\(\int \frac {(a+b \log (c (d+e x)^n))^3}{(f+g x)^2} \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e f-d g) (f+g x)}-\frac {3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)}-\frac {6 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)}+\frac {6 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)} \]

[Out]

(e*x+d)*(a+b*ln(c*(e*x+d)^n))^3/(-d*g+e*f)/(g*x+f)-3*b*e*n*(a+b*ln(c*(e*x+d)^n))^2*ln(e*(g*x+f)/(-d*g+e*f))/g/
(-d*g+e*f)-6*b^2*e*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(2,-g*(e*x+d)/(-d*g+e*f))/g/(-d*g+e*f)+6*b^3*e*n^3*polylog
(3,-g*(e*x+d)/(-d*g+e*f))/g/(-d*g+e*f)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2444, 2443, 2481, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=-\frac {6 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)}-\frac {3 b e n \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g (e f-d g)}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (e f-d g)}+\frac {6 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)} \]

[In]

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

[Out]

((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/((e*f - d*g)*(f + g*x)) - (3*b*e*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(
e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)) - (6*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x)
)/(e*f - d*g))])/(g*(e*f - d*g)) + (6*b^3*e*n^3*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*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 2444

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[(d + e
*x)*((a + b*Log[c*(d + e*x)^n])^p/((e*f - d*g)*(f + g*x))), x] - Dist[b*e*n*(p/(e*f - d*g)), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

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 {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e f-d g) (f+g x)}-\frac {(3 b e n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx}{e f-d g} \\ & = \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e f-d g) (f+g x)}-\frac {3 b e n \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 (e f-d g)}+\frac {\left (6 b^2 e^2 n^2\right ) \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 (e f-d g)} \\ & = \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e f-d g) (f+g x)}-\frac {3 b e n \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 (e f-d g)}+\frac {\left (6 b^2 e n^2\right ) \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 (e f-d g)} \\ & = \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e f-d g) (f+g x)}-\frac {3 b e n \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 (e f-d g)}-\frac {6 b^2 e n^2 \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 (e f-d g)}+\frac {\left (6 b^3 e n^3\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 (e f-d g)} \\ & = \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e f-d g) (f+g x)}-\frac {3 b e n \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 (e f-d g)}-\frac {6 b^2 e n^2 \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 (e f-d g)}+\frac {6 b^3 e n^3 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(410\) vs. \(2(190)=380\).

Time = 0.25 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\frac {-3 b (e f-d g) n \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+3 b e n (f+g x) \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-(e f-d g) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3-3 b e n (f+g x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+3 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (d+e x) \left (g (d+e x) \log (d+e x)-2 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-2 e (f+g x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^3 n^3 \left (\log ^2(d+e x) \left (g (d+e x) \log (d+e x)-3 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-6 e (f+g x) \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )+6 e (f+g x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{g (e f-d g) (f+g x)} \]

[In]

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

[Out]

(-3*b*(e*f - d*g)*n*Log[d + e*x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 3*b*e*n*(f + g*x)*Log[d + e
*x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - (e*f - d*g)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n]
)^3 - 3*b*e*n*(f + g*x)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x] + 3*b^2*n^2*(a - b*n*Log[
d + e*x] + b*Log[c*(d + e*x)^n])*(Log[d + e*x]*(g*(d + e*x)*Log[d + e*x] - 2*e*(f + g*x)*Log[(e*(f + g*x))/(e*
f - d*g)]) - 2*e*(f + g*x)*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) + b^3*n^3*(Log[d + e*x]^2*(g*(d + e*x)*Lo
g[d + e*x] - 3*e*(f + g*x)*Log[(e*(f + g*x))/(e*f - d*g)]) - 6*e*(f + g*x)*Log[d + e*x]*PolyLog[2, (g*(d + e*x
))/(-(e*f) + d*g)] + 6*e*(f + g*x)*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)]))/(g*(e*f - d*g)*(f + g*x))

Maple [C] (warning: unable to verify)

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

Time = 2.02 (sec) , antiderivative size = 1268, normalized size of antiderivative = 6.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(1268\)

[In]

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)^2,x,method=_RETURNVERBOSE)

[Out]

-b^3*ln((e*x+d)^n)^3/(g*x+f)/g+3*b^3/g*n^3*e/(d*g-e*f)*ln(g*(e*x+d)-d*g+e*f)*ln(e*x+d)^2-6*b^3/g*n^2*e/(d*g-e*
f)*ln(g*(e*x+d)-d*g+e*f)*ln((e*x+d)^n)*ln(e*x+d)+3*b^3/g*n*e/(d*g-e*f)*ln(g*(e*x+d)-d*g+e*f)*ln((e*x+d)^n)^2+3
*b^3/g*n^2*e/(d*g-e*f)*ln(e*x+d)^2*ln((e*x+d)^n)-3*b^3/g*n*e/(d*g-e*f)*ln(e*x+d)*ln((e*x+d)^n)^2+3*b^3/g*n^3*e
/(d*g-e*f)*ln(e*x+d)^2*ln(1+g*(e*x+d)/(-d*g+e*f))+6*b^3/g*n^3*e/(d*g-e*f)*ln(e*x+d)*polylog(2,-g*(e*x+d)/(-d*g
+e*f))-6*b^3/g*n^3*e/(d*g-e*f)*polylog(3,-g*(e*x+d)/(-d*g+e*f))+b^3/g*n^3*e/(-d*g+e*f)*ln(e*x+d)^3-6*b^3/g*n^3
*e/(d*g-e*f)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln(e*x+d)+6*b^3/g*n^2*e/(d*g-e*f)*dilog((g*(e*x+d)-d*g+e*f)
/(-d*g+e*f))*ln((e*x+d)^n)-6*b^3/g*n^3*e/(d*g-e*f)*ln(e*x+d)^2*ln((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))+6*b^3/g*n^2*
e/(d*g-e*f)*ln(e*x+d)*ln((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln((e*x+d)^n)-1/8*(-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)^3/(g*x+f)/g+3/2*(-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)*b^2*(-ln((e*x+d)^n)^2/(g*x+f)/g+2/g*n*e*(-ln((e*x+d)^n)/(d*g-e*f)*ln(e*x+d)+ln((e*x+d
)^n)/(d*g-e*f)*ln(g*x+f)-e*n*(1/(d*g-e*f)*(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)-1/2/(d*g-e*f)/e*ln(e*x+d)^2)))+3/4*(-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*b*(-ln((e*x+d)^n)/(g*x+f)/g+1/g*n*e*(-1/(d*g-e*f)*ln(e*x+d)+1/(d*g-e*f)*ln(g*x+f)))

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)^2,x, algorithm="fricas")

[Out]

integral((b^3*log((e*x + d)^n*c)^3 + 3*a*b^2*log((e*x + d)^n*c)^2 + 3*a^2*b*log((e*x + d)^n*c) + a^3)/(g^2*x^2
+ 2*f*g*x + f^2), x)

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{\left (f + g x\right )^{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*log(c*(d + e*x)**n))**3/(f + g*x)**2, x)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)^2,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^3/(g*x + f)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{{\left (f+g\,x\right )}^2} \,d x \]

[In]

int((a + b*log(c*(d + e*x)^n))^3/(f + g*x)^2,x)

[Out]

int((a + b*log(c*(d + e*x)^n))^3/(f + g*x)^2, x)