\(\int \frac {A+B x^2}{(d+e x^2) \sqrt {b x^2+c (\frac {a}{c}+x^4)}} \, dx\) [184]

Optimal result

Integrand size = 38, antiderivative size = 439 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {b x^2+c \left (\frac {a}{c}+x^4\right )}} \, dx=-\frac {(B d-A e) \arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {c d^2-b d e+a e^2}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+b x^2+c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {b x^2+c \left (\frac {a}{c}+x^4\right )}} \, dx=-\frac {i \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (B d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+(-B d+A e) \operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d e \sqrt {a+b x^2+c x^4}} \] Input:

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

Output:

((-I)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt 
[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(B*d*EllipticF[I*ArcSinh[Sqrt[2]*S 
qrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 
 4*a*c])] + (-(B*d) + A*e)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), 
 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[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d 
*e*Sqrt[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2091, 2226, 27, 1416, 2220}

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.

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

\(\Big \downarrow \) 2091

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

\(\Big \downarrow \) 2226

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 2220

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

Input:

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

Output:

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

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 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 2091

Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex 
pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && PolyQ[Px, x] && BinomialQ[z, x 
] && TrinomialQ[u, x] &&  !(BinomialMatchQ[z, x] && TrinomialMatchQ[u, x])
 

rule 2220

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 
rcTan[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]))*El 
lipticPi[-(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] && PosQ[-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]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {B \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (A e -B d \right ) \sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{e d \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\)	\(359\)
elliptic	\(\frac {B \sqrt {2}\, \sqrt {4+\frac {2 b \,x^{2}}{a}-\frac {2 x^{2} \sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {4+\frac {2 b \,x^{2}}{a}+\frac {2 x^{2} \sqrt {-4 a c +b^{2}}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right ) A}{d \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \operatorname {EllipticPi}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right ) B}{e \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\)	\(565\)

Input:

int((B*x^2+A)/(e*x^2+d)/(b*x^2+c*(a/c+x^4))^(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/d*2^(1/2)/(-b/a+1/a*(-4* 
a*c+b^2)^(1/2))^(1/2)*(1+1/2*b/a*x^2-1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)*( 
1+1/2*b/a*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*El 
lipticPi(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*e,(-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. \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {b x^2+c \left (\frac {a}{c}+x^4\right )}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a + b*x**2 + c*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d* 
x**4 + c*e*x**6),x)*a + int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a*d + a*e*x* 
*2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*b