3.107 \(\int \frac{x^2 \left (b+2 c x^3\right )}{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{3} \log \left (a+b x^3+c x^6\right ) \]

[Out]

Log[a + b*x^3 + c*x^6]/3

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Rubi [A]  time = 0.0109252, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{1}{3} \log \left (a+b x^3+c x^6\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x^3 + c*x^6]/3

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Rubi in Sympy [A]  time = 4.9955, size = 14, normalized size = 0.82 \[ \frac{\log{\left (a + b x^{3} + c x^{6} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**3 + c*x**6)/3

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Mathematica [A]  time = 0.010969, size = 17, normalized size = 1. \[ \frac{1}{3} \log \left (a+b x^3+c x^6\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x^3 + c*x^6]/3

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*ln(c*x^6+b*x^3+a)

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Maxima [A]  time = 0.772063, size = 20, normalized size = 1.18 \[ \frac{1}{3} \, \log \left (c x^{6} + b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(c*x^6 + b*x^3 + a)

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Fricas [A]  time = 0.282079, size = 20, normalized size = 1.18 \[ \frac{1}{3} \, \log \left (c x^{6} + b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(c*x^6 + b*x^3 + a)

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Sympy [A]  time = 1.98869, size = 14, normalized size = 0.82 \[ \frac{\log{\left (a + b x^{3} + c x^{6} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**3 + c*x**6)/3

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GIAC/XCAS [A]  time = 0.265643, size = 20, normalized size = 1.18 \[ \frac{1}{3} \,{\rm ln}\left (c x^{6} + b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*ln(c*x^6 + b*x^3 + a)