3.45 \(\int x (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \]

[Out]

-1/4*b*n*x^2+1/2*x^2*(a+b*ln(c*x^n))

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2304} \[ \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*Log[c*x^n]),x]

[Out]

-(b*n*x^2)/4 + (x^2*(a + b*Log[c*x^n]))/2

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 \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {1}{4} b n x^2+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.19 \[ \frac {a x^2}{2}+\frac {1}{2} b x^2 \log \left (c x^n\right )-\frac {1}{4} b n x^2 \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*Log[c*x^n]),x]

[Out]

(a*x^2)/2 - (b*n*x^2)/4 + (b*x^2*Log[c*x^n])/2

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fricas [A]  time = 0.45, size = 30, normalized size = 1.11 \[ \frac {1}{2} \, b n x^{2} \log \relax (x) + \frac {1}{2} \, b x^{2} \log \relax (c) - \frac {1}{4} \, {\left (b n - 2 \, a\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/2*b*n*x^2*log(x) + 1/2*b*x^2*log(c) - 1/4*(b*n - 2*a)*x^2

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giac [A]  time = 0.25, size = 31, normalized size = 1.15 \[ \frac {1}{2} \, b n x^{2} \log \relax (x) - \frac {1}{4} \, b n x^{2} + \frac {1}{2} \, b x^{2} \log \relax (c) + \frac {1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/2*b*n*x^2*log(x) - 1/4*b*n*x^2 + 1/2*b*x^2*log(c) + 1/2*a*x^2

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maple [A]  time = 0.05, size = 29, normalized size = 1.07 \[ -\frac {b n \,x^{2}}{4}+\frac {b \,x^{2} \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )}{2}+\frac {a \,x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*ln(c*x^n)),x)

[Out]

1/2*a*x^2+1/2*b*x^2*ln(c*exp(n*ln(x)))-1/4*b*n*x^2

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maxima [A]  time = 0.61, size = 26, normalized size = 0.96 \[ -\frac {1}{4} \, b n x^{2} + \frac {1}{2} \, b x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/4*b*n*x^2 + 1/2*b*x^2*log(c*x^n) + 1/2*a*x^2

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mupad [B]  time = 3.26, size = 25, normalized size = 0.93 \[ x^2\,\left (\frac {a}{2}-\frac {b\,n}{4}\right )+\frac {b\,x^2\,\ln \left (c\,x^n\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*log(c*x^n)),x)

[Out]

x^2*(a/2 - (b*n)/4) + (b*x^2*log(c*x^n))/2

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sympy [A]  time = 0.46, size = 36, normalized size = 1.33 \[ \frac {a x^{2}}{2} + \frac {b n x^{2} \log {\relax (x )}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\relax (c )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*ln(c*x**n)),x)

[Out]

a*x**2/2 + b*n*x**2*log(x)/2 - b*n*x**2/4 + b*x**2*log(c)/2

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