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

Optimal. Leaf size=63 \[ \frac {\log \left (-\frac {e (f x+g)}{d f-e g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}+\frac {b n \text {Li}_2\left (\frac {f (d+e x)}{d f-e g}\right )}{f} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2412, 2394, 2393, 2391} \[ \frac {b n \text {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f}+\frac {\log \left (-\frac {e (f x+g)}{d f-e g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2412

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 62, normalized size = 0.98 \[ \frac {\log \left (\frac {e (f x+g)}{e g-d f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}+\frac {b n \text {Li}_2\left (\frac {f (d+e x)}{d f-e g}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{f x + g}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.28, size = 261, normalized size = 4.14 \[ -\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \left (f x +g \right )}{2 f}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (f x +g \right )}{2 f}+\frac {i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (f x +g \right )}{2 f}-\frac {i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (f x +g \right )}{2 f}-\frac {b n \ln \left (\frac {d f -e g +\left (f x +g \right ) e}{d f -e g}\right ) \ln \left (f x +g \right )}{f}-\frac {b n \dilog \left (\frac {d f -e g +\left (f x +g \right ) e}{d f -e g}\right )}{f}+\frac {b \ln \relax (c ) \ln \left (f x +g \right )}{f}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (f x +g \right )}{f}+\frac {a \ln \left (f x +g \right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x\,\left (f+\frac {g}{x}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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