\(\int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx\) [201]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {416, 418, 1232} \[ \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx=\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c} \]

[In]

Int[(a + b*x^4)^(1/4)/(c + d*x^4),x]

[Out]

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

Rule 416

Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[Sqrt[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 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 1232

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]

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

Mathematica [C] (warning: unable to verify)

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

Time = 10.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx=\frac {5 a c x \sqrt [4]{a+b x^4} \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{\left (c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+x^4 \left (-4 a d \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )} \]

[In]

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

[Out]

(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*Appell
F1[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)])))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*x^4)^(1/4)/(c + d*x^4),x)

[Out]

int((a + b*x^4)^(1/4)/(c + d*x^4), x)