\(\int (g+h x)^3 (a+b \log (c (d (e+f x)^p)^q))^2 \, dx\) [428]

Optimal result

Integrand size = 28, antiderivative size = 409 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {2 b^2 (f g-e h)^3 p^2 q^2 x}{f^3}+\frac {3 b^2 h (f g-e h)^2 p^2 q^2 (e+f x)^2}{4 f^4}+\frac {2 b^2 h^2 (f g-e h) p^2 q^2 (e+f x)^3}{9 f^4}+\frac {b^2 h^3 p^2 q^2 (e+f x)^4}{32 f^4}+\frac {b^2 (f g-e h)^4 p^2 q^2 \log ^2(e+f x)}{4 f^4 h}-\frac {2 b (f g-e h)^3 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^4}-\frac {3 b h (f g-e h)^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4}-\frac {2 b h^2 (f g-e h) p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^4}-\frac {b h^3 p q (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{8 f^4}-\frac {b (f g-e h)^4 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h} \]

[Out]

2*b^2*(-e*h+f*g)^3*p^2*q^2*x/f^3+3/4*b^2*h*(-e*h+f*g)^2*p^2*q^2*(f*x+e)^2/f^4+2/9*b^2*h^2*(-e*h+f*g)*p^2*q^2*(
f*x+e)^3/f^4+1/32*b^2*h^3*p^2*q^2*(f*x+e)^4/f^4+1/4*b^2*(-e*h+f*g)^4*p^2*q^2*ln(f*x+e)^2/f^4/h-2*b*(-e*h+f*g)^
3*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^4-3/2*b*h*(-e*h+f*g)^2*p*q*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))/f
^4-2/3*b*h^2*(-e*h+f*g)*p*q*(f*x+e)^3*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^4-1/8*b*h^3*p*q*(f*x+e)^4*(a+b*ln(c*(d*(f*
x+e)^p)^q))/f^4-1/2*b*(-e*h+f*g)^4*p*q*ln(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^4/h+1/4*(h*x+g)^4*(a+b*ln(c*(d*
(f*x+e)^p)^q))^2/h

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2445, 2458, 45, 2372, 12, 2338, 2495} \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=-\frac {2 b h^2 p q (e+f x)^3 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^4}-\frac {b p q (f g-e h)^4 \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4 h}-\frac {2 b p q (e+f x) (f g-e h)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^4}-\frac {3 b h p q (e+f x)^2 (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4}-\frac {b h^3 p q (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{8 f^4}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}+\frac {2 b^2 h^2 p^2 q^2 (e+f x)^3 (f g-e h)}{9 f^4}+\frac {3 b^2 h p^2 q^2 (e+f x)^2 (f g-e h)^2}{4 f^4}+\frac {b^2 p^2 q^2 (f g-e h)^4 \log ^2(e+f x)}{4 f^4 h}+\frac {b^2 h^3 p^2 q^2 (e+f x)^4}{32 f^4}+\frac {2 b^2 p^2 q^2 x (f g-e h)^3}{f^3} \]

[In]

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

[Out]

(2*b^2*(f*g - e*h)^3*p^2*q^2*x)/f^3 + (3*b^2*h*(f*g - e*h)^2*p^2*q^2*(e + f*x)^2)/(4*f^4) + (2*b^2*h^2*(f*g -
e*h)*p^2*q^2*(e + f*x)^3)/(9*f^4) + (b^2*h^3*p^2*q^2*(e + f*x)^4)/(32*f^4) + (b^2*(f*g - e*h)^4*p^2*q^2*Log[e
+ f*x]^2)/(4*f^4*h) - (2*b*(f*g - e*h)^3*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/f^4 - (3*b*h*(f*g - e
*h)^2*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^4) - (2*b*h^2*(f*g - e*h)*p*q*(e + f*x)^3*(a + b*
Log[c*(d*(e + f*x)^p)^q]))/(3*f^4) - (b*h^3*p*q*(e + f*x)^4*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(8*f^4) - (b*(f*
g - e*h)^4*p*q*Log[e + f*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^4*h) + ((g + h*x)^4*(a + b*Log[c*(d*(e + f*
x)^p)^q])^2)/(4*h)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -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]]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (g+h x)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}-\text {Subst}\left (\frac {(b f p q) \int \frac {(g+h x)^4 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x} \, dx}{2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}-\text {Subst}\left (\frac {(b p q) \text {Subst}\left (\int \frac {\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^4 \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, 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 {2 b (f g-e h)^3 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^4}-\frac {3 b h (f g-e h)^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4}-\frac {2 b h^2 (f g-e h) p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^4}-\frac {b h^3 p q (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{8 f^4}-\frac {b (f g-e h)^4 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}+\text {Subst}\left (\frac {\left (b^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {48 h (f g-e h)^3+36 h^2 (f g-e h)^2 x+16 h^3 (f g-e h) x^2+3 h^4 x^3+\frac {12 (f g-e h)^4 \log (x)}{x}}{12 f^4} \, 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 {2 b (f g-e h)^3 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^4}-\frac {3 b h (f g-e h)^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4}-\frac {2 b h^2 (f g-e h) p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^4}-\frac {b h^3 p q (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{8 f^4}-\frac {b (f g-e h)^4 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}+\text {Subst}\left (\frac {\left (b^2 p^2 q^2\right ) \text {Subst}\left (\int \left (48 h (f g-e h)^3+36 h^2 (f g-e h)^2 x+16 h^3 (f g-e h) x^2+3 h^4 x^3+\frac {12 (f g-e h)^4 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{24 f^4 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {2 b^2 (f g-e h)^3 p^2 q^2 x}{f^3}+\frac {3 b^2 h (f g-e h)^2 p^2 q^2 (e+f x)^2}{4 f^4}+\frac {2 b^2 h^2 (f g-e h) p^2 q^2 (e+f x)^3}{9 f^4}+\frac {b^2 h^3 p^2 q^2 (e+f x)^4}{32 f^4}-\frac {2 b (f g-e h)^3 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^4}-\frac {3 b h (f g-e h)^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4}-\frac {2 b h^2 (f g-e h) p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^4}-\frac {b h^3 p q (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{8 f^4}-\frac {b (f g-e h)^4 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}+\text {Subst}\left (\frac {\left (b^2 (f g-e h)^4 p^2 q^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{2 f^4 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {2 b^2 (f g-e h)^3 p^2 q^2 x}{f^3}+\frac {3 b^2 h (f g-e h)^2 p^2 q^2 (e+f x)^2}{4 f^4}+\frac {2 b^2 h^2 (f g-e h) p^2 q^2 (e+f x)^3}{9 f^4}+\frac {b^2 h^3 p^2 q^2 (e+f x)^4}{32 f^4}+\frac {b^2 (f g-e h)^4 p^2 q^2 \log ^2(e+f x)}{4 f^4 h}-\frac {2 b (f g-e h)^3 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^4}-\frac {3 b h (f g-e h)^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4}-\frac {2 b h^2 (f g-e h) p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^4}-\frac {b h^3 p q (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{8 f^4}-\frac {b (f g-e h)^4 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.98 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {288 (f g-e h)^3 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+432 h (f g-e h)^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+288 h^2 (f g-e h) (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+72 h^3 (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-576 b (f g-e h)^3 p q \left (f (a-b p q) x+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+216 b h (f g-e h)^2 p q \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+64 b h^2 (f g-e h) p q \left (b f p q x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+9 b h^3 p q \left (b f p q x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )-4 (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{288 f^4} \]

