\(\int \frac {(1+2 x^2)^3 (4-7 x^2+x^4)}{(2+5 x^2+3 x^4)^{3/2}} \, dx\) [211]

Optimal result

Integrand size = 36, antiderivative size = 211 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {3437 x \left (2+3 x^2\right )}{135 \sqrt {2+5 x^2+3 x^4}}-\frac {x \left (689+1013 x^2\right )}{27 \sqrt {2+5 x^2+3 x^4}}-\frac {68}{27} x \sqrt {2+5 x^2+3 x^4}+\frac {8}{45} x^3 \sqrt {2+5 x^2+3 x^4}-\frac {3437 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{135 \sqrt {2+5 x^2+3 x^4}}+\frac {293 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{9 \sqrt {2} \sqrt {2+5 x^2+3 x^4}} \] Output:

3437/135*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)-1/27*x*(1013*x^2+689)/(3*x^4+5* 
x^2+2)^(1/2)-68/27*x*(3*x^4+5*x^2+2)^(1/2)+8/45*x^3*(3*x^4+5*x^2+2)^(1/2)- 
3437/135*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)+293/18*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 = 10.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.63 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {-4125 x-6717 x^3-900 x^5+72 x^7-3437 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+1972 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{135 \sqrt {2+5 x^2+3 x^4}} \] Input:

Integrate[((1 + 2*x^2)^3*(4 - 7*x^2 + x^4))/(2 + 5*x^2 + 3*x^4)^(3/2),x]
 

Output:

(-4125*x - 6717*x^3 - 900*x^5 + 72*x^7 - (3437*I)*Sqrt[3]*Sqrt[1 + x^2]*Sq 
rt[2 + 3*x^2]*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (1972*I)*Sqrt[3]*Sq 
rt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/(135*S 
qrt[2 + 5*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2206, 27, 2207, 27, 2207, 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.

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

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {1}{2} \int -\frac {2 \left (72 x^6-516 x^4+1411 x^2+743\right )}{27 \sqrt {3 x^4+5 x^2+2}}dx-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \int \frac {72 x^6-516 x^4+1411 x^2+743}{\sqrt {3 x^4+5 x^2+2}}dx-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{27} \left (\frac {1}{15} \int \frac {3 \left (-3060 x^4+6911 x^2+3715\right )}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {24}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \left (\frac {1}{5} \int \frac {-3060 x^4+6911 x^2+3715}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {24}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {1}{27} \left (\frac {1}{5} \left (\frac {1}{9} \int \frac {27 \left (3437 x^2+1465\right )}{\sqrt {3 x^4+5 x^2+2}}dx-340 x \sqrt {3 x^4+5 x^2+2}\right )+\frac {24}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \left (\frac {1}{5} \left (3 \int \frac {3437 x^2+1465}{\sqrt {3 x^4+5 x^2+2}}dx-340 x \sqrt {3 x^4+5 x^2+2}\right )+\frac {24}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{27} \left (\frac {1}{5} \left (3 \left (1465 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+3437 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )-340 x \sqrt {3 x^4+5 x^2+2}\right )+\frac {24}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{27} \left (\frac {1}{5} \left (3 \left (3437 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {1465 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}}\right )-340 x \sqrt {3 x^4+5 x^2+2}\right )+\frac {24}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{27} \left (\frac {1}{5} \left (3 \left (\frac {1465 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}}+3437 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )-340 x \sqrt {3 x^4+5 x^2+2}\right )+\frac {24}{5} \sqrt {3 x^4+5 x^2+2} x^3\right )-\frac {x \left (1013 x^2+689\right )}{27 \sqrt {3 x^4+5 x^2+2}}\)

Input:

Int[((1 + 2*x^2)^3*(4 - 7*x^2 + x^4))/(2 + 5*x^2 + 3*x^4)^(3/2),x]
 

Output:

-1/27*(x*(689 + 1013*x^2))/Sqrt[2 + 5*x^2 + 3*x^4] + ((24*x^3*Sqrt[2 + 5*x 
^2 + 3*x^4])/5 + (-340*x*Sqrt[2 + 5*x^2 + 3*x^4] + 3*(3437*((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])) + (1465*(1 + 
 x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqr 
t[2 + 5*x^2 + 3*x^4])))/5)/27
 

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 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]
 

