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]

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

________________________________________________________________________________________

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 = 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 = 0.00, 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]

-1/2*a/x^2 - (b*n)/(4*x^2) - (b*Log[c*x^n])/(2*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 23, normalized size = 0.85 \[ -\frac {2 \, b n \log \relax (x) + b n + 2 \, b \log \relax (c) + 2 \, a}{4 \, x^{2}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.25, size = 27, normalized size = 1.00 \[ -\frac {b n \log \relax (x)}{2 \, x^{2}} - \frac {b n + 2 \, b \log \relax (c) + 2 \, a}{4 \, x^{2}} \]

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

________________________________________________________________________________________

maple [C]  time = 0.11, size = 111, normalized size = 4.11 \[ -\frac {b \ln \left (x^{n}\right )}{2 x^{2}}-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+b n +2 b \ln \relax (c )+2 a}{4 x^{2}} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 26, normalized size = 0.96 \[ -\frac {b n}{4 \, x^{2}} - \frac {b \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \]

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

________________________________________________________________________________________

mupad [B]  time = 3.49, size = 26, normalized size = 0.96 \[ -\frac {\frac {a}{2}+\frac {b\,n}{4}}{x^2}-\frac {b\,\ln \left (c\,x^n\right )}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/x^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.94, size = 37, normalized size = 1.37 \[ - \frac {a}{2 x^{2}} - \frac {b n \log {\relax (x )}}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\relax (c )}}{2 x^{2}} \]

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)

________________________________________________________________________________________