3.2.8 \(\int \frac {x^2 (a+b \log (c x^n))^2}{(d+e x)^3} \, dx\) [108]

Optimal. Leaf size=232 \[ -\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \log (d+e x)}{e^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {3 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3} \]

[Out]

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

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Rubi [A]
time = 0.28, antiderivative size = 262, normalized size of antiderivative = 1.13, number of steps used = 14, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2395, 2356, 2389, 2379, 2438, 2351, 31, 2355, 2354, 2421, 6724} \begin {gather*} \frac {2 b n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {b^2 n^2 \text {PolyLog}\left (2,-\frac {d}{e x}\right )}{e^3}+\frac {4 b^2 n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {4 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \log (d+e x)}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*n*x*(a + b*Log[c*x^n]))/(e^2*(d + e*x))) - (b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/e^3 - (d^2*(a + b*Lo
g[c*x^n])^2)/(2*e^3*(d + e*x)^2) - (2*x*(a + b*Log[c*x^n])^2)/(e^2*(d + e*x)) + (b^2*n^2*Log[d + e*x])/e^3 + (
4*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 + ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^3 + (b^2*n^2*PolyLo
g[2, -(d/(e*x))])/e^3 + (4*b^2*n^2*PolyLog[2, -((e*x)/d)])/e^3 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/
d)])/e^3 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^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 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 {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^3}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^2}-\frac {(2 d) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^2}+\frac {d^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^2}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^3}+\frac {\left (b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^3}+\frac {(4 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^2}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {(b d n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^3}-\frac {(b d n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^2}-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^3}-\frac {\left (4 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^3}\\ &=-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {4 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{e^3}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^2}+\frac {\left (b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{e^2}\\ &=-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \log (d+e x)}{e^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {4 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3}+\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^3}\\ &=-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \log (d+e x)}{e^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {3 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 212, normalized size = 0.91 \begin {gather*} \frac {\frac {2 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-2 b^2 n^2 (\log (x)-\log (d+e x))+6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+6 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )+4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )-4 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{2 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 3831, normalized size = 16.51

