\(\int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx\) [248]

Optimal result

Integrand size = 24, antiderivative size = 491 \[ \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx=\frac {\left (5 b c d^2-2 b^2 e^2+6 a c e^2\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {d e \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}+\frac {x \left (5 c d^2+b e^2+3 c e^2 x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\left (b^2-4 a c\right ) d e \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}-\frac {\sqrt [4]{a} \left (5 b c d^2-2 b^2 e^2+6 a c e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (5 c d^2-2 b e^2+3 \sqrt {a} \sqrt {c} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{7/4} \sqrt {a+b x^2+c x^4}} \] Output:

1/15*(6*a*c*e^2-2*b^2*e^2+5*b*c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/c^(3/2)/(a^(1 
/2)+c^(1/2)*x^2)+1/4*d*e*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c+1/15*x*(3*c*e 
^2*x^2+b*e^2+5*c*d^2)*(c*x^4+b*x^2+a)^(1/2)/c-1/8*(-4*a*c+b^2)*d*e*arctanh 
(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(3/2)-1/15*a^(1/4)*(6*a* 
c*e^2-2*b^2*e^2+5*b*c*d^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2) 
+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b 
/a^(1/2)/c^(1/2))^(1/2))/c^(7/4)/(c*x^4+b*x^2+a)^(1/2)+1/30*a^(1/4)*(b+2*a 
^(1/2)*c^(1/2))*(5*c*d^2-2*b*e^2+3*a^(1/2)*c^(1/2)*e^2)*(a^(1/2)+c^(1/2)*x 
^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arct 
an(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(7/4)/(c*x^4+b*x^ 
2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.80 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.30 \[ \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx=\frac {-2 i \left (-b+\sqrt {b^2-4 a c}\right ) \left (-5 b c d^2+2 b^2 e^2-6 a c e^2\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+2 i \left (-2 b^3 e^2+b c \left (-5 \sqrt {b^2-4 a c} d^2+8 a e^2\right )+b^2 \left (5 c d^2+2 \sqrt {b^2-4 a c} e^2\right )-2 a c \left (10 c d^2+3 \sqrt {b^2-4 a c} e^2\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+\sqrt {c} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (2 \sqrt {c} \left (a+b x^2+c x^4\right ) \left (b e (15 d+4 e x)+2 c x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+15 \left (b^2-4 a c\right ) d e \sqrt {a+b x^2+c x^4} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{120 c^2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:

Integrate[(d + e*x)^2*Sqrt[a + b*x^2 + c*x^4],x]
 

Output:

((-2*I)*(-b + Sqrt[b^2 - 4*a*c])*(-5*b*c*d^2 + 2*b^2*e^2 - 6*a*c*e^2)*Sqrt 
[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2* 
Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[ 
Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - S 
qrt[b^2 - 4*a*c])] + (2*I)*(-2*b^3*e^2 + b*c*(-5*Sqrt[b^2 - 4*a*c]*d^2 + 8 
*a*e^2) + b^2*(5*c*d^2 + 2*Sqrt[b^2 - 4*a*c]*e^2) - 2*a*c*(10*c*d^2 + 3*Sq 
rt[b^2 - 4*a*c]*e^2))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 
 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a 
*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + 
 Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + Sqrt[c]*Sqrt[c/(b + Sqrt[b^ 
2 - 4*a*c])]*(2*Sqrt[c]*(a + b*x^2 + c*x^4)*(b*e*(15*d + 4*e*x) + 2*c*x*(1 
0*d^2 + 15*d*e*x + 6*e^2*x^2)) + 15*(b^2 - 4*a*c)*d*e*Sqrt[a + b*x^2 + c*x 
^4]*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]]))/(120*c^2*Sqrt[c 
/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 474, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {2202, 27, 1432, 1087, 1092, 219, 1490, 1511, 27, 1416, 1509}

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.

\(\displaystyle \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx\)

\(\Big \downarrow \) 2202

\(\displaystyle \int \left (d^2+e^2 x^2\right ) \sqrt {c x^4+b x^2+a}dx+\int 2 d e x \sqrt {c x^4+b x^2+a}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \left (d^2+e^2 x^2\right ) \sqrt {c x^4+b x^2+a}dx+2 d e \int x \sqrt {c x^4+b x^2+a}dx\)

\(\Big \downarrow \) 1432

\(\displaystyle \int \left (d^2+e^2 x^2\right ) \sqrt {c x^4+b x^2+a}dx+d e \int \sqrt {c x^4+b x^2+a}dx^2\)

\(\Big \downarrow \) 1087

\(\displaystyle d e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )+\int \left (d^2+e^2 x^2\right ) \sqrt {c x^4+b x^2+a}dx\)

\(\Big \downarrow \) 1092

\(\displaystyle d e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )+\int \left (d^2+e^2 x^2\right ) \sqrt {c x^4+b x^2+a}dx\)

