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

Optimal. Leaf size=677 \[ -\frac{b \tan ^{-1}\left (\frac{x \sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a^2 \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{b \tan ^{-1}\left (\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{b \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}-\frac{b \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )}-\frac{d^{3/4} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} (a d+4 b c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{6 a c^{5/4} \sqrt{c+d x^4} (a d+b c)}-\frac{\sqrt{c+d x^4}}{3 a c x^3} \]

[Out]

-Sqrt[c + d*x^4]/(3*a*c*x^3) - (b*ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]
*x)/Sqrt[c + d*x^4]])/(4*a^2*Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) - (b*ArcTa
n[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*a^2*Sqrt[(b*c -
a*d)/(Sqrt[-a]*Sqrt[b])]) - (d^(3/4)*(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])/(6*a*c^(5/4)*(b*c + a*d)*Sqrt[c + d*x^4]) - (b*(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]*El
lipticPi[-(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqr
t[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sq
rt[-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^4]) - (b*(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*a^2*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*S
qrt[d])*d^(1/4)*Sqrt[c + d*x^4])

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Rubi [A]  time = 1.57011, antiderivative size = 901, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{b \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^2 \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{b \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 a^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\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 )}{6 a c^{5/4} \sqrt{d x^4+c}}+\frac{b \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 a \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt{d x^4+c}}-\frac{b \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 a \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt{d x^4+c}}-\frac{b \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 a^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^4+c}}-\frac{b \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 a^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^4+c}}-\frac{\sqrt{d x^4+c}}{3 a c x^3} \]

Warning: Unable to verify antiderivative.

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

[Out]

-Sqrt[c + d*x^4]/(3*a*c*x^3) - (b*ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]
*x)/Sqrt[c + d*x^4]])/(4*a^2*Sqrt[-((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) - (b*ArcTa
n[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*a^2*Sqrt[(b*c -
a*d)/(Sqrt[-a]*Sqrt[b])]) + (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*a*c
^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*Sqrt[c + d*x^4]) - (b*d^(1/4)*(Sqr
t[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcT
an[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*a*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d
])*Sqrt[c + d*x^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])/(6*a*c^(5/4)*S
qrt[c + d*x^4]) - (b*(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] - Sqr
t[-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^2*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*Sqrt[c +
d*x^4]) - (b*(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]*Sq
rt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/
2])/(8*a^2*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.488468, size = 344, normalized size = 0.51 \[ \frac{\frac{25 x^4 (a d+3 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (2 x^4 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}+\frac{9 b d x^8 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (2 x^4 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{5 \left (c+d x^4\right )}{a c}}{15 x^3 \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-5*(c + d*x^4))/(a*c) + (25*(3*b*c + a*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*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))) + (9*b
*d*x^8*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])/((a + b*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*Appell
F1[9/4, 1/2, 2, 13/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[9/4, 3/2, 1, 13
/4, -((d*x^4)/c), -((b*x^4)/a)]))))/(15*x^3*Sqrt[c + d*x^4])

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Maple [C]  time = 0.023, size = 288, normalized size = 0.4 \[{\frac{1}{a} \left ( -{\frac{1}{3\,c{x}^{3}}\sqrt{d{x}^{4}+c}}-{\frac{d}{3\,c}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ) }-{\frac{1}{8\,a}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{1}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{1\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a*(-1/3/c*(d*x^4+c)^(1/2)/x^3-1/3*d/c/(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*sum(1/_alpha^3*(-1/((-a*d+b*c)/b)^(1/2)*arctan
h(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. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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