\(\int \frac {(a+b \log (c x^n))^2}{x (d+e x)^3} \, dx\) [111]

Optimal result

Integrand size = 23, antiderivative size = 257 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^3} \, dx=\frac {b e n x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b d^3 n}-\frac {b^2 n^2 \log (d+e x)}{d^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^3}+\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^3} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {b e n x \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}+\frac {2 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^3}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^3}-\frac {b^2 n^2 \log (d+e x)}{d^3} \]

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

Rule 31

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

Rule 2351

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 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 2356

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 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 2389

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^3} \, dx=\frac {-\frac {6 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-9 \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {6 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}+6 b^2 n^2 (\log (x)-\log (d+e x))+18 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+18 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+12 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{6 d^3} \]

[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)

Maple [C] (warning: unable to verify)

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

Time = 0.52 (sec) , antiderivative size = 793, normalized size of antiderivative = 3.09

method	result	size
risch	\(\text {Expression too large to display}\)	\(793\)

[In]

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

[Out]

-b^2*ln(x^n)^2/d^3*ln(e*x+d)+b^2*ln(x^n)^2/d^2/(e*x+d)+1/2*b^2*ln(x^n)^2/d/(e*x+d)^2+b^2*ln(x^n)^2/d^3*ln(x)-b
^2*n*ln(x^n)/d^2/(e*x+d)+3*b^2*n*ln(x^n)/d^3*ln(e*x+d)-3*b^2*n*ln(x^n)/d^3*ln(x)-b^2*n^2*ln(e*x+d)/d^3+b^2/d^3
*n^2*ln(x)+3/2*b^2/d^3*n^2*ln(x)^2-3*b^2/d^3*n^2*ln(e*x+d)*ln(-e*x/d)-3*b^2/d^3*n^2*dilog(-e*x/d)-b^2*n/d^3*ln
(x^n)*ln(x)^2+1/3*b^2/d^3*ln(x)^3*n^2-2*b^2/d^3*ln(e*x+d)*ln(-e*x/d)*ln(x)*n^2+2*b^2*n/d^3*ln(x^n)*ln(e*x+d)*l
n(-e*x/d)-2*b^2/d^3*dilog(-e*x/d)*ln(x)*n^2+2*b^2*n/d^3*ln(x^n)*dilog(-e*x/d)+b^2/d^3*n^2*ln(e*x+d)*ln(x)^2-b^
2/d^3*n^2*ln(x)^2*ln(1+e*x/d)-2*b^2/d^3*n^2*ln(x)*polylog(2,-e*x/d)+2*b^2*n^2*polylog(3,-e*x/d)/d^3+(-I*b*Pi*c
sgn(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*(-ln(x^n)/d^3*ln(e*x+d)+ln(x^n)/d^2/(e*x+d)+1/2*ln(x^n)/d/(e*x+d)^2+ln(x^n)/d
^3*ln(x)-1/2*n*(1/d^2/(e*x+d)-3/d^3*ln(e*x+d)+3/d^3*ln(x)+1/d^3*ln(x)^2-2/d^3*ln(e*x+d)*ln(-e*x/d)-2/d^3*dilog
(-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*(-1/d^3*ln(e*x+d)+1/d^2/(e*x+d)+1/2/d/(e*x+d)^2+1/d^
3*ln(x))

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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