[In]

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

[Out]

(288*(f*g - e*h)^3*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + 432*h*(f*g - e*h)^2*(e + f*x)^2*(a + b*Log[c
*(d*(e + f*x)^p)^q])^2 + 288*h^2*(f*g - e*h)*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + 72*h^3*(e + f*x)
^4*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 - 576*b*(f*g - e*h)^3*p*q*(f*(a - b*p*q)*x + b*(e + f*x)*Log[c*(d*(e + f
*x)^p)^q]) + 216*b*h*(f*g - e*h)^2*p*q*(b*f*p*q*x*(2*e + f*x) - 2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])
) + 64*b*h^2*(f*g - e*h)*p*q*(b*f*p*q*x*(3*e^2 + 3*e*f*x + f^2*x^2) - 3*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^
p)^q])) + 9*b*h^3*p*q*(b*f*p*q*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) - 4*(e + f*x)^4*(a + b*Log[c*(d*(
e + f*x)^p)^q])))/(288*f^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1536\) vs. \(2(391)=782\).

Time = 16.24 (sec) , antiderivative size = 1537, normalized size of antiderivative = 3.76

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1537\)

[In]

int((h*x+g)^3*(a+b*ln(c*(d*(f*x+e)^p)^q))^2,x,method=_RETURNVERBOSE)

[Out]

