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

Optimal. Leaf size=44 \[ \frac{b n \text{PolyLog}\left (2,-\frac{e}{d x}\right )}{e}-\frac{\log \left (\frac{e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.0660136, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2337, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{e}{d x}\right )}{e}-\frac{\log \left (\frac{e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

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^2} \, dx &=-\frac{\log \left (1+\frac{e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{(b n) \int \frac{\log \left (1+\frac{e}{d x}\right )}{x} \, dx}{e}\\ &=-\frac{\log \left (1+\frac{e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{b n \text{Li}_2\left (-\frac{e}{d x}\right )}{e}\\ \end{align*}

Mathematica [A]  time = 0.0349509, size = 63, normalized size = 1.43 \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac{d x}{e}+1\right )\right )}{2 b e n}-\frac{b n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.134, size = 336, normalized size = 7.6 \begin{align*}{\frac{b\ln \left ({x}^{n} \right ) \ln \left ( x \right ) }{e}}-{\frac{b\ln \left ({x}^{n} \right ) \ln \left ( dx+e \right ) }{e}}-{\frac{bn \left ( \ln \left ( x \right ) \right ) ^{2}}{2\,e}}+{\frac{bn\ln \left ( dx+e \right ) }{e}\ln \left ( -{\frac{dx}{e}} \right ) }+{\frac{bn}{e}{\it dilog} \left ( -{\frac{dx}{e}} \right ) }-{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) \ln \left ( x \right ) }{e}}-{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}\ln \left ( dx+e \right ) }{e}}+{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) \ln \left ( dx+e \right ) }{e}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ( x \right ) }{e}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ( dx+e \right ) }{e}}+{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}\ln \left ( x \right ) }{e}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ( x \right ) }{e}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ( dx+e \right ) }{e}}+{\frac{b\ln \left ( c \right ) \ln \left ( x \right ) }{e}}-{\frac{b\ln \left ( c \right ) \ln \left ( dx+e \right ) }{e}}+{\frac{a\ln \left ( x \right ) }{e}}-{\frac{a\ln \left ( dx+e \right ) }{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a{\left (\frac{\log \left (d x + e\right )}{e} - \frac{\log \left (x\right )}{e}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{d x^{2} + e x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(log(d*x + e)/e - log(x)/e) + b*integrate((log(c) + log(x^n))/(d*x^2 + e*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c x^{n}\right ) + a}{d x^{2} + e x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)/(d*x^2 + e*x), x)

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Sympy [C]  time = 11.2964, size = 156, normalized size = 3.55 \begin{align*} \frac{2 a d \left (\begin{cases} - \frac{x}{e} - \frac{1}{2 d} & \text{for}\: d = 0 \\\frac{\log{\left (2 d x \right )}}{2 d} & \text{otherwise} \end{cases}\right )}{e} - \frac{2 a d \left (\begin{cases} \frac{x}{e} + \frac{1}{2 d} & \text{for}\: d = 0 \\\frac{\log{\left (2 d x + 2 e \right )}}{2 d} & \text{otherwise} \end{cases}\right )}{e} + b n \left (\begin{cases} - \frac{1}{d x} & \text{for}\: e = 0 \\\frac{\begin{cases} \log{\left (d \right )} \log{\left (x \right )} + \operatorname{Li}_{2}\left (\frac{e e^{i \pi }}{d x}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (d \right )} \log{\left (\frac{1}{x} \right )} + \operatorname{Li}_{2}\left (\frac{e e^{i \pi }}{d x}\right ) & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} \log{\left (d \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (d \right )} + \operatorname{Li}_{2}\left (\frac{e e^{i \pi }}{d x}\right ) & \text{otherwise} \end{cases}}{e} & \text{otherwise} \end{cases}\right ) - b \left (\begin{cases} \frac{1}{d x} & \text{for}\: e = 0 \\\frac{\log{\left (d + \frac{e}{x} \right )}}{e} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*d*Piecewise((-x/e - 1/(2*d), Eq(d, 0)), (log(2*d*x)/(2*d), True))/e - 2*a*d*Piecewise((x/e + 1/(2*d), Eq(d
, 0)), (log(2*d*x + 2*e)/(2*d), True))/e + b*n*Piecewise((-1/(d*x), Eq(e, 0)), (Piecewise((log(d)*log(x) + pol
ylog(2, e*exp_polar(I*pi)/(d*x)), Abs(x) < 1), (-log(d)*log(1/x) + polylog(2, e*exp_polar(I*pi)/(d*x)), 1/Abs(
x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) + pol
ylog(2, e*exp_polar(I*pi)/(d*x)), True))/e, True)) - b*Piecewise((1/(d*x), Eq(e, 0)), (log(d + e/x)/e, True))*
log(c*x**n)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (d + \frac{e}{x}\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)/((d + e/x)*x^2), x)