3.630 \(\int \frac{\sqrt{c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx\)

Optimal. Leaf size=809 \[ \frac{b \sqrt [4]{d} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right ) c^{5/4}}{a (b c+a d) \sqrt{d x^4+c}}-\frac{\sqrt [4]{d} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right ) \sqrt [4]{c}}{a \sqrt{d x^4+c}}-\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 a}-\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 a}-\frac{\sqrt{d x^4+c}}{a x}+\frac{\sqrt{d} x \sqrt{d x^4+c}}{a \left (\sqrt{d} x^2+\sqrt{c}\right )}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{b} \left (\sqrt{d} a^2+(-a)^{3/2} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^4+c} \sqrt [4]{c}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \Pi \left (-\frac{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a \sqrt{b} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^4+c} \sqrt [4]{c}} \]

[Out]

-(Sqrt[c + d*x^4]/(a*x)) + (Sqrt[d]*x*Sqrt[c + d*x^4])/(a*(Sqrt[c] + Sqrt[d]*x^2
)) - (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])]
*ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*a) - (c^(1
/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*
EllipticE[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(a*Sqrt[c + d*x^4]) + (b*c^(5/4)*
d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Elli
pticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(a*(b*c + a*d)*Sqrt[c + d*x^4]) - ((S
qrt[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*Sqrt[b]*c^(1/4)*((-a)^(3/2)*Sqrt[b]*Sqrt[c] + a^2*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[c]*(Sqrt[b] - (
Sqrt[-a]*Sqrt[d])/Sqrt[c])^2)/(4*Sqrt[-a]*Sqrt[b]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)
/c^(1/4)], 1/2])/(8*a*Sqrt[b]*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*d^(
1/4)*Sqrt[c + d*x^4])

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Rubi [A]  time = 2.64682, antiderivative size = 1062, normalized size of antiderivative = 1.31, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt{d} \sqrt{d x^4+c} x}{a \left (\sqrt{d} x^2+\sqrt{c}\right )}-\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 a}-\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 a}-\frac{\sqrt [4]{c} \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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{a \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 a \sqrt{b} \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-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 \sqrt{b} \sqrt [4]{c} \left (\sqrt{d} (-a)^{3/2}+a \sqrt{b} \sqrt{c}\right ) \sqrt{d x^4+c}}+\frac{\sqrt [4]{c} \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 )}{2 a \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 \sqrt{b} \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-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{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a \sqrt{b} \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^4+c}}-\frac{\sqrt{d x^4+c}}{a x} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(Sqrt[c + d*x^4]/(a*x)) + (Sqrt[d]*x*Sqrt[c + d*x^4])/(a*(Sqrt[c] + Sqrt[d]*x^2
)) - (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])]
*ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*a) - (c^(1
/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*
EllipticE[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(a*Sqrt[c + d*x^4]) + (c^(1/4)*d^
(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ellipt
icF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*a*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]*Ellipt
icF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*a*Sqrt[b]*c^(1/4)*(Sqrt[b]*Sqrt[c] +
Sqrt[-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*Sqrt[b]*c^(1/4)*(a*Sqrt[b]*Sqrt[c] + (-a)^(3/2)*Sqrt[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*Sqrt[b]*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - 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[c
]*(Sqrt[b] - (Sqrt[-a]*Sqrt[d])/Sqrt[c])^2)/(4*Sqrt[-a]*Sqrt[b]*Sqrt[d]), 2*ArcT
an[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*a*Sqrt[b]*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] +
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((d*x**4+c)**(1/2)/x**2/(b*x**4+a),x)

[Out]

Timed out

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Mathematica [C]  time = 0.446987, size = 343, normalized size = 0.42 \[ \frac{\frac{49 c x^4 (b c-2 a d) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (2 x^4 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{33 b c d x^8 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (2 x^4 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{21 \left (c+d x^4\right )}{a}}{21 x \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-21*(c + d*x^4))/a + (49*c*(b*c - 2*a*d)*x^4*AppellF1[3/4, 1/2, 1, 7/4, -((d*x
^4)/c), -((b*x^4)/a)])/((a + b*x^4)*(-7*a*c*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)
/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[7/4, 1/2, 2, 11/4, -((d*x^4)/c), -((b
*x^4)/a)] + a*d*AppellF1[7/4, 3/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)]))) - (33
*b*c*d*x^8*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/((a + b*x^4)
*(-11*a*c*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c
*AppellF1[11/4, 1/2, 2, 15/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[11/4, 3
/2, 1, 15/4, -((d*x^4)/c), -((b*x^4)/a)]))))/(21*x*Sqrt[c + d*x^4])

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Maple [C]  time = 0.02, size = 421, normalized size = 0.5 \[{\frac{1}{a} \left ( -{\frac{1}{x}\sqrt{d{x}^{4}+c}}+{2\,i\sqrt{c}\sqrt{d}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ) }-{\frac{b}{a} \left ({\frac{i}{b}\sqrt{c}\sqrt{d}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}}-{\frac{1}{8\,{b}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{ad-bc}{{\it \_alpha}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{1\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a*(-1/x*(d*x^4+c)^(1/2)+2*I*d^(1/2)*c^(1/2)/(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)*(Ellipti
cF(x*(I/c^(1/2)*d^(1/2))^(1/2),I)-EllipticE(x*(I/c^(1/2)*d^(1/2))^(1/2),I)))-b/a
*(I/b*d^(1/2)*c^(1/2)/(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)-EllipticE(x*(I/c^(1/2)*d^(1/2))^(1/2),I))-1/8/b^2*sum((a*d-b*c)/_alpha
*(-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. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^2), 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^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{x^{2} \left (a + b x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c + d*x**4)/(x**2*(a + b*x**4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^2),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^2), x)