\(\int \frac {\log ^2(c (a+b x)^n)}{d x+e x^2} \, dx\) [344]

Optimal result

Integrand size = 24, antiderivative size = 168 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {\log \left (-\frac {b x}{a}\right ) \log ^2\left (c (a+b x)^n\right )}{d}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{d}+\frac {2 n^2 \operatorname {PolyLog}\left (3,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {2 n^2 \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1607, 2463, 2443, 2481, 2421, 6724} \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=-\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {2 n \operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right ) \log \left (c (a+b x)^n\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log ^2\left (c (a+b x)^n\right )}{d}+\frac {2 n^2 \operatorname {PolyLog}\left (3,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {2 n^2 \operatorname {PolyLog}\left (3,\frac {b x}{a}+1\right )}{d} \]

[In]

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

[Out]

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

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 2443

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((
f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Dist[b*e*n*(p/g), Int[Log[(e*(f + g*x))/(e*f - d
*g)]*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.74 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {\log (x) \left (-n \log (a+b x)+\log \left (c (a+b x)^n\right )\right )^2-\left (-n \log (a+b x)+\log \left (c (a+b x)^n\right )\right )^2 \log (d+e x)-2 n \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right ) \left (\log (x) \left (\log (a+b x)-\log \left (1+\frac {b x}{a}\right )\right )-\log (a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-\operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )\right )+n^2 \left (\log \left (-\frac {b x}{a}\right ) \log ^2(a+b x)-\log ^2(a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-2 \log (a+b x) \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )+2 \log (a+b x) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+2 \operatorname {PolyLog}\left (3,\frac {e (a+b x)}{-b d+a e}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )\right )}{d} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 1.28 (sec) , antiderivative size = 761, normalized size of antiderivative = 4.53

method	result	size
risch	\(\frac {{\left (\ln \left (\left (b x +a \right )^{n}\right )-n \ln \left (b x +a \right )\right )}^{2} \ln \left (b x \right )}{d}-\frac {{\left (\ln \left (\left (b x +a \right )^{n}\right )-n \ln \left (b x +a \right )\right )}^{2} \ln \left (e \left (b x +a \right )-a e +b d \right )}{d}+\frac {n^{2} \ln \left (b x +a \right )^{2} \ln \left (1-\frac {b x +a}{a}\right )}{d}+\frac {2 n^{2} \ln \left (b x +a \right ) \operatorname {Li}_{2}\left (\frac {b x +a}{a}\right )}{d}-\frac {2 n^{2} \operatorname {Li}_{3}\left (\frac {b x +a}{a}\right )}{d}-\frac {n^{2} \ln \left (b x +a \right )^{2} \ln \left (1+\frac {e \left (b x +a \right )}{-a e +b d}\right )}{d}-\frac {2 n^{2} \ln \left (b x +a \right ) \operatorname {Li}_{2}\left (-\frac {e \left (b x +a \right )}{-a e +b d}\right )}{d}+\frac {2 n^{2} \operatorname {Li}_{3}\left (-\frac {e \left (b x +a \right )}{-a e +b d}\right )}{d}+2 b n \left (\ln \left (\left (b x +a \right )^{n}\right )-n \ln \left (b x +a \right )\right ) \left (\frac {\operatorname {dilog}\left (-\frac {x b}{a}\right )+\ln \left (b x +a \right ) \ln \left (-\frac {x b}{a}\right )}{b d}-\frac {e \left (\frac {\operatorname {dilog}\left (\frac {e \left (b x +a \right )-a e +b d}{-a e +b d}\right )}{e}+\frac {\ln \left (b x +a \right ) \ln \left (\frac {e \left (b x +a \right )-a e +b d}{-a e +b d}\right )}{e}\right )}{b d}\right )+\left (-i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{3}+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right )+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,\operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (-\frac {\ln \left (e x +d \right ) \ln \left (\left (b x +a \right )^{n}\right )}{d}+\frac {\ln \left (\left (b x +a \right )^{n}\right ) \ln \left (x \right )}{d}-b n \left (\frac {\operatorname {dilog}\left (\frac {b x +a}{a}\right )}{d b}+\frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{d b}-\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{d b}-\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{d b}\right )\right )+\frac {{\left (-i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{3}+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right )+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,\operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2} \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )}{4}\)	\(761\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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