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

Optimal. Leaf size=786 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-a d) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) (b c-a d) \Pi \left (-\frac{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right )}+\frac{a \sqrt [4]{c} d^{5/4} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{b \sqrt{c+d x^4} (a d+b c)}+\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{x \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 b}+\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 b}+\frac{\sqrt{d} x \sqrt{c+d x^4}}{b \left (\sqrt{c}+\sqrt{d} x^2\right )}-\frac{\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{b \sqrt{c+d x^4}} \]

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^4])/(b*(Sqrt[c] + Sqrt[d]*x^2)) + (Sqrt[-((b*c - a*d)/(S
qrt[-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])]*ArcTan[(Sqrt[(b*c - a*d)/(
Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*b) - (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])/(b*Sqrt[c + d*x^4]) + (a*c^(1/4)*d^(5/4)*(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*(b*c + a*d)*Sqrt[c + d*x^4]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*S
qrt[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]*Sq
rt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*b^(3/2)*c^(1/4)*(Sqrt[-a
]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sq
rt[-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*b^(3/2)*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 = 1.72677, antiderivative size = 1036, normalized size of antiderivative = 1.32, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{d} \sqrt{d x^4+c} x}{b \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 b}+\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 b}-\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 )}{b \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^{3/2} \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 b^{3/2} \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\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 b \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 b^{3/2} \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 b^{3/2} \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^4+c}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^4])/(b*(Sqrt[c] + Sqrt[d]*x^2)) + (Sqrt[-((b*c - a*d)/(S
qrt[-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])]*ArcTan[(Sqrt[(b*c - a*d)/(
Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*b) - (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])/(b*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]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^
(1/4)], 1/2])/(2*b*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^(3/2)*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*b^(3/2)*c^(1
/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*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*b^(3/2)*c^(1/4
)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sq
rt[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])/Sqr
t[c])^2)/(4*Sqrt[-a]*Sqrt[b]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*b
^(3/2)*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 [A]  time = 154.848, size = 945, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c**(1/4)*d**(1/4)*sqrt((c + d*x**4)/(sqrt(c) + sqrt(d)*x**2)**2)*(sqrt(c) + sqr
t(d)*x**2)*elliptic_e(2*atan(d**(1/4)*x/c**(1/4)), 1/2)/(b*sqrt(c + d*x**4)) + c
**(1/4)*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)/(2*b*sqrt(c + d*x**4)) + s
qrt(d)*x*sqrt(c + d*x**4)/(b*(sqrt(c) + sqrt(d)*x**2)) + (a*d - b*c)*atan(x*sqrt
((a*d - b*c)/(sqrt(b)*sqrt(-a)))/sqrt(c + d*x**4))/(4*b**(3/2)*sqrt(-a)*sqrt((a*
d - b*c)/(sqrt(b)*sqrt(-a)))) - (a*d - b*c)*atan(x*sqrt((-a*d + b*c)/(sqrt(b)*sq
rt(-a)))/sqrt(c + d*x**4))/(4*b**(3/2)*sqrt(-a)*sqrt((-a*d + b*c)/(sqrt(b)*sqrt(
-a)))) + 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**(3/2)*c
**(1/4)*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)*(a*d - b*c)*el
liptic_f(2*atan(d**(1/4)*x/c**(1/4)), 1/2)/(4*b**(3/2)*c**(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(c)*(sqrt(b) - sqrt(d)*sqrt(-a)/sqrt(c))**2/(4*sqrt(b)*sqrt(d
)*sqrt(-a)), 2*atan(d**(1/4)*x/c**(1/4)), 1/2)/(8*b**(3/2)*c**(1/4)*d**(1/4)*sqr
t(c + d*x**4)*(a*sqrt(d) + sqrt(b)*sqrt(c)*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) - s
qrt(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*b**(3/2)*c**(1/4
)*d**(1/4)*sqrt(c + d*x**4)*(a*sqrt(d) - sqrt(b)*sqrt(c)*sqrt(-a)))

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Mathematica [C]  time = 0.243723, size = 165, normalized size = 0.21 \[ \frac{7 a c x^3 \sqrt{c+d x^4} F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{3 \left (a+b x^4\right ) \left (2 x^4 \left (a d F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-2 b c F_1\left (\frac{7}{4};-\frac{1}{2},2;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.01, size = 299, normalized size = 0.4 \[{\frac{i}{b}\sqrt{d}\sqrt{c}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}}-{\frac{1}{8\,{b}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{ad-bc}{{\it \_alpha}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{1\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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. \[ \int \frac{\sqrt{d x^{4} + c} x^{2}}{b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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