3.2.5 \(\int \frac {(a+b \log (c x^n))^2}{x^2 (d+e x)^2} \, dx\) [105]

Optimal. Leaf size=211 \[ -\frac {2 b^2 n^2}{d^2 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}+\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)}+\frac {2 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac {2 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}-\frac {4 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}-\frac {2 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^3}-\frac {4 b^2 e n^2 \text {Li}_3\left (-\frac {d}{e x}\right )}{d^3} \]

[Out]

-2*b^2*n^2/d^2/x-2*b*n*(a+b*ln(c*x^n))/d^2/x-(a+b*ln(c*x^n))^2/d^2/x+e^2*x*(a+b*ln(c*x^n))^2/d^3/(e*x+d)+2*e*l
n(1+d/e/x)*(a+b*ln(c*x^n))^2/d^3-2*b*e*n*(a+b*ln(c*x^n))*ln(1+e*x/d)/d^3-4*b*e*n*(a+b*ln(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 = 0.22, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2395, 2342, 2341, 2355, 2354, 2438, 2379, 2421, 6724} \begin {gather*} -\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} \end {gather*}

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*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

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*} \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) \text {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*}

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Mathematica [A]
time = 0.21, size = 223, normalized size = 1.06 \begin {gather*} -\frac {\frac {6 b^2 d n^2}{x}+\frac {6 b d n \left (a+b \log \left (c x^n\right )\right )}{x}-3 e \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {3 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}+6 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+6 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )-12 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )+12 b^2 e n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((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])*Lo
g[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)])/d^3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.22, size = 4586, normalized size = 21.73

method	result	size
risch	\(\text {Expression too large to display}\)	\(4586\)

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,method=_RETURNVERBOSE)

[Out]

