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

Optimal. Leaf size=114 \[ -\frac{2 b e n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^3}+\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}+\frac{2 e \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{a+b \log \left (c x^n\right )}{d^2 x}-\frac{b e n \log (d+e x)}{d^3}-\frac{b n}{d^2 x} \]

[Out]

-((b*n)/(d^2*x)) - (a + b*Log[c*x^n])/(d^2*x) + (e^2*x*(a + b*Log[c*x^n]))/(d^3*(d + e*x)) + (2*e*Log[1 + d/(e
*x)]*(a + b*Log[c*x^n]))/d^3 - (b*e*n*Log[d + e*x])/d^3 - (2*b*e*n*PolyLog[2, -(d/(e*x))])/d^3

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Rubi [A]  time = 0.179155, antiderivative size = 134, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {44, 2351, 2304, 2301, 2314, 31, 2317, 2391} \[ \frac{2 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}+\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{b d^3 n}+\frac{2 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{a+b \log \left (c x^n\right )}{d^2 x}-\frac{b e n \log (d+e x)}{d^3}-\frac{b n}{d^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*n)/(d^2*x)) - (a + b*Log[c*x^n])/(d^2*x) + (e^2*x*(a + b*Log[c*x^n]))/(d^3*(d + e*x)) - (e*(a + b*Log[c*x
^n])^2)/(b*d^3*n) - (b*e*n*Log[d + e*x])/d^3 + (2*e*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^3 + (2*b*e*n*PolyLo
g[2, -((e*x)/d)])/d^3

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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
]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2314

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 )}{x^2 (d+e x)^2} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{d^2 x^2}-\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)^2}+\frac{2 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d^2}-\frac{(2 e) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^3}+\frac{\left (2 e^2\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^3}+\frac{e^2 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^2}\\ &=-\frac{b n}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{d^2 x}+\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{b d^3 n}+\frac{2 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}-\frac{(2 b e n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^3}-\frac{\left (b e^2 n\right ) \int \frac{1}{d+e x} \, dx}{d^3}\\ &=-\frac{b n}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{d^2 x}+\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{b d^3 n}-\frac{b e n \log (d+e x)}{d^3}+\frac{2 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{2 b e n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.137759, size = 120, normalized size = 1.05 \[ -\frac{-2 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right )-2 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{d e \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{b n}-b e n (\log (x)-\log (d+e x))+\frac{b d n}{x}}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*d*n)/x + (d*(a + b*Log[c*x^n]))/x + (d*e*(a + b*Log[c*x^n]))/(d + e*x) + (e*(a + b*Log[c*x^n])^2)/(b*n)
- b*e*n*(Log[x] - Log[d + e*x]) - 2*e*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 2*b*e*n*PolyLog[2, -((e*x)/d)])/d^
3)

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Maple [C]  time = 0.169, size = 703, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a{\left (\frac{2 \, e x + d}{d^{2} e x^{2} + d^{3} x} - \frac{2 \, e \log \left (e x + d\right )}{d^{3}} + \frac{2 \, e \log \left (x\right )}{d^{3}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{2} x^{4} + 2 \, d e x^{3} + d^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*((2*e*x + d)/(d^2*e*x^2 + d^3*x) - 2*e*log(e*x + d)/d^3 + 2*e*log(x)/d^3) + b*integrate((log(c) + log(x^n))
/(e^2*x^4 + 2*d*e*x^3 + d^2*x^2), 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}{e^{2} x^{4} + 2 \, d e x^{3} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)/(e^2*x^4 + 2*d*e*x^3 + d^2*x^2), x)

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Sympy [A]  time = 71.7971, size = 299, normalized size = 2.62 \begin{align*} \frac{a e^{2} \left (\begin{cases} \frac{x}{d^{2}} & \text{for}\: e = 0 \\- \frac{1}{d e + e^{2} x} & \text{otherwise} \end{cases}\right )}{d^{2}} - \frac{a}{d^{2} x} + \frac{2 a e^{2} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right )}{d^{3}} - \frac{2 a e \log{\left (x \right )}}{d^{3}} - \frac{b e^{2} n \left (\begin{cases} \frac{x}{d^{2}} & \text{for}\: e = 0 \\- \frac{\log{\left (x \right )}}{d e} + \frac{\log{\left (\frac{d}{e} + x \right )}}{d e} & \text{otherwise} \end{cases}\right )}{d^{2}} + \frac{b e^{2} \left (\begin{cases} \frac{x}{d^{2}} & \text{for}\: e = 0 \\- \frac{1}{d e + e^{2} x} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{d^{2}} - \frac{b n}{d^{2} x} - \frac{b \log{\left (c x^{n} \right )}}{d^{2} x} - \frac{2 b e^{2} n \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\begin{cases} \log{\left (d \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (d \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\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 x e^{i \pi }}{d}\right ) & \text{otherwise} \end{cases}}{e} & \text{otherwise} \end{cases}\right )}{d^{3}} + \frac{2 b e^{2} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{d^{3}} + \frac{b e n \log{\left (x \right )}^{2}}{d^{3}} - \frac{2 b e \log{\left (x \right )} \log{\left (c x^{n} \right )}}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*e**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/d**2 - a/(d**2*x) + 2*a*e**2*Piecewise((x/d, E
q(e, 0)), (log(d + e*x)/e, True))/d**3 - 2*a*e*log(x)/d**3 - b*e**2*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(
d*e) + log(d/e + x)/(d*e), True))/d**2 + b*e**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c
*x**n)/d**2 - b*n/(d**2*x) - b*log(c*x**n)/(d**2*x) - 2*b*e**2*n*Piecewise((x/d, Eq(e, 0)), (Piecewise((log(d)
*log(x) - polylog(2, e*x*exp_polar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/d
), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log
(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True))/d**3 + 2*b*e**2*Piecewise((x/d, Eq(e, 0)), (log(d +
e*x)/e, True))*log(c*x**n)/d**3 + b*e*n*log(x)**2/d**3 - 2*b*e*log(x)*log(c*x**n)/d**3

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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 (e x + d\right )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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