3.442 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^4}{g+h x} \, dx\)

Optimal. Leaf size=231 \[ \frac {24 b^3 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {12 b^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac {4 b p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{h}-\frac {24 b^4 p^4 q^4 \text {Li}_5\left (-\frac {h (e+f x)}{f g-e h}\right )}{h} \]

[Out]

(a+b*ln(c*(d*(f*x+e)^p)^q))^4*ln(f*(h*x+g)/(-e*h+f*g))/h+4*b*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))^3*polylog(2,-h*(f
*x+e)/(-e*h+f*g))/h-12*b^2*p^2*q^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2*polylog(3,-h*(f*x+e)/(-e*h+f*g))/h+24*b^3*p^3
*q^3*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(4,-h*(f*x+e)/(-e*h+f*g))/h-24*b^4*p^4*q^4*polylog(5,-h*(f*x+e)/(-e*h+
f*g))/h

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Rubi [A]  time = 0.53, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2396, 2433, 2374, 2383, 6589, 2445} \[ \frac {24 b^3 p^3 q^3 \text {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {12 b^2 p^2 q^2 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac {4 b p q \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}-\frac {24 b^4 p^4 q^4 \text {PolyLog}\left (5,-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{h} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^4/(g + h*x),x]

[Out]

((a + b*Log[c*(d*(e + f*x)^p)^q])^4*Log[(f*(g + h*x))/(f*g - e*h)])/h + (4*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^
q])^3*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h - (12*b^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog
[3, -((h*(e + f*x))/(f*g - e*h))])/h + (24*b^3*p^3*q^3*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[4, -((h*(e + f
*x))/(f*g - e*h))])/h - (24*b^4*p^4*q^4*PolyLog[5, -((h*(e + f*x))/(f*g - e*h))])/h

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 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 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]]

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx &=\operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4}{g+h x} \, 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 )^4 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}-\operatorname {Subst}\left (\frac {(4 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{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 )^4 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}-\operatorname {Subst}\left (\frac {(4 b p q) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \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},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 )^4 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {4 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\operatorname {Subst}\left (\frac {\left (12 b^2 p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{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 )^4 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {4 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {12 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}+\operatorname {Subst}\left (\frac {\left (24 b^3 p^3 q^3\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text {Li}_3\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{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 )^4 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {4 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {12 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {24 b^3 p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\operatorname {Subst}\left (\frac {\left (24 b^4 p^4 q^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{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 )^4 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {4 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {12 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {24 b^3 p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {24 b^4 p^4 q^4 \text {Li}_5\left (-\frac {h (e+f x)}{f g-e h}\right )}{h}\\ \end {align*}

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Mathematica [B]  time = 0.45, size = 1095, normalized size = 4.74 \[ \frac {\log (g+h x) a^4-4 b p q \log (e+f x) \log (g+h x) a^3+4 b \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x) a^3+4 b p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right ) a^3+6 b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x) a^2+6 b^2 \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x) a^2-12 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x) a^2-6 b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right ) a^2+12 b^2 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 ) a^2-4 b^3 p^3 q^3 \log ^3(e+f x) \log (g+h x) a+4 b^3 \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x) a-12 b^3 p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x) a+12 b^3 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) a+4 b^3 p^3 q^3 \log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right ) a+12 b^3 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 ) a-12 b^3 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 ) a+24 b^3 p^3 q^3 \text {Li}_4\left (\frac {h (e+f x)}{e h-f g}\right ) a+b^4 p^4 q^4 \log ^4(e+f x) \log (g+h x)+b^4 \log ^4\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-4 b^4 p q \log (e+f x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+6 b^4 p^2 q^2 \log ^2(e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-4 b^4 p^3 q^3 \log ^3(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-b^4 p^4 q^4 \log ^4(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+4 b^4 p q \log (e+f x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-6 b^4 p^2 q^2 \log ^2(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 )+4 b^4 p^3 q^3 \log ^3(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 )+4 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right )-12 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_3\left (\frac {h (e+f x)}{e h-f g}\right )+24 b^4 p^3 q^3 \log \left (c \left (d (e+f x)^p\right )^q\right ) \text {Li}_4\left (\frac {h (e+f x)}{e h-f g}\right )-24 b^4 p^4 q^4 \text {Li}_5\left (\frac {h (e+f x)}{e h-f g}\right )}{h} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^4/(g + h*x),x]

[Out]

