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

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 = 0.0032196, 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, Rules used = {260} \[ \frac{\log \left (a+c x^4\right )}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + c*x^4),x]

[Out]

Log[a + c*x^4]/(4*c)

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{x^3}{a+c x^4} \, dx &=\frac{\log \left (a+c x^4\right )}{4 c}\\ \end{align*}

Mathematica [A]  time = 0.0034486, 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 = 0.001, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,c}} \end{align*}

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 = 0.963588, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (c x^{4} + a\right )}{4 \, c} \end{align*}

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 = 1.59236, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (c x^{4} + a\right )}{4 \, c} \end{align*}

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 = 0.192819, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (a + c x^{4} \right )}}{4 c} \end{align*}

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 [A]  time = 1.12585, size = 19, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(abs(c*x^4 + a))/c