3.536 \(\int \frac{(i+j x) (a+b \log (c (d (e+f x)^p)^q))^3}{g+h x} \, dx\)

Optimal. Leaf size=349 \[ -\frac{6 b^2 p^2 q^2 (h i-g j) \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac{3 b p q (h i-g j) \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^2}+\frac{6 b^3 p^3 q^3 (h i-g j) \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\frac{6 a b^2 j p^2 q^2 x}{h}+\frac{(h i-g j) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^2}-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac{6 b^3 j p^3 q^3 x}{h} \]

[Out]

(6*a*b^2*j*p^2*q^2*x)/h - (6*b^3*j*p^3*q^3*x)/h + (6*b^3*j*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h) -
(3*b*j*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h) + (j*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q]
)^3)/(f*h) + ((h*i - g*j)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h^2 + (3*b*(h*i -
g*j)*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^2 - (6*b^2*(h*i - g*j
)*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^2 + (6*b^3*(h*i - g*j)*
p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/h^2

________________________________________________________________________________________

Rubi [A]  time = 0.893341, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {2418, 2389, 2296, 2295, 2396, 2433, 2374, 2383, 6589, 2445} \[ -\frac{6 b^2 p^2 q^2 (h i-g j) \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac{3 b p q (h i-g j) \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^2}+\frac{6 b^3 p^3 q^3 (h i-g j) \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\frac{6 a b^2 j p^2 q^2 x}{h}+\frac{(h i-g j) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^2}-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac{6 b^3 j p^3 q^3 x}{h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a*b^2*j*p^2*q^2*x)/h - (6*b^3*j*p^3*q^3*x)/h + (6*b^3*j*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h) -
(3*b*j*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h) + (j*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q]
)^3)/(f*h) + ((h*i - g*j)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h^2 + (3*b*(h*i -
g*j)*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^2 - (6*b^2*(h*i - g*j
)*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^2 + (6*b^3*(h*i - g*j)*
p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/h^2

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2396

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 2433

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

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]

Rule 2445

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]]

Rubi steps

\begin{align*} \int \frac{(536+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx &=\operatorname{Subst}\left (\int \frac{(536+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{h}+\frac{(536 h-g j) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{h (g+h x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{j \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(536 h-g j) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(536 h-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h^2}+\operatorname{Subst}\left (\frac{j \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(3 b f (536 h-g j) p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{(536 h-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h^2}-\operatorname{Subst}\left (\frac{(3 b j p q) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(3 b (536 h-g j) p q) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \log \left (\frac{f \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{(536 h-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h^2}+\frac{3 b (536 h-g j) p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\operatorname{Subst}\left (\frac{\left (6 b^2 j p^2 q^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (6 b^2 (536 h-g j) p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text{Li}_2\left (-\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{6 a b^2 j p^2 q^2 x}{h}-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{(536 h-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h^2}+\frac{3 b (536 h-g j) p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^2}-\frac{6 b^2 (536 h-g j) p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\operatorname{Subst}\left (\frac{\left (6 b^3 j p^2 q^2\right ) \operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 b^3 (536 h-g j) p^3 q^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{6 a b^2 j p^2 q^2 x}{h}-\frac{6 b^3 j p^3 q^3 x}{h}+\frac{6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{(536 h-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h^2}+\frac{3 b (536 h-g j) p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^2}-\frac{6 b^2 (536 h-g j) p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\frac{6 b^3 (536 h-g j) p^3 q^3 \text{Li}_4\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^2}\\ \end{align*}

