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

Optimal. Leaf size=638 \[ \frac{d^{3/4} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{c} \sqrt{c+d x^4} (a d+b c)}+\frac{\tan ^{-1}\left (\frac{x \sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\tan ^{-1}\left (\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )} \]

[Out]

ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x)/Sqrt[c + d*x^4]]/(4*a*Sqrt[-((
b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) + ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*
x)/Sqrt[c + d*x^4]]/(4*a*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]) + (d^(3/4)*(Sqrt[
c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan
[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + ((Sqrt[b]
*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] +
Sqrt[d]*x^2)^2]*EllipticPi[-(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*
Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a*c^(1/4)*(Sqr
t[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] -
Sqrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^
2)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt
[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a*c^(1/4)*(Sqrt[b]*Sqrt[c]
+ Sqrt[-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^4])

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Rubi [A]  time = 0.817015, antiderivative size = 774, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\tan ^{-1}\left (\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right )}+\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x)/Sqrt[c + d*x^4]]/(4*a*Sqrt[-((
b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) + ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*
x)/Sqrt[c + d*x^4]]/(4*a*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]) - (d^(1/4)*(Sqrt[
c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan
[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*S
qrt[c + d*x^4]) + (d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + S
qrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*c^(1/4)*(Sqrt[-
a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*
Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ell
ipticPi[-(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt
[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[
-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(S
qrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[(Sq
rt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcT
an[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])
*d^(1/4)*Sqrt[c + d*x^4])

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Rubi in Sympy [A]  time = 98.0022, size = 690, normalized size = 1.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.082906, size = 161, normalized size = 0.25 \[ -\frac{5 a c x F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \sqrt{c+d x^4} \left (2 x^4 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.007, size = 191, normalized size = 0.3 \[{\frac{1}{8\,b}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{1}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{1\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/b*sum(1/_alpha^3*(-1/((-a*d+b*c)/b)^(1/2)*arctanh(1/2*(2*_alpha^2*d*x^2+2*c)
/((-a*d+b*c)/b)^(1/2)/(d*x^4+c)^(1/2))+2/(I/c^(1/2)*d^(1/2))^(1/2)*_alpha^3*b/a*
(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1/2)/(d*x^4+c)^(1/2)*
EllipticPi(x*(I/c^(1/2)*d^(1/2))^(1/2),I*c^(1/2)/d^(1/2)*_alpha^2/a*b,(-I/c^(1/2
)*d^(1/2))^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=RootOf(_Z^4*b+a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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