-1/288*(1632*ln(f*x+e)*b^2*e^3*f*g*h^2*p^2*q^2-864*x*ln(c*(d*(f*x+e)^p)^q)*b^2*e*f^3*g^2*h*p*q+576*x*a*b*e^2*f
^2*g*h^2*p*q-864*x*a*b*e*f^3*g^2*h*p*q-576*ln(f*x+e)*a*b*e^3*f*g*h^2*p*q+864*ln(f*x+e)*a*b*e^2*f^2*g^2*h*p*q-2
88*x^2*ln(c*(d*(f*x+e)^p)^q)*b^2*e*f^3*g*h^2*p*q-288*x^2*a*b*e*f^3*g*h^2*p*q+576*x*ln(c*(d*(f*x+e)^p)^q)*b^2*e
^2*f^2*g*h^2*p*q-2160*ln(f*x+e)*b^2*e^2*f^2*g^2*h*p^2*q^2-1152*ln(f*x+e)*a*b*e*f^3*g^3*p*q-48*x^3*ln(c*(d*(f*x
+e)^p)^q)*b^2*e*f^3*h^3*p*q+192*x^3*ln(c*(d*(f*x+e)^p)^q)*b^2*f^4*g*h^2*p*q+240*x^2*b^2*e*f^3*g*h^2*p^2*q^2-48
*x^3*a*b*e*f^3*h^3*p*q+192*x^3*a*b*f^4*g*h^2*p*q+72*x^2*ln(c*(d*(f*x+e)^p)^q)*b^2*e^2*f^2*h^3*p*q+432*x^2*ln(c
*(d*(f*x+e)^p)^q)*b^2*f^4*g^2*h*p*q-1056*x*b^2*e^2*f^2*g*h^2*p^2*q^2+864*a*b*e^2*f^2*g^2*h*p*q-576*a*b*e^3*f*g
*h^2*p*q+1296*x*b^2*e*f^3*g^2*h*p^2*q^2+72*x^2*a*b*e^2*f^2*h^3*p*q+432*x^2*a*b*f^4*g^2*h*p*q-144*x*ln(c*(d*(f*
x+e)^p)^q)*b^2*e^3*f*h^3*p*q-144*x*a*b*e^3*f*h^3*p*q-576*ln(c*(d*(f*x+e)^p)^q)*b^2*e^3*f*g*h^2*p*q+864*ln(c*(d
*(f*x+e)^p)^q)*b^2*e^2*f^2*g^2*h*p*q+36*x^4*ln(c*(d*(f*x+e)^p)^q)*b^2*f^4*h^3*p*q+28*x^3*b^2*e*f^3*h^3*p^2*q^2
-64*x^3*b^2*f^4*g*h^2*p^2*q^2-576*a*b*e*f^3*g^3*p*q+1056*b^2*e^3*f*g*h^2*p^2*q^2-1296*b^2*e^2*f^2*g^2*h*p^2*q^
2+36*x^4*a*b*f^4*h^3*p*q-78*x^2*b^2*e^2*f^2*h^3*p^2*q^2-216*x^2*b^2*f^4*g^2*h*p^2*q^2+300*x*b^2*e^3*f*h^3*p^2*
q^2-576*x^3*ln(c*(d*(f*x+e)^p)^q)*a*b*f^4*g*h^2+576*x*ln(c*(d*(f*x+e)^p)^q)*b^2*f^4*g^3*p*q-864*x^2*ln(c*(d*(f
*x+e)^p)^q)*a*b*f^4*g^2*h+576*x*a*b*f^4*g^3*p*q-576*ln(c*(d*(f*x+e)^p)^q)*b^2*e*f^3*g^3*p*q+1152*ln(f*x+e)*b^2
*e*f^3*g^3*p^2*q^2+144*ln(f*x+e)*a*b*e^4*h^3*p*q-300*b^2*e^4*h^3*p^2*q^2+144*a*b*e^4*h^3*p*q+576*b^2*e*f^3*g^3
*p^2*q^2+288*a^2*e*f^3*g^3-9*x^4*b^2*f^4*h^3*p^2*q^2-144*x^4*ln(c*(d*(f*x+e)^p)^q)*a*b*f^4*h^3-288*x^3*ln(c*(d
*(f*x+e)^p)^q)^2*b^2*f^4*g*h^2-72*x^4*ln(c*(d*(f*x+e)^p)^q)^2*b^2*f^4*h^3-288*x^3*a^2*f^4*g*h^2-288*x*ln(c*(d*
(f*x+e)^p)^q)^2*b^2*f^4*g^3-432*x^2*a^2*f^4*g^2*h-288*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e*f^3*g^3-288*x*a^2*f^4*g^3-
72*x^4*a^2*f^4*h^3+72*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e^4*h^3-576*x*b^2*f^4*g^3*p^2*q^2-432*x^2*ln(c*(d*(f*x+e)^p)
^q)^2*b^2*f^4*g^2*h+144*ln(c*(d*(f*x+e)^p)^q)*b^2*e^4*h^3*p*q-576*x*ln(c*(d*(f*x+e)^p)^q)*a*b*f^4*g^3-288*ln(c
*(d*(f*x+e)^p)^q)^2*b^2*e^3*f*g*h^2+432*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e^2*f^2*g^2*h+576*ln(c*(d*(f*x+e)^p)^q)*a*
b*e*f^3*g^3-444*ln(f*x+e)*b^2*e^4*h^3*p^2*q^2)/f^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (391) = 782\).

Time = 0.35 (sec) , antiderivative size = 1742, normalized size of antiderivative = 4.26 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((h*x+g)^3*(a+b*log(c*(d*(f*x+e)^p)^q))^2,x, algorithm="fricas")

[Out]

