\(\int \frac {1}{\sqrt {a+b x^4} (c-d x^4)} \, dx\) [50]

Optimal result

Integrand size = 22, antiderivative size = 571 \[ \int \frac {1}{\sqrt {a+b x^4} \left (c-d x^4\right )} \, dx=\frac {\sqrt [4]{d} \arctan \left (\frac {\sqrt {b c+a d} x}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 c^{3/4} \sqrt {b c+a d}}+\frac {\sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt {b c+a d} x}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 c^{3/4} \sqrt {b c+a d}}+\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} (b c-a d) \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d) \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right )^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d) \sqrt {a+b x^4}} \] Output:

1/4*d^(1/4)*arctan((a*d+b*c)^(1/2)*x/c^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/c^(3 
/4)/(a*d+b*c)^(1/2)+1/4*d^(1/4)*arctanh((a*d+b*c)^(1/2)*x/c^(1/4)/d^(1/4)/ 
(b*x^4+a)^(1/2))/c^(3/4)/(a*d+b*c)^(1/2)+1/2*b^(3/4)*(a^(1/2)+b^(1/2)*x^2) 
*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4 
)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/(-a*d+b*c)/(b*x^4+a)^(1/2)-1/8*(b^(1/2)* 
c^(1/2)+a^(1/2)*d^(1/2))^2*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/ 
2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(b^(1/4)*x/a^(1/4))),-1/4*(b^(1/2) 
*c^(1/2)-a^(1/2)*d^(1/2))^2/a^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))/a 
^(1/4)/b^(1/4)/c/(-a*d+b*c)/(b*x^4+a)^(1/2)-1/8*(b^(1/2)*c^(1/2)-a^(1/2)*d 
^(1/2))^2*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)* 
EllipticPi(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/4*(b^(1/2)*c^(1/2)+a^(1/2)*d 
^(1/2))^2/a^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))/a^(1/4)/b^(1/4)/c/( 
-a*d+b*c)/(b*x^4+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt {a+b x^4} \left (c-d x^4\right )} \, dx=-\frac {5 a c x \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )}{\sqrt {a+b x^4} \left (c-d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (-2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )} \] Input:

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

Output:

(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), (d*x^4)/c])/(Sqrt[a + b 
*x^4]*(c - d*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), (d*x^4) 
/c] + 2*x^4*(-2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), (d*x^4)/c] + 
b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), (d*x^4)/c])))
 

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.41, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {925, 27, 1541, 27, 761, 2221, 2223}

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.

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

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

\(\Big \downarrow \) 2223

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

Input:

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

Output:

((b^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2) 
^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[b]*Sqr 
t[c] - Sqrt[a]*Sqrt[d])*Sqrt[a + b*x^4]) - (Sqrt[d]*(-1/2*((Sqrt[b]*Sqrt[c 
] - Sqrt[a]*Sqrt[d])*ArcTan[(Sqrt[b*c + a*d]*x)/(c^(1/4)*d^(1/4)*Sqrt[a + 
b*x^4])])/(c^(1/4)*d^(1/4)*Sqrt[b*c + a*d]) + ((Sqrt[a]/Sqrt[c] + Sqrt[b]/ 
Sqrt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^ 
2]*EllipticPi[-1/4*(Sqrt[a]*((Sqrt[b]*Sqrt[c])/Sqrt[a] - Sqrt[d])^2)/(Sqrt 
[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*b^(1 
/4)*Sqrt[a + b*x^4])))/(Sqrt[b]*Sqrt[c] - Sqrt[a]*Sqrt[d]))/(2*Sqrt[c]) + 
((b^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2) 
^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[b]*Sqr 
t[c] + Sqrt[a]*Sqrt[d])*Sqrt[a + b*x^4]) + (Sqrt[d]*(((Sqrt[b]*Sqrt[c] + S 
qrt[a]*Sqrt[d])*ArcTanh[(Sqrt[b*c + a*d]*x)/(c^(1/4)*d^(1/4)*Sqrt[a + b*x^ 
4])])/(2*c^(1/4)*d^(1/4)*Sqrt[b*c + a*d]) + ((Sqrt[a]/Sqrt[c] - Sqrt[b]/Sq 
rt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2] 
*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[a]*Sqrt[d])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[ 
c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*Sqrt[ 
a + b*x^4])))/(Sqrt[b]*Sqrt[c] + Sqrt[a]*Sqrt[d]))/(2*Sqrt[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 761

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

rule 925

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

rule 1541

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

rule 2221

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

rule 2223

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

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.58 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.33

method	result	size
default	\(-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d +b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d +b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d}\)	\(191\)
elliptic	\(-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d +b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d +b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d}\)	\(191\)

Input:

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

Output:

-1/8/d*sum(1/_alpha^3*(-1/((a*d+b*c)/d)^(1/2)*arctanh(1/2*(2*_alpha^2*b*x^ 
2+2*a)/((a*d+b*c)/d)^(1/2)/(b*x^4+a)^(1/2))-2/(I/a^(1/2)*b^(1/2))^(1/2)*_a 
lpha^3*d/c*(1-I*b^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*b^(1/2)*x^2/a^(1/2))^(1/2) 
/(b*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*b^(1/2))^(1/2),-I*a^(1/2)/b^(1/2) 
*_alpha^2/c*d,(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2))),_alph 
a=RootOf(_Z^4*d-c))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^4} \left (c-d x^4\right )} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x^4} \left (c-d x^4\right )} \, dx=- \int \frac {1}{- c \sqrt {a + b x^{4}} + d x^{4} \sqrt {a + b x^{4}}}\, dx \] Input:

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

Output:

-Integral(1/(-c*sqrt(a + b*x**4) + d*x**4*sqrt(a + b*x**4)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x^4} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{\sqrt {b x^{4} + a} {\left (d x^{4} - c\right )}} \,d x } \] Input:

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

Output:

-integrate(1/(sqrt(b*x^4 + a)*(d*x^4 - c)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {a+b x^4} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{\sqrt {b x^{4} + a} {\left (d x^{4} - c\right )}} \,d x } \] Input:

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

Output:

integrate(-1/(sqrt(b*x^4 + a)*(d*x^4 - c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^4} \left (c-d x^4\right )} \, dx=\int \frac {1}{\sqrt {b\,x^4+a}\,\left (c-d\,x^4\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {a+b x^4} \left (c-d x^4\right )} \, dx=\int \frac {\sqrt {b \,x^{4}+a}}{-b d \,x^{8}-a d \,x^{4}+b c \,x^{4}+a c}d x \] Input:

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

Output:

int(sqrt(a + b*x**4)/(a*c - a*d*x**4 + b*c*x**4 - b*d*x**8),x)