\(\int \frac {(a+b \log (c x^n))^2}{x^3 (d+e x)^2} \, dx\) [106]

Optimal result

Integrand size = 23, antiderivative size = 285 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=-\frac {b^2 n^2}{4 d^2 x^2}+\frac {4 b^2 e n^2}{d^3 x}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {3 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {6 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {6 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4} \]

[Out]

-1/4*b^2*n^2/d^2/x^2+4*b^2*e*n^2/d^3/x-1/2*b*n*(a+b*ln(c*x^n))/d^2/x^2+4*b*e*n*(a+b*ln(c*x^n))/d^3/x-1/2*(a+b*
ln(c*x^n))^2/d^2/x^2+2*e*(a+b*ln(c*x^n))^2/d^3/x-e^3*x*(a+b*ln(c*x^n))^2/d^4/(e*x+d)-3*e^2*ln(1+d/e/x)*(a+b*ln
(c*x^n))^2/d^4+2*b*e^2*n*(a+b*ln(c*x^n))*ln(1+e*x/d)/d^4+6*b*e^2*n*(a+b*ln(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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2395, 2342, 2341, 2355, 2354, 2438, 2379, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=-\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {6 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {3 e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {2 b e^2 n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {6 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4}+\frac {4 b^2 e n^2}{d^3 x}-\frac {b^2 n^2}{4 d^2 x^2} \]

[In]

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

[Out]

-1/4*(b^2*n^2)/(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])*
Log[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

Rule 2341

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 2342

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 2354

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 2355

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 2379

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

Rule 2395

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2438

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 6724

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\frac {-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}+\frac {8 d e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {4 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}+\frac {16 b d e n \left (a+b n+b \log \left (c x^n\right )\right )}{x}-\frac {b d^2 n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{x^2}-12 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+4 e^2 \left (-\left (\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (1+\frac {e x}{d}\right )\right )\right )+2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )-24 b e^2 n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{4 d^4} \]

[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)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.48 (sec) , antiderivative size = 924, normalized size of antiderivative = 3.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(924\)

[In]

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

[Out]

-1/2*b^2*ln(x^n)^2/d^2/x^2+2*b^2*ln(x^n)^2/d^3*e/x-6*b^2/d^4*e^2*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2+6*b^2*n/d^4*e^
2*ln(x^n)*ln(e*x+d)*ln(-e*x/d)+b^2*ln(x^n)^2/d^3*e^2/(e*x+d)+4*b^2*n*ln(x^n)/d^3*e/x+4*b^2*e*n^2/d^3/x+(-I*b*P
i*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*
Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(-3*ln(x^n)/d^4*e^2*ln(e*x+d)+ln(x^n)/d^3*e^2/(e*x+d)-1/2*ln(x^n)/d^2/x^2+
3*ln(x^n)/d^4*e^2*ln(x)+2*ln(x^n)/d^3*e/x-1/2*n*(-2/d^4*e^2*ln(e*x+d)+1/2/d^2/x^2-4/d^3*e/x+2/d^4*e^2*ln(x)+3/
d^4*e^2*ln(x)^2-6/d^4*e^2*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))))+2*b^2*n*ln(x^n)/d^4*e^2*ln(e*x+d)-2*b^2*n*ln(
x^n)/d^4*e^2*ln(x)-2*b^2/d^4*n^2*e^2*ln(e*x+d)*ln(-e*x/d)-3*b^2*n/d^4*e^2*ln(x^n)*ln(x)^2-6*b^2/d^4*e^2*ln(x)*
dilog(-e*x/d)*n^2+6*b^2*n/d^4*e^2*ln(x^n)*dilog(-e*x/d)+3*b^2/d^4*e^2*n^2*ln(e*x+d)*ln(x)^2-3*b^2/d^4*e^2*n^2*
ln(x)^2*ln(1+e*x/d)-6*b^2/d^4*e^2*n^2*ln(x)*polylog(2,-e*x/d)+1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)
+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*(
-3/d^4*e^2*ln(e*x+d)+1/d^3*e^2/(e*x+d)-1/2/d^2/x^2+3/d^4*e^2*ln(x)+2/d^3*e/x)-3*b^2*ln(x^n)^2/d^4*e^2*ln(e*x+d
)+3*b^2*ln(x^n)^2/d^4*e^2*ln(x)-1/2*b^2*n*ln(x^n)/d^2/x^2+b^2/d^4*n^2*e^2*ln(x)^2-2*b^2/d^4*n^2*e^2*dilog(-e*x
/d)+b^2/d^4*e^2*ln(x)^3*n^2+6*b^2/d^4*e^2*n^2*polylog(3,-e*x/d)-1/4*b^2*n^2/d^2/x^2

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{3}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{3} \left (d + e x\right )^{2}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3\,{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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