3.48 \(\int \frac {a+b \log (c x^n)}{x^2} \, dx\)

Optimal. Leaf size=23 \[ -\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, 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} \[ -\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/x^2,x]

[Out]

-((b*n)/x) - (a + b*Log[c*x^n])/x

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

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.13 \[ -\frac {a}{x}-\frac {b \log \left (c x^n\right )}{x}-\frac {b n}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/x^2,x]

[Out]

-(a/x) - (b*n)/x - (b*Log[c*x^n])/x

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fricas [A]  time = 0.46, size = 19, normalized size = 0.83 \[ -\frac {b n \log \relax (x) + b n + b \log \relax (c) + a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2,x, algorithm="fricas")

[Out]

-(b*n*log(x) + b*n + b*log(c) + a)/x

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giac [A]  time = 0.25, size = 24, normalized size = 1.04 \[ -\frac {b n \log \relax (x)}{x} - \frac {b n + b \log \relax (c) + a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.12, size = 112, normalized size = 4.87 \[ -\frac {b \ln \left (x^{n}\right )}{x}-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b n +2 b \ln \relax (c )+2 a}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^2,x)

[Out]

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

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maxima [A]  time = 0.55, size = 26, normalized size = 1.13 \[ -\frac {b n}{x} - \frac {b \log \left (c x^{n}\right )}{x} - \frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2,x, algorithm="maxima")

[Out]

-b*n/x - b*log(c*x^n)/x - a/x

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mupad [B]  time = 3.56, size = 23, normalized size = 1.00 \[ -\frac {a+b\,n}{x}-\frac {b\,\ln \left (c\,x^n\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/x^2,x)

[Out]

- (a + b*n)/x - (b*log(c*x^n))/x

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sympy [A]  time = 0.46, size = 24, normalized size = 1.04 \[ - \frac {a}{x} - \frac {b n \log {\relax (x )}}{x} - \frac {b n}{x} - \frac {b \log {\relax (c )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**2,x)

[Out]

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

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