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

Optimal. Leaf size=932 \[ \text{result too large to display} \]

[Out]

-((b*c - a*d)*ArcTan[(Sqrt[-((Sqrt[-c]*(b - (a*d)/c))/Sqrt[d])]*x)/Sqrt[a + b*x^
4]])/(4*c*d*Sqrt[(b*c - a*d)/(Sqrt[-c]*Sqrt[d])]) - ((b*c - a*d)*ArcTan[(Sqrt[(S
qrt[-c]*(b - (a*d)/c))/Sqrt[d]]*x)/Sqrt[a + b*x^4]])/(4*c*d*Sqrt[-((b*c - a*d)/(
Sqrt[-c]*Sqrt[d]))]) + (b^(3/4)*(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])/(2*a^(1/4)*d*
Sqrt[a + b*x^4]) - (b^(1/4)*(b*c - a*d)*(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)*(Sqrt[b]*c - Sqrt[a]*Sqrt[-c]*Sqrt[d])*d*Sqrt[a + b*x^4]) - (b^(1/4)*(b*c
- a*d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellip
ticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*(Sqrt[b]*c + Sqrt[a]*Sqrt[-
c]*Sqrt[d])*d*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*(b*c - a*
d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticP
i[-(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*(Sqrt[b]*Sqrt[-c] - Sq
rt[a]*Sqrt[d])*d*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*(b*c -
a*d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellipt
icPi[(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*(Sqrt[b]*Sqrt[-c] +
Sqrt[a]*Sqrt[d])*d*Sqrt[a + b*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 1.76572, antiderivative size = 932, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right )}{4 c d \sqrt{\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right )}{4 c d \sqrt{-\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}-\frac{\sqrt [4]{b} (b c-a d) \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\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} \left (\sqrt{b} c-\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d \sqrt{b x^4+a}}-\frac{\sqrt [4]{b} (b c-a d) \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\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} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d \sqrt{b x^4+a}}+\frac{b^{3/4} \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{b x^4+a}}+\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) (b c-a d) \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\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 \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) d \sqrt{b x^4+a}}+\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) (b c-a d) \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\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 \left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) d \sqrt{b x^4+a}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((b*c - a*d)*ArcTan[(Sqrt[-((Sqrt[-c]*(b - (a*d)/c))/Sqrt[d])]*x)/Sqrt[a + b*x^
4]])/(4*c*d*Sqrt[(b*c - a*d)/(Sqrt[-c]*Sqrt[d])]) - ((b*c - a*d)*ArcTan[(Sqrt[(S
qrt[-c]*(b - (a*d)/c))/Sqrt[d]]*x)/Sqrt[a + b*x^4]])/(4*c*d*Sqrt[-((b*c - a*d)/(
Sqrt[-c]*Sqrt[d]))]) + (b^(3/4)*(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])/(2*a^(1/4)*d*
Sqrt[a + b*x^4]) - (b^(1/4)*(b*c - a*d)*(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)*(Sqrt[b]*c - Sqrt[a]*Sqrt[-c]*Sqrt[d])*d*Sqrt[a + b*x^4]) - (b^(1/4)*(b*c
- a*d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellip
ticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*(Sqrt[b]*c + Sqrt[a]*Sqrt[-
c]*Sqrt[d])*d*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*(b*c - a*
d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticP
i[-(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*(Sqrt[b]*Sqrt[-c] - Sq
rt[a]*Sqrt[d])*d*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*(b*c -
a*d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellipt
icPi[(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*(Sqrt[b]*Sqrt[-c] +
Sqrt[a]*Sqrt[d])*d*Sqrt[a + b*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 149.324, size = 814, normalized size = 0.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/2)/(d*x**4+c),x)

[Out]

(a*d - b*c)*atan(x*sqrt(sqrt(-c)*(-a*d + b*c)/(c*sqrt(d)))/sqrt(a + b*x**4))/(4*
c*d*sqrt(-sqrt(-c)*(a*d - b*c)/(c*sqrt(d)))) + (a*d - b*c)*atan(x*sqrt(sqrt(-c)*
(a*d - b*c)/(c*sqrt(d)))/sqrt(a + b*x**4))/(4*c*d*sqrt(sqrt(-c)*(a*d - b*c)/(c*s
qrt(d)))) + b**(3/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + s
qrt(b)*x**2)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(2*a**(1/4)*d*sqrt(a +
b*x**4)) + b**(1/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + s
qrt(b)*x**2)*(a*d - b*c)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(4*a**(1/4
)*d*sqrt(a + b*x**4)*(sqrt(a)*sqrt(d)*sqrt(-c) + sqrt(b)*c)) + b**(1/4)*sqrt((a
+ b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*(a*d - b*c)*elli
ptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(4*a**(1/4)*d*sqrt(a + b*x**4)*(-sqrt(a
)*sqrt(d)*sqrt(-c) + sqrt(b)*c)) + sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2
)*(sqrt(a) + sqrt(b)*x**2)*(sqrt(a)*sqrt(d) - sqrt(b)*sqrt(-c))*(a*d - b*c)*elli
ptic_pi((sqrt(a)*sqrt(d) + sqrt(b)*sqrt(-c))**2/(4*sqrt(a)*sqrt(b)*sqrt(d)*sqrt(
-c)), 2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(8*a**(1/4)*b**(1/4)*c*d*sqrt(a + b*x**4
)*(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(-c))) + sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x
**2)**2)*(sqrt(a) + sqrt(b)*x**2)*(sqrt(a)*sqrt(d) + sqrt(b)*sqrt(-c))*(a*d - b*
c)*elliptic_pi(-(sqrt(a)*sqrt(d) - sqrt(b)*sqrt(-c))**2/(4*sqrt(a)*sqrt(b)*sqrt(
d)*sqrt(-c)), 2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(8*a**(1/4)*b**(1/4)*c*d*sqrt(a
+ b*x**4)*(sqrt(a)*sqrt(d) - sqrt(b)*sqrt(-c)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.237976, size = 161, normalized size = 0.17 \[ \frac{5 a c x \sqrt{a+b x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (2 x^4 \left (b c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-2 a d F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(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*Appe
llF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))

_______________________________________________________________________________________

Maple [C]  time = 0.023, size = 273, normalized size = 0.3 \[{\frac{b}{d}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{1}{8\,{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d+c \right ) }{\frac{-ad+bc}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{b{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b/d/(I/a^(1/2)*b^(1/2))^(1/2)*(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)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-1/8/d^2*s
um((-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/(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))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{4} + a}}{d x^{4} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)/(d*x^4 + c),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)/(d*x^4 + c),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{4}}}{c + d x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{4} + a}}{d x^{4} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^4 + a)/(d*x^4 + c),x, algorithm="giac")

[Out]

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