1/288*(9*(b^2*f^4*h^3*p^2*q^2 - 4*a*b*f^4*h^3*p*q + 8*a^2*f^4*h^3)*x^4 + 4*(72*a^2*f^4*g*h^2 + (16*b^2*f^4*g*h
^2 - 7*b^2*e*f^3*h^3)*p^2*q^2 - 12*(4*a*b*f^4*g*h^2 - a*b*e*f^3*h^3)*p*q)*x^3 + 6*(72*a^2*f^4*g^2*h + (36*b^2*
f^4*g^2*h - 40*b^2*e*f^3*g*h^2 + 13*b^2*e^2*f^2*h^3)*p^2*q^2 - 12*(6*a*b*f^4*g^2*h - 4*a*b*e*f^3*g*h^2 + a*b*e
^2*f^2*h^3)*p*q)*x^2 + 72*(b^2*f^4*h^3*p^2*q^2*x^4 + 4*b^2*f^4*g*h^2*p^2*q^2*x^3 + 6*b^2*f^4*g^2*h*p^2*q^2*x^2
+ 4*b^2*f^4*g^3*p^2*q^2*x + (4*b^2*e*f^3*g^3 - 6*b^2*e^2*f^2*g^2*h + 4*b^2*e^3*f*g*h^2 - b^2*e^4*h^3)*p^2*q^2
)*log(f*x + e)^2 + 72*(b^2*f^4*h^3*x^4 + 4*b^2*f^4*g*h^2*x^3 + 6*b^2*f^4*g^2*h*x^2 + 4*b^2*f^4*g^3*x)*log(c)^2
+ 72*(b^2*f^4*h^3*q^2*x^4 + 4*b^2*f^4*g*h^2*q^2*x^3 + 6*b^2*f^4*g^2*h*q^2*x^2 + 4*b^2*f^4*g^3*q^2*x)*log(d)^2
+ 12*(24*a^2*f^4*g^3 + (48*b^2*f^4*g^3 - 108*b^2*e*f^3*g^2*h + 88*b^2*e^2*f^2*g*h^2 - 25*b^2*e^3*f*h^3)*p^2*q
^2 - 12*(4*a*b*f^4*g^3 - 6*a*b*e*f^3*g^2*h + 4*a*b*e^2*f^2*g*h^2 - a*b*e^3*f*h^3)*p*q)*x - 12*((48*b^2*e*f^3*g
^3 - 108*b^2*e^2*f^2*g^2*h + 88*b^2*e^3*f*g*h^2 - 25*b^2*e^4*h^3)*p^2*q^2 + 3*(b^2*f^4*h^3*p^2*q^2 - 4*a*b*f^4
*h^3*p*q)*x^4 - 4*(12*a*b*f^4*g*h^2*p*q - (4*b^2*f^4*g*h^2 - b^2*e*f^3*h^3)*p^2*q^2)*x^3 - 12*(4*a*b*e*f^3*g^3
- 6*a*b*e^2*f^2*g^2*h + 4*a*b*e^3*f*g*h^2 - a*b*e^4*h^3)*p*q - 6*(12*a*b*f^4*g^2*h*p*q - (6*b^2*f^4*g^2*h - 4
*b^2*e*f^3*g*h^2 + b^2*e^2*f^2*h^3)*p^2*q^2)*x^2 - 12*(4*a*b*f^4*g^3*p*q - (4*b^2*f^4*g^3 - 6*b^2*e*f^3*g^2*h
+ 4*b^2*e^2*f^2*g*h^2 - b^2*e^3*f*h^3)*p^2*q^2)*x - 12*(b^2*f^4*h^3*p*q*x^4 + 4*b^2*f^4*g*h^2*p*q*x^3 + 6*b^2*
f^4*g^2*h*p*q*x^2 + 4*b^2*f^4*g^3*p*q*x + (4*b^2*e*f^3*g^3 - 6*b^2*e^2*f^2*g^2*h + 4*b^2*e^3*f*g*h^2 - b^2*e^4
*h^3)*p*q)*log(c) - 12*(b^2*f^4*h^3*p*q^2*x^4 + 4*b^2*f^4*g*h^2*p*q^2*x^3 + 6*b^2*f^4*g^2*h*p*q^2*x^2 + 4*b^2*
f^4*g^3*p*q^2*x + (4*b^2*e*f^3*g^3 - 6*b^2*e^2*f^2*g^2*h + 4*b^2*e^3*f*g*h^2 - b^2*e^4*h^3)*p*q^2)*log(d))*log
(f*x + e) - 12*(3*(b^2*f^4*h^3*p*q - 4*a*b*f^4*h^3)*x^4 - 4*(12*a*b*f^4*g*h^2 - (4*b^2*f^4*g*h^2 - b^2*e*f^3*h
^3)*p*q)*x^3 - 6*(12*a*b*f^4*g^2*h - (6*b^2*f^4*g^2*h - 4*b^2*e*f^3*g*h^2 + b^2*e^2*f^2*h^3)*p*q)*x^2 - 12*(4*
a*b*f^4*g^3 - (4*b^2*f^4*g^3 - 6*b^2*e*f^3*g^2*h + 4*b^2*e^2*f^2*g*h^2 - b^2*e^3*f*h^3)*p*q)*x)*log(c) - 12*(3
*(b^2*f^4*h^3*p*q^2 - 4*a*b*f^4*h^3*q)*x^4 - 4*(12*a*b*f^4*g*h^2*q - (4*b^2*f^4*g*h^2 - b^2*e*f^3*h^3)*p*q^2)*
x^3 - 6*(12*a*b*f^4*g^2*h*q - (6*b^2*f^4*g^2*h - 4*b^2*e*f^3*g*h^2 + b^2*e^2*f^2*h^3)*p*q^2)*x^2 - 12*(4*a*b*f
^4*g^3*q - (4*b^2*f^4*g^3 - 6*b^2*e*f^3*g^2*h + 4*b^2*e^2*f^2*g*h^2 - b^2*e^3*f*h^3)*p*q^2)*x - 12*(b^2*f^4*h^
3*q*x^4 + 4*b^2*f^4*g*h^2*q*x^3 + 6*b^2*f^4*g^2*h*q*x^2 + 4*b^2*f^4*g^3*q*x)*log(c))*log(d))/f^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (394) = 788\).

Time = 5.39 (sec) , antiderivative size = 1421, normalized size of antiderivative = 3.47 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((h*x+g)**3*(a+b*ln(c*(d*(f*x+e)**p)**q))**2,x)

[Out]

