3.7.47 \(\int \frac {x^3}{a+c x^4} \, dx\) [647]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {266} \begin {gather*} \frac {\log \left (a+c x^4\right )}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + c*x^4),x]

[Out]

Log[a + c*x^4]/(4*c)

Rule 266

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*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (a+c x^4\right )}{4 c} \end {gather*}

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 = 0.13, size = 14, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {\ln \left (x^{4} c +a \right )}{4 c}\)	\(14\)
default	\(\frac {\ln \left (x^{4} c +a \right )}{4 c}\)	\(14\)
norman	\(\frac {\ln \left (x^{4} c +a \right )}{4 c}\)	\(14\)
risch	\(\frac {\ln \left (x^{4} c +a \right )}{4 c}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (c x^{4} + a\right )}{4 \, c} \end {gather*}

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 = 0.36, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (c x^{4} + a\right )}{4 \, c} \end {gather*}

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 = 0.06, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log {\left (a + c x^{4} \right )}}{4 c} \end {gather*}

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 = 0.57, size = 14, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, c} \end {gather*}

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

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Mupad [B]
time = 0.03, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (c\,x^4+a\right )}{4\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + c*x^4),x)

[Out]

log(a + c*x^4)/(4*c)

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