3.49 \(\int \frac{a+b \log (c x^n)}{x^3} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0125244, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2304} \[ -\frac{a+b \log \left (c x^n\right )}{2 x^2}-\frac{b n}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/x^3,x]

[Out]

-(b*n)/(4*x^2) - (a + b*Log[c*x^n])/(2*x^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 \frac{a+b \log \left (c x^n\right )}{x^3} \, dx &=-\frac{b n}{4 x^2}-\frac{a+b \log \left (c x^n\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0012246, size = 32, normalized size = 1.19 \[ -\frac{a}{2 x^2}-\frac{b \log \left (c x^n\right )}{2 x^2}-\frac{b n}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.073, size = 111, normalized size = 4.1 \begin{align*} -{\frac{b\ln \left ({x}^{n} \right ) }{2\,{x}^{2}}}-{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +bn+2\,a}{4\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^3,x)

[Out]

-1/2*b/x^2*ln(x^n)-1/4*(I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*b*Pi*c
sgn(I*c*x^n)^3+I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+b*n+2*a)/x^2

________________________________________________________________________________________

Maxima [A]  time = 1.13792, size = 35, normalized size = 1.3 \begin{align*} -\frac{b n}{4 \, x^{2}} - \frac{b \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.834715, size = 68, normalized size = 2.52 \begin{align*} -\frac{2 \, b n \log \left (x\right ) + b n + 2 \, b \log \left (c\right ) + 2 \, a}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.2403, size = 37, normalized size = 1.37 \begin{align*} - \frac{a}{2 x^{2}} - \frac{b n \log{\left (x \right )}}{2 x^{2}} - \frac{b n}{4 x^{2}} - \frac{b \log{\left (c \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.14973, size = 36, normalized size = 1.33 \begin{align*} -\frac{b n \log \left (x\right )}{2 \, x^{2}} - \frac{b n + 2 \, b \log \left (c\right ) + 2 \, a}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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