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 = 15.97 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.64

method	result	size
risch	\(\frac {x \left (24 x^{6}-300 x^{4}-2239 x^{2}-1375\right )}{45 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {293 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{18 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {3437 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{135 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(135\)
elliptic	\(-\frac {6 \left (\frac {1013}{162} x^{3}+\frac {689}{162} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {8 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{45}-\frac {68 x \sqrt {3 x^{4}+5 x^{2}+2}}{27}-\frac {293 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{18 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {3437 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{135 \sqrt {3 x^{4}+5 x^{2}+2}}\)	\(162\)
default	\(-\frac {24 \left (-\frac {5}{4} x^{3}-\frac {13}{12} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {293 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{18 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {3437 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{135 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {102 \left (x^{3}+\frac {5}{6} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {42 \left (-\frac {5}{6} x^{3}-\frac {2}{3} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {\frac {598}{3} x^{3}+\frac {460}{3} x}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {-\frac {1540}{9} x^{3}-\frac {1144}{9} x}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {68 x \sqrt {3 x^{4}+5 x^{2}+2}}{27}-\frac {48 \left (\frac {97}{162} x^{3}+\frac {35}{81} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {8 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{45}\)	\(285\)

Input:

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

Output:

1/45*x*(24*x^6-300*x^4-2239*x^2-1375)/(3*x^4+5*x^2+2)^(1/2)-293/18*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))+ 
3437/135*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 = 121, normalized size of antiderivative = 0.57 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {13748 \, \sqrt {3} \sqrt {-\frac {2}{3}} {\left (3 \, x^{5} + 5 \, x^{3} + 2 \, x\right )} E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 26933 \, \sqrt {3} \sqrt {-\frac {2}{3}} {\left (3 \, x^{5} + 5 \, x^{3} + 2 \, x\right )} F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 12 \, {\left (36 \, x^{8} - 450 \, x^{6} + 1797 \, x^{4} + 6530 \, x^{2} + 3437\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{810 \, {\left (3 \, x^{5} + 5 \, x^{3} + 2 \, x\right )}} \] Input:

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

Output:

-1/810*(13748*sqrt(3)*sqrt(-2/3)*(3*x^5 + 5*x^3 + 2*x)*elliptic_e(arcsin(s 
qrt(-2/3)/x), 3/2) - 26933*sqrt(3)*sqrt(-2/3)*(3*x^5 + 5*x^3 + 2*x)*ellipt 
ic_f(arcsin(sqrt(-2/3)/x), 3/2) - 12*(36*x^8 - 450*x^6 + 1797*x^4 + 6530*x 
^2 + 3437)*sqrt(3*x^4 + 5*x^2 + 2))/(3*x^5 + 5*x^3 + 2*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {72 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}-900 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}+3594 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}+8665 \sqrt {3 x^{4}+5 x^{2}+2}\, x -50370 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right ) x^{4}-83950 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right ) x^{2}-33580 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right )-57807 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right ) x^{4}-96345 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right ) x^{2}-38538 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{9 x^{8}+30 x^{6}+37 x^{4}+20 x^{2}+4}d x \right )}{405 x^{4}+675 x^{2}+270} \] Input:

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

Output:

(72*sqrt(3*x**4 + 5*x**2 + 2)*x**7 - 900*sqrt(3*x**4 + 5*x**2 + 2)*x**5 + 
3594*sqrt(3*x**4 + 5*x**2 + 2)*x**3 + 8665*sqrt(3*x**4 + 5*x**2 + 2)*x - 5 
0370*int(sqrt(3*x**4 + 5*x**2 + 2)/(9*x**8 + 30*x**6 + 37*x**4 + 20*x**2 + 
 4),x)*x**4 - 83950*int(sqrt(3*x**4 + 5*x**2 + 2)/(9*x**8 + 30*x**6 + 37*x 
**4 + 20*x**2 + 4),x)*x**2 - 33580*int(sqrt(3*x**4 + 5*x**2 + 2)/(9*x**8 + 
 30*x**6 + 37*x**4 + 20*x**2 + 4),x) - 57807*int((sqrt(3*x**4 + 5*x**2 + 2 
)*x**2)/(9*x**8 + 30*x**6 + 37*x**4 + 20*x**2 + 4),x)*x**4 - 96345*int((sq 
rt(3*x**4 + 5*x**2 + 2)*x**2)/(9*x**8 + 30*x**6 + 37*x**4 + 20*x**2 + 4),x 
)*x**2 - 38538*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(9*x**8 + 30*x**6 + 37 
*x**4 + 20*x**2 + 4),x))/(135*(3*x**4 + 5*x**2 + 2))