4*b^2/d^3*e*ln(x)*dilog(-e*x/d)*n^2-2*b^2/d^3*e*n^2*ln(x)^2*ln(e*x+d)+2*b^2/d^3*e*n^2*ln(x)^2*ln(1+e*x/d)+4*b^
2/d^3*e*n^2*ln(x)*polylog(2,-e*x/d)-2/3*b^2/d^3*e*ln(x)^3*n^2-4*b^2/d^3*e*n^2*polylog(3,-e*x/d)-b^2/d^3*n^2*e*
ln(x)^2+2*b^2/d^3*n^2*e*dilog(-e*x/d)+2*I/d^3*n*e*ln(e*x+d)*ln(-e*x/d)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x
^n)+2*I/d^3*e*ln(e*x+d)*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I*e/d^2/(e*x+d)*Pi*a*b*csgn(I*c)*csgn(I*c*x^n
)^2-1/2/d^3*e*ln(e*x+d)*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+1/4/d^2/x*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*cs
gn(I*c*x^n)^2+1/4*e/d^2/(e*x+d)*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4+1/2/d^3*e*ln(x)*Pi^2*b^2*csgn(I*x^n)^2*cs
gn(I*c*x^n)^4+2*b^2/d^3*n^2*e*ln(e*x+d)*ln(-e*x/d)-2/d^2*ln(x^n)/x*b^2*ln(c)-I/d^3*n*e*ln(x)^2*b^2*Pi*csgn(I*c
*x^n)^3+2*I/d^3*n*e*dilog(-e*x/d)*b^2*Pi*csgn(I*c*x^n)^3-I/d^2*n/x*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-a^2/d^2/x-
1/d^2/x*ln(c)^2*b^2-1/2/d^2/x*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+1/4/d^2/x*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n
)^4-1/2/d^2/x*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-1/2*e/d^2/(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5-1/2/d^
2/x*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+1/d^2/x*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+1/
d^3*e*ln(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5-2*ln(x^n)*e/d^2/(e*x+d)*b^2*ln(c)+2*b^2*n/d^3*ln(x^n)*e*ln(
x)-2*b*ln(x^n)*e/d^2/(e*x+d)*a+4*b/d^3*ln(x^n)*e*ln(e*x+d)*a-4*b/d^3*ln(x^n)*e*ln(x)*a+2*I/d^3*e*ln(x)*ln(c)*P
i*b^2*csgn(I*c*x^n)^3+2*I/d^3*e*ln(x)*Pi*a*b*csgn(I*c*x^n)^3+e/d^2/(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn
(I*c*x^n)^4-I/d^2/x*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^n)^2+2*b^2*n/d^3*e*ln(x)^2*ln(x^n)-4*b^2*n/d^3*e*ln(x^n)
*dilog(-e*x/d)-2*b^2*n/d^3*ln(x^n)*e*ln(e*x+d)+2*b^2*ln(x^n)^2/d^3*e*ln(e*x+d)+4*b^2/d^3*e*ln(x)*ln(e*x+d)*ln(
-e*x/d)*n^2-2*e/d^2/(e*x+d)*ln(c)*a*b+4/d^3*e*ln(e*x+d)*ln(c)*a*b-4/d^3*e*ln(x)*ln(c)*a*b+2/d^3*n*e*ln(x)^2*b^
2*ln(c)-4/d^3*n*e*dilog(-e*x/d)*b^2*ln(c)-2*n/d^3*e*ln(e*x+d)*b^2*ln(c)+2*n/d^3*e*ln(x)*b^2*ln(c)+1/2/d^3*e*ln
(x)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-1/2/d^3*e*ln(e*x+d)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*
csgn(I*c*x^n)^2+2/d^3*e*ln(x)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4-4*b/d^3*n*e*dilog(-e*x/d)*a-2*b*n
/d^3*e*ln(e*x+d)*a+2*b*n/d^3*e*ln(x)*a+2*b/d^3*n*e*ln(x)^2*a-a^2*e/d^2/(e*x+d)+2*a^2/d^3*e*ln(e*x+d)-2*a^2/d^3
*e*ln(x)+1/4*e/d^2/(e*x+d)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-1/2*e/d^2/(e*x+d)*Pi^2*b^2*csgn(
I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-1/2*e/d^2/(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3+I*e/d^2/
(e*x+d)*ln(c)*Pi*b^2*csgn(I*c*x^n)^3+I*e/d^2/(e*x+d)*Pi*a*b*csgn(I*c*x^n)^3-2/d^3*e*ln(e*x+d)*Pi^2*b^2*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)^4+1/d^3*e*ln(e*x+d)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+1/d^3*e*ln(e
*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-I/d^2*n/x*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*n/d^3*e*
ln(x)*b^2*Pi*csgn(I*c*x^n)^3+1/4*e/d^2/(e*x+d)*Pi^2*b^2*csgn(I*c*x^n)^6-4*b/d^3*n*e*ln(e*x+d)*ln(-e*x/d)*a-4/d
^3*n*e*ln(e*x+d)*ln(-e*x/d)*b^2*ln(c)-b^2*ln(x^n)^2/d^2/x-I/d^2/x*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I/d^3*e
*ln(e*x+d)*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-2*I/d^3*e*ln(e*x+d)*Pi*a*b*csgn(I*c*x^n)^3-I/d^2*ln(x^n)/x*b^2*Pi*csgn
(I*x^n)*csgn(I*c*x^n)^2-I/d^2/x*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I/d^2/x*Pi*a*b*csgn(I*c)*csgn(I*c*x^n
)^2-2*b/d^2*n/x*a-2/d^2*n/x*b^2*ln(c)+1/d^3*e*ln(e*x+d)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+1/2/d^3*e*ln(x)*P
i^2*b^2*csgn(I*c*x^n)^6+4/d^3*ln(x^n)*e*ln(e*x+d)*b^2*ln(c)-4/d^3*ln(x^n)*e*ln(x)*b^2*ln(c)-b^2*ln(x^n)^2*e/d^
2/(e*x+d)+I*ln(x^n)*e/d^2/(e*x+d)*b^2*Pi*csgn(I*c*x^n)^3+I*n/d^3*e*ln(e*x+d)*b^2*Pi*csgn(I*c*x^n)^3-1/d^3*e*ln
(x)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-1/d^3*e*ln(x)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x
^n)^3-2*I/d^3*ln(x^n)*e*ln(e*x+d)*b^2*Pi*csgn(I*c*x^n)^3-I/d^2*ln(x^n)/x*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*I/
d^3*ln(x^n)*e*ln(x)*b^2*Pi*csgn(I*c*x^n)^3-1/2/d^3*e*ln(e*x+d)*Pi^2*b^2*csgn(I*c*x^n)^6+1/4/d^2/x*Pi^2*b^2*csg
n(I*c)^2*csgn(I*c*x^n)^4+I/d^2*n/x*b^2*Pi*csgn(I*c*x^n)^3-I*e/d^2/(e*x+d)*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^
n)^2-I*ln(x^n)*e/d^2/(e*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*b^2*ln(x^n)^2/d^3*e*ln(x)-2*b^2*n/d^2*ln(x^n
)/x-I*e/d^2/(e*x+d)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^n)^2+2*I/d^3*e*ln(x)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*x^n)*
csgn(I*c*x^n)+2*I/d^3*e*ln(x)*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*I/d^3*e*ln(e*x+d)*Pi*a*b*csgn(I*c)*
csgn(I*c*x^n)^2+2*I/d^3*e*ln(e*x+d)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*e/d^2/(e*x+d)*Pi*a*b*csgn(I*x^n)*csgn
(I*c*x^n)^2-2*I/d^3*n*e*dilog(-e*x/d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+I/d^3*n*e*ln(x)^2*b^2*Pi*csgn(I*c)*cs
gn(I*c*x^n)^2+I/d^2*n/x*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I/d^2/x*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I
*c*x^n)+I/d^2/x*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*n/d^3*e*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x
^n)^2+I/d^3*n*e*ln(x)^2*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I/d^3*ln(x^n)*e*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*c
*x^n)^2+2*I/d^3*ln(x^n)*e*ln(e*x+d)*b^2*Pi*csgn...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*x*e + d)/(d^2*x^2*e + d^3*x) - 2*e*log(x*e + 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))/(x^4*e^2 + 2*d*x^3*e + d^2*x^2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)/(x^4*e^2 + 2*d*x^3*e + d^2*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{2} \left (d + e x\right )^{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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/((x*e + d)^2*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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