\(\int (d+e x) \sqrt {a+b x^2+c x^4} \, dx\) [249]

Optimal result
Mathematica [C] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [A] (verification not implemented)
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 22, antiderivative size = 397 \[ \int (d+e x) \sqrt {a+b x^2+c x^4} \, dx=\frac {1}{3} d x \sqrt {a+b x^2+c x^4}+\frac {b d x \sqrt {a+b x^2+c x^4}}{3 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c}-\frac {\left (b^2-4 a c\right ) e \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2}}-\frac {\sqrt [4]{a} b d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 c^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 c^{3/4} \sqrt {a+b x^2+c x^4}} \] Output:

1/3*d*x*(c*x^4+b*x^2+a)^(1/2)+1/3*b*d*x*(c*x^4+b*x^2+a)^(1/2)/c^(1/2)/(a^( 
1/2)+c^(1/2)*x^2)+1/8*e*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c-1/16*(-4*a*c+b 
^2)*e*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(3/2)-1/3*a 
^(1/4)*b*d*(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)+1/6*a^(1/4)*(b+2*a^(1/2)*c^(1/2))*d 
*(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)*Inv 
erseJacobiAM(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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.87 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.46 \[ \int (d+e x) \sqrt {a+b x^2+c x^4} \, dx=\frac {1}{48} \left (\frac {6 b e \sqrt {a+b x^2+c x^4}}{c}+16 d x \sqrt {a+b x^2+c x^4}+12 e x^2 \sqrt {a+b x^2+c x^4}-\frac {16 i \sqrt {2} a b \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{c \sqrt {a+b x^2+c x^4}}-\frac {16 i \sqrt {2} a \sqrt {b^2-4 a c} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{c \sqrt {a+b x^2+c x^4}}+\frac {3 b^2 e \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{c^{3/2}}-\frac {12 a e \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{\sqrt {c}}\right ) \] Input:

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

Output:

((6*b*e*Sqrt[a + b*x^2 + c*x^4])/c + 16*d*x*Sqrt[a + b*x^2 + c*x^4] + 12*e 
*x^2*Sqrt[a + b*x^2 + c*x^4] - ((16*I)*Sqrt[2]*a*b*Sqrt[c/(b + Sqrt[b^2 - 
4*a*c])]*d*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])] 
*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*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])])/(c*Sqrt[a + b*x^2 + c*x^4]) - ((16*I)*Sqrt[ 
2]*a*Sqrt[b^2 - 4*a*c]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*Sqrt[(b - Sqrt[b^ 
2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] 
 + 2*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]) 
])/(c*Sqrt[a + b*x^2 + c*x^4]) + (3*b^2*e*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt 
[a + b*x^2 + c*x^4]])/c^(3/2) - (12*a*e*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a 
 + b*x^2 + c*x^4]])/Sqrt[c])/48
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2202, 27, 1404, 1432, 1087, 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.

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1404

\(\displaystyle d \left (\frac {1}{3} \int \frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}dx+\frac {1}{3} x \sqrt {a+b x^2+c x^4}\right )+e \int x \sqrt {c x^4+b x^2+a}dx\)

\(\Big \downarrow \) 1432

\(\displaystyle d \left (\frac {1}{3} \int \frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}dx+\frac {1}{3} x \sqrt {a+b x^2+c x^4}\right )+\frac {1}{2} e \int \sqrt {c x^4+b x^2+a}dx^2\)

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} e \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )+d \left (\frac {1}{3} \int \frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}dx+\frac {1}{3} x \sqrt {a+b x^2+c x^4}\right )\)

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

(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))))/2 + d*( 
(x*Sqrt[a + b*x^2 + c*x^4])/3 + (-((b*(-((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]) 
 + (a^(1/4)*(2*Sqrt[a] + b/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*c^(1/4)*Sqrt[a + b*x^2 + c*x^4]))/3)
 

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 1404
Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b 
*x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1))   Int[(2*a + b*x^2)*( 
a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* 
c, 0] && GtQ[p, 0] && IntegerQ[2*p]
 