Piecewise((a**2*g**3*x + 3*a**2*g**2*h*x**2/2 + a**2*g*h**2*x**3 + a**2*h**3*x**4/4 - a*b*e**4*h**3*log(c*(d*(
e + f*x)**p)**q)/(2*f**4) + 2*a*b*e**3*g*h**2*log(c*(d*(e + f*x)**p)**q)/f**3 + a*b*e**3*h**3*p*q*x/(2*f**3) -
3*a*b*e**2*g**2*h*log(c*(d*(e + f*x)**p)**q)/f**2 - 2*a*b*e**2*g*h**2*p*q*x/f**2 - a*b*e**2*h**3*p*q*x**2/(4*
f**2) + 2*a*b*e*g**3*log(c*(d*(e + f*x)**p)**q)/f + 3*a*b*e*g**2*h*p*q*x/f + a*b*e*g*h**2*p*q*x**2/f + a*b*e*h
**3*p*q*x**3/(6*f) - 2*a*b*g**3*p*q*x + 2*a*b*g**3*x*log(c*(d*(e + f*x)**p)**q) - 3*a*b*g**2*h*p*q*x**2/2 + 3*
a*b*g**2*h*x**2*log(c*(d*(e + f*x)**p)**q) - 2*a*b*g*h**2*p*q*x**3/3 + 2*a*b*g*h**2*x**3*log(c*(d*(e + f*x)**p
)**q) - a*b*h**3*p*q*x**4/8 + a*b*h**3*x**4*log(c*(d*(e + f*x)**p)**q)/2 + 25*b**2*e**4*h**3*p*q*log(c*(d*(e +
f*x)**p)**q)/(24*f**4) - b**2*e**4*h**3*log(c*(d*(e + f*x)**p)**q)**2/(4*f**4) - 11*b**2*e**3*g*h**2*p*q*log(
c*(d*(e + f*x)**p)**q)/(3*f**3) + b**2*e**3*g*h**2*log(c*(d*(e + f*x)**p)**q)**2/f**3 - 25*b**2*e**3*h**3*p**2
*q**2*x/(24*f**3) + b**2*e**3*h**3*p*q*x*log(c*(d*(e + f*x)**p)**q)/(2*f**3) + 9*b**2*e**2*g**2*h*p*q*log(c*(d
*(e + f*x)**p)**q)/(2*f**2) - 3*b**2*e**2*g**2*h*log(c*(d*(e + f*x)**p)**q)**2/(2*f**2) + 11*b**2*e**2*g*h**2*
p**2*q**2*x/(3*f**2) - 2*b**2*e**2*g*h**2*p*q*x*log(c*(d*(e + f*x)**p)**q)/f**2 + 13*b**2*e**2*h**3*p**2*q**2*
x**2/(48*f**2) - b**2*e**2*h**3*p*q*x**2*log(c*(d*(e + f*x)**p)**q)/(4*f**2) - 2*b**2*e*g**3*p*q*log(c*(d*(e +
f*x)**p)**q)/f + b**2*e*g**3*log(c*(d*(e + f*x)**p)**q)**2/f - 9*b**2*e*g**2*h*p**2*q**2*x/(2*f) + 3*b**2*e*g
**2*h*p*q*x*log(c*(d*(e + f*x)**p)**q)/f - 5*b**2*e*g*h**2*p**2*q**2*x**2/(6*f) + b**2*e*g*h**2*p*q*x**2*log(c
*(d*(e + f*x)**p)**q)/f - 7*b**2*e*h**3*p**2*q**2*x**3/(72*f) + b**2*e*h**3*p*q*x**3*log(c*(d*(e + f*x)**p)**q
)/(6*f) + 2*b**2*g**3*p**2*q**2*x - 2*b**2*g**3*p*q*x*log(c*(d*(e + f*x)**p)**q) + b**2*g**3*x*log(c*(d*(e + f
*x)**p)**q)**2 + 3*b**2*g**2*h*p**2*q**2*x**2/4 - 3*b**2*g**2*h*p*q*x**2*log(c*(d*(e + f*x)**p)**q)/2 + 3*b**2
*g**2*h*x**2*log(c*(d*(e + f*x)**p)**q)**2/2 + 2*b**2*g*h**2*p**2*q**2*x**3/9 - 2*b**2*g*h**2*p*q*x**3*log(c*(
d*(e + f*x)**p)**q)/3 + b**2*g*h**2*x**3*log(c*(d*(e + f*x)**p)**q)**2 + b**2*h**3*p**2*q**2*x**4/32 - b**2*h*
*3*p*q*x**4*log(c*(d*(e + f*x)**p)**q)/8 + b**2*h**3*x**4*log(c*(d*(e + f*x)**p)**q)**2/4, Ne(f, 0)), ((a + b*
log(c*(d*e**p)**q))**2*(g**3*x + 3*g**2*h*x**2/2 + g*h**2*x**3 + h**3*x**4/4), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (391) = 782\).

Time = 0.23 (sec) , antiderivative size = 895, normalized size of antiderivative = 2.19 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((h*x+g)^3*(a+b*log(c*(d*(f*x+e)^p)^q))^2,x, algorithm="maxima")

[Out]

1/4*b^2*h^3*x^4*log(((f*x + e)^p*d)^q*c)^2 + 1/2*a*b*h^3*x^4*log(((f*x + e)^p*d)^q*c) + b^2*g*h^2*x^3*log(((f*
x + e)^p*d)^q*c)^2 + 1/4*a^2*h^3*x^4 - 2*a*b*f*g^3*p*q*(x/f - e*log(f*x + e)/f^2) - 1/24*a*b*f*h^3*p*q*(12*e^4
*log(f*x + e)/f^5 + (3*f^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/f^4) + 1/3*a*b*f*g*h^2*p*q*(6*e^3*log(f
*x + e)/f^4 - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/f^3) - 3/2*a*b*f*g^2*h*p*q*(2*e^2*log(f*x + e)/f^3 + (f*x^2 -
2*e*x)/f^2) + 2*a*b*g*h^2*x^3*log(((f*x + e)^p*d)^q*c) + 3/2*b^2*g^2*h*x^2*log(((f*x + e)^p*d)^q*c)^2 + a^2*g*
h^2*x^3 + 3*a*b*g^2*h*x^2*log(((f*x + e)^p*d)^q*c) + b^2*g^3*x*log(((f*x + e)^p*d)^q*c)^2 + 3/2*a^2*g^2*h*x^2
+ 2*a*b*g^3*x*log(((f*x + e)^p*d)^q*c) - (2*f*p*q*(x/f - e*log(f*x + e)/f^2)*log(((f*x + e)^p*d)^q*c) + (e*log
(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*p^2*q^2/f)*b^2*g^3 - 3/4*(2*f*p*q*(2*e^2*log(f*x + e)/f^3 + (f*x^2 - 2
*e*x)/f^2)*log(((f*x + e)^p*d)^q*c) - (f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(f*x + e))*p^2*q^2/
f^2)*b^2*g^2*h + 1/18*(6*f*p*q*(6*e^3*log(f*x + e)/f^4 - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/f^3)*log(((f*x + e)
^p*d)^q*c) + (4*f^3*x^3 - 15*e*f^2*x^2 - 18*e^3*log(f*x + e)^2 + 66*e^2*f*x - 66*e^3*log(f*x + e))*p^2*q^2/f^3
)*b^2*g*h^2 - 1/288*(12*f*p*q*(12*e^4*log(f*x + e)/f^5 + (3*f^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/f^
4)*log(((f*x + e)^p*d)^q*c) - (9*f^4*x^4 - 28*e*f^3*x^3 + 78*e^2*f^2*x^2 + 72*e^4*log(f*x + e)^2 - 300*e^3*f*x
+ 300*e^4*log(f*x + e))*p^2*q^2/f^4)*b^2*h^3 + a^2*g^3*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3738 vs. \(2 (391) = 782\).

