\(\int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx\) [111]

Optimal result

Integrand size = 21, antiderivative size = 166 \[ \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx=\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c} \] Output:

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

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 = 160, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx=\frac {5 a c x \sqrt [4]{a+b x^4} \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{\left (c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+x^4 \left (-4 a d \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {923, 925, 27, 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.

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

\(\Big \downarrow \) 923

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*(EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b] 
*Sqrt[c])), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1]/(2*b^(1/4)*c) + Ell 
ipticPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^ 
(1/4)], -1]/(2*b^(1/4)*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 923

Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Simp[Sq 
rt[a + b*x^4]*Sqrt[a/(a + b*x^4)]   Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c 
- a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)], 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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