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

Optimal. Leaf size=19 \[ \frac{1}{3} \log \left (a-b x^3-c x^6\right ) \]

[Out]

Log[a - b*x^3 - c*x^6]/3

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Rubi [A]  time = 0.010457, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{1}{3} \log \left (a-b x^3-c x^6\right ) \]

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 = 5.4267, size = 14, normalized size = 0.74 \[ \frac{\log{\left (- a + b x^{3} + c x^{6} \right )}}{3} \]

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 = 0.0114698, size = 19, 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 = 18, normalized size = 1. \[{\frac{\ln \left ( c{x}^{6}+b{x}^{3}-a \right ) }{3}} \]

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 = 0.759191, size = 23, normalized size = 1.21 \[ \frac{1}{3} \, \log \left (c x^{6} + b x^{3} - a\right ) \]

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 = 0.256263, size = 23, normalized size = 1.21 \[ \frac{1}{3} \, \log \left (c x^{6} + b x^{3} - a\right ) \]

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 = 1.93796, size = 14, normalized size = 0.74 \[ \frac{\log{\left (- a + b x^{3} + c x^{6} \right )}}{3} \]

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 = 0.266664, size = 23, normalized size = 1.21 \[ \frac{1}{3} \,{\rm ln}\left (c x^{6} + b x^{3} - a\right ) \]

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)