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

Optimal. Leaf size=775 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{-\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}}}{\sqrt{a+b x^4}}\right )}{4 c \sqrt{\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}+\frac{\tan ^{-1}\left (\frac{x \sqrt{\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}}}{\sqrt{a+b x^4}}\right )}{4 c \sqrt{-\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt{a+b x^4} \left (\sqrt{b} c-\sqrt{a} \sqrt{-c} \sqrt{d}\right )}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt{a+b x^4} \left (\sqrt{a} \sqrt{-c} \sqrt{d}+\sqrt{b} c\right )}-\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{-c}\right ) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt{a+b x^4} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )}-\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt{a+b x^4} \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{-c}\right )} \]

[Out]

ArcTan[(Sqrt[-((Sqrt[-c]*(b - (a*d)/c))/Sqrt[d])]*x)/Sqrt[a + b*x^4]]/(4*c*Sqrt[
(b*c - a*d)/(Sqrt[-c]*Sqrt[d])]) + ArcTan[(Sqrt[(Sqrt[-c]*(b - (a*d)/c))/Sqrt[d]
]*x)/Sqrt[a + b*x^4]]/(4*c*Sqrt[-((b*c - a*d)/(Sqrt[-c]*Sqrt[d]))]) + (b^(1/4)*(
Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*A
rcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*(Sqrt[b]*c - Sqrt[a]*Sqrt[-c]*Sqrt[
d])*Sqrt[a + b*x^4]) + (b^(1/4)*(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])/(4*a^(1/4)*(S
qrt[b]*c + Sqrt[a]*Sqrt[-c]*Sqrt[d])*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] + Sqr
t[a]*Sqrt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2
]*EllipticPi[-(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]*Sq
rt[-c] - Sqrt[a]*Sqrt[d])*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[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] + Sqr
t[a]*Sqrt[d])*Sqrt[a + b*x^4])

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Rubi [A]  time = 1.23431, antiderivative size = 775, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{-\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}}}{\sqrt{a+b x^4}}\right )}{4 c \sqrt{\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}+\frac{\tan ^{-1}\left (\frac{x \sqrt{\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}}}{\sqrt{a+b x^4}}\right )}{4 c \sqrt{-\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt{a+b x^4} \left (\sqrt{b} c-\sqrt{a} \sqrt{-c} \sqrt{d}\right )}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt{a+b x^4} \left (\sqrt{a} \sqrt{-c} \sqrt{d}+\sqrt{b} c\right )}-\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{-c}\right ) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt{a+b x^4} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )}-\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt{a+b x^4} \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{-c}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

ArcTan[(Sqrt[-((Sqrt[-c]*(b - (a*d)/c))/Sqrt[d])]*x)/Sqrt[a + b*x^4]]/(4*c*Sqrt[
(b*c - a*d)/(Sqrt[-c]*Sqrt[d])]) + ArcTan[(Sqrt[(Sqrt[-c]*(b - (a*d)/c))/Sqrt[d]
]*x)/Sqrt[a + b*x^4]]/(4*c*Sqrt[-((b*c - a*d)/(Sqrt[-c]*Sqrt[d]))]) + (b^(1/4)*(
Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*A
rcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*(Sqrt[b]*c - Sqrt[a]*Sqrt[-c]*Sqrt[
d])*Sqrt[a + b*x^4]) + (b^(1/4)*(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])/(4*a^(1/4)*(S
qrt[b]*c + Sqrt[a]*Sqrt[-c]*Sqrt[d])*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] + Sqr
t[a]*Sqrt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2
]*EllipticPi[-(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]*Sq
rt[-c] - Sqrt[a]*Sqrt[d])*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[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] + Sqr
t[a]*Sqrt[d])*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 100.558, size = 690, normalized size = 0.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x*sqrt(sqrt(-c)*(a*d - b*c)/(c*sqrt(d)))/sqrt(a + b*x**4))/(4*c*sqrt(sqrt(-
c)*(a*d - b*c)/(c*sqrt(d)))) + atan(x*sqrt(sqrt(-c)*(-a*d + b*c)/(c*sqrt(d)))/sq
rt(a + b*x**4))/(4*c*sqrt(sqrt(-c)*(-a*d + b*c)/(c*sqrt(d)))) - b**(1/4)*sqrt((a
+ b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*elliptic_f(2*at
an(b**(1/4)*x/a**(1/4)), 1/2)/(4*a**(1/4)*sqrt(-c)*sqrt(a + b*x**4)*(sqrt(a)*sqr
t(d) + sqrt(b)*sqrt(-c))) + b**(1/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)*
*2)*(sqrt(a) + sqrt(b)*x**2)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(4*a**
(1/4)*sqrt(-c)*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))*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*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))*elliptic_pi(-(sqrt(a)*sqrt(d) - sqrt(b)*sqrt(-c))**2/(4*sqrt(a)*sq
rt(b)*sqrt(d)*sqrt(-c)), 2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(8*a**(1/4)*b**(1/4)*
c*sqrt(a + b*x**4)*(sqrt(a)*sqrt(d) - sqrt(b)*sqrt(-c)))

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Mathematica [C]  time = 0.0810482, size = 161, normalized size = 0.21 \[ -\frac{5 a c x F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\sqrt{a+b x^4} \left (c+d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/(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*Appell
F1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))

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Maple [C]  time = 0.021, size = 191, normalized size = 0.3 \[{\frac{1}{8\,d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d+c \right ) }{\frac{1}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+2\,{\frac{d{{\it \_alpha}}^{3}}{c\sqrt{b{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/d*sum(1/_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)*El
lipticPi(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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x^{4}} \left (c + d x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a}{\left (d x^{4} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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