3.645 \(\int \frac{x^3}{a+c x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log \left (a+c x^4\right )}{4 c} \]

[Out]

Log[a + c*x^4]/(4*c)

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Rubi [A]  time = 0.00909488, 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 \[ \frac{\log \left (a+c x^4\right )}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + c*x^4),x]

[Out]

Log[a + c*x^4]/(4*c)

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Rubi in Sympy [A]  time = 2.16837, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a + c x^{4} \right )}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(c*x**4+a),x)

[Out]

log(a + c*x**4)/(4*c)

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Mathematica [A]  time = 0.00617343, size = 15, normalized size = 1. \[ \frac{\log \left (a+c x^4\right )}{4 c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + c*x^4),x]

[Out]

Log[a + c*x^4]/(4*c)

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \[{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^4+a),x)

[Out]

1/4*ln(c*x^4+a)/c

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Maxima [A]  time = 1.44464, size = 18, normalized size = 1.2 \[ \frac{\log \left (c x^{4} + a\right )}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + a),x, algorithm="maxima")

[Out]

1/4*log(c*x^4 + a)/c

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Fricas [A]  time = 0.219767, size = 18, normalized size = 1.2 \[ \frac{\log \left (c x^{4} + a\right )}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + a),x, algorithm="fricas")

[Out]

1/4*log(c*x^4 + a)/c

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Sympy [A]  time = 0.363051, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a + c x^{4} \right )}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(c*x**4+a),x)

[Out]

log(a + c*x**4)/(4*c)

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GIAC/XCAS [A]  time = 0.221671, size = 19, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + a),x, algorithm="giac")

[Out]

1/4*ln(abs(c*x^4 + a))/c