\(\int \frac {(f+g x) (a+b \log (c x^n))^3}{(d+e x)^3} \, dx\) [456]

Optimal result

Integrand size = 25, antiderivative size = 295 \[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e f-d g) (d+e x)^2}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b (e f+d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}+\frac {3 b^3 (e f-d g) n^3 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b^2 (e f+d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2 e^2}+\frac {3 b^3 (e f+d g) n^3 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2 e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2398, 2404, 2339, 30, 2355, 2354, 2438, 2421, 6724} \[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx=-\frac {3 b^2 n^2 (d g+e f) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}+\frac {3 b^2 n^2 (e f-d g) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}-\frac {3 b n (d g+e f) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 (e f-d g)}-\frac {3 b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (d+e x)^2 (e f-d g)}+\frac {3 b^3 n^3 (e f-d g) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2 e^2}+\frac {3 b^3 n^3 (d g+e f) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2 e^2} \]

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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 2398

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

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, 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 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}& = -\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e f-d g) (d+e x)^2}+\frac {(3 b n) \int \frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx}{2 (e f-d g)} \\ & = -\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e f-d g) (d+e x)^2}+\frac {(3 b n) \int \left (\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac {(-e f+d g)^2 \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)^2}+\frac {\left (-e^2 f^2+d^2 g^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 e (d+e x)}\right ) \, dx}{2 (e f-d g)} \\ & = -\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e f-d g) (d+e x)^2}+\frac {\left (3 b f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 d^2 (e f-d g)}-\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{2 d e}-\frac {(3 b (e f+d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{2 d^2 e} \\ & = -\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e f-d g) (d+e x)^2}-\frac {3 b (e f+d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}+\frac {\left (3 f^2\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 d^2 (e f-d g)}+\frac {\left (3 b^2 (e f-d g) n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2 e}+\frac {\left (3 b^2 (e f+d g) n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2} \\ & = -\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e f-d g) (d+e x)^2}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b (e f+d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}-\frac {3 b^2 (e f+d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}-\frac {\left (3 b^3 (e f-d g) n^3\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}+\frac {\left (3 b^3 (e f+d g) n^3\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2} \\ & = -\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e f-d g) (d+e x)^2}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b (e f+d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}+\frac {3 b^3 (e f-d g) n^3 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b^2 (e f+d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}+\frac {3 b^3 (e f+d g) n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^2 e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.15 \[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx=\frac {-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2}-\frac {2 g \left (a+b \log \left (c x^n\right )\right )^3}{d+e x}+\frac {2 g \left (\left (a+b \log \left (c x^n\right )\right )^2 \left (a+b \log \left (c x^n\right )-3 b n \log \left (1+\frac {e x}{d}\right )\right )-6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{d}+\frac {(e f-d g) \left (3 b d n \left (a+b \log \left (c x^n\right )\right )^2+(d+e x) \left (a+b \log \left (c x^n\right )\right )^3-3 b n (d+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-3 b n (d+e x) \left (\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (1+\frac {e x}{d}\right )\right )-2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )-6 b^2 n^2 (d+e x) \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )\right )}{d^2 (d+e x)}}{2 e^2} \]

[In]

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

[Out]

(-(((e*f - d*g)*(a + b*Log[c*x^n])^3)/(d + e*x)^2) - (2*g*(a + b*Log[c*x^n])^3)/(d + e*x) + (2*g*((a + b*Log[c
*x^n])^2*(a + b*Log[c*x^n] - 3*b*n*Log[1 + (e*x)/d]) - 6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 6
*b^3*n^3*PolyLog[3, -((e*x)/d)]))/d + ((e*f - d*g)*(3*b*d*n*(a + b*Log[c*x^n])^2 + (d + e*x)*(a + b*Log[c*x^n]
)^3 - 3*b*n*(d + e*x)*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] - 3*b*n*(d + e*x)*((a + b*Log[c*x^n])*(a + b*Log[c
*x^n] - 2*b*n*Log[1 + (e*x)/d]) - 2*b^2*n^2*PolyLog[2, -((e*x)/d)]) - 6*b^2*n^2*(d + e*x)*((a + b*Log[c*x^n])*
PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)])))/(d^2*(d + e*x)))/(2*e^2)

Maple [C] (warning: unable to verify)

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

Time = 1.20 (sec) , antiderivative size = 1652, normalized size of antiderivative = 5.60

method	result	size
risch	\(\text {Expression too large to display}\)	\(1652\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

3/2*a^2*b*f*n*(1/(d*e^2*x + d^2*e) - log(e*x + d)/(d^2*e) + log(x)/(d^2*e)) - 3/2*a^2*b*g*n*(1/(e^3*x + d*e^2)
+ log(e*x + d)/(d*e^2) - log(x)/(d*e^2)) - 3/2*(2*e*x + d)*a^2*b*g*log(c*x^n)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)
- 1/2*(2*e*x + d)*a^3*g/(e^4*x^2 + 2*d*e^3*x + d^2*e^2) - 3/2*a^2*b*f*log(c*x^n)/(e^3*x^2 + 2*d*e^2*x + d^2*e
) - 1/2*a^3*f/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 1/2*(2*b^3*e*g*x + (e*f + d*g)*b^3)*log(x^n)^3/(e^4*x^2 + 2*d*e^
3*x + d^2*e^2) + integrate(1/2*(2*(b^3*e^2*g*log(c)^3 + 3*a*b^2*e^2*g*log(c)^2)*x^2 + 3*((d*e*f*n + d^2*g*n)*b
^3 + 2*(a*b^2*e^2*g + (e^2*g*n + e^2*g*log(c))*b^3)*x^2 + (2*a*b^2*e^2*f + (e^2*f*n + 3*d*e*g*n + 2*e^2*f*log(
c))*b^3)*x)*log(x^n)^2 + 2*(b^3*e^2*f*log(c)^3 + 3*a*b^2*e^2*f*log(c)^2)*x + 6*((b^3*e^2*g*log(c)^2 + 2*a*b^2*
e^2*g*log(c))*x^2 + (b^3*e^2*f*log(c)^2 + 2*a*b^2*e^2*f*log(c))*x)*log(x^n))/(e^5*x^4 + 3*d*e^4*x^3 + 3*d^2*e^
3*x^2 + d^3*e^2*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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