\(\int \frac {(a+b \log (c x^n))^2}{d+e x} \, dx\) [95]

Optimal result

Integrand size = 20, antiderivative size = 72 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2354, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e} \]

[In]

Int[(a + b*Log[c*x^n])^2/(d + e*x),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {d+e x}{d}\right )}{e}-\frac {2 b n \left (-\left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{e} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.37 (sec) , antiderivative size = 445, normalized size of antiderivative = 6.18

method	result	size
risch	\(\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (e x +d \right )}{e}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e}-\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e}+\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{e}-\frac {2 b^{2} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{e}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e}-\frac {n \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{e}\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \ln \left (e x +d \right )}{4 e}\)	\(445\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*log(c*x^n))^2/(d + e*x),x)

[Out]

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