3.320 \(\int \frac{x^2}{a+b x^3} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log \left (a+b x^3\right )}{3 b} \]

[Out]

Log[a + b*x^3]/(3*b)

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Rubi [A]  time = 0.00897648, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\log \left (a+b x^3\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^3),x]

[Out]

Log[a + b*x^3]/(3*b)

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Rubi in Sympy [A]  time = 2.21244, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a + b x^{3} \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**3)/(3*b)

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Mathematica [A]  time = 0.00437929, size = 15, normalized size = 1. \[ \frac{\log \left (a+b x^3\right )}{3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^3),x]

[Out]

Log[a + b*x^3]/(3*b)

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \[{\frac{\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.43207, size = 18, normalized size = 1.2 \[ \frac{\log \left (b x^{3} + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(b*x^3 + a)/b

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Fricas [A]  time = 0.2163, size = 18, normalized size = 1.2 \[ \frac{\log \left (b x^{3} + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(b*x^3 + a)/b

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Sympy [A]  time = 0.296338, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a + b x^{3} \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**3+a),x)

[Out]

log(a + b*x**3)/(3*b)

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GIAC/XCAS [A]  time = 0.214722, size = 19, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*ln(abs(b*x^3 + a))/b