method	result	size
risch	\(\text {Expression too large to display}\)	\(3831\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b/e^3*n*ln(e*x+d)*a-2*b/e^3*n*dilog(-e*x/d)*a+2*b/e^3*ln(x^n)*ln(e*x+d)*a+1/8*d^2/e^3/(e*x+d)^2*Pi^2*b^2*csg
n(I*x^n)^2*csgn(I*c*x^n)^4-1/4*d^2/e^3/(e*x+d)^2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-3*n/e^3*ln(e*x)*b^2*ln(c
)+I/e^3*n*dilog(-e*x/d)*b^2*Pi*csgn(I*c*x^n)^3-1/4/e^3*ln(e*x+d)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x
^n)^2+a^2/e^3*ln(e*x+d)+1/2*I*ln(x^n)*d^2/e^3/(e*x+d)^2*b^2*Pi*csgn(I*c*x^n)^3+d/e^3/(e*x+d)*Pi^2*b^2*csgn(I*c
)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+d/e^3/(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3+3/2*I/e^3*n*ln(
e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+4*d/e^3/(e*x+d)*ln(c)*a*b-d^2/e^3/(e*x+d)^2*ln(c)*a*b-2/e^3*n*ln(e*x+d
)*ln(-e*x/d)*b^2*ln(c)+n*d/e^3/(e*x+d)*b^2*ln(c)+4*b*d/e^3*ln(x^n)/(e*x+d)*a-2*b^2*n/e^3*ln(e*x+d)*ln(x^n)*ln(
-e*x/d)+b^2*n*d/e^3*ln(x^n)/(e*x+d)+2*a^2*d/e^3/(e*x+d)-1/2*a^2*d^2/e^3/(e*x+d)^2-2*b/e^3*n*ln(e*x+d)*ln(-e*x/
d)*a-ln(x^n)*d^2/e^3/(e*x+d)^2*b^2*ln(c)+4*d/e^3*ln(x^n)/(e*x+d)*b^2*ln(c)+I/e^3*ln(e*x+d)*ln(c)*Pi*b^2*csgn(I
*x^n)*csgn(I*c*x^n)^2+I/e^3*ln(e*x+d)*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2-2*I*d/e^3/(e*x+d)*ln(c)*Pi*b^2*csgn(I*c
*x^n)^3-b*ln(x^n)*d^2/e^3/(e*x+d)^2*a+I/e^3*ln(e*x+d)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+I/e^3*n*ln(e*x+d)*ln(
-e*x/d)*b^2*Pi*csgn(I*c*x^n)^3-1/2*d/e^3/(e*x+d)*Pi^2*b^2*csgn(I*c*x^n)^6-1/4/e^3*ln(e*x+d)*Pi^2*b^2*csgn(I*x^
n)^2*csgn(I*c*x^n)^4+1/2/e^3*ln(e*x+d)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+1/8*d^2/e^3/(e*x+d)^2*Pi^2*b^2*csg
n(I*c*x^n)^6+1/2/e^3*ln(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+2/e^3*ln(x^n)*ln(e*x+d)*b^2*ln(c)-2*I*d/e^3/
(e*x+d)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I/e^3*n*ln(e*x+d)*ln(-e*x/d)*b^2*Pi*csgn(I*c)*csgn(I*
x^n)*csgn(I*c*x^n)+1/e^3*ln(e*x+d)*ln(c)^2*b^2+b^2*ln(x^n)^2/e^3*ln(e*x+d)-1/4*d^2/e^3/(e*x+d)^2*Pi^2*b^2*csgn
(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-3*b*n/e^3*ln(e*x)*a-1/2*d/e^3/(e*x+d)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*c
sgn(I*c*x^n)^2+d/e^3/(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+d/e^3/(e*x+d)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^
n)^5+1/2*I*d^2/e^3/(e*x+d)^2*Pi*a*b*csgn(I*c*x^n)^3+1/2*I*d^2/e^3/(e*x+d)^2*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-2*I*d
/e^3/(e*x+d)*Pi*a*b*csgn(I*c*x^n)^3+b*n*d/e^3/(e*x+d)*a-3/2*I*n/e^3*ln(e*x)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2
-I/e^3*n*dilog(-e*x/d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*d/e^3*ln(x^n)/(e*x+d)*b^2*Pi*csgn(I*c*x^n)^3-3/2
*I*n/e^3*ln(e*x)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*b^2/e^3*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2-1/4/e^3*ln(e*x+d)*P
i^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-I/e^3*n*dilog(-e*x/d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+3/2*I/e^3*n*ln(e*x+
d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+3/e^3*n*ln(e*x+d)*b^2*ln(c)-2/e^3*n*dilog(-e*x/d)*b^2*ln(c)+2*d/e^3/(e*x
+d)*ln(c)^2*b^2-1/2*d^2/e^3/(e*x+d)^2*ln(c)^2*b^2+2/e^3*ln(e*x+d)*ln(c)*a*b-I/e^3*ln(e*x+d)*Pi*a*b*csgn(I*c*x^
n)^3-I/e^3*ln(x^n)*ln(e*x+d)*b^2*Pi*csgn(I*c*x^n)^3+3/2*I*n/e^3*ln(e*x)*b^2*Pi*csgn(I*c*x^n)^3+1/8*d^2/e^3/(e*
x+d)^2*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4+I/e^3*ln(x^n)*ln(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I/e^3*ln(
x^n)*ln(e*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*b^2/e^3*n^2*ln(e*x+d)*ln(-e*x/d)+2*b^2/e^3*ln(x)*dilog(-e*
x/d)*n^2-b^2/e^3*n^2*ln(x)^2*ln(e*x+d)+b^2/e^3*n^2*ln(x)^2*ln(1+e*x/d)+2*b^2/e^3*n^2*ln(x)*polylog(2,-e*x/d)-b
^2/e^3*n^2*ln(x)+3/2*b^2/e^3*n^2*ln(x)^2-3*b^2/e^3*n^2*dilog(-e*x/d)-1/2*I*n*d/e^3/(e*x+d)*b^2*Pi*csgn(I*c*x^n
)^3-1/4*d^2/e^3/(e*x+d)^2*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3+1/2*d^2/e^3/(e*x+d)^2*Pi^2*b^2*csgn
(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4-2*d/e^3/(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+1/8*d^2/e^3/(
e*x+d)^2*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2+I/e^3*ln(e*x+d)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^
n)^2-2*I*d/e^3/(e*x+d)*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*ln(x^n)*ln(e*x+d)/e^3*b^2*n+1/2*I*n*d/e^3/
(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*d/e^3/(e*x+d)*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-I/e^3*ln(e*x
+d)*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-3/2*I/e^3*n*ln(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*d^2/e^
3/(e*x+d)^2*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*b^2*ln(x^n)^2*d^2/e^3/(e*x+d)^2-3*b^2*n/e^3*ln(x^n)*l
n(x)+2*I*d/e^3/(e*x+d)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^n)^2-I/e^3*n*ln(e*x+d)*ln(-e*x/d)*b^2*Pi*csgn(I*x^n)*
csgn(I*c*x^n)^2-1/e^3*ln(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4-I/e^3*ln(e*x+d)*ln(c)*Pi*b^2*cs
gn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*d^2/e^3/(e*x+d)^2*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*d^2/e^3/(e*x+
d)^2*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+2*I*d/e^3*ln(x^n)/(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*d^2/e
^3/(e*x+d)^2*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^n)^2-2*b^2*n/e^3*ln(x^n)*dilog(-e*x/d)-I/e^3*n*ln(e*x+d)*ln(-e*
x/d)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-2*b^2*n^2*polylog(3,-e*x/d)/e^3+b^2*n^2*ln(e*x+d)/e^3+2*b^2*ln(x^n)^2*d/
e^3/(e*x+d)-1/4/e^3*ln(e*x+d)*Pi^2*b^2*csgn(I*c*x^n)^6+1/2*I*n*d/e^3/(e*x+d)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^
2-1/4*d^2/e^3/(e*x+d)^2*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+1/2/e^3*ln(e*x+d)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*
csgn(I*c*x^n)^3+1/2*I*d^2/e^3/(e*x+d)^2*ln(c)*P...

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

[Out]

1/2*(2*e^(-3)*log(x*e + d) + (4*d*x*e + 3*d^2)/(x^2*e^5 + 2*d*x*e^4 + d^2*e^3))*a^2 + integrate((b^2*x^2*log(x
^n)^2 + 2*(b^2*log(c) + a*b)*x^2*log(x^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x^2)/(x^3*e^3 + 3*d*x^2*e^2 + 3*d^2*
x*e + d^3), 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(x^2*(a+b*log(c*x^n))^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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