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 = 1.62 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.14

method result size
risch \(\frac {\left (6 c e \,x^{2}+8 x c d +3 e b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{24 c}+\frac {\frac {3 e \left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}+\frac {4 a c d \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {4 c b d a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}}{24 c}\) \(453\)
default \(d \left (\frac {x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+e \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}\right )\) \(456\)
elliptic \(\frac {e \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4}+\frac {d x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}+\frac {e b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c}+\frac {a d \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (\frac {a e}{2}-\frac {b^{2} e}{8 c}\right ) \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {b d a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) \(472\)

Input:

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

Output:

1/24*(6*c*e*x^2+8*c*d*x+3*b*e)/c*(c*x^4+b*x^2+a)^(1/2)+1/24/c*(3/2*e*(4*a* 
c-b^2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)+4*a*c*d*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)*Elli 
pticF(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))-4*c*b*d*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 = 0.19 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.98 \[ \int (d+e x) \sqrt {a+b x^2+c x^4} \, dx=-\frac {3 \, {\left (b^{2} - 4 \, a c\right )} \sqrt {c} e x \log \left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} + 4 \, a c\right ) - 16 \, \sqrt {\frac {1}{2}} {\left (b c d x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b^{2} d x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 16 \, \sqrt {\frac {1}{2}} {\left ({\left (b c - 2 \, c^{2}\right )} d x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (b^{2} + 2 \, b c\right )} d x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 4 \, {\left (6 \, c^{2} e x^{3} + 8 \, c^{2} d x^{2} + 3 \, b c e x + 8 \, b c d\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, c^{2} x} \] Input:

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

Output:

-1/96*(3*(b^2 - 4*a*c)*sqrt(c)*e*x*log(8*c^2*x^4 + 8*b*c*x^2 + b^2 + 4*sqr 
t(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) + 4*a*c) - 16*sqrt(1/2)*(b*c*d* 
x*sqrt((b^2 - 4*a*c)/c^2) - b^2*d*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)) + 16*sqrt( 
1/2)*((b*c - 2*c^2)*d*x*sqrt((b^2 - 4*a*c)/c^2) - (b^2 + 2*b*c)*d*x)*sqrt( 
c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sqrt(1/2)*sqr 
t((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*(6*c^2*e*x^3 + 8*c^2*d*x^2 + 3*b*c*e*x + 8*b*c*d 
)*sqrt(c*x^4 + b*x^2 + a))/(c^2*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((e*x+d)*(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), x)
 

Giac [F]

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

integrate((e*x+d)*(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), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x) \sqrt {a+b x^2+c x^4} \, dx=\frac {6 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b c e +16 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2} d x +12 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2} e \,x^{2}-12 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+b \,x^{2}+a}-\sqrt {c}\, x^{2}\right ) a c e +3 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+b \,x^{2}+a}-\sqrt {c}\, x^{2}\right ) b^{2} e +12 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+b \,x^{2}+a}+\sqrt {c}\, x^{2}\right ) a c e -3 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+b \,x^{2}+a}+\sqrt {c}\, x^{2}\right ) b^{2} e +32 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{2} c^{2} d +16 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{4}}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b^{2} c^{2} d +48 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b \,c^{2} d -24 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{2} c^{2} e +6 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x}{b c \,x^{6}+a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a \,b^{2} c e}{48 c^{2}} \] Input:

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

Output:

(6*sqrt(a + b*x**2 + c*x**4)*b*c*e + 16*sqrt(a + b*x**2 + c*x**4)*c**2*d*x 
 + 12*sqrt(a + b*x**2 + c*x**4)*c**2*e*x**2 - 12*sqrt(c)*log(sqrt(a + b*x* 
*2 + c*x**4) - sqrt(c)*x**2)*a*c*e + 3*sqrt(c)*log(sqrt(a + b*x**2 + c*x** 
4) - sqrt(c)*x**2)*b**2*e + 12*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqr 
t(c)*x**2)*a*c*e - 3*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2) 
*b**2*e + 32*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 + 16*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 
 + 48*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 - 24*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**2*e + 6 
*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*b**2*c*e)/(48*c**2)