Time = 0.39 (sec) , antiderivative size = 3738, normalized size of antiderivative = 9.14 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((h*x+g)^3*(a+b*log(c*(d*(f*x+e)^p)^q))^2,x, algorithm="giac")

[Out]

(f*x + e)*b^2*g^3*p^2*q^2*log(f*x + e)^2/f + 3/2*(f*x + e)^2*b^2*g^2*h*p^2*q^2*log(f*x + e)^2/f^2 - 3*(f*x + e
)*b^2*e*g^2*h*p^2*q^2*log(f*x + e)^2/f^2 + (f*x + e)^3*b^2*g*h^2*p^2*q^2*log(f*x + e)^2/f^3 - 3*(f*x + e)^2*b^
2*e*g*h^2*p^2*q^2*log(f*x + e)^2/f^3 + 3*(f*x + e)*b^2*e^2*g*h^2*p^2*q^2*log(f*x + e)^2/f^3 + 1/4*(f*x + e)^4*
b^2*h^3*p^2*q^2*log(f*x + e)^2/f^4 - (f*x + e)^3*b^2*e*h^3*p^2*q^2*log(f*x + e)^2/f^4 + 3/2*(f*x + e)^2*b^2*e^
2*h^3*p^2*q^2*log(f*x + e)^2/f^4 - (f*x + e)*b^2*e^3*h^3*p^2*q^2*log(f*x + e)^2/f^4 - 2*(f*x + e)*b^2*g^3*p^2*
q^2*log(f*x + e)/f - 3/2*(f*x + e)^2*b^2*g^2*h*p^2*q^2*log(f*x + e)/f^2 + 6*(f*x + e)*b^2*e*g^2*h*p^2*q^2*log(
f*x + e)/f^2 - 2/3*(f*x + e)^3*b^2*g*h^2*p^2*q^2*log(f*x + e)/f^3 + 3*(f*x + e)^2*b^2*e*g*h^2*p^2*q^2*log(f*x
+ e)/f^3 - 6*(f*x + e)*b^2*e^2*g*h^2*p^2*q^2*log(f*x + e)/f^3 - 1/8*(f*x + e)^4*b^2*h^3*p^2*q^2*log(f*x + e)/f
^4 + 2/3*(f*x + e)^3*b^2*e*h^3*p^2*q^2*log(f*x + e)/f^4 - 3/2*(f*x + e)^2*b^2*e^2*h^3*p^2*q^2*log(f*x + e)/f^4
+ 2*(f*x + e)*b^2*e^3*h^3*p^2*q^2*log(f*x + e)/f^4 + 2*(f*x + e)*b^2*g^3*p*q^2*log(f*x + e)*log(d)/f + 3*(f*x
+ e)^2*b^2*g^2*h*p*q^2*log(f*x + e)*log(d)/f^2 - 6*(f*x + e)*b^2*e*g^2*h*p*q^2*log(f*x + e)*log(d)/f^2 + 2*(f
*x + e)^3*b^2*g*h^2*p*q^2*log(f*x + e)*log(d)/f^3 - 6*(f*x + e)^2*b^2*e*g*h^2*p*q^2*log(f*x + e)*log(d)/f^3 +
6*(f*x + e)*b^2*e^2*g*h^2*p*q^2*log(f*x + e)*log(d)/f^3 + 1/2*(f*x + e)^4*b^2*h^3*p*q^2*log(f*x + e)*log(d)/f^
4 - 2*(f*x + e)^3*b^2*e*h^3*p*q^2*log(f*x + e)*log(d)/f^4 + 3*(f*x + e)^2*b^2*e^2*h^3*p*q^2*log(f*x + e)*log(d
)/f^4 - 2*(f*x + e)*b^2*e^3*h^3*p*q^2*log(f*x + e)*log(d)/f^4 + 2*(f*x + e)*b^2*g^3*p^2*q^2/f + 3/4*(f*x + e)^
2*b^2*g^2*h*p^2*q^2/f^2 - 6*(f*x + e)*b^2*e*g^2*h*p^2*q^2/f^2 + 2/9*(f*x + e)^3*b^2*g*h^2*p^2*q^2/f^3 - 3/2*(f
*x + e)^2*b^2*e*g*h^2*p^2*q^2/f^3 + 6*(f*x + e)*b^2*e^2*g*h^2*p^2*q^2/f^3 + 1/32*(f*x + e)^4*b^2*h^3*p^2*q^2/f
^4 - 2/9*(f*x + e)^3*b^2*e*h^3*p^2*q^2/f^4 + 3/4*(f*x + e)^2*b^2*e^2*h^3*p^2*q^2/f^4 - 2*(f*x + e)*b^2*e^3*h^3
*p^2*q^2/f^4 + 2*(f*x + e)*b^2*g^3*p*q*log(f*x + e)*log(c)/f + 3*(f*x + e)^2*b^2*g^2*h*p*q*log(f*x + e)*log(c)
/f^2 - 6*(f*x + e)*b^2*e*g^2*h*p*q*log(f*x + e)*log(c)/f^2 + 2*(f*x + e)^3*b^2*g*h^2*p*q*log(f*x + e)*log(c)/f
^3 - 6*(f*x + e)^2*b^2*e*g*h^2*p*q*log(f*x + e)*log(c)/f^3 + 6*(f*x + e)*b^2*e^2*g*h^2*p*q*log(f*x + e)*log(c)
/f^3 + 1/2*(f*x + e)^4*b^2*h^3*p*q*log(f*x + e)*log(c)/f^4 - 2*(f*x + e)^3*b^2*e*h^3*p*q*log(f*x + e)*log(c)/f
^4 + 3*(f*x + e)^2*b^2*e^2*h^3*p*q*log(f*x + e)*log(c)/f^4 - 2*(f*x + e)*b^2*e^3*h^3*p*q*log(f*x + e)*log(c)/f
^4 - 2*(f*x + e)*b^2*g^3*p*q^2*log(d)/f - 3/2*(f*x + e)^2*b^2*g^2*h*p*q^2*log(d)/f^2 + 6*(f*x + e)*b^2*e*g^2*h
*p*q^2*log(d)/f^2 - 2/3*(f*x + e)^3*b^2*g*h^2*p*q^2*log(d)/f^3 + 3*(f*x + e)^2*b^2*e*g*h^2*p*q^2*log(d)/f^3 -
6*(f*x + e)*b^2*e^2*g*h^2*p*q^2*log(d)/f^3 - 1/8*(f*x + e)^4*b^2*h^3*p*q^2*log(d)/f^4 + 2/3*(f*x + e)^3*b^2*e*
h^3*p*q^2*log(d)/f^4 - 3/2*(f*x + e)^2*b^2*e^2*h^3*p*q^2*log(d)/f^4 + 2*(f*x + e)*b^2*e^3*h^3*p*q^2*log(d)/f^4
+ (f*x + e)*b^2*g^3*q^2*log(d)^2/f + 3/2*(f*x + e)^2*b^2*g^2*h*q^2*log(d)^2/f^2 - 3*(f*x + e)*b^2*e*g^2*h*q^2
*log(d)^2/f^2 + (f*x + e)^3*b^2*g*h^2*q^2*log(d)^2/f^3 - 3*(f*x + e)^2*b^2*e*g*h^2*q^2*log(d)^2/f^3 + 3*(f*x +
