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

Optimal. Leaf size=211 \[ -\frac{4 b e n \text{PolyLog}\left (2,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{2 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}-\frac{4 b^2 e n^2 \text{PolyLog}\left (3,-\frac{d}{e x}\right )}{d^3}+\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}-\frac{2 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{2 e \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac{2 b^2 n^2}{d^2 x} \]

[Out]

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

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Rubi [A]  time = 0.314867, antiderivative size = 231, normalized size of antiderivative = 1.09, number of steps used = 12, 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} \[ \frac{4 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{2 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}-\frac{4 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^3}+\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}-\frac{2 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^3 n}+\frac{2 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac{2 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac{2 b^2 n^2}{d^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.307402, size = 223, normalized size = 1.06 \[ -\frac{-12 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b^2 e n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+12 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )-6 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+6 b e n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{6 b d n \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}-3 e \left (a+b \log \left (c x^n\right )\right )^2+\frac{6 b^2 d n^2}{x}}{3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.717, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{{x}^{2} \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^2/(e*x+d)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a^{2}{\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 )} + \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^{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))^2/x^2/(e*x+d)^2,x, algorithm="maxima")

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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