3.314 \(\int \frac{x^3}{a+b x^4+c x^8} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c}} \]

[Out]

-ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]]/(2*Sqrt[b^2 - 4*a*c])

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Rubi [A]  time = 0.0724077, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^4 + c*x^8),x]

[Out]

-ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]]/(2*Sqrt[b^2 - 4*a*c])

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Rubi in Sympy [A]  time = 11.4714, size = 36, normalized size = 0.95 \[ - \frac{\operatorname{atanh}{\left (\frac{b + 2 c x^{4}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(c*x**8+b*x**4+a),x)

[Out]

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

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Mathematica [A]  time = 0.0149854, size = 42, normalized size = 1.11 \[ \frac{\tan ^{-1}\left (\frac{b+2 c x^4}{\sqrt{4 a c-b^2}}\right )}{2 \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x^4 + c*x^8),x]

[Out]

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

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Maple [A]  time = 0.002, size = 37, normalized size = 1. \[{\frac{1}{2}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^8+b*x^4+a),x)

[Out]

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

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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(x^3/(c*x^8 + b*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263406, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + b^{3} - 4 \, a b c -{\left (2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right )}{4 \, \sqrt{b^{2} - 4 \, a c}}, \frac{\arctan \left (-\frac{{\left (2 \, c x^{4} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^8 + b*x^4 + a),x, algorithm="fricas")

[Out]

[1/4*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))/sqrt(b^2 - 4*a*c), 1/2*arctan
(-(2*c*x^4 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c))/sqrt(-b^2 + 4*a*c)]

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Sympy [A]  time = 2.33962, size = 131, normalized size = 3.45 \[ - \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{4} + \frac{- 4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 c} \right )}}{4} + \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{4} + \frac{4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 c} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(c*x**8+b*x**4+a),x)

[Out]

-sqrt(-1/(4*a*c - b**2))*log(x**4 + (-4*a*c*sqrt(-1/(4*a*c - b**2)) + b**2*sqrt(
-1/(4*a*c - b**2)) + b)/(2*c))/4 + sqrt(-1/(4*a*c - b**2))*log(x**4 + (4*a*c*sqr
t(-1/(4*a*c - b**2)) - b**2*sqrt(-1/(4*a*c - b**2)) + b)/(2*c))/4

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GIAC/XCAS [A]  time = 0.259801, size = 49, normalized size = 1.29 \[ \frac{\arctan \left (\frac{2 \, c x^{4} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^8 + b*x^4 + a),x, algorithm="giac")

[Out]

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