\(\int \frac {1}{\sqrt {a+b x^4} (c+d x^4)} \, dx\) [181]

Optimal result

Integrand size = 21, antiderivative size = 742 \[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )} \, dx=-\frac {\sqrt [4]{d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt {b c-a d}}-\frac {\sqrt [4]{d} \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt {-b c+a d}}+\frac {\sqrt [4]{b} \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} (b c+a d) \sqrt {a+b x^4}}+\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} \sqrt {-c} \sqrt {d}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c+a d) \sqrt {a+b x^4}} \]

[Out]

-1/4*d^(1/4)*arctan(x*(-a*d+b*c)^(1/2)/(-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)*arctan(x*(a*d-b*c)^(1/2)/(-c)^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/(-c)^(3/4)/(a*d-b*c)^(1/2)+1/8*(cos(2*arctan
(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*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^(1/2)+x^2*b^(1/2))*
(b^(1/2)*(-c)^(1/2)-a^(1/2)*d^(1/2))^2*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/b^(1/4)/c/(a*d+b*c)/(
b*x^4+a)^(1/2)+1/8*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*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^(1/2)+x^2*b^(1/2))*(b^(1/2)*(-c)^(1/2)+a^(1/2)*d^(1/2))^2*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1
/2)/a^(1/4)/b^(1/4)/c/(a*d+b*c)/(b*x^4+a)^(1/2)+1/4*b^(1/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*a
rctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*(b^(1/
2)+a^(1/2)*d^(1/2)/(-c)^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/(a*d+b*c)/(b*x^4+a)^(1/2)+1/4
*b^(1/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^
(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*(c*b^(1/2)+a^(1/2)*(-c)^(1/2)*d^(1/2))*((b*x^4+a)/(a^(1/2
)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/c/(a*d+b*c)/(b*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {418, 1231, 226, 1721} \[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )} \, dx=-\frac {\sqrt [4]{d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt {b c-a d}}-\frac {\sqrt [4]{d} \arctan \left (\frac {x \sqrt {a d-b c}}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt {a d-b c}}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt {a+b x^4} (a d+b c)}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c \sqrt {a+b x^4} (a d+b c)}+\frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2 \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt {a+b x^4} (a d+b c)}+\frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right )^2 \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt {a+b x^4} (a d+b c)} \]

[In]

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

[Out]

-1/4*(d^(1/4)*ArcTan[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/((-c)^(3/4)*Sqrt[b*c - a*d]) -
(d^(1/4)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/(4*(-c)^(3/4)*Sqrt[-(b*c) + a*d
]) + (b^(1/4)*(Sqrt[b] + (Sqrt[a]*Sqrt[d])/Sqrt[-c])*(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])/(4*a^(1/4)*(b*c + a*d)*Sqrt[a + b*x^4]) + (b^(1/4)*(
Sqrt[b]*c + Sqrt[a]*Sqrt[-c]*Sqrt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Elli
pticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] + Sq
rt[a]*Sqrt[d])^2*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[-1/4*(Sqrt[b]*
Sqrt[-c] - Sqrt[a]*Sqrt[d])^2/(Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(8*a^(1
/4)*b^(1/4)*c*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])^2*(Sqrt[a] + Sqrt[b]*x^2)*S
qrt[(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])/(8*a^(1/4)*b^(1/4)*c*(b*c + a*d)*Sqrt[a + b*x^4])

Rule 226

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 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[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 1231

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(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 1721

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)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {a+b x^4}} \, dx}{2 c}+\frac {\int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {a+b x^4}} \, dx}{2 c} \\ & = \frac {\left (\sqrt {b} \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 (b c+a d)}+\frac {\left (\sqrt {b} \left (\sqrt {b} c+\sqrt {a} \sqrt {-c} \sqrt {d}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 c (b c+a d)}-\frac {\left (\sqrt {a} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \sqrt {d}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {a+b x^4}} \, dx}{2 c (b c+a d)}+\frac {\left (\sqrt {a} \left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \sqrt {d}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {a+b x^4}} \, dx}{2 c (b c+a d)} \\ & = -\frac {\sqrt [4]{d} \tan ^{-1}\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} \tan ^{-1}\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]{b} \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} (b c+a d) \sqrt {a+b x^4}}+\frac {\sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} \sqrt {-c} \sqrt {d}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} 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}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}};2 \tan ^{-1}\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}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}};2 \tan ^{-1}\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}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 10.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )} \, dx=-\frac {5 a c x \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{\sqrt {a+b x^4} \left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )} \]

[In]

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

[Out]

(-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*AppellF
1[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)])))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.26

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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