3.726 \(\int \frac{1}{x^5 \left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{b \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^8}}{4 a c x^4} \]

[Out]

-Sqrt[c + d*x^8]/(4*a*c*x^4) - (b*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c +
d*x^8])])/(4*a^(3/2)*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.248567, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{b \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^8}}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

-Sqrt[c + d*x^8]/(4*a*c*x^4) - (b*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c +
d*x^8])])/(4*a^(3/2)*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 34.1354, size = 68, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{8}}}{4 a c x^{4}} - \frac{b \operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{4 a^{\frac{3}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

-sqrt(c + d*x**8)/(4*a*c*x**4) - b*atanh(x**4*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c +
d*x**8)))/(4*a**(3/2)*sqrt(a*d - b*c))

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Mathematica [A]  time = 1.33288, size = 129, normalized size = 1.61 \[ \frac{\sqrt{c+d x^8} \left (-\frac{b x^8 \sin ^{-1}\left (\frac{\sqrt{x^8 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^8}{a}+1}}\right )}{\sqrt{\frac{b x^8}{a}+1} \sqrt{x^8 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^8\right )}{c \left (a+b x^8\right )}}}-a\right )}{4 a^2 c x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

(Sqrt[c + d*x^8]*(-a - (b*x^8*ArcSin[Sqrt[(b/a - d/c)*x^8]/Sqrt[1 + (b*x^8)/a]])
/(Sqrt[(b/a - d/c)*x^8]*Sqrt[1 + (b*x^8)/a]*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))
])))/(4*a^2*c*x^4)

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5} \left ( b{x}^{8}+a \right ) }{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^5/(b*x^8+a)/(d*x^8+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^5), x)

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Fricas [A]  time = 0.297959, size = 1, normalized size = 0.01 \[ \left [\frac{b c x^{4} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{12} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{4}\right )} \sqrt{d x^{8} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right ) - 4 \, \sqrt{d x^{8} + c} \sqrt{-a b c + a^{2} d}}{16 \, \sqrt{-a b c + a^{2} d} a c x^{4}}, -\frac{b c x^{4} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{8} - a c}{2 \, \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} x^{4}}\right ) + 2 \, \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d}}{8 \, \sqrt{a b c - a^{2} d} a c x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^5),x, algorithm="fricas")

[Out]

[1/16*(b*c*x^4*log(-(4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^12 - (a^2*b*c^2
- a^3*c*d)*x^4)*sqrt(d*x^8 + c) - ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^16 - 2*(3
*a*b*c^2 - 4*a^2*c*d)*x^8 + a^2*c^2)*sqrt(-a*b*c + a^2*d))/(b^2*x^16 + 2*a*b*x^8
+ a^2)) - 4*sqrt(d*x^8 + c)*sqrt(-a*b*c + a^2*d))/(sqrt(-a*b*c + a^2*d)*a*c*x^4
), -1/8*(b*c*x^4*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)/(sqrt(d*x^8 + c)*sqrt(a*b*
c - a^2*d)*x^4)) + 2*sqrt(d*x^8 + c)*sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a
*c*x^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228714, size = 86, normalized size = 1.08 \[ \frac{\frac{b c \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d} a} - \frac{\sqrt{d + \frac{c}{x^{8}}}}{a}}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)*x^5),x, algorithm="giac")

[Out]

1/4*(b*c*arctan(a*sqrt(d + c/x^8)/sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a) -
sqrt(d + c/x^8)/a)/c