\(\int \frac {(a+b \log (c (d+e x)^n))^2}{f+g x} \, dx\) [48]

Optimal result

Integrand size = 24, antiderivative size = 111 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2443, 2481, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx=\frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^2 n^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{g} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 1.27 (sec) , antiderivative size = 737, normalized size of antiderivative = 6.64

method	result	size
risch	\(\frac {b^{2} \ln \left (g \left (e x +d \right )-d g +e f \right ) \ln \left (e x +d \right )^{2} n^{2}}{g}-\frac {2 b^{2} \ln \left (g \left (e x +d \right )-d g +e f \right ) \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (e x +d \right ) n}{g}+\frac {b^{2} \ln \left (g \left (e x +d \right )-d g +e f \right ) \ln \left (\left (e x +d \right )^{n}\right )^{2}}{g}+\frac {b^{2} n^{2} \ln \left (e x +d \right )^{2} \ln \left (1-\frac {g \left (e x +d \right )}{d g -e f}\right )}{g}+\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \operatorname {Li}_{2}\left (\frac {g \left (e x +d \right )}{d g -e f}\right )}{g}-\frac {2 b^{2} n^{2} \operatorname {Li}_{3}\left (\frac {g \left (e x +d \right )}{d g -e f}\right )}{g}-\frac {2 b^{2} n^{2} \operatorname {dilog}\left (\frac {g \left (e x +d \right )-d g +e f}{-d g +e f}\right ) \ln \left (e x +d \right )}{g}+\frac {2 b^{2} n \operatorname {dilog}\left (\frac {g \left (e x +d \right )-d g +e f}{-d g +e f}\right ) \ln \left (\left (e x +d \right )^{n}\right )}{g}-\frac {2 b^{2} n^{2} \ln \left (e x +d \right )^{2} \ln \left (\frac {g \left (e x +d \right )-d g +e f}{-d g +e f}\right )}{g}+\frac {2 b^{2} n \ln \left (e x +d \right ) \ln \left (\frac {g \left (e x +d \right )-d g +e f}{-d g +e f}\right ) \ln \left (\left (e x +d \right )^{n}\right )}{g}+\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g}-\frac {n e \left (\frac {\operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{e}+\frac {\ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{e}\right )}{g}\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 a \right )}^{2} \ln \left (g x +f \right )}{4 g}\)	\(737\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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