\(\int \frac {(d+e x)^2}{\sqrt {a+b x^2+c x^4}} \, dx\) [261]

Optimal result

Integrand size = 24, antiderivative size = 334 \[ \int \frac {(d+e x)^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {e^2 x \sqrt {a+b x^2+c x^4}}{\sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {d e \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {c}}-\frac {\sqrt [4]{a} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{c^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} c^{3/4} \sqrt {a+b x^2+c x^4}} \] Output:

e^2*x*(c*x^4+b*x^2+a)^(1/2)/c^(1/2)/(a^(1/2)+c^(1/2)*x^2)+d*e*arctanh(1/2* 
(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(1/2)-a^(1/4)*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)*EllipticE(sin(2 
*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(3/4)/(c*x^ 
4+b*x^2+a)^(1/2)+1/2*(c^(1/2)*d^2+a^(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*arctan(c^(1/4 
)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(1/4)/c^(3/4)/(c*x^4+b*x^2 
+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.20 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {i \left (\left (-b+\sqrt {b^2-4 a c}\right ) e^2 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\left (2 c d^2+\left (-b+\sqrt {b^2-4 a c}\right ) e^2\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+4 i \sqrt {c} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d e \sqrt {a+b x^2+c x^4} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:

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

Output:

((I/4)*((-b + Sqrt[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])]*EllipticE[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])] - (2*c*d^2 + ( 
-b + Sqrt[b^2 - 4*a*c])*e^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + S 
qrt[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])] + (4*I)*Sqrt[c]*Sqrt[c 
/(b + Sqrt[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]]))/(c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt 
[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2202, 27, 1432, 1092, 219, 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.

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1432

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

(d*e*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c] - 
 (e^2*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*( 
Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[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] + ((d^2 + (Sqrt[a]*e^2)/Sqrt[c]) 
*(Sqrt[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*a^(1/4)*c^(1/4)*Sqrt[a + b*x^2 + c*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 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 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 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 = 1.38 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {d^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {e^{2} a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {d e \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{\sqrt {c}}\)	\(401\)
elliptic	\(\frac {d^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {d e \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{\sqrt {c}}-\frac {e^{2} a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(402\)

Input:

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

Output:

1/4*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*e^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))*(Ell 
ipticF(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)))+d*e*ln((1/2*b+ 
c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {a c^{\frac {3}{2}} d e x \log \left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} + 4 \, a c\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} a c e^{2} + \sqrt {\frac {1}{2}} {\left (a c e^{2} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - a b e^{2} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left (c^{2} d^{2} - a c e^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + {\left (b c d^{2} + a b e^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c})}{2 \, a c^{2} x} \] Input:

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

Output:

1/2*(a*c^(3/2)*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) + 2*sqrt(c*x^4 + b*x^2 + a)*a*c*e^2 
+ sqrt(1/2)*(a*c*e^2*x*sqrt((b^2 - 4*a*c)/c^2) - a*b*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)) + sqrt(1/2)*((c^2*d^2 - a*c*e^2)*x*sqrt((b^2 - 4*a*c)/c^2) + 
(b*c*d^2 + a*b*e^2)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*ell 
iptic_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)))/(a*c^2*x)
 

Sympy [F]

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

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \frac {(d+e x)^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {-\sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+b \,x^{2}+a}-\sqrt {c}\, x^{2}\right ) d e +\sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+b \,x^{2}+a}+\sqrt {c}\, x^{2}\right ) d e +\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a c \,d^{2}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{4}}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b c \,e^{2}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a c \,e^{2}+\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b c \,d^{2}-2 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a c d e}{c} \] Input:

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

Output:

( - sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*d*e + sqrt(c)*lo 
g(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*d*e + 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*c*d**2 + 
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*c*e**2 + 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*c*e**2 + 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)*b*c*d**2 - 2*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*c*d*e)/c