e)*b^2*e^2*g*h^2*q^2*log(d)^2/f^3 + 1/4*(f*x + e)^4*b^2*h^3*q^2*log(d)^2/f^4 - (f*x + e)^3*b^2*e*h^3*q^2*log(
d)^2/f^4 + 3/2*(f*x + e)^2*b^2*e^2*h^3*q^2*log(d)^2/f^4 - (f*x + e)*b^2*e^3*h^3*q^2*log(d)^2/f^4 + 2*(f*x + e)
*a*b*g^3*p*q*log(f*x + e)/f + 3*(f*x + e)^2*a*b*g^2*h*p*q*log(f*x + e)/f^2 - 6*(f*x + e)*a*b*e*g^2*h*p*q*log(f
*x + e)/f^2 + 2*(f*x + e)^3*a*b*g*h^2*p*q*log(f*x + e)/f^3 - 6*(f*x + e)^2*a*b*e*g*h^2*p*q*log(f*x + e)/f^3 +
6*(f*x + e)*a*b*e^2*g*h^2*p*q*log(f*x + e)/f^3 + 1/2*(f*x + e)^4*a*b*h^3*p*q*log(f*x + e)/f^4 - 2*(f*x + e)^3*
a*b*e*h^3*p*q*log(f*x + e)/f^4 + 3*(f*x + e)^2*a*b*e^2*h^3*p*q*log(f*x + e)/f^4 - 2*(f*x + e)*a*b*e^3*h^3*p*q*
log(f*x + e)/f^4 - 2*(f*x + e)*b^2*g^3*p*q*log(c)/f - 3/2*(f*x + e)^2*b^2*g^2*h*p*q*log(c)/f^2 + 6*(f*x + e)*b
^2*e*g^2*h*p*q*log(c)/f^2 - 2/3*(f*x + e)^3*b^2*g*h^2*p*q*log(c)/f^3 + 3*(f*x + e)^2*b^2*e*g*h^2*p*q*log(c)/f^
3 - 6*(f*x + e)*b^2*e^2*g*h^2*p*q*log(c)/f^3 - 1/8*(f*x + e)^4*b^2*h^3*p*q*log(c)/f^4 + 2/3*(f*x + e)^3*b^2*e*
h^3*p*q*log(c)/f^4 - 3/2*(f*x + e)^2*b^2*e^2*h^3*p*q*log(c)/f^4 + 2*(f*x + e)*b^2*e^3*h^3*p*q*log(c)/f^4 + 2*(
f*x + e)*b^2*g^3*q*log(c)*log(d)/f + 3*(f*x + e)^2*b^2*g^2*h*q*log(c)*log(d)/f^2 - 6*(f*x + e)*b^2*e*g^2*h*q*l
og(c)*log(d)/f^2 + 2*(f*x + e)^3*b^2*g*h^2*q*log(c)*log(d)/f^3 - 6*(f*x + e)^2*b^2*e*g*h^2*q*log(c)*log(d)/f^3
+ 6*(f*x + e)*b^2*e^2*g*h^2*q*log(c)*log(d)/f^3 + 1/2*(f*x + e)^4*b^2*h^3*q*log(c)*log(d)/f^4 - 2*(f*x + e)^3
*b^2*e*h^3*q*log(c)*log(d)/f^4 + 3*(f*x + e)^2*b^2*e^2*h^3*q*log(c)*log(d)/f^4 - 2*(f*x + e)*b^2*e^3*h^3*q*log
(c)*log(d)/f^4 - 2*(f*x + e)*a*b*g^3*p*q/f - 3/2*(f*x + e)^2*a*b*g^2*h*p*q/f^2 + 6*(f*x + e)*a*b*e*g^2*h*p*q/f
^2 - 2/3*(f*x + e)^3*a*b*g*h^2*p*q/f^3 + 3*(f*x + e)^2*a*b*e*g*h^2*p*q/f^3 - 6*(f*x + e)*a*b*e^2*g*h^2*p*q/f^3
- 1/8*(f*x + e)^4*a*b*h^3*p*q/f^4 + 2/3*(f*x + e)^3*a*b*e*h^3*p*q/f^4 - 3/2*(f*x + e)^2*a*b*e^2*h^3*p*q/f^4 +
2*(f*x + e)*a*b*e^3*h^3*p*q/f^4 + (f*x + e)*b^2*g^3*log(c)^2/f + 3/2*(f*x + e)^2*b^2*g^2*h*log(c)^2/f^2 - 3*(
f*x + e)*b^2*e*g^2*h*log(c)^2/f^2 + (f*x + e)^3*b^2*g*h^2*log(c)^2/f^3 - 3*(f*x + e)^2*b^2*e*g*h^2*log(c)^2/f^
3 + 3*(f*x + e)*b^2*e^2*g*h^2*log(c)^2/f^3 + 1/4*(f*x + e)^4*b^2*h^3*log(c)^2/f^4 - (f*x + e)^3*b^2*e*h^3*log(
c)^2/f^4 + 3/2*(f*x + e)^2*b^2*e^2*h^3*log(c)^2/f^4 - (f*x + e)*b^2*e^3*h^3*log(c)^2/f^4 + 2*(f*x + e)*a*b*g^3
*q*log(d)/f + 3*(f*x + e)^2*a*b*g^2*h*q*log(d)/f^2 - 6*(f*x + e)*a*b*e*g^2*h*q*log(d)/f^2 + 2*(f*x + e)^3*a*b*
g*h^2*q*log(d)/f^3 - 6*(f*x + e)^2*a*b*e*g*h^2*q*log(d)/f^3 + 6*(f*x + e)*a*b*e^2*g*h^2*q*log(d)/f^3 + 1/2*(f*
x + e)^4*a*b*h^3*q*log(d)/f^4 - 2*(f*x + e)^3*a*b*e*h^3*q*log(d)/f^4 + 3*(f*x + e)^2*a*b*e^2*h^3*q*log(d)/f^4
- 2*(f*x + e)*a*b*e^3*h^3*q*log(d)/f^4 + 2*(f*x + e)*a*b*g^3*log(c)/f + 3*(f*x + e)^2*a*b*g^2*h*log(c)/f^2 - 6
*(f*x + e)*a*b*e*g^2*h*log(c)/f^2 + 2*(f*x + e)^3*a*b*g*h^2*log(c)/f^3 - 6*(f*x + e)^2*a*b*e*g*h^2*log(c)/f^3
+ 6*(f*x + e)*a*b*e^2*g*h^2*log(c)/f^3 + 1/2*(f*x + e)^4*a*b*h^3*log(c)/f^4 - 2*(f*x + e)^3*a*b*e*h^3*log(c)/f
^4 + 3*(f*x + e)^2*a*b*e^2*h^3*log(c)/f^4 - 2*(f*x + e)*a*b*e^3*h^3*log(c)/f^4 + (f*x + e)*a^2*g^3/f + 3/2*(f*
x + e)^2*a^2*g^2*h/f^2 - 3*(f*x + e)*a^2*e*g^2*h/f^2 + (f*x + e)^3*a^2*g*h^2/f^3 - 3*(f*x + e)^2*a^2*e*g*h^2/f
^3 + 3*(f*x + e)*a^2*e^2*g*h^2/f^3 + 1/4*(f*x + e)^4*a^2*h^3/f^4 - (f*x + e)^3*a^2*e*h^3/f^4 + 3/2*(f*x + e)^2
*a^2*e^2*h^3/f^4 - (f*x + e)*a^2*e^3*h^3/f^4

