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

Optimal. Leaf size=679 \[ \frac{c^{3/4} d^{3/4} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{\sqrt{c+d x^4} (a d+b c)}+\frac{(b c-a d) \tan ^{-1}\left (\frac{x \sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a b \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a b \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) (b c-a d) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a b \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-a d) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a b \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )} \]

[Out]

((b*c - a*d)*ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x)/Sqrt[c + d*x^4]])
/(4*a*b*Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) + ((b*c - a*d)*ArcTan[(Sqrt[(b*
c - a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*a*b*Sqrt[(b*c - a*d)/(Sqrt[
-a]*Sqrt[b])]) + (c^(3/4)*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])/((b*c + a*d
)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(b*c - a*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*b*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])*(b*c - a*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*b*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 = 1.54601, antiderivative size = 931, normalized size of antiderivative = 1.37, 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{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 a b \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 a b \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\sqrt [4]{d} (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt{d x^4+c}}+\frac{\sqrt [4]{d} (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt{d x^4+c}}+\frac{d^{3/4} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{2 b \sqrt [4]{c} \sqrt{d x^4+c}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\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]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a b \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^4+c}}+\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\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]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a b \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^4+c}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((b*c - a*d)*ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x)/Sqrt[c + d*x^4]])
/(4*a*b*Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) + ((b*c - a*d)*ArcTan[(Sqrt[(b*
c - a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*a*b*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] + Sq
rt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*b*c^(1/4)*Sqrt[c
+ d*x^4]) - (d^(1/4)*(b*c - a*d)*(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*b*c^(1/4
)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*Sqrt[c + d*x^4]) + (d^(1/4)*(b*c - a*d)
*(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*b*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*S
qrt[d])*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(b*c - a*d)*(Sq
rt[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*b*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])*(b*c - a*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*Ar
cTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a*b*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 = 146.886, size = 813, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**(3/4)*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)/(2*b*c**(1/4)*sqrt(c + d*x**4)) -
d**(1/4)*sqrt((c + d*x**4)/(sqrt(c) + sqrt(d)*x**2)**2)*(sqrt(c) + sqrt(d)*x**2)
*(a*d - b*c)*elliptic_f(2*atan(d**(1/4)*x/c**(1/4)), 1/2)/(4*b*c**(1/4)*sqrt(c +
d*x**4)*(a*sqrt(d) + sqrt(b)*sqrt(c)*sqrt(-a))) - d**(1/4)*sqrt((c + d*x**4)/(s
qrt(c) + sqrt(d)*x**2)**2)*(sqrt(c) + sqrt(d)*x**2)*(a*d - b*c)*elliptic_f(2*ata
n(d**(1/4)*x/c**(1/4)), 1/2)/(4*b*c**(1/4)*sqrt(c + d*x**4)*(a*sqrt(d) - sqrt(b)
*sqrt(c)*sqrt(-a))) - (a*d - b*c)*atan(x*sqrt(sqrt(-a)*(a*d - b*c)/(a*sqrt(b)))/
sqrt(c + d*x**4))/(4*a*b*sqrt(sqrt(-a)*(a*d - b*c)/(a*sqrt(b)))) - (a*d - b*c)*a
tan(x*sqrt(sqrt(-a)*(-a*d + b*c)/(a*sqrt(b)))/sqrt(c + d*x**4))/(4*a*b*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)*(a*d - b*c)*(sqrt(b)*sqrt(c) - sqrt(d)*sqrt(-a))*ellip
tic_pi((sqrt(b)*sqrt(c) + sqrt(d)*sqrt(-a))**2/(4*sqrt(b)*sqrt(c)*sqrt(d)*sqrt(-
a)), 2*atan(d**(1/4)*x/c**(1/4)), 1/2)/(8*a*b*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)*(a*d - b*c)*(sqrt(b)*sqrt(c) + sqrt(d)*sqrt(-a)
)*elliptic_pi(-(sqrt(b)*sqrt(c) - sqrt(d)*sqrt(-a))**2/(4*sqrt(b)*sqrt(c)*sqrt(d
)*sqrt(-a)), 2*atan(d**(1/4)*x/c**(1/4)), 1/2)/(8*a*b*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 = 0.238726, size = 161, normalized size = 0.24 \[ \frac{5 a c x \sqrt{c+d x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (2 x^4 \left (a d F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-2 b c F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*a*c*x*Sqrt[c + d*x^4]*AppellF1[1/4, -1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)]
)/((a + b*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*Appe
llF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))

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Maple [C]  time = 0.01, size = 273, normalized size = 0.4 \[{\frac{d}{b}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}}-{\frac{1}{8\,{b}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{ad-bc}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{1\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b*d/(I/c^(1/2)*d^(1/2))^(1/2)*(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)*EllipticF(x*(I/c^(1/2)*d^(1/2))^(1/2),I)-1/8/b^2
*sum((a*d-b*c)/_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. \[ \int \frac{\sqrt{d x^{4} + c}}{b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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