\(\int \frac {(i+j x) (a+b \log (c (d (e+f x)^p)^q))^3}{g+h x} \, dx\) [536]

Optimal result

Integrand size = 33, antiderivative size = 349 \[ \int \frac {(i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\frac {6 a b^2 j p^2 q^2 x}{h}-\frac {6 b^3 j p^3 q^3 x}{h}+\frac {6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac {3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac {(h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^2}+\frac {3 b (h i-g j) p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^2}-\frac {6 b^2 (h i-g j) p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h^2}+\frac {6 b^3 (h i-g j) p^3 q^3 \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h^2} \]

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {2465, 2436, 2333, 2332, 2443, 2481, 2421, 2430, 6724, 2495} \[ \int \frac {(i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=-\frac {6 b^2 p^2 q^2 (h i-g j) \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac {6 a b^2 j p^2 q^2 x}{h}+\frac {3 b p q (h i-g j) \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^2}+\frac {(h i-g j) \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^2}-\frac {3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac {6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}+\frac {6 b^3 p^3 q^3 (h i-g j) \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h^2}-\frac {6 b^3 j p^3 q^3 x}{h} \]

[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 2332

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

Rule 2333

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 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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 2436

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 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 2465

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 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 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*} \text {integral}& = \text {Subst}\left (\int \frac {(i+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 ) \\ & = \text {Subst}\left (\int \left (\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{h}+\frac {(h i-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 ) \\ & = \text {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 )+\text {Subst}\left (\frac {(h i-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 {(h i-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}+\text {Subst}\left (\frac {j \text {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 )-\text {Subst}\left (\frac {(3 b f (h i-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 {(h i-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}-\text {Subst}\left (\frac {(3 b j p q) \text {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 )-\text {Subst}\left (\frac {(3 b (h i-g j) p q) \text {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 {(h i-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 (h i-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}+\text {Subst}\left (\frac {\left (6 b^2 j p^2 q^2\right ) \text {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 )-\text {Subst}\left (\frac {\left (6 b^2 (h i-g j) p^2 q^2\right ) \text {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 {(h i-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 (h i-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 (h i-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}+\text {Subst}\left (\frac {\left (6 b^3 j p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (6 b^3 (h i-g j) p^3 q^3\right ) \text {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 {(h i-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 (h i-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 (h i-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 (h i-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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1769\) vs. \(2(349)=698\).

Time = 0.48 (sec) , antiderivative size = 1769, normalized size of antiderivative = 5.07 \[ \int \frac {(i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx=\frac {-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 ^2(e+f x)+b^3 e h j p^3 q^3 \log ^3(e+f x)-6 a b^2 e h j p q \log \left (c \left (d (e+f x)^p\right )^q\right )+3 a^2 b f h j x \log \left (c \left (d (e+f x)^p\right )^q\right )-6 a b^2 f h j p q x \log \left (c \left (d (e+f x)^p\right )^q\right )+6 b^3 f h j p^2 q^2 x \log \left (c \left (d (e+f x)^p\right )^q\right )+6 a b^2 e h j p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 e h j p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 e h j p q \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+3 a b^2 f h j x \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 f h j p q x \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+3 b^3 e h j p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+b^3 f h j x \log ^3\left (c \left (d (e+f x)^p\right )^q\right )+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 ^2(e+f x) \log (g+h x)-3 a b^2 f g j p^2 q^2 \log ^2(e+f x) \log (g+h x)-b^3 f h i p^3 q^3 \log ^3(e+f x) \log (g+h x)+b^3 f g j p^3 q^3 \log ^3(e+f x) \log (g+h x)+3 a^2 b f h i \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 a^2 b f g j \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-6 a b^2 f h i p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+6 a b^2 f g j p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 b^3 f h i p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 b^3 f g j p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 a b^2 f h i \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 a b^2 f g j \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 b^3 f h i p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 b^3 f g j p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+b^3 f h i \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-b^3 f g j \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 a^2 b f h i p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-3 a^2 b f g j p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-3 a b^2 f h i p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+3 a b^2 f g j p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+b^3 f h i p^3 q^3 \log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-b^3 f g j p^3 q^3 \log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+6 a b^2 f h i p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-6 a b^2 f g j p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-3 b^3 f h i p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+3 b^3 f g j p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+3 b^3 f h i p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-3 b^3 f g j p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+3 b f (h i-g j) p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-6 b^2 f (h i-g j) p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )+6 b^3 f h i p^3 q^3 \operatorname {PolyLog}\left (4,\frac {h (e+f x)}{-f g+e h}\right )-6 b^3 f g j p^3 q^3 \operatorname {PolyLog}\left (4,\frac {h (e+f x)}{-f g+e h}\right )}{f h^2} \]

[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)

Maple [F]

\[\int \frac {\left (j x +i \right ) {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{3}}{h x +g}d x\]

[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)

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

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

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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