Mathematica [B]  time = 0.677263, size = 1769, normalized size = 5.07 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^2*b*e*h*j*p*q + a^3*f*h*j*x - 3*a^2*b*f*h*j*p*q*x + 6*a*b^2*f*h*j*p^2*q^2*x - 6*b^3*f*h*j*p^3*q^3*x + 3*
a^2*b*e*h*j*p*q*Log[e + f*x] + 6*b^3*e*h*j*p^3*q^3*Log[e + f*x] - 3*a*b^2*e*h*j*p^2*q^2*Log[e + f*x]^2 + b^3*e
*h*j*p^3*q^3*Log[e + f*x]^3 - 6*a*b^2*e*h*j*p*q*Log[c*(d*(e + f*x)^p)^q] + 3*a^2*b*f*h*j*x*Log[c*(d*(e + f*x)^
p)^q] - 6*a*b^2*f*h*j*p*q*x*Log[c*(d*(e + f*x)^p)^q] + 6*b^3*f*h*j*p^2*q^2*x*Log[c*(d*(e + f*x)^p)^q] + 6*a*b^
2*e*h*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q] - 3*b^3*e*h*j*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q
] - 3*b^3*e*h*j*p*q*Log[c*(d*(e + f*x)^p)^q]^2 + 3*a*b^2*f*h*j*x*Log[c*(d*(e + f*x)^p)^q]^2 - 3*b^3*f*h*j*p*q*
x*Log[c*(d*(e + f*x)^p)^q]^2 + 3*b^3*e*h*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2 + b^3*f*h*j*x*Log[c*(d*
(e + f*x)^p)^q]^3 + a^3*f*h*i*Log[g + h*x] - a^3*f*g*j*Log[g + h*x] - 3*a^2*b*f*h*i*p*q*Log[e + f*x]*Log[g + h
*x] + 3*a^2*b*f*g*j*p*q*Log[e + f*x]*Log[g + h*x] + 3*a*b^2*f*h*i*p^2*q^2*Log[e + f*x]^2*Log[g + h*x] - 3*a*b^
2*f*g*j*p^2*q^2*Log[e + f*x]^2*Log[g + h*x] - b^3*f*h*i*p^3*q^3*Log[e + f*x]^3*Log[g + h*x] + b^3*f*g*j*p^3*q^
3*Log[e + f*x]^3*Log[g + h*x] + 3*a^2*b*f*h*i*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] - 3*a^2*b*f*g*j*Log[c*(d*(
e + f*x)^p)^q]*Log[g + h*x] - 6*a*b^2*f*h*i*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 6*a*b^2*f
*g*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 3*b^3*f*h*i*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e +
f*x)^p)^q]*Log[g + h*x] - 3*b^3*f*g*j*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 3*a*b^2*
f*h*i*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] - 3*a*b^2*f*g*j*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] - 3*b^3*
f*h*i*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] + 3*b^3*f*g*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*
x)^p)^q]^2*Log[g + h*x] + b^3*f*h*i*Log[c*(d*(e + f*x)^p)^q]^3*Log[g + h*x] - b^3*f*g*j*Log[c*(d*(e + f*x)^p)^
q]^3*Log[g + h*x] + 3*a^2*b*f*h*i*p*q*Log[e + f*x]*Log[(f*(g + h*x))/(f*g - e*h)] - 3*a^2*b*f*g*j*p*q*Log[e +
f*x]*Log[(f*(g + h*x))/(f*g - e*h)] - 3*a*b^2*f*h*i*p^2*q^2*Log[e + f*x]^2*Log[(f*(g + h*x))/(f*g - e*h)] + 3*
a*b^2*f*g*j*p^2*q^2*Log[e + f*x]^2*Log[(f*(g + h*x))/(f*g - e*h)] + b^3*f*h*i*p^3*q^3*Log[e + f*x]^3*Log[(f*(g
+ h*x))/(f*g - e*h)] - b^3*f*g*j*p^3*q^3*Log[e + f*x]^3*Log[(f*(g + h*x))/(f*g - e*h)] + 6*a*b^2*f*h*i*p*q*Lo
g[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] - 6*a*b^2*f*g*j*p*q*Log[e + f*x]*Log[c*(d*(
e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] - 3*b^3*f*h*i*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*Lo
g[(f*(g + h*x))/(f*g - e*h)] + 3*b^3*f*g*j*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(
f*g - e*h)] + 3*b^3*f*h*i*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[(f*(g + h*x))/(f*g - e*h)] - 3*b^3*f
*g*j*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[(f*(g + h*x))/(f*g - e*h)] + 3*b*f*(h*i - g*j)*p*q*(a + b
*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)] - 6*b^2*f*(h*i - g*j)*p^2*q^2*(a + b*Log
[c*(d*(e + f*x)^p)^q])*PolyLog[3, (h*(e + f*x))/(-(f*g) + e*h)] + 6*b^3*f*h*i*p^3*q^3*PolyLog[4, (h*(e + f*x))
/(-(f*g) + e*h)] - 6*b^3*f*g*j*p^3*q^3*PolyLog[4, (h*(e + f*x))/(-(f*g) + e*h)])/(f*h^2)

