3.5.40 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^3}{(g+h x)^3} \, dx\) [440]

Optimal. Leaf size=376 \[ -\frac {3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (f g-e h)^2 (g+h x)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}+\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)^2}-\frac {3 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (1+\frac {f g-e h}{h (e+f x)}\right )}{2 h (f g-e h)^2}+\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \text {Li}_3\left (-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2} \]

[Out]

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

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Rubi [A]
time = 0.86, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2495} \begin {gather*} \frac {3 b^2 f^2 p^2 q^2 \text {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \text {PolyLog}\left (3,-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2}+\frac {3 b^2 f^2 p^2 q^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)^2}-\frac {3 b f^2 p q \log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (f g-e h)^2}-\frac {3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (g+h x) (f g-e h)^2}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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 2445

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

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

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*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(g+h x)^3} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}+\text {Subst}\left (\frac {(3 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(e+f x) (g+h x)^2} \, dx}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}+\text {Subst}\left (\frac {(3 b p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2}{x \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^2} \, dx,x,e+f x\right )}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}-\text {Subst}\left (\frac {(3 b p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2}{\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^2} \, dx,x,e+f x\right )}{2 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(3 b f p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2}{x \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )} \, dx,x,e+f x\right )}{2 h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (f g-e h)^2 (g+h x)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}-\text {Subst}\left (\frac {(3 b f p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2}{\frac {f g-e h}{f}+\frac {h x}{f}} \, dx,x,e+f x\right )}{2 (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b f^2 p q\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2}{x} \, dx,x,e+f x\right )}{2 h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 f p^2 q^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{\frac {f g-e h}{f}+\frac {h x}{f}} \, dx,x,e+f x\right )}{(f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (f g-e h)^2 (g+h x)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}+\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)^2}-\frac {3 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{2 h (f g-e h)^2}+\text {Subst}\left (\frac {\left (3 f^2\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{2 h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 f^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (1+\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (3 b^3 f^2 p^3 q^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (f g-e h)^2 (g+h x)}+\frac {f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (f g-e h)^2}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}+\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)^2}-\frac {3 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{2 h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}-\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}+\text {Subst}\left (\frac {\left (3 b^3 f^2 p^3 q^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (f g-e h)^2 (g+h x)}+\frac {f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (f g-e h)^2}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}+\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)^2}-\frac {3 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{2 h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}-\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}\\ \end {align*}

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Mathematica [A]
time = 0.52, size = 660, normalized size = 1.76 \begin {gather*} -\frac {-3 b f (f g-e h) p q (g+h x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+3 b (f g-e h)^2 p q \log (e+f x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-3 b f^2 p q (g+h x)^2 \log (e+f x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+(f g-e h)^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3+3 b f^2 p q (g+h x)^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log (g+h x)+3 b^2 p^2 q^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (h (e+f x) (e h-f (2 g+h x)) \log ^2(e+f x)-2 f^2 (g+h x)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 f (g+h x) \log (e+f x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )+2 f^2 (g+h x)^2 \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )\right )+b^3 p^3 q^3 \left (h (e+f x) (e h-f (2 g+h x)) \log ^3(e+f x)+3 f (g+h x) \log ^2(e+f x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )-6 f^2 (g+h x)^2 \log (e+f x) \left (\log \left (\frac {f (g+h x)}{f g-e h}\right )-\text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )\right )-6 f^2 (g+h x)^2 \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )-6 f^2 (g+h x)^2 \text {Li}_3\left (\frac {h (e+f x)}{-f g+e h}\right )\right )}{2 h (f g-e h)^2 (g+h x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{3}}{\left (h x +g \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^3/(h*x+g)^3,x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^3/(h*x+g)^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a^2*b*f*p*q*(f*log(f*x + e)/(f^2*g^2*h - 2*f*g*h^2*e + h^3*e^2) - f*log(h*x + g)/(f^2*g^2*h - 2*f*g*h^2*e
+ h^3*e^2) + 1/(f*g^2*h - g*h^2*e + (f*g*h^2 - h^3*e)*x)) - 1/2*b^3*log(((f*x + e)^p)^q)^3/(h^3*x^2 + 2*g*h^2*
x + g^2*h) - 3/2*a^2*b*log(((f*x + e)^p*d)^q*c)/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 1/2*a^3/(h^3*x^2 + 2*g*h^2*x +
g^2*h) + integrate(1/2*(3*(b^3*f*g*p*q + (2*a*b^2*f*h + (f*h*p*q + 2*f*h*q*log(d) + 2*f*h*log(c))*b^3)*x + 2*
((h*q*log(d) + h*log(c))*b^3 + a*b^2*h)*e)*log(((f*x + e)^p)^q)^2 + 2*(3*(f*h*q^2*log(d)^2 + 2*f*h*q*log(c)*lo
g(d) + f*h*log(c)^2)*a*b^2 + (f*h*q^3*log(d)^3 + 3*f*h*q^2*log(c)*log(d)^2 + 3*f*h*q*log(c)^2*log(d) + f*h*log
(c)^3)*b^3)*x + 2*(3*(h*q^2*log(d)^2 + 2*h*q*log(c)*log(d) + h*log(c)^2)*a*b^2 + (h*q^3*log(d)^3 + 3*h*q^2*log
(c)*log(d)^2 + 3*h*q*log(c)^2*log(d) + h*log(c)^3)*b^3)*e + 6*((2*(f*h*q*log(d) + f*h*log(c))*a*b^2 + (f*h*q^2
*log(d)^2 + 2*f*h*q*log(c)*log(d) + f*h*log(c)^2)*b^3)*x + (2*(h*q*log(d) + h*log(c))*a*b^2 + (h*q^2*log(d)^2
+ 2*h*q*log(c)*log(d) + h*log(c)^2)*b^3)*e)*log(((f*x + e)^p)^q))/(f*h^4*x^4 + g^3*h*e + (3*f*g*h^3 + h^4*e)*x
^3 + 3*(f*g^2*h^2 + g*h^3*e)*x^2 + (f*g^3*h + 3*g^2*h^2*e)*x), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3}}{\left (g + h x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{{\left (g+h\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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