3.1.93 \(\int \frac {x^2 (a+b \log (c x^n))^2}{d+e x} \, dx\) [93]

Optimal. Leaf size=200 \[ \frac {2 a b d n x}{e^2}-\frac {2 b^2 d n^2 x}{e^2}+\frac {b^2 n^2 x^2}{4 e}+\frac {2 b^2 d n x \log \left (c x^n\right )}{e^2}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 d^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2395, 2333, 2332, 2342, 2341, 2354, 2421, 6724} \begin {gather*} \frac {2 b d^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {2 b^2 d^2 n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3}+\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {2 a b d n x}{e^2}+\frac {2 b^2 d n x \log \left (c x^n\right )}{e^2}-\frac {2 b^2 d n^2 x}{e^2}+\frac {b^2 n^2 x^2}{4 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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

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Mathematica [A]
time = 0.07, size = 158, normalized size = 0.79 \begin {gather*} \frac {-4 d e x \left (a+b \log \left (c x^n\right )\right )^2+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+8 b d e n x \left (a-b n+b \log \left (c x^n\right )\right )+b e^2 n x^2 \left (b n-2 \left (a+b \log \left (c x^n\right )\right )\right )+4 d^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+8 b d^2 n \left (\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )-b n \text {Li}_3\left (-\frac {e x}{d}\right )\right )}{4 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*d*e*x*(a + b*Log[c*x^n])^2 + 2*e^2*x^2*(a + b*Log[c*x^n])^2 + 8*b*d*e*n*x*(a - b*n + b*Log[c*x^n]) + b*e^2
*n*x^2*(b*n - 2*(a + b*Log[c*x^n])) + 4*d^2*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 8*b*d^2*n*((a + b*Log[c*x^
n])*PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)]))/(4*e^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.23, size = 3479, normalized size = 17.40

method	result	size
risch	\(\text {Expression too large to display}\)	\(3479\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

1/2*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2*d*x)*e^(-2))*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*e + d), 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),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*e + d), 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}}{d + e x}\, 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),x)

[Out]

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

[Out]

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

[Out]

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

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