3.854 \(\int \frac{1}{x^5 \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=114 \[ \frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c}}-\frac{\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

[Out]

-1/(4*a*x^4) + b/(2*a^2*x^2) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 -
4*a*c]])/(2*a^3*Sqrt[b^2 - 4*a*c]) + ((b^2 - a*c)*Log[x])/a^3 - ((b^2 - a*c)*Lo
g[a + b*x^2 + c*x^4])/(4*a^3)

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Rubi [A]  time = 0.427216, antiderivative size = 114, 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 \[ \frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c}}-\frac{\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

-1/(4*a*x^4) + b/(2*a^2*x^2) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 -
4*a*c]])/(2*a^3*Sqrt[b^2 - 4*a*c]) + ((b^2 - a*c)*Log[x])/a^3 - ((b^2 - a*c)*Lo
g[a + b*x^2 + c*x^4])/(4*a^3)

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Rubi in Sympy [A]  time = 44.8793, size = 109, normalized size = 0.96 \[ - \frac{1}{4 a x^{4}} + \frac{b}{2 a^{2} x^{2}} + \frac{b \left (- 3 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{3} \sqrt{- 4 a c + b^{2}}} + \frac{\left (- a c + b^{2}\right ) \log{\left (x^{2} \right )}}{2 a^{3}} - \frac{\left (- a c + b^{2}\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(c*x**4+b*x**2+a),x)

[Out]

-1/(4*a*x**4) + b/(2*a**2*x**2) + b*(-3*a*c + b**2)*atanh((b + 2*c*x**2)/sqrt(-4
*a*c + b**2))/(2*a**3*sqrt(-4*a*c + b**2)) + (-a*c + b**2)*log(x**2)/(2*a**3) -
(-a*c + b**2)*log(a + b*x**2 + c*x**4)/(4*a**3)

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Mathematica [A]  time = 0.559158, size = 188, normalized size = 1.65 \[ \frac{-\frac{a^2}{x^4}+4 \log (x) \left (b^2-a c\right )-\frac{\left (b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\sqrt{b^2-4 a c}}+\frac{\left (-b^2 \sqrt{b^2-4 a c}+a c \sqrt{b^2-4 a c}-3 a b c+b^3\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\sqrt{b^2-4 a c}}+\frac{2 a b}{x^2}}{4 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

(-(a^2/x^4) + (2*a*b)/x^2 + 4*(b^2 - a*c)*Log[x] - ((b^3 - 3*a*b*c + b^2*Sqrt[b^
2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b
^2 - 4*a*c] + ((b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sqrt[b^2 - 4*a*c])*L
og[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*a^3)

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Maple [A]  time = 0.013, size = 159, normalized size = 1.4 \[{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}}{4\,{a}^{3}}}+{\frac{3\,bc}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}}{2\,{a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{4\,a{x}^{4}}}-{\frac{c\ln \left ( x \right ) }{{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(c*x^4+b*x^2+a),x)

[Out]

1/4/a^2*c*ln(c*x^4+b*x^2+a)-1/4/a^3*ln(c*x^4+b*x^2+a)*b^2+3/2/a^2/(4*a*c-b^2)^(1
/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*c-1/2/a^3/(4*a*c-b^2)^(1/2)*arctan((
2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^3-1/4/a/x^4-1/a^2*ln(x)*c+1/a^3*ln(x)*b^2+1/2*b/
a^2/x^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^4 + b*x^2 + a)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299061, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{3} - 3 \, a b c\right )} x^{4} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left ({\left (b^{2} - a c\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \,{\left (b^{2} - a c\right )} x^{4} \log \left (x\right ) - 2 \, a b x^{2} + a^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{4 \, \sqrt{b^{2} - 4 \, a c} a^{3} x^{4}}, -\frac{2 \,{\left (b^{3} - 3 \, a b c\right )} x^{4} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (b^{2} - a c\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \,{\left (b^{2} - a c\right )} x^{4} \log \left (x\right ) - 2 \, a b x^{2} + a^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \, \sqrt{-b^{2} + 4 \, a c} a^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="fricas")

[Out]

[-1/4*((b^3 - 3*a*b*c)*x^4*log(-(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x^2 - (2*c^
2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) + ((b^2
- a*c)*x^4*log(c*x^4 + b*x^2 + a) - 4*(b^2 - a*c)*x^4*log(x) - 2*a*b*x^2 + a^2)
*sqrt(b^2 - 4*a*c))/(sqrt(b^2 - 4*a*c)*a^3*x^4), -1/4*(2*(b^3 - 3*a*b*c)*x^4*arc
tan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + ((b^2 - a*c)*x^4*log(c*x^
4 + b*x^2 + a) - 4*(b^2 - a*c)*x^4*log(x) - 2*a*b*x^2 + a^2)*sqrt(-b^2 + 4*a*c))
/(sqrt(-b^2 + 4*a*c)*a^3*x^4)]

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Sympy [A]  time = 26.4883, size = 423, normalized size = 3.71 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) \log{\left (x^{2} + \frac{8 a^{4} c \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) - 2 a^{3} b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) - 2 a^{2} c^{2} + 4 a b^{2} c - b^{4}}{3 a b c^{2} - b^{3} c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) \log{\left (x^{2} + \frac{8 a^{4} c \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) - 2 a^{3} b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) - 2 a^{2} c^{2} + 4 a b^{2} c - b^{4}}{3 a b c^{2} - b^{3} c} \right )} + \frac{- a + 2 b x^{2}}{4 a^{2} x^{4}} - \frac{\left (a c - b^{2}\right ) \log{\left (x \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(c*x**4+b*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.291943, size = 170, normalized size = 1.49 \[ -\frac{{\left (b^{2} - a c\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac{{\left (b^{2} - a c\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} - \frac{{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{3}} - \frac{3 \, b^{2} x^{4} - 3 \, a c x^{4} - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="giac")

[Out]

-1/4*(b^2 - a*c)*ln(c*x^4 + b*x^2 + a)/a^3 + 1/2*(b^2 - a*c)*ln(x^2)/a^3 - 1/2*(
b^3 - 3*a*b*c)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^3)
- 1/4*(3*b^2*x^4 - 3*a*c*x^4 - 2*a*b*x^2 + a^2)/(a^3*x^4)