\(\int \frac {x^2 (a+b \log (c x^n))^2}{(d+e x)^2} \, dx\) [101]

Optimal result

Integrand size = 23, antiderivative size = 203 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=-\frac {2 a b n x}{e^2}+\frac {2 b^2 n^2 x}{e^2}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {4 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {4 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2395, 2333, 2332, 2355, 2354, 2438, 2421, 6724} \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=-\frac {4 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {2 b d n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {2 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {2 a b n x}{e^2}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^2}-\frac {2 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {4 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3}+\frac {2 b^2 n^2 x}{e^2} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {d \left (a+b \log \left (c x^n\right )\right )^2+e x \left (a+b \log \left (c x^n\right )\right )^2-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-2 b e n x \left (a-b n+b \log \left (c x^n\right )\right )-2 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-2 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-4 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+4 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.51 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.45

method	result	size
risch	\(\frac {b^{2} \ln \left (x^{n}\right )^{2} x}{e^{2}}-\frac {2 b^{2} \ln \left (x^{n}\right )^{2} d \ln \left (e x +d \right )}{e^{3}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d^{2}}{e^{3} \left (e x +d \right )}+\frac {2 b^{2} n \ln \left (x \right ) \ln \left (x^{n}\right ) d}{e^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) x}{e^{2}}+\frac {2 b^{2} n^{2} x}{e^{2}}+\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) d}{e^{3}}+\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right ) d}{e^{3}}-\frac {b^{2} n^{2} d \ln \left (x \right )^{2}}{e^{3}}-\frac {4 b^{2} d \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{e^{3}}-\frac {4 b^{2} d \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{e^{3}}+\frac {4 b^{2} n d \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}+\frac {4 b^{2} n d \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}+\frac {2 b^{2} d \,n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{3}}-\frac {2 b^{2} d \,n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{3}}-\frac {4 b^{2} d \,n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{e^{3}}+\frac {4 b^{2} d \,n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{e^{3}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (x^{n}\right ) x}{e^{2}}-\frac {2 \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{3}}-\frac {\ln \left (x^{n}\right ) d^{2}}{e^{3} \left (e x +d \right )}-n \left (\frac {e x +d +d \ln \left (e x +d \right )-d \ln \left (e x \right )}{e^{3}}-\frac {2 d \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{e^{3}}\right )\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (\frac {x}{e^{2}}-\frac {2 d \ln \left (e x +d \right )}{e^{3}}-\frac {d^{2}}{e^{3} \left (e x +d \right )}\right )}{4}\)	\(700\)

[In]

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

[Out]

b^2*ln(x^n)^2/e^2*x-2*b^2*ln(x^n)^2/e^3*d*ln(e*x+d)-b^2*ln(x^n)^2/e^3*d^2/(e*x+d)+2*b^2*n/e^3*ln(x)*ln(x^n)*d-
2*b^2*n*ln(x^n)/e^3*d*ln(e*x+d)-2*b^2*n*ln(x^n)/e^2*x+2*b^2*n^2*x/e^2+2*b^2/e^3*n^2*ln(e*x+d)*ln(-e*x/d)*d+2*b
^2/e^3*n^2*dilog(-e*x/d)*d-b^2/e^3*n^2*d*ln(x)^2-4*b^2/e^3*d*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2-4*b^2/e^3*d*ln(x)*
dilog(-e*x/d)*n^2+4*b^2*n/e^3*d*ln(x^n)*ln(e*x+d)*ln(-e*x/d)+4*b^2*n/e^3*d*ln(x^n)*dilog(-e*x/d)+2*b^2/e^3*d*n
^2*ln(e*x+d)*ln(x)^2-2*b^2/e^3*d*n^2*ln(x)^2*ln(1+e*x/d)-4*b^2/e^3*d*n^2*ln(x)*polylog(2,-e*x/d)+4*b^2*d*n^2*p
olylog(3,-e*x/d)/e^3+(-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)*b*(ln(x^n)/e^2*x-2*ln(x^n)/e^3*d*ln(e*x+d)-ln(x^
n)/e^3*d^2/(e*x+d)-n*(1/e^3*(e*x+d+d*ln(e*x+d)-d*ln(e*x))-2/e^3*d*(dilog(-e*x/d)+ln(e*x+d)*ln(-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*(x/e^2-2/e^3*d*ln(e*x+d)-1/e^3*d^2/(e*x+d))

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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