(a^4*Log[g + h*x] - 4*a^3*b*p*q*Log[e + f*x]*Log[g + h*x] + 6*a^2*b^2*p^2*q^2*Log[e + f*x]^2*Log[g + h*x] - 4*
a*b^3*p^3*q^3*Log[e + f*x]^3*Log[g + h*x] + b^4*p^4*q^4*Log[e + f*x]^4*Log[g + h*x] + 4*a^3*b*Log[c*(d*(e + f*
x)^p)^q]*Log[g + h*x] - 12*a^2*b^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 12*a*b^3*p^2*q^2*L
og[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] - 4*b^4*p^3*q^3*Log[e + f*x]^3*Log[c*(d*(e + f*x)^p)^q]*Lo
g[g + h*x] + 6*a^2*b^2*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] - 12*a*b^3*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^
p)^q]^2*Log[g + h*x] + 6*b^4*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] + 4*a*b^3*Log[c*(d
*(e + f*x)^p)^q]^3*Log[g + h*x] - 4*b^4*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^3*Log[g + h*x] + b^4*Log[c*(
d*(e + f*x)^p)^q]^4*Log[g + h*x] + 4*a^3*b*p*q*Log[e + f*x]*Log[(f*(g + h*x))/(f*g - e*h)] - 6*a^2*b^2*p^2*q^2
*Log[e + f*x]^2*Log[(f*(g + h*x))/(f*g - e*h)] + 4*a*b^3*p^3*q^3*Log[e + f*x]^3*Log[(f*(g + h*x))/(f*g - e*h)]
- b^4*p^4*q^4*Log[e + f*x]^4*Log[(f*(g + h*x))/(f*g - e*h)] + 12*a^2*b^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^
p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] - 12*a*b^3*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)] + 4*b^4*p^3*q^3*Log[e + f*x]^3*Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] + 12*a
*b^3*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[(f*(g + h*x))/(f*g - e*h)] - 6*b^4*p^2*q^2*Log[e + f*x]^2
*Log[c*(d*(e + f*x)^p)^q]^2*Log[(f*(g + h*x))/(f*g - e*h)] + 4*b^4*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^3
*Log[(f*(g + h*x))/(f*g - e*h)] + 4*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*PolyLog[2, (h*(e + f*x))/(-(f*g)
+ e*h)] - 12*b^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[3, (h*(e + f*x))/(-(f*g) + e*h)] + 24*a*b^
3*p^3*q^3*PolyLog[4, (h*(e + f*x))/(-(f*g) + e*h)] + 24*b^4*p^3*q^3*Log[c*(d*(e + f*x)^p)^q]*PolyLog[4, (h*(e
+ f*x))/(-(f*g) + e*h)] - 24*b^4*p^4*q^4*PolyLog[5, (h*(e + f*x))/(-(f*g) + e*h)])/h

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{4} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{4} + 4 \, a b^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 6 \, a^{2} b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 4 \, a^{3} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{4}}{h x + g}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{4}}{h x + g}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g),x, algorithm="giac")

[Out]

integrate((b*log(((f*x + e)^p*d)^q*c) + a)^4/(h*x + g), x)

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maple [F]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{4}}{h x +g}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*(d*(f*x+e)^p)^q)+a)^4/(h*x+g),x)

[Out]

int((b*ln(c*(d*(f*x+e)^p)^q)+a)^4/(h*x+g),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{4} \log \left (h x + g\right )}{h} + \int \frac {b^{4} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{4} + 4 \, {\left (q \log \relax (d) + \log \relax (c)\right )} a^{3} b + 6 \, {\left (q^{2} \log \relax (d)^{2} + 2 \, q \log \relax (c) \log \relax (d) + \log \relax (c)^{2}\right )} a^{2} b^{2} + 4 \, {\left (q^{3} \log \relax (d)^{3} + 3 \, q^{2} \log \relax (c) \log \relax (d)^{2} + 3 \, q \log \relax (c)^{2} \log \relax (d) + \log \relax (c)^{3}\right )} a b^{3} + {\left (q^{4} \log \relax (d)^{4} + 4 \, q^{3} \log \relax (c) \log \relax (d)^{3} + 6 \, q^{2} \log \relax (c)^{2} \log \relax (d)^{2} + 4 \, q \log \relax (c)^{3} \log \relax (d) + \log \relax (c)^{4}\right )} b^{4} + 4 \, {\left ({\left (q \log \relax (d) + \log \relax (c)\right )} b^{4} + a b^{3}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{3} + 6 \, {\left (2 \, {\left (q \log \relax (d) + \log \relax (c)\right )} a b^{3} + {\left (q^{2} \log \relax (d)^{2} + 2 \, q \log \relax (c) \log \relax (d) + \log \relax (c)^{2}\right )} b^{4} + a^{2} b^{2}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + 4 \, {\left (3 \, {\left (q \log \relax (d) + \log \relax (c)\right )} a^{2} b^{2} + 3 \, {\left (q^{2} \log \relax (d)^{2} + 2 \, q \log \relax (c) \log \relax (d) + \log \relax (c)^{2}\right )} a b^{3} + {\left (q^{3} \log \relax (d)^{3} + 3 \, q^{2} \log \relax (c) \log \relax (d)^{2} + 3 \, q \log \relax (c)^{2} \log \relax (d) + \log \relax (c)^{3}\right )} b^{4} + a^{3} b\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h x + g}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^4}{g+h\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d*(e + f*x)^p)^q))^4/(g + h*x),x)

[Out]

int((a + b*log(c*(d*(e + f*x)^p)^q))^4/(g + h*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{4}}{g + h x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**4/(h*x+g),x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**4/(g + h*x), x)

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