\(\int \frac {1}{\sqrt {a-b x^4} (c-d x^4)^2} \, dx\) [57]

Optimal result

Integrand size = 23, antiderivative size = 310 \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )^2} \, dx=-\frac {d x \sqrt {a-b x^4}}{4 c (b c-a d) \left (c-d x^4\right )}-\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c-a d) \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (5 b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d) \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (5 b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d) \sqrt {a-b x^4}} \] Output:

-1/4*d*x*(-b*x^4+a)^(1/2)/c/(-a*d+b*c)/(-d*x^4+c)-1/4*a^(1/4)*b^(3/4)*(1-b 
*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/c/(-a*d+b*c)/(-b*x^4+a)^(1/2) 
+1/8*a^(1/4)*(-3*a*d+5*b*c)*(1-b*x^4/a)^(1/2)*EllipticPi(b^(1/4)*x/a^(1/4) 
,-a^(1/2)*d^(1/2)/b^(1/2)/c^(1/2),I)/b^(1/4)/c^2/(-a*d+b*c)/(-b*x^4+a)^(1/ 
2)+1/8*a^(1/4)*(-3*a*d+5*b*c)*(1-b*x^4/a)^(1/2)*EllipticPi(b^(1/4)*x/a^(1/ 
4),a^(1/2)*d^(1/2)/b^(1/2)/c^(1/2),I)/b^(1/4)/c^2/(-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 = 10.34 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )^2} \, dx=\frac {5 a c x \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right ) \left (-5 c \left (4 b c-4 a d+b d x^4\right )+b d x^4 \sqrt {1-\frac {b x^4}{a}} \left (-c+d x^4\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+2 d x^5 \left (5 c \left (a-b x^4\right )+b x^4 \sqrt {1-\frac {b x^4}{a}} \left (-c+d x^4\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}{20 c^2 (b c-a d) \sqrt {a-b x^4} \left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )} \] Input:

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

Output:

(5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c]*(-5*c*(4*b*c - 4 
*a*d + b*d*x^4) + b*d*x^4*Sqrt[1 - (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1 
/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]) + 2*d*x^5*(5*c*(a - b*x^4) + b*x^4*Sqrt 
[1 - (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4) 
/c])*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF 
1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))/(20*c^2*(b*c - a*d)*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*App 
ellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {931, 25, 1021, 765, 762, 925, 27, 1543, 1542}

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.

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

\(\Big \downarrow \) 931

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1021

\(\displaystyle \frac {(5 b c-3 a d) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx-b \int \frac {1}{\sqrt {a-b x^4}}dx}{4 c (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {(5 b c-3 a d) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx-\frac {b \sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {(5 b c-3 a d) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx-\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {(5 b c-3 a d) \left (\frac {\int \frac {\sqrt {c}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 c}+\frac {\int \frac {\sqrt {c}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 c}\right )-\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(5 b c-3 a d) \left (\frac {\int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}\right )-\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 1543

\(\displaystyle \frac {(5 b c-3 a d) \left (\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}+\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}\right )-\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {(5 b c-3 a d) \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}\right )-\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)}\)

Input:

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

Output:

-1/4*(d*x*Sqrt[a - b*x^4])/(c*(b*c - a*d)*(c - d*x^4)) + (-((a^(1/4)*b^(3/ 
4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/Sqrt[a 
- b*x^4]) + (5*b*c - 3*a*d)*((a^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sq 
rt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^ 
(1/4)*c*Sqrt[a - b*x^4]) + (a^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a 
]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4) 
*c*Sqrt[a - b*x^4])))/(4*c*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

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 931

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d))   Int[(a + b*x^n)^(p + 1)*(c 
 + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, 
 x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, 
-1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, 
 c, d, n, p, q, x]
 

rule 1021

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x 
_)^(n_)]), x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* 
e - a*f)/b   Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, 
 d, e, f, n}, x]
 

rule 1542

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

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

Maple [C] (verified)

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

Time = 2.23 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.04

method	result	size
default	\(\frac {d x \sqrt {-b \,x^{4}+a}}{4 c \left (a d -b c \right ) \left (-d \,x^{4}+c \right )}+\frac {b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{4 c \left (a d -b c \right ) \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (3 a d -5 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c d}\)	\(321\)
elliptic	\(\frac {d x \sqrt {-b \,x^{4}+a}}{4 c \left (a d -b c \right ) \left (-d \,x^{4}+c \right )}+\frac {b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{4 c \left (a d -b c \right ) \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (3 a d -5 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c d}\)	\(321\)

Input:

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

Output:

1/4/c*d/(a*d-b*c)*x*(-b*x^4+a)^(1/2)/(-d*x^4+c)+1/4*b/c/(a*d-b*c)/(1/a^(1/ 
2)*b^(1/2))^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1 
/2)/(-b*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-1/32/c/d*sum 
((3*a*d-5*b*c)/(a*d-b*c)/_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/(1/a^(1/2)*b^( 
1/2))^(1/2)*_alpha^3*d/c*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1 
/2))^(1/2)/(-b*x^4+a)^(1/2)*EllipticPi(x*(1/a^(1/2)*b^(1/2))^(1/2),a^(1/2) 
/b^(1/2)*_alpha^2/c*d,(-1/a^(1/2)*b^(1/2))^(1/2)/(1/a^(1/2)*b^(1/2))^(1/2) 
)),_alpha=RootOf(_Z^4*d-c))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )^2} \, dx=\int \frac {1}{\sqrt {a - b x^{4}} \left (- c + d x^{4}\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )^2} \, dx=\int { \frac {1}{\sqrt {-b x^{4} + a} {\left (d x^{4} - c\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )^2} \, dx=\int { \frac {1}{\sqrt {-b x^{4} + a} {\left (d x^{4} - c\right )}^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )^2} \, dx=\int \frac {1}{\sqrt {a-b\,x^4}\,{\left (c-d\,x^4\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )^2} \, dx=\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,d^{2} x^{12}+a \,d^{2} x^{8}+2 b c d \,x^{8}-2 a c d \,x^{4}-b \,c^{2} x^{4}+a \,c^{2}}d x \] Input:

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

Output:

int(sqrt(a - b*x**4)/(a*c**2 - 2*a*c*d*x**4 + a*d**2*x**8 - b*c**2*x**4 + 
2*b*c*d*x**8 - b*d**2*x**12),x)