3.318 \(\int \frac{1}{x^5 \left (a+b x^4+c x^8\right )} \, dx\)

Optimal. Leaf size=89 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b x^4+c x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{4 a x^4} \]

[Out]

-1/(4*a*x^4) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*a^2*S
qrt[b^2 - 4*a*c]) - (b*Log[x])/a^2 + (b*Log[a + b*x^4 + c*x^8])/(8*a^2)

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Rubi [A]  time = 0.265326, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{4 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b x^4+c x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^4 + c*x^8)),x]

[Out]

-1/(4*a*x^4) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*a^2*S
qrt[b^2 - 4*a*c]) - (b*Log[x])/a^2 + (b*Log[a + b*x^4 + c*x^8])/(8*a^2)

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Rubi in Sympy [A]  time = 44.0345, size = 87, normalized size = 0.98 \[ - \frac{1}{4 a x^{4}} - \frac{b \log{\left (x^{4} \right )}}{4 a^{2}} + \frac{b \log{\left (a + b x^{4} + c x^{8} \right )}}{8 a^{2}} - \frac{\left (- 2 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{4}}{\sqrt{- 4 a c + b^{2}}} \right )}}{4 a^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(c*x**8+b*x**4+a),x)

[Out]

-1/(4*a*x**4) - b*log(x**4)/(4*a**2) + b*log(a + b*x**4 + c*x**8)/(8*a**2) - (-2
*a*c + b**2)*atanh((b + 2*c*x**4)/sqrt(-4*a*c + b**2))/(4*a**2*sqrt(-4*a*c + b**
2))

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Mathematica [C]  time = 0.0480938, size = 92, normalized size = 1.03 \[ \frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b c \log (x-\text{$\#$1})-a c \log (x-\text{$\#$1})+b^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+b}\&\right ]}{4 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^4 + c*x^8)),x]

[Out]

-1/(4*a*x^4) - (b*Log[x])/a^2 + RootSum[a + b*#1^4 + c*#1^8 & , (b^2*Log[x - #1]
- a*c*Log[x - #1] + b*c*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(4*a^2)

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Maple [A]  time = 0.011, size = 119, normalized size = 1.3 \[{\frac{b\ln \left ( c{x}^{8}+b{x}^{4}+a \right ) }{8\,{a}^{2}}}-{\frac{c}{2\,a}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}}{4\,{a}^{2}}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{4\,a{x}^{4}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(c*x^8+b*x^4+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^8 + b*x^4 + a)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.418975, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 2 \, a c\right )} x^{4} \log \left (\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + b^{3} - 4 \, a b c +{\left (2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) -{\left (b x^{4} \log \left (c x^{8} + b x^{4} + a\right ) - 8 \, b x^{4} \log \left (x\right ) - 2 \, a\right )} \sqrt{b^{2} - 4 \, a c}}{8 \, \sqrt{b^{2} - 4 \, a c} a^{2} x^{4}}, \frac{2 \,{\left (b^{2} - 2 \, a c\right )} x^{4} \arctan \left (-\frac{{\left (2 \, c x^{4} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (b x^{4} \log \left (c x^{8} + b x^{4} + a\right ) - 8 \, b x^{4} \log \left (x\right ) - 2 \, a\right )} \sqrt{-b^{2} + 4 \, a c}}{8 \, \sqrt{-b^{2} + 4 \, a c} a^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^8 + b*x^4 + a)*x^5),x, algorithm="fricas")

[Out]

[-1/8*((b^2 - 2*a*c)*x^4*log((2*(b^2*c - 4*a*c^2)*x^4 + b^3 - 4*a*b*c + (2*c^2*x
^8 + 2*b*c*x^4 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^8 + b*x^4 + a)) - (b*x^4*l
og(c*x^8 + b*x^4 + a) - 8*b*x^4*log(x) - 2*a)*sqrt(b^2 - 4*a*c))/(sqrt(b^2 - 4*a
*c)*a^2*x^4), 1/8*(2*(b^2 - 2*a*c)*x^4*arctan(-(2*c*x^4 + b)*sqrt(-b^2 + 4*a*c)/
(b^2 - 4*a*c)) + (b*x^4*log(c*x^8 + b*x^4 + a) - 8*b*x^4*log(x) - 2*a)*sqrt(-b^2
+ 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^2*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/(c*x**8+b*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.291142, size = 127, normalized size = 1.43 \[ \frac{b{\rm ln}\left (c x^{8} + b x^{4} + a\right )}{8 \, a^{2}} - \frac{b{\rm ln}\left (x^{4}\right )}{4 \, a^{2}} + \frac{{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac{2 \, c x^{4} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} a^{2}} + \frac{b x^{4} - a}{4 \, a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^8 + b*x^4 + a)*x^5),x, algorithm="giac")

[Out]

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