3.333 \(\int \frac {a+b \log (c x^n)}{(d+\frac {e}{x}) x} \, dx\)

Optimal. Leaf size=39 \[ \frac {\log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {b n \text {Li}_2\left (-\frac {d x}{e}\right )}{d} \]

[Out]

(a+b*ln(c*x^n))*ln(1+d*x/e)/d+b*n*polylog(2,-d*x/e)/d

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Rubi [A]  time = 0.08, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2333, 2317, 2391} \[ \frac {b n \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{d}+\frac {\log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/((d + e/x)*x),x]

[Out]

((a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d + (b*n*PolyLog[2, -((d*x)/e)])/d

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*
x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx &=\int \frac {a+b \log \left (c x^n\right )}{e+d x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d}-\frac {(b n) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d}+\frac {b n \text {Li}_2\left (-\frac {d x}{e}\right )}{d}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 0.95 \[ \frac {\log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )+b n \text {Li}_2\left (-\frac {d x}{e}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/((d + e/x)*x),x]

[Out]

((a + b*Log[c*x^n])*Log[1 + (d*x)/e] + b*n*PolyLog[2, -((d*x)/e)])/d

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fricas [F]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{d x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)/(d*x + e), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)/((d + e/x)*x), x)

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maple [C]  time = 0.19, size = 195, normalized size = 5.00 \[ -\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (d x +e \right )}{2 d}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d x +e \right )}{2 d}+\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d x +e \right )}{2 d}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (d x +e \right )}{2 d}-\frac {b n \ln \left (-\frac {d x}{e}\right ) \ln \left (d x +e \right )}{d}-\frac {b n \dilog \left (-\frac {d x}{e}\right )}{d}+\frac {b \ln \relax (c ) \ln \left (d x +e \right )}{d}+\frac {b \ln \left (x^{n}\right ) \ln \left (d x +e \right )}{d}+\frac {a \ln \left (d x +e \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{d x + e}\,{d x} + \frac {a \log \left (d x + e\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate((log(c) + log(x^n))/(d*x + e), x) + a*log(d*x + e)/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (d+\frac {e}{x}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(x*(d + e/x)),x)

[Out]

int((a + b*log(c*x^n))/(x*(d + e/x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \log {\left (c x^{n} \right )}}{d x + e}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*x**n))/(d*x + e), x)

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