\(\int \frac {A+B x}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx\) [230]

Optimal result

Integrand size = 29, antiderivative size = 553 \[ \int \frac {A+B x}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=-\frac {(B d-A e) \text {arctanh}\left (\frac {\sqrt {c d^4+b d^2 e^2+a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c d^4+b d^2 e^2+a e^4}}+\frac {(B d-A e) \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b d^2 e^2+a e^4} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c d^4+b d^2 e^2+a e^4}}+\frac {\left (A \sqrt {c} d+\sqrt {a} B e\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {(B d-A e) \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}} \] Output:

-1/2*(-A*e+B*d)*arctanh((a*e^4+b*d^2*e^2+c*d^4)^(1/2)*x/d/e/(c*x^4+b*x^2+a 
)^(1/2))/(a*e^4+b*d^2*e^2+c*d^4)^(1/2)+1/2*(-A*e+B*d)*arctanh(1/2*(b*d^2+2 
*a*e^2+(b*e^2+2*c*d^2)*x^2)/(a*e^4+b*d^2*e^2+c*d^4)^(1/2)/(c*x^4+b*x^2+a)^ 
(1/2))/(a*e^4+b*d^2*e^2+c*d^4)^(1/2)+1/2*(A*c^(1/2)*d+a^(1/2)*B*e)*(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)*InverseJaco 
biAM(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^(1/4)/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)+1/4*(-A*e+B*d)*(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)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(c^(1/2 
)*d^2+a^(1/2)*e^2)^2/a^(1/2)/c^(1/2)/d^2/e^2,1/2*(2-b/a^(1/2)/c^(1/2))^(1/ 
2))/a^(1/4)/c^(1/4)/d/e/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 18.26 (sec) , antiderivative size = 3652, normalized size of antiderivative = 6.60 \[ \int \frac {A+B x}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=\text {Result too large to show} \] Input:

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

Output:

((-I)*B*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (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])])/(Sqr 
t[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*e*Sqrt[a + b*x^2 + c*x^4]) + (2*A 
*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4* 
a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x)^2*Sqr 
t[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/ 
Sqrt[2]) + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) 
+ Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt 
[2]) + x))]*Sqrt[(Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c]*(Sqrt[-(b/c) + Sqrt[b^2 
 - 4*a*c]/c]/Sqrt[2] + x))/((Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - 
Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4* 
a*c]/c]/Sqrt[2]) + x))]*Sqrt[((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - Sqrt[(-b 
 + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + 2*x) 
)/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*( 
Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x))]*((-d + (Sqrt[-(b/c) - Sq 
rt[b^2 - 4*a*c]/c]*e)/Sqrt[2])*EllipticF[ArcSin[Sqrt[((Sqrt[(-b - Sqrt[b^2 
 - 4*a*c])/c] - Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt 
[b^2 - 4*a*c])/c] + 2*x))/((Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] + Sqrt[(-b + 
Sqrt[b^2 - 4*a*c])/c])*(Sqrt[2]*Sqrt[(-b - Sqrt[b^2 - 4*a*c])/c] - 2*x)...
 

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2279, 27, 1576, 1154, 219, 2226, 27, 1416, 2222}

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.

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

\(\Big \downarrow \) 2279

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1576

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 2226

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 2222

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

Input:

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

Output:

((B*d - A*e)*ArcTanh[(b*d^2 + 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*Sqrt[c*d 
^4 + b*d^2*e^2 + a*e^4]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[c*d^4 + b*d^2*e 
^2 + a*e^4]) + ((A*Sqrt[c]*d + Sqrt[a]*B*e)*(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[c]*d^2 
 + Sqrt[a]*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (d*e*(B*d - A*e)*(((Sqrt[c]*d^2 
 + Sqrt[a]*e^2)*ArcTanh[(Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]*x)/(d*e*Sqrt[a + 
b*x^2 + c*x^4])])/(2*d*e*Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]) + ((Sqrt[a]/d^2 
- Sqrt[c]/e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + 
 Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[ 
c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/ 
(4*a^(1/4)*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/(Sqrt[c]*d^2 + Sqrt[a]*e^2)
 

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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, 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 1576

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

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

rule 2226

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

rule 2279

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x 
_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x 
, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e 
^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4 
)/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e}, 
x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4 
, 0]
 

Maple [A] (verified)

Time = 1.51 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {B \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (A e -B d \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{2}}\)	\(437\)
elliptic	\(\frac {B \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (A e -B d \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{2}}\)	\(437\)

Input:

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

Output:

1/4*B/e*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))+(A*e-B*d)/e^2*(-1/2/(c*d^4/e^4+b*d^2 
/e^2+a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+b*d^2/e^2+b*x^2+2*a)/(c*d^4/e^4 
+b*d^2/e^2+a)^(1/2)/(c*x^4+b*x^2+a)^(1/2))+2^(1/2)/((-b+(-4*a*c+b^2)^(1/2) 
)/a)^(1/2)/d*e*(1-1/2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(1+1/2*(b+(-4*a 
*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2) 
*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),2/(-b+(-4*a*c+b^2)^(1/2))*a/d^2*e^2,(-1 
/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/ 
2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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