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]

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

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Rubi [A]  time = 0.0072916, antiderivative size = 27, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0010958, 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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Maple [A]  time = 0.052, size = 29, normalized size = 1.1 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{{x}^{2}b\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{2}}-{\frac{bn{x}^{2}}{4}} \end{align*}

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

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Maxima [A]  time = 1.1695, size = 35, normalized size = 1.3 \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 0.803031, size = 82, normalized size = 3.04 \begin{align*} \frac{1}{2} \, b n x^{2} \log \left (x\right ) + \frac{1}{2} \, b x^{2} \log \left (c\right ) - \frac{1}{4} \,{\left (b n - 2 \, a\right )} x^{2} \end{align*}

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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Sympy [A]  time = 0.460422, size = 36, normalized size = 1.33 \begin{align*} \frac{a x^{2}}{2} + \frac{b n x^{2} \log{\left (x \right )}}{2} - \frac{b n x^{2}}{4} + \frac{b x^{2} \log{\left (c \right )}}{2} \end{align*}

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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Giac [A]  time = 1.17518, size = 42, normalized size = 1.56 \begin{align*} \frac{1}{2} \, b n x^{2} \log \left (x\right ) - \frac{1}{4} \, b n x^{2} + \frac{1}{2} \, b x^{2} \log \left (c\right ) + \frac{1}{2} \, a x^{2} \end{align*}

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