3.4.6 \(\int \frac {a+b \log (c (d+e x)^n)}{(f+\frac {g}{x}) x} \, dx\) [306]

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

[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 = 0.07, 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 = {2459, 2441, 2440, 2438} \begin {gather*} \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} \end {gather*}

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 2438

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 2440

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 2441

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 2459

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) \text {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 \begin {gather*} \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 {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 261, normalized size = 4.14

method	result	size
risch	\(\frac {b \ln \left (f x +g \right ) \ln \left (\left (e x +d \right )^{n}\right )}{f}-\frac {b n \dilog \left (\frac {\left (f x +g \right ) e +d f -e g}{d f -e g}\right )}{f}-\frac {b n \ln \left (f x +g \right ) \ln \left (\frac {\left (f x +g \right ) e +d f -e g}{d f -e g}\right )}{f}-\frac {i \ln \left (f x +g \right ) b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2 f}+\frac {i \ln \left (f x +g \right ) b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 f}+\frac {i \ln \left (f x +g \right ) b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 f}-\frac {i \ln \left (f x +g \right ) b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2 f}+\frac {\ln \left (f x +g \right ) b \ln \left (c \right )}{f}+\frac {a \ln \left (f x +g \right )}{f}\)	\(261\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e + d)^n*c) + a)/(f*x + g), x)

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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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