\(\Big \downarrow \) 219

\(\displaystyle \int \left (d^2+e^2 x^2\right ) \sqrt {c x^4+b x^2+a}dx+d e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {\int \frac {\left (5 b c d^2-2 b^2 e^2+6 a c e^2\right ) x^2+a \left (10 c d^2-b e^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+d e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )+\frac {x \sqrt {a+b x^2+c x^4} \left (b e^2+5 c d^2+3 c e^2 x^2\right )}{15 c}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (3 \sqrt {a} \sqrt {c} e^2-2 b e^2+5 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (6 a c e^2-2 b^2 e^2+5 b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}+d e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )+\frac {x \sqrt {a+b x^2+c x^4} \left (b e^2+5 c d^2+3 c e^2 x^2\right )}{15 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (3 \sqrt {a} \sqrt {c} e^2-2 b e^2+5 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (6 a c e^2-2 b^2 e^2+5 b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}+d e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )+\frac {x \sqrt {a+b x^2+c x^4} \left (b e^2+5 c d^2+3 c e^2 x^2\right )}{15 c}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} e^2-2 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c e^2-2 b^2 e^2+5 b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}+d e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )+\frac {x \sqrt {a+b x^2+c x^4} \left (b e^2+5 c d^2+3 c e^2 x^2\right )}{15 c}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} \sqrt {c} e^2-2 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (6 a c e^2-2 b^2 e^2+5 b c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}+d e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )+\frac {x \sqrt {a+b x^2+c x^4} \left (b e^2+5 c d^2+3 c e^2 x^2\right )}{15 c}\)

Input:

Int[(d + e*x)^2*Sqrt[a + b*x^2 + c*x^4],x]
 

Output:

(x*(5*c*d^2 + b*e^2 + 3*c*e^2*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*c) + d*e*( 
((b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b 
+ 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^(3/2))) + (-(((5*b*c 
*d^2 - 2*b^2*e^2 + 6*a*c*e^2)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sq 
rt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(S 
qrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/( 
Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a^(1 
/4)*(b + 2*Sqrt[a]*Sqrt[c])*(5*c*d^2 - 2*b*e^2 + 3*Sqrt[a]*Sqrt[c]*e^2)*(S 
qrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]* 
EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2* 
c^(3/4)*Sqrt[a + b*x^2 + c*x^4]))/(15*c)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1432

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 
 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
 

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 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 2202

Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n 
 = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b 
*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 
1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] 
 &&  !PolyQ[Pn, x^2]
 

Maple [A] (verified)

Time = 3.39 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.14

method	result	size
elliptic	\(\frac {e^{2} x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {d e \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2}+\frac {\left (\frac {b \,e^{2}}{5}+c \,d^{2}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {b d e \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (a \,d^{2}-\frac {a \left (\frac {b \,e^{2}}{5}+c \,d^{2}\right )}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (a d e -\frac {b^{2} d e}{4 c}\right ) \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {\left (\frac {2 a \,e^{2}}{5}+b \,d^{2}-\frac {2 b \left (\frac {b \,e^{2}}{5}+c \,d^{2}\right )}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(561\)
risch	\(\frac {\left (12 c \,e^{2} x^{3}+30 c d e \,x^{2}+4 b \,e^{2} x +20 c \,d^{2} x +15 b d e \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{60 c}-\frac {-\frac {15 \left (4 a c -b^{2}\right ) d e \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}+\frac {\left (24 a c \,e^{2}-8 b^{2} e^{2}+20 b c \,d^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {a b \,e^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {10 a c \,d^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}}{60 c}\)	\(643\)
default	\(d^{2} \left (\frac {x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+e^{2} \left (\frac {x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {b x \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{60 c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {2 a}{5}-\frac {2 b^{2}}{15 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+2 d e \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}\right )\)	\(880\)

Input:

int((e*x+d)^2*(c*x^4+b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/5*e^2*x^3*(c*x^4+b*x^2+a)^(1/2)+1/2*d*e*x^2*(c*x^4+b*x^2+a)^(1/2)+1/3*(1 
/5*b*e^2+c*d^2)/c*x*(c*x^4+b*x^2+a)^(1/2)+1/4*b*d*e/c*(c*x^4+b*x^2+a)^(1/2 
)+1/4*(a*d^2-1/3*a/c*(1/5*b*e^2+c*d^2))*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a 
)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/ 
2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a* 
c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/2* 
(a*d*e-1/4*b^2/c*d*e)*ln((2*c*x^2+b)/c^(1/2)+2*(c*x^4+b*x^2+a)^(1/2))/c^(1 
/2)-1/2*(2/5*a*e^2+b*d^2-2/3*b/c*(1/5*b*e^2+c*d^2))*a*2^(1/2)/((-b+(-4*a*c 
+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(- 
4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2) 
)*(EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*( 
b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2 
)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.12 \[ \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx=-\frac {15 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {c} d e x \log \left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} + 4 \, a c\right ) - 8 \, \sqrt {\frac {1}{2}} {\left ({\left (5 \, b c^{2} d^{2} - 2 \, {\left (b^{2} c - 3 \, a c^{2}\right )} e^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (5 \, b^{2} c d^{2} - 2 \, {\left (b^{3} - 3 \, a b c\right )} e^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 8 \, \sqrt {\frac {1}{2}} {\left ({\left (5 \, {\left (b c^{2} - 2 \, c^{3}\right )} d^{2} - {\left (2 \, b^{2} c - {\left (6 \, a + b\right )} c^{2}\right )} e^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (5 \, {\left (b^{2} c + 2 \, b c^{2}\right )} d^{2} - {\left (2 \, b^{3} - {\left (6 \, a b - b^{2}\right )} c\right )} e^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 4 \, {\left (12 \, c^{3} e^{2} x^{4} + 30 \, c^{3} d e x^{3} + 15 \, b c^{2} d e x + 20 \, b c^{2} d^{2} - 8 \, {\left (b^{2} c - 3 \, a c^{2}\right )} e^{2} + 4 \, {\left (5 \, c^{3} d^{2} + b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{240 \, c^{3} x} \] Input:

