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

Optimal. Leaf size=656 \[ -\frac{\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{2 \sqrt{c+d x^4} (a d+b c)}+\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{x \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 (b c-a d)}+\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 (b c-a d)}-\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) \Pi \left (-\frac{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right )} \]

[Out]

(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 - a*d)) + (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 - a*d)) - (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*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*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])*
(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*Sqrt[b]*c^(1/4)*(Sq
rt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^4])

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Rubi [A]  time = 0.709563, antiderivative size = 756, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {490, 1217, 220, 1707} \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{4 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{b c-a d}}-\frac{\tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{4 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{a d-b c}}-\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} \sqrt{c+d x^4} (a d+b c)}-\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} \sqrt{c+d x^4} (a d+b c)}-\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} (a d+b c)}+\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )^2 \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} (a d+b c)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

ArcTan[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])]/(4*(-a)^(1/4)*b^(1/4)*Sqrt[b*c - a*d]) - ArcT
an[(Sqrt[-(b*c) + a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])]/(4*(-a)^(1/4)*b^(1/4)*Sqrt[-(b*c) + a*d]) - ((
Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[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*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) - ((Sqrt[c] + (Sqrt
[-a]*Sqrt[d])/Sqrt[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*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqr
t[d])^2*(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*Sqrt[-a]*Sqrt[b
]*c^(1/4)*d^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(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]*S
qrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*Sqrt[-a]*Sqrt[b]*c^(1/4)*d^(1/4)*(b*c + a*d)*
Sqrt[c + d*x^4])

Rule 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1217

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d +
e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
&& PosQ[c/a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1707

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, -Simp[((B*d - A*e)*ArcTan[(Rt[(c*d)/e + (a*e)/d, 2]*x)/Sqrt[a + c*x^4]])/(2*d*e*Rt[(c*d)/e + (a*e)/d, 2]),
x] + Simp[((B*d + A*e)*(A + B*x^2)*Sqrt[(A^2*(a + c*x^4))/(a*(A + B*x^2)^2)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx &=-\frac{\int \frac{1}{\left (\sqrt{-a}-\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{2 \sqrt{b}}+\frac{\int \frac{1}{\left (\sqrt{-a}+\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{2 \sqrt{b}}\\ &=-\frac{\left (\sqrt{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )\right ) \int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (\sqrt{-a}-\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{2 (b c+a d)}+\frac{\left (\sqrt{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )\right ) \int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (\sqrt{-a}+\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{2 (b c+a d)}-\frac{\left (\left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt{d}\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{2 (b c+a d)}-\frac{\left (\left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt{d}\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{2 (b c+a d)}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{4 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{b c-a d}}-\frac{\tan ^{-1}\left (\frac{\sqrt{-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{4 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{-b c+a d}}-\frac{\left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} (b c+a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{c} (b c+a d) \sqrt{c+d x^4}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\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{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\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{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^4}}\\ \end{align*}

Mathematica [C]  time = 0.032795, size = 65, normalized size = 0.1 \[ \frac{x^3 \sqrt{\frac{c+d x^4}{c}} F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{3 a \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.007, size = 191, normalized size = 0.3 \begin{align*}{\frac{1}{8\,b}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{1}{{\it \_alpha}} \left ( -{{\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}}},{\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 ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/b*sum(1/_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. \begin{align*} \int \frac{x^{2}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")

[Out]

Timed out