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

Optimal. Leaf size=285 \[ \frac{6 b e^2 n \text{PolyLog}\left (2,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac{2 b^2 e^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^4}+\frac{6 b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{d}{e x}\right )}{d^4}-\frac{e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac{3 e^2 \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac{2 b e^2 n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac{4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac{4 b^2 e n^2}{d^3 x}-\frac{b^2 n^2}{4 d^2 x^2} \]

[Out]

-(b^2*n^2)/(4*d^2*x^2) + (4*b^2*e*n^2)/(d^3*x) - (b*n*(a + b*Log[c*x^n]))/(2*d^2*x^2) + (4*b*e*n*(a + b*Log[c*
x^n]))/(d^3*x) - (a + b*Log[c*x^n])^2/(2*d^2*x^2) + (2*e*(a + b*Log[c*x^n])^2)/(d^3*x) - (e^3*x*(a + b*Log[c*x
^n])^2)/(d^4*(d + e*x)) - (3*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^4 + (2*b*e^2*n*(a + b*Log[c*x^n])*Lo
g[1 + (e*x)/d])/d^4 + (6*b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^4 + (2*b^2*e^2*n^2*PolyLog[2, -(
(e*x)/d)])/d^4 + (6*b^2*e^2*n^2*PolyLog[3, -(d/(e*x))])/d^4

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Rubi [A]  time = 0.377446, antiderivative size = 304, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {2353, 2305, 2304, 2302, 30, 2318, 2317, 2391, 2374, 6589} \[ -\frac{6 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac{2 b^2 e^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^4}+\frac{6 b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^4}-\frac{e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}-\frac{3 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac{2 b e^2 n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac{4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac{4 b^2 e n^2}{d^3 x}-\frac{b^2 n^2}{4 d^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*n^2)/(4*d^2*x^2) + (4*b^2*e*n^2)/(d^3*x) - (b*n*(a + b*Log[c*x^n]))/(2*d^2*x^2) + (4*b*e*n*(a + b*Log[c*
x^n]))/(d^3*x) - (a + b*Log[c*x^n])^2/(2*d^2*x^2) + (2*e*(a + b*Log[c*x^n])^2)/(d^3*x) - (e^3*x*(a + b*Log[c*x
^n])^2)/(d^4*(d + e*x)) + (e^2*(a + b*Log[c*x^n])^3)/(b*d^4*n) + (2*b*e^2*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d
])/d^4 - (3*e^2*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^4 + (2*b^2*e^2*n^2*PolyLog[2, -((e*x)/d)])/d^4 - (6*b
*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^4 + (6*b^2*e^2*n^2*PolyLog[3, -((e*x)/d)])/d^4

Rule 2353

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

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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 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 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 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]

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx &=\int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x^3}-\frac{2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x^2}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}-\frac{3 e^3 \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx}{d^2}-\frac{(2 e) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{d^3}+\frac{\left (3 e^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{d^4}-\frac{\left (3 e^3\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^4}-\frac{e^3 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{d^3}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac{e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^4}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d^4 n}+\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{d^2}-\frac{(4 b e n) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d^3}+\frac{\left (6 b e^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^4}+\frac{\left (2 b e^3 n\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^4}\\ &=-\frac{b^2 n^2}{4 d^2 x^2}+\frac{4 b^2 e n^2}{d^3 x}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac{4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac{e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}+\frac{2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^4}-\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^4}-\frac{6 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^4}-\frac{\left (2 b^2 e^2 n^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^4}+\frac{\left (6 b^2 e^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d^4}\\ &=-\frac{b^2 n^2}{4 d^2 x^2}+\frac{4 b^2 e n^2}{d^3 x}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac{4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac{e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}+\frac{2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^4}-\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^4}+\frac{2 b^2 e^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^4}-\frac{6 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^4}+\frac{6 b^2 e^2 n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^4}\\ \end{align*}

Mathematica [A]  time = 0.181271, size = 268, normalized size = 0.94 \[ \frac{4 e^2 \left (2 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )-\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac{e x}{d}+1\right )\right )\right )-24 b e^2 n \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{PolyLog}\left (3,-\frac{e x}{d}\right )\right )-\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac{b d^2 n \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{x^2}-12 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{4 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac{8 d e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{16 b d e n \left (a+b \log \left (c x^n\right )+b n\right )}{x}+\frac{4 e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}}{4 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*d^2*(a + b*Log[c*x^n])^2)/x^2 + (8*d*e*(a + b*Log[c*x^n])^2)/x + (4*d*e^2*(a + b*Log[c*x^n])^2)/(d + e*x)
+ (4*e^2*(a + b*Log[c*x^n])^3)/(b*n) + (16*b*d*e*n*(a + b*n + b*Log[c*x^n]))/x - (b*d^2*n*(2*a + b*n + 2*b*Lo
g[c*x^n]))/x^2 - 12*e^2*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 4*e^2*(-((a + b*Log[c*x^n])*(a + b*Log[c*x^n]
- 2*b*n*Log[1 + (e*x)/d])) + 2*b^2*n^2*PolyLog[2, -((e*x)/d)]) - 24*b*e^2*n*((a + b*Log[c*x^n])*PolyLog[2, -((
e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)]))/(4*d^4)

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Maple [F]  time = 0.875, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{{x}^{3} \left ( ex+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{6 \, e^{2} x^{2} + 3 \, d e x - d^{2}}{d^{3} e x^{3} + d^{4} x^{2}} - \frac{6 \, e^{2} \log \left (e x + d\right )}{d^{4}} + \frac{6 \, e^{2} \log \left (x\right )}{d^{4}}\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^{2} x^{5} + 2 \, d e x^{4} + d^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*((6*e^2*x^2 + 3*d*e*x - d^2)/(d^3*e*x^3 + d^4*x^2) - 6*e^2*log(e*x + d)/d^4 + 6*e^2*log(x)/d^4) + inte
grate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(e^2*x^5 + 2*d*e*x^4 + d^
2*x^3), 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^{2} x^{5} + 2 \, d e x^{4} + d^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^2*x^5 + 2*d*e*x^4 + d^2*x^3), 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^{3} \left (d + e x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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