3.223 \(\int \frac{x^2}{\sqrt{a+b x^3+c x^6}} \, dx\)

Optimal. Leaf size=43 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0727881, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.5233, size = 37, normalized size = 0.86 \[ \frac{\operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0245715, size = 41, normalized size = 0.95 \[ \frac{\log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{3 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

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

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \[ \int{{x}^{2}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(x^2/(c*x^6+b*x^3+a)^(1/2),x)

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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^2/sqrt(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266205, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{6 \, \sqrt{c}}, \frac{\arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{3 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(c*x^6 + b*x^3 + a),x, algorithm="fricas")

[Out]

[1/6*log(-4*sqrt(c*x^6 + b*x^3 + a)*(2*c^2*x^3 + b*c) - (8*c^2*x^6 + 8*b*c*x^3 +
b^2 + 4*a*c)*sqrt(c))/sqrt(c), 1/3*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^
6 + b*x^3 + a)*c))/sqrt(-c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + b x^{3} + c x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(x**2/sqrt(a + b*x**3 + c*x**6), x)

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GIAC/XCAS [A]  time = 0.292777, size = 54, normalized size = 1.26 \[ -\frac{{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{3} - \sqrt{c x^{6} + b x^{3} + a}\right )} \sqrt{c} - b \right |}\right )}{3 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

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