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Maple [F]  time = 0.628, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( jx+i \right ) \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{3}}{hx+g}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} j{\left (\frac{x}{h} - \frac{g \log \left (h x + g\right )}{h^{2}}\right )} + \frac{a^{3} i \log \left (h x + g\right )}{h} + \int \frac{3 \,{\left (i \log \left (c\right ) + i \log \left (d^{q}\right )\right )} a^{2} b + 3 \,{\left (i \log \left (c\right )^{2} + 2 \, i \log \left (c\right ) \log \left (d^{q}\right ) + i \log \left (d^{q}\right )^{2}\right )} a b^{2} +{\left (i \log \left (c\right )^{3} + 3 \, i \log \left (c\right )^{2} \log \left (d^{q}\right ) + 3 \, i \log \left (c\right ) \log \left (d^{q}\right )^{2} + i \log \left (d^{q}\right )^{3}\right )} b^{3} +{\left (b^{3} j x + b^{3} i\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{3} + 3 \,{\left ({\left (i \log \left (c\right ) + i \log \left (d^{q}\right )\right )} b^{3} + a b^{2} i +{\left ({\left (j \log \left (c\right ) + j \log \left (d^{q}\right )\right )} b^{3} + a b^{2} j\right )} x\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} +{\left (3 \,{\left (j \log \left (c\right ) + j \log \left (d^{q}\right )\right )} a^{2} b + 3 \,{\left (j \log \left (c\right )^{2} + 2 \, j \log \left (c\right ) \log \left (d^{q}\right ) + j \log \left (d^{q}\right )^{2}\right )} a b^{2} +{\left (j \log \left (c\right )^{3} + 3 \, j \log \left (c\right )^{2} \log \left (d^{q}\right ) + 3 \, j \log \left (c\right ) \log \left (d^{q}\right )^{2} + j \log \left (d^{q}\right )^{3}\right )} b^{3}\right )} x + 3 \,{\left (2 \,{\left (i \log \left (c\right ) + i \log \left (d^{q}\right )\right )} a b^{2} +{\left (i \log \left (c\right )^{2} + 2 \, i \log \left (c\right ) \log \left (d^{q}\right ) + i \log \left (d^{q}\right )^{2}\right )} b^{3} + a^{2} b i +{\left (2 \,{\left (j \log \left (c\right ) + j \log \left (d^{q}\right )\right )} a b^{2} +{\left (j \log \left (c\right )^{2} + 2 \, j \log \left (c\right ) \log \left (d^{q}\right ) + j \log \left (d^{q}\right )^{2}\right )} b^{3} + a^{2} b j\right )} x\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h x + g}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*j*(x/h - g*log(h*x + g)/h^2) + a^3*i*log(h*x + g)/h + integrate((3*(i*log(c) + i*log(d^q))*a^2*b + 3*(i*lo
g(c)^2 + 2*i*log(c)*log(d^q) + i*log(d^q)^2)*a*b^2 + (i*log(c)^3 + 3*i*log(c)^2*log(d^q) + 3*i*log(c)*log(d^q)
^2 + i*log(d^q)^3)*b^3 + (b^3*j*x + b^3*i)*log(((f*x + e)^p)^q)^3 + 3*((i*log(c) + i*log(d^q))*b^3 + a*b^2*i +
((j*log(c) + j*log(d^q))*b^3 + a*b^2*j)*x)*log(((f*x + e)^p)^q)^2 + (3*(j*log(c) + j*log(d^q))*a^2*b + 3*(j*l
og(c)^2 + 2*j*log(c)*log(d^q) + j*log(d^q)^2)*a*b^2 + (j*log(c)^3 + 3*j*log(c)^2*log(d^q) + 3*j*log(c)*log(d^q
)^2 + j*log(d^q)^3)*b^3)*x + 3*(2*(i*log(c) + i*log(d^q))*a*b^2 + (i*log(c)^2 + 2*i*log(c)*log(d^q) + i*log(d^
q)^2)*b^3 + a^2*b*i + (2*(j*log(c) + j*log(d^q))*a*b^2 + (j*log(c)^2 + 2*j*log(c)*log(d^q) + j*log(d^q)^2)*b^3
+ a^2*b*j)*x)*log(((f*x + e)^p)^q))/(h*x + g), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{3} j x + a^{3} i +{\left (b^{3} j x + b^{3} i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 3 \,{\left (a b^{2} j x + a b^{2} i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 3 \,{\left (a^{2} b j x + a^{2} b i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{h x + g}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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