3.111 \(\int \frac{(a+b \log (c x^n))^2}{x (d+e x)^3} \, dx\)

Optimal. Leaf size=257 \[ -\frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{3 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^3}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac{3 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{b e n x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 b d^3 n}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac{b^2 n^2 \log (d+e x)}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.593848, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31} \[ -\frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{3 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^3}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac{3 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{b e n x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 b d^3 n}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac{b^2 n^2 \log (d+e x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

Rule 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[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]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^3} \, dx &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx}{d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{d^2}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{d^2}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{d^3}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^3}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^2}+\frac{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^3}+\frac{(b e n) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^2}\\ &=\frac{b e n x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d^3 n}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^3}+\frac{(2 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^3}+\frac{(b e n) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^3}-\frac{\left (2 b^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^3}-\frac{\left (b^2 e n^2\right ) \int \frac{1}{d+e x} \, dx}{d^3}\\ &=\frac{b e n x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 b d^3 n}-\frac{b^2 n^2 \log (d+e x)}{d^3}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{2 b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}-\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^3}+\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d^3}\\ &=\frac{b e n x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 b d^3 n}-\frac{b^2 n^2 \log (d+e x)}{d^3}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{3 b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}+\frac{2 b^2 n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.251811, size = 232, normalized size = 0.9 \[ \frac{-12 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+18 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+12 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-6 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{6 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+18 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac{2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}-9 \left (a+b \log \left (c x^n\right )\right )^2+6 b^2 n^2 (\log (x)-\log (d+e x))}{6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*b*d*n*(a + b*Log[c*x^n]))/(d + e*x) - 9*(a + b*Log[c*x^n])^2 + (3*d^2*(a + b*Log[c*x^n])^2)/(d + e*x)^2 +
(6*d*(a + b*Log[c*x^n])^2)/(d + e*x) + (2*(a + b*Log[c*x^n])^3)/(b*n) + 6*b^2*n^2*(Log[x] - Log[d + e*x]) + 1
8*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 6*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 18*b^2*n^2*PolyLog[2, -(
(e*x)/d)] - 12*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 12*b^2*n^2*PolyLog[3, -((e*x)/d)])/(6*d^3)

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Maple [F]  time = 0.718, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{x \left ( ex+d \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^2/x/(e*x+d)^3,x)

[Out]

int((a+b*ln(c*x^n))^2/x/(e*x+d)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{2 \, e x + 3 \, d}{d^{2} e^{2} x^{2} + 2 \, d^{3} e x + d^{4}} - \frac{2 \, \log \left (e x + d\right )}{d^{3}} + \frac{2 \, \log \left (x\right )}{d^{3}}\right )} + \int \frac{b^{2} \log \left (c\right )^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \left (c\right ) + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x^{n}\right )}{e^{3} x^{4} + 3 \, d e^{2} x^{3} + 3 \, d^{2} e x^{2} + d^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/x/(e*x+d)^3,x, algorithm="maxima")

[Out]

1/2*a^2*((2*e*x + 3*d)/(d^2*e^2*x^2 + 2*d^3*e*x + d^4) - 2*log(e*x + d)/d^3 + 2*log(x)/d^3) + integrate((b^2*l
og(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(e^3*x^4 + 3*d*e^2*x^3 + 3*d^2*e*x^2
+ d^3*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/x/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**2/x/(e*x+d)**3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2/x/(e*x+d)^3,x, algorithm="giac")

[Out]

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