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

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

[Out]

-ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x)/Sqrt[c + d*x^4]]/(4*b*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*b*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]) + (c^(3/4)*(Sqrt
[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTa
n[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*d^(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*b*c^(1/4)*(Sq
rt[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]*Sqr
t[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*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 = 1.30484, antiderivative size = 873, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\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 b \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 b \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{a \sqrt [4]{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{a \sqrt [4]{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{\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 [4]{d} \sqrt{d x^4+c}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \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 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 ) \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 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[x^4/((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*b*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*b*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]) + ((Sqrt[c] + Sq
rt[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*b*c^(1/4)*d^(1/4)*Sqrt[c + d*x^4]) + (a*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*b*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*
Sqrt[c + d*x^4]) - (a*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*b*c^(1/4)*(S
qrt[-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
]*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*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
])*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*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[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*b*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqr
t[d])*d^(1/4)*Sqrt[c + d*x^4])

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Rubi in Sympy [A]  time = 140.487, size = 768, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0831277, size = 165, normalized size = 0.26 \[ -\frac{9 a c x^5 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{5 \left (a+b x^4\right ) \sqrt{c+d x^4} \left (2 x^4 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-9*a*c*x^5*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])/(5*(a + b*x^
4)*Sqrt[c + d*x^4]*(-9*a*c*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)
] + 2*x^4*(2*b*c*AppellF1[9/4, 1/2, 2, 13/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*A
ppellF1[9/4, 3/2, 1, 13/4, -((d*x^4)/c), -((b*x^4)/a)])))

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Maple [C]  time = 0.012, size = 265, normalized size = 0.4 \[{\frac{1}{b}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}}-{\frac{a}{8\,{b}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{1}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{1\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b/(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*a/b^2
*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)*Ellip
ticPi(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. \[ \int \frac{x^{4}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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