3.650 \(\int \frac{x^6}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=804 \[ \frac{\sqrt{d x^4+c} x}{b \sqrt{d} \left (\sqrt{d} x^2+\sqrt{c}\right )}-\frac{a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 b (b c-a d)}-\frac{a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 b (b c-a d)}-\frac{\sqrt [4]{c} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{b d^{3/4} \sqrt{d x^4+c}}+\frac{\sqrt [4]{c} (b c+2 a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{2 b d^{3/4} (b c+a d) \sqrt{d x^4+c}}+\frac{a \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^4+c}}-\frac{a \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \Pi \left (-\frac{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^4+c}} \]

[Out]

(x*Sqrt[c + d*x^4])/(b*Sqrt[d]*(Sqrt[c] + Sqrt[d]*x^2)) - (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*b*(b*c - a*d)) - (a*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*ArcTan[(Sq
rt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*b*(b*c - a*d)) - (c^(
1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ellipti
cE[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(b*d^(3/4)*Sqrt[c + d*x^4]) + (c^(1/4)*(
b*c + 2*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])/(2*b*d^(3/4)*(b*c + a*d)*Sqrt[c
+ d*x^4]) + (a*(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]*S
qrt[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]) - (a*(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*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 = 1.87129, antiderivative size = 1030, normalized size of antiderivative = 1.28, 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 x^4+c} x}{b \sqrt{d} \left (\sqrt{d} x^2+\sqrt{c}\right )}-\frac{a \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 (b c-a d)}-\frac{a \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 (b c-a d)}-\frac{\sqrt [4]{c} \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 d^{3/4} \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^{3/2} \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-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^{3/2} \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt{d x^4+c}}+\frac{\sqrt [4]{c} \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 d^{3/4} \sqrt{d x^4+c}}+\frac{a \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^{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{a \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{\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^6/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

(x*Sqrt[c + d*x^4])/(b*Sqrt[d]*(Sqrt[c] + Sqrt[d]*x^2)) - (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*b*(b*c - a*d)) - (a*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*ArcTan[(Sq
rt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(4*b*(b*c - a*d)) - (c^(
1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ellipti
cE[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(b*d^(3/4)*Sqrt[c + d*x^4]) + (c^(1/4)*(
Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*A
rcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*b*d^(3/4)*Sqrt[c + d*x^4]) + (a*d^(1/4)*(Sq
rt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*Arc
Tan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*b^(3/2)*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*S
qrt[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^(3
/2)*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*Sqrt[c + d*x^4]) + (a*(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]*Sq
rt[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]) - (a*(Sqrt[b]*Sq
rt[c] + Sqrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sq
rt[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 in Sympy [A]  time = 152.809, size = 911, normalized size = 1.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(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**(3/2)*c**(1/4)*sqrt(c + d*
x**4)*(sqrt(b)*sqrt(c) + sqrt(d)*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**(3/2)*c**(1/4)*sqrt(c + d*x**4)*(sqrt(b)*sqrt(c) - sqrt(d)*s
qrt(-a))) - a*sqrt((c + d*x**4)/(sqrt(c) + sqrt(d)*x**2)**2)*(sqrt(c) + sqrt(d)*
x**2)*(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)*sqrt(c + d*x**4)*(a*sqrt(d) + sqrt(b)*sqrt
(c)*sqrt(-a))) - a*sqrt((c + d*x**4)/(sqrt(c) + sqrt(d)*x**2)**2)*(sqrt(c) + sqr
t(d)*x**2)*(sqrt(b)*sqrt(c) - sqrt(d)*sqrt(-a))*elliptic_pi((sqrt(b)*sqrt(c) + s
qrt(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))) - c**(1/4)*sqrt((c + d*x**4)/(sqrt(c) + sqrt(d)*x**2)**2)*(sq
rt(c) + sqrt(d)*x**2)*elliptic_e(2*atan(d**(1/4)*x/c**(1/4)), 1/2)/(b*d**(3/4)*s
qrt(c + d*x**4)) + c**(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*d**(3/4)*s
qrt(c + d*x**4)) + x*sqrt(c + d*x**4)/(b*sqrt(d)*(sqrt(c) + sqrt(d)*x**2)) - sqr
t(-a)*atan(x*sqrt((a*d - b*c)/(sqrt(b)*sqrt(-a)))/sqrt(c + d*x**4))/(4*b**(3/2)*
sqrt((a*d - b*c)/(sqrt(b)*sqrt(-a)))) + sqrt(-a)*atan(x*sqrt((-a*d + b*c)/(sqrt(
b)*sqrt(-a)))/sqrt(c + d*x**4))/(4*b**(3/2)*sqrt((-a*d + b*c)/(sqrt(b)*sqrt(-a))
))

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Mathematica [C]  time = 0.100905, size = 165, normalized size = 0.21 \[ -\frac{11 a c x^7 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{7 \left (a+b x^4\right ) \sqrt{c+d x^4} \left (2 x^4 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^6/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

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

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Maple [C]  time = 0.029, size = 292, normalized size = 0.4 \[{\frac{i}{b}\sqrt{c}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}{\frac{1}{\sqrt{d}}}}-{\frac{a}{8\,{b}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{1}{{\it \_alpha}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{1\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

I/b*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)/d^(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*a/b^2*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. \[ \int \frac{x^{6}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(x^6/((b*x^4 + a)*sqrt(d*x^4 + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

integral(x^6/((b*x^4 + a)*sqrt(d*x^4 + c)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral(x**6/((a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

integrate(x^6/((b*x^4 + a)*sqrt(d*x^4 + c)), x)