\(\int \frac {\sqrt {a-b x^4}}{c-d x^4} \, dx\) [44]

Optimal result

Integrand size = 23, antiderivative size = 240 \[ \int \frac {\sqrt {a-b x^4}}{c-d x^4} \, dx=\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d \sqrt {a-b x^4}} \] Output:

a^(1/4)*b^(3/4)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/d/(-b*x^4 
+a)^(1/2)-1/2*a^(1/4)*(-a*d+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/d/(-b*x^4+a)^(1/2)-1/2 
*a^(1/4)*(-a*d+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/d/(-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 = 10.21 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {a-b x^4}}{c-d x^4} \, dx=-\frac {5 a c x \sqrt {a-b x^4} \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{\left (c-d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (-2 a d \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )} \] Input:

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

Output:

(-5*a*c*x*Sqrt[a - b*x^4]*AppellF1[1/4, -1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c 
])/((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*Ap 
pellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {922, 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.

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

\(\Big \downarrow \) 922

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

(a^(1/4)*b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)] 
, -1])/(d*Sqrt[a - b*x^4]) - ((b*c - a*d)*((a^(1/4)*Sqrt[1 - (b*x^4)/a]*El 
lipticPi[-((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]) + (a^(1/4)*Sqrt[1 - (b*x^4)/a]*Elli 
pticPi[(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])))/d
 

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 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 922

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

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

Time = 1.58 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.08

method	result	size
default	\(\frac {b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (a d -b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{2}}\)	\(259\)
elliptic	\(\frac {b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (a d -b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{2}}\)	\(259\)

Input:

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

Output:

b/d/(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)- 
1/8/d^2*sum((a*d-b*c)/_alpha^3*(-1/((a*d-b*c)/d)^(1/2)*arctanh(1/2*(-2*_al 
pha^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. \[ \int \frac {\sqrt {a-b x^4}}{c-d x^4} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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