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

Optimal. Leaf size=786 \[ \frac{\tan ^{-1}\left (\frac{x^2 \sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}}}{\sqrt{c+d x^8}}\right )}{8 a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\tan ^{-1}\left (\frac{x^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^8}}\right )}{8 a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} \sqrt{c+d x^8} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right )}+\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} \sqrt{c+d x^8} \left (\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 a \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 a \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )} \]

[Out]

ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]]/(8*a*Sqrt[-
((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) + ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])
]*x^2)/Sqrt[c + d*x^8]]/(8*a*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]) - (d^(1/4)*(S
qrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*Ar
cTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt
[d])*Sqrt[c + d*x^8]) + (d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[
c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*c^(1/4)
*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + S
qrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4
)^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^2)/c^(1/4)], 1/2])/(16*a*c^(1/4)*(Sqrt[b]*Sqrt
[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*
Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ell
ipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[
d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(16*a*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqr
t[-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^8])

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Rubi [A]  time = 1.32898, antiderivative size = 786, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\tan ^{-1}\left (\frac{x^2 \sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}}}{\sqrt{c+d x^8}}\right )}{8 a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\tan ^{-1}\left (\frac{x^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^8}}\right )}{8 a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} \sqrt{c+d x^8} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right )}+\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} \sqrt{c+d x^8} \left (\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 a \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 a \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

ArcTan[(Sqrt[(Sqrt[-a]*((b*c)/a - d))/Sqrt[b]]*x^2)/Sqrt[c + d*x^8]]/(8*a*Sqrt[-
((b*c - a*d)/(Sqrt[-a]*Sqrt[b]))]) + ArcTan[(Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])
]*x^2)/Sqrt[c + d*x^8]]/(8*a*Sqrt[(b*c - a*d)/(Sqrt[-a]*Sqrt[b])]) - (d^(1/4)*(S
qrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*EllipticF[2*Ar
cTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*c^(1/4)*(Sqrt[-a]*Sqrt[b]*Sqrt[c] - a*Sqrt
[d])*Sqrt[c + d*x^8]) + (d^(1/4)*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[
c] + Sqrt[d]*x^4)^2]*EllipticF[2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(8*c^(1/4)
*(Sqrt[-a]*Sqrt[b]*Sqrt[c] + a*Sqrt[d])*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] + S
qrt[-a]*Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4
)^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^2)/c^(1/4)], 1/2])/(16*a*c^(1/4)*(Sqrt[b]*Sqrt
[c] - Sqrt[-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^8]) + ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*
Sqrt[d])*(Sqrt[c] + Sqrt[d]*x^4)*Sqrt[(c + d*x^8)/(Sqrt[c] + Sqrt[d]*x^4)^2]*Ell
ipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[
d]), 2*ArcTan[(d^(1/4)*x^2)/c^(1/4)], 1/2])/(16*a*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqr
t[-a]*Sqrt[d])*d^(1/4)*Sqrt[c + d*x^8])

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Rubi in Sympy [A]  time = 142.019, size = 690, normalized size = 0.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**(1/4)*sqrt((c + d*x**8)/(sqrt(c) + sqrt(d)*x**4)**2)*(sqrt(c) + sqrt(d)*x**4)
*elliptic_f(2*atan(d**(1/4)*x**2/c**(1/4)), 1/2)/(8*c**(1/4)*sqrt(c + d*x**8)*(a
*sqrt(d) + sqrt(b)*sqrt(c)*sqrt(-a))) + d**(1/4)*sqrt((c + d*x**8)/(sqrt(c) + sq
rt(d)*x**4)**2)*(sqrt(c) + sqrt(d)*x**4)*elliptic_f(2*atan(d**(1/4)*x**2/c**(1/4
)), 1/2)/(8*c**(1/4)*sqrt(c + d*x**8)*(a*sqrt(d) - sqrt(b)*sqrt(c)*sqrt(-a))) +
atan(x**2*sqrt(sqrt(-a)*(a*d - b*c)/(a*sqrt(b)))/sqrt(c + d*x**8))/(8*a*sqrt(sqr
t(-a)*(a*d - b*c)/(a*sqrt(b)))) + atan(x**2*sqrt(sqrt(-a)*(-a*d + b*c)/(a*sqrt(b
)))/sqrt(c + d*x**8))/(8*a*sqrt(sqrt(-a)*(-a*d + b*c)/(a*sqrt(b)))) + sqrt((c +
d*x**8)/(sqrt(c) + sqrt(d)*x**4)**2)*(sqrt(c) + sqrt(d)*x**4)*(sqrt(b)*sqrt(c) -
sqrt(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**2/c**(1/4)), 1/2)/(16*a*c**(1/4)
*d**(1/4)*sqrt(c + d*x**8)*(sqrt(b)*sqrt(c) + sqrt(d)*sqrt(-a))) + sqrt((c + d*x
**8)/(sqrt(c) + sqrt(d)*x**4)**2)*(sqrt(c) + sqrt(d)*x**4)*(sqrt(b)*sqrt(c) + sq
rt(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**2/c**(1/4)), 1/2)/(16*a*c**(1/4)*d
**(1/4)*sqrt(c + d*x**8)*(sqrt(b)*sqrt(c) - sqrt(d)*sqrt(-a)))

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Mathematica [C]  time = 0.270581, size = 165, normalized size = 0.21 \[ -\frac{5 a c x^2 F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{2 \left (a+b x^8\right ) \sqrt{c+d x^8} \left (2 x^8 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*a*c*x^2*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)])/(2*(a + b*x^
8)*Sqrt[c + d*x^8]*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)
] + 2*x^8*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^8)/c), -((b*x^8)/a)] + a*d*Ap
pellF1[5/4, 3/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{x}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/((b*x^8 + a)*sqrt(d*x^8 + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/((a + b*x**8)*sqrt(c + d*x**8)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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