3.665 \(\int \frac{x^8}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=1043 \[ \text{result too large to display} \]

[Out]

(a*x*Sqrt[c + d*x^4])/(4*b*(b*c - a*d)*(a + b*x^4)) + ((5*b*c - 3*a*d)*ArcTan[(S
qrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x)/Sqrt[c + d*x^4]])/(16*Sqrt[-a]*b^(5/2)*
(-((b*c - a*d)/(Sqrt[-a]*Sqrt[b])))^(3/2)) - ((5*b*c - 3*a*d)*ArcTan[(Sqrt[(b*c
- a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(16*Sqrt[-a]*b^(5/2)*((b*c - a*d
)/(Sqrt[-a]*Sqrt[b]))^(3/2)) + ((4*b*c - 3*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
])/(8*b^2*c^(1/4)*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^4]) + (a*d^(1/4)*(5*b*c - 3*a
*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Elliptic
F[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(16*b^2*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c]
- a*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^4]) - (a*d^(1/4)*(5*b*c - 3*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])/(16*b^2*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d]
)*(b*c - a*d)*Sqrt[c + d*x^4]) - ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(5*b*c -
3*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ellip
ticPi[-(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])/(32*b^2*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt
[-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^4]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a
]*Sqrt[d])*(5*b*c - 3*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + S
qrt[d]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqr
t[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(32*b^2*c^(1/4)*(Sqr
t[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^4])

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Rubi [A]  time = 3.35699, antiderivative size = 1043, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{a \sqrt{d x^4+c} x}{4 b (b c-a d) \left (b x^4+a\right )}+\frac{(5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{16 \sqrt{-a} b^{5/2} \left (-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}\right )^{3/2}}-\frac{(5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{16 \sqrt{-a} b^{5/2} \left (\frac{b c-a d}{\sqrt{-a} \sqrt{b}}\right )^{3/2}}+\frac{(4 b c-3 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 )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) \sqrt{d x^4+c}}+\frac{a \sqrt [4]{d} (5 b c-3 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 )}{16 b^2 \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) (b c-a d) \sqrt{d x^4+c}}-\frac{a \sqrt [4]{d} (5 b c-3 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 )}{16 b^2 \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) (b c-a d) \sqrt{d x^4+c}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (5 b c-3 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 )}{32 b^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \sqrt{d x^4+c}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (5 b c-3 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 )}{32 b^2 \sqrt [4]{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \sqrt{d x^4+c}} \]

Warning: Unable to verify antiderivative.

[In]  Int[x^8/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

(a*x*Sqrt[c + d*x^4])/(4*b*(b*c - a*d)*(a + b*x^4)) + ((5*b*c - 3*a*d)*ArcTan[(S
qrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x)/Sqrt[c + d*x^4]])/(16*Sqrt[-a]*b^(5/2)*
(-((b*c - a*d)/(Sqrt[-a]*Sqrt[b])))^(3/2)) - ((5*b*c - 3*a*d)*ArcTan[(Sqrt[(b*c
- a*d)/(Sqrt[-a]*Sqrt[b])]*x)/Sqrt[c + d*x^4]])/(16*Sqrt[-a]*b^(5/2)*((b*c - a*d
)/(Sqrt[-a]*Sqrt[b]))^(3/2)) + ((4*b*c - 3*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
])/(8*b^2*c^(1/4)*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^4]) + (a*d^(1/4)*(5*b*c - 3*a
*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Elliptic
F[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(16*b^2*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c]
- a*Sqrt[d])*(b*c - a*d)*Sqrt[c + d*x^4]) - (a*d^(1/4)*(5*b*c - 3*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])/(16*b^2*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d]
)*(b*c - a*d)*Sqrt[c + d*x^4]) - ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*(5*b*c -
3*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Ellip
ticPi[-(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])/(32*b^2*c^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt
[-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*Sqrt[c + d*x^4]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a
]*Sqrt[d])*(5*b*c - 3*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + S
qrt[d]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqr
t[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(32*b^2*c^(1/4)*(Sqr
t[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*d^(1/4)*(b*c - a*d)*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(x**8/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.574204, size = 420, normalized size = 0.4 \[ \frac{a x \left (\frac{25 a c^2 F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{2 x^4 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}+\frac{10 x^4 \left (c+d x^4\right ) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-9 c \left (5 a c+2 a d x^4+4 b c x^4\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{2 x^4 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}\right )}{20 b \left (a+b x^4\right ) \sqrt{c+d x^4} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^8/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

(a*x*((25*a*c^2*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/(-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*c*(5*a*c + 4*b*c*x^4 + 2*a*d*x^4)*AppellF1[5/4
, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + 10*x^4*(c + d*x^4)*(2*b*c*AppellF1[
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)]))/(-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*AppellF1[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)]))))/(20*b*(b*
c - a*d)*(a + b*x^4)*Sqrt[c + d*x^4])

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Maple [C]  time = 0.044, size = 604, normalized size = 0.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)

[Out]

1/b^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)+a^2/b^2
*(-1/4*b/a/(a*d-b*c)*x*(d*x^4+c)^(1/2)/(b*x^4+a)-1/4*d/(a*d-b*c)/a/(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)-1/32/b/a*sum((-5*a*d+3*b*c
)/(a*d-b*c)/_alpha^3*(-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)))-1/4*a/b^3
*sum(1/_alpha^3*(-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)*Ellip
ticPi(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. \[ \int \frac{x^{8}}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(x^8/((b*x^4 + a)^2*sqrt(d*x^4 + c)), 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(x^8/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

integrate(x^8/((b*x^4 + a)^2*sqrt(d*x^4 + c)), x)