3.1 \(\int x^3 \log (c x) \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{4} x^4 \log (c x)-\frac {x^4}{16} \]

[Out]

-1/16*x^4+1/4*x^4*ln(c*x)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2304} \[ \frac {1}{4} x^4 \log (c x)-\frac {x^4}{16} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Log[c*x],x]

[Out]

-x^4/16 + (x^4*Log[c*x])/4

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {align*} \int x^3 \log (c x) \, dx &=-\frac {x^4}{16}+\frac {1}{4} x^4 \log (c x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {1}{4} x^4 \log (c x)-\frac {x^4}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Log[c*x],x]

[Out]

-1/16*x^4 + (x^4*Log[c*x])/4

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 15, normalized size = 0.79 \[ \frac {1}{4} \, x^{4} \log \left (c x\right ) - \frac {1}{16} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(c*x),x, algorithm="fricas")

[Out]

1/4*x^4*log(c*x) - 1/16*x^4

________________________________________________________________________________________

giac [A]  time = 0.30, size = 15, normalized size = 0.79 \[ \frac {1}{4} \, x^{4} \log \left (c x\right ) - \frac {1}{16} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(c*x),x, algorithm="giac")

[Out]

1/4*x^4*log(c*x) - 1/16*x^4

________________________________________________________________________________________

maple [A]  time = 0.03, size = 16, normalized size = 0.84 \[ \frac {x^{4} \ln \left (c x \right )}{4}-\frac {x^{4}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*ln(c*x),x)

[Out]

-1/16*x^4+1/4*x^4*ln(c*x)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 15, normalized size = 0.79 \[ \frac {1}{4} \, x^{4} \log \left (c x\right ) - \frac {1}{16} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(c*x),x, algorithm="maxima")

[Out]

1/4*x^4*log(c*x) - 1/16*x^4

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 11, normalized size = 0.58 \[ \frac {x^4\,\left (\ln \left (c\,x\right )-\frac {1}{4}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*log(c*x),x)

[Out]

(x^4*(log(c*x) - 1/4))/4

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 14, normalized size = 0.74 \[ \frac {x^{4} \log {\left (c x \right )}}{4} - \frac {x^{4}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*ln(c*x),x)

[Out]

x**4*log(c*x)/4 - x**4/16

________________________________________________________________________________________