3.139 \(\int \frac{x^5}{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=63 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^3+c x^6\right )}{6 c} \]

[Out]

(b*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*c*Sqrt[b^2 - 4*a*c]) + Log[a + b
*x^3 + c*x^6]/(6*c)

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Rubi [A]  time = 0.112803, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^3+c x^6\right )}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^3 + c*x^6),x]

[Out]

(b*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*c*Sqrt[b^2 - 4*a*c]) + Log[a + b
*x^3 + c*x^6]/(6*c)

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Rubi in Sympy [A]  time = 19.092, size = 54, normalized size = 0.86 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{\sqrt{- 4 a c + b^{2}}} \right )}}{3 c \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (a + b x^{3} + c x^{6} \right )}}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**6+b*x**3+a),x)

[Out]

b*atanh((b + 2*c*x**3)/sqrt(-4*a*c + b**2))/(3*c*sqrt(-4*a*c + b**2)) + log(a +
b*x**3 + c*x**6)/(6*c)

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Mathematica [A]  time = 0.0423322, size = 62, normalized size = 0.98 \[ \frac{\log \left (a+b x^3+c x^6\right )-\frac{2 b \tan ^{-1}\left (\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{6 c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^3 + c*x^6),x]

[Out]

((-2*b*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + b*
x^3 + c*x^6])/(6*c)

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Maple [A]  time = 0.004, size = 60, normalized size = 1. \[{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) }{6\,c}}-{\frac{b}{3\,c}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^6+b*x^3+a),x)

[Out]

1/6*ln(c*x^6+b*x^3+a)/c-1/3*b/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x^3+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^5/(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275205, size = 1, normalized size = 0.02 \[ \left [\frac{b \log \left (\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + b^{3} - 4 \, a b c +{\left (2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) + \sqrt{b^{2} - 4 \, a c} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, \sqrt{b^{2} - 4 \, a c} c}, -\frac{2 \, b \arctan \left (-\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, \sqrt{-b^{2} + 4 \, a c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^6 + b*x^3 + a),x, algorithm="fricas")

[Out]

[1/6*(b*log((2*(b^2*c - 4*a*c^2)*x^3 + b^3 - 4*a*b*c + (2*c^2*x^6 + 2*b*c*x^3 +
b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^6 + b*x^3 + a)) + sqrt(b^2 - 4*a*c)*log(c*x
^6 + b*x^3 + a))/(sqrt(b^2 - 4*a*c)*c), -1/6*(2*b*arctan(-(2*c*x^3 + b)*sqrt(-b^
2 + 4*a*c)/(b^2 - 4*a*c)) - sqrt(-b^2 + 4*a*c)*log(c*x^6 + b*x^3 + a))/(sqrt(-b^
2 + 4*a*c)*c)]

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Sympy [A]  time = 4.17719, size = 223, normalized size = 3.54 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 c \left (4 a c - b^{2}\right )} + \frac{1}{6 c}\right ) \log{\left (x^{3} + \frac{- 12 a c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 c \left (4 a c - b^{2}\right )} + \frac{1}{6 c}\right ) + 2 a + 3 b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 c \left (4 a c - b^{2}\right )} + \frac{1}{6 c}\right )}{b} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 c \left (4 a c - b^{2}\right )} + \frac{1}{6 c}\right ) \log{\left (x^{3} + \frac{- 12 a c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 c \left (4 a c - b^{2}\right )} + \frac{1}{6 c}\right ) + 2 a + 3 b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 c \left (4 a c - b^{2}\right )} + \frac{1}{6 c}\right )}{b} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**6+b*x**3+a),x)

[Out]

(-b*sqrt(-4*a*c + b**2)/(6*c*(4*a*c - b**2)) + 1/(6*c))*log(x**3 + (-12*a*c*(-b*
sqrt(-4*a*c + b**2)/(6*c*(4*a*c - b**2)) + 1/(6*c)) + 2*a + 3*b**2*(-b*sqrt(-4*a
*c + b**2)/(6*c*(4*a*c - b**2)) + 1/(6*c)))/b) + (b*sqrt(-4*a*c + b**2)/(6*c*(4*
a*c - b**2)) + 1/(6*c))*log(x**3 + (-12*a*c*(b*sqrt(-4*a*c + b**2)/(6*c*(4*a*c -
b**2)) + 1/(6*c)) + 2*a + 3*b**2*(b*sqrt(-4*a*c + b**2)/(6*c*(4*a*c - b**2)) +
1/(6*c)))/b)

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GIAC/XCAS [A]  time = 0.310209, size = 80, normalized size = 1.27 \[ -\frac{b \arctan \left (\frac{2 \, c x^{3} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} c} + \frac{{\rm ln}\left (c x^{6} + b x^{3} + a\right )}{6 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

-1/3*b*arctan((2*c*x^3 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c) + 1/6*ln(
c*x^6 + b*x^3 + a)/c