arcsin(sqrt(-2/3)/x), 3/2) - 3* 
(8505*x^10 - 53865*x^8 - 108135*x^6 + 19143*x^4 + 119190*x^2 + 104954)*sqr 
t(3*x^4 + 5*x^2 + 2))/x
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2} \, dx=\frac {3 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{9}}{11}-\frac {19 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}}{11}-\frac {267 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}}{77}+\frac {709 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}}{1155}+\frac {7946 \sqrt {3 x^{4}+5 x^{2}+2}\, x}{2079}+\frac {17372 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{3 x^{4}+5 x^{2}+2}d x \right )}{2079}+\frac {104954 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{3 x^{4}+5 x^{2}+2}d x \right )}{10395} \] Input:

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

Output:

(2835*sqrt(3*x**4 + 5*x**2 + 2)*x**9 - 17955*sqrt(3*x**4 + 5*x**2 + 2)*x** 
7 - 36045*sqrt(3*x**4 + 5*x**2 + 2)*x**5 + 6381*sqrt(3*x**4 + 5*x**2 + 2)* 
x**3 + 39730*sqrt(3*x**4 + 5*x**2 + 2)*x + 86860*int(sqrt(3*x**4 + 5*x**2 
+ 2)/(3*x**4 + 5*x**2 + 2),x) + 104954*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2 
)/(3*x**4 + 5*x**2 + 2),x))/10395