integrate((e*x+d)^2*(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

-1/240*(15*(b^2*c - 4*a*c^2)*sqrt(c)*d*e*x*log(8*c^2*x^4 + 8*b*c*x^2 + b^2 
 + 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) + 4*a*c) - 8*sqrt(1/2)* 
((5*b*c^2*d^2 - 2*(b^2*c - 3*a*c^2)*e^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (5*b^ 
2*c*d^2 - 2*(b^3 - 3*a*b*c)*e^2)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2 
) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b) 
/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 8*sqrt(1/ 
2)*((5*(b*c^2 - 2*c^3)*d^2 - (2*b^2*c - (6*a + b)*c^2)*e^2)*x*sqrt((b^2 - 
4*a*c)/c^2) - (5*(b^2*c + 2*b*c^2)*d^2 - (2*b^3 - (6*a*b - b^2)*c)*e^2)*x) 
*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sqrt(1/ 
2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c) 
/c^2) + b^2 - 2*a*c)/(a*c)) - 4*(12*c^3*e^2*x^4 + 30*c^3*d*e*x^3 + 15*b*c^ 
2*d*e*x + 20*b*c^2*d^2 - 8*(b^2*c - 3*a*c^2)*e^2 + 4*(5*c^3*d^2 + b*c^2*e^ 
2)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(c^3*x)
 

Sympy [F]

\[ \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx=\int \left (d + e x\right )^{2} \sqrt {a + b x^{2} + c x^{4}}\, dx \] Input:

integrate((e*x+d)**2*(c*x**4+b*x**2+a)**(1/2),x)
 

Output:

Integral((d + e*x)**2*sqrt(a + b*x**2 + c*x**4), x)
 

Maxima [F]

\[ \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (e x + d\right )}^{2} \,d x } \] Input:

integrate((e*x+d)^2*(c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(e*x + d)^2, x)
 

Giac [F]

\[ \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (e x + d\right )}^{2} \,d x } \] Input:

integrate((e*x+d)^2*(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(e*x + d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx=\int {\left (d+e\,x\right )}^2\,\sqrt {c\,x^4+b\,x^2+a} \,d x \] Input:

int((d + e*x)^2*(a + b*x^2 + c*x^4)^(1/2),x)
 

Output:

int((d + e*x)^2*(a + b*x^2 + c*x^4)^(1/2), x)
 

Reduce [F]

\[ \int (d+e x)^2 \sqrt {a+b x^2+c x^4} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^2*(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

(30*sqrt(a + b*x**2 + c*x**4)*b*c*d*e + 8*sqrt(a + b*x**2 + c*x**4)*b*c*e* 
*2*x + 40*sqrt(a + b*x**2 + c*x**4)*c**2*d**2*x + 60*sqrt(a + b*x**2 + c*x 
**4)*c**2*d*e*x**2 + 24*sqrt(a + b*x**2 + c*x**4)*c**2*e**2*x**3 - 60*sqrt 
(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*a*c*d*e + 15*sqrt(c)*log 
(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*b**2*d*e + 60*sqrt(c)*log(sqrt( 
a + b*x**2 + c*x**4) + sqrt(c)*x**2)*a*c*d*e - 15*sqrt(c)*log(sqrt(a + b*x 
**2 + c*x**4) + sqrt(c)*x**2)*b**2*d*e - 8*int(sqrt(a + b*x**2 + c*x**4)/( 
a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a**2*b*c*e**2 + 80 
*int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + 
 b*c*x**6),x)*a**2*c**2*d**2 + 48*int((sqrt(a + b*x**2 + c*x**4)*x**4)/(a* 
*2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a*b*c**2*e**2 - 16*i 
nt((sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x 
**4 + b*c*x**6),x)*b**3*c*e**2 + 40*int((sqrt(a + b*x**2 + c*x**4)*x**4)/( 
a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*b**2*c**2*d**2 + 4 
8*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b** 
2*x**4 + b*c*x**6),x)*a**2*c**2*e**2 - 24*int((sqrt(a + b*x**2 + c*x**4)*x 
**2)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a*b**2*c*e** 
2 + 120*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a**2 + 2*a*b*x**2 + a*c*x**4 
 + b**2*x**4 + b*c*x**6),x)*a*b*c**2*d**2 - 120*int((sqrt(a + b*x**2 + c*x 
**4)*x)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a**2*c...