Mupad [B] (verification not implemented)

Time = 1.96 (sec) , antiderivative size = 1154, normalized size of antiderivative = 2.82 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=x^3\,\left (\frac {h^2\,\left (6\,a^2\,e\,h+18\,a^2\,f\,g-b^2\,e\,h\,p^2\,q^2+4\,b^2\,f\,g\,p^2\,q^2-12\,a\,b\,f\,g\,p\,q\right )}{18\,f}-\frac {e\,h^3\,\left (8\,a^2-4\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{24\,f}\right )+\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (\frac {x\,\left (\frac {e\,\left (\frac {e\,\left (\frac {4\,b\,h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {b\,e\,h^3\,\left (4\,a-b\,p\,q\right )}{f}\right )}{f}-\frac {6\,b\,g\,h\,\left (2\,a\,e\,h+2\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}\right )}{f}+\frac {4\,b\,g^2\,\left (3\,a\,e\,h+a\,f\,g-b\,f\,g\,p\,q\right )}{f}\right )}{2}+\frac {x^3\,\left (\frac {4\,b\,h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{3\,f}-\frac {b\,e\,h^3\,\left (4\,a-b\,p\,q\right )}{3\,f}\right )}{2}-\frac {x^2\,\left (\frac {e\,\left (\frac {4\,b\,h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {b\,e\,h^3\,\left (4\,a-b\,p\,q\right )}{f}\right )}{2\,f}-\frac {3\,b\,g\,h\,\left (2\,a\,e\,h+2\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}\right )}{2}+\frac {b\,h^3\,x^4\,\left (4\,a-b\,p\,q\right )}{8}\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (b^2\,g^3\,x-\frac {e\,\left (b^2\,e^3\,h^3-4\,b^2\,e^2\,f\,g\,h^2+6\,b^2\,e\,f^2\,g^2\,h-4\,b^2\,f^3\,g^3\right )}{4\,f^4}+\frac {b^2\,h^3\,x^4}{4}+\frac {3\,b^2\,g^2\,h\,x^2}{2}+b^2\,g\,h^2\,x^3\right )+x\,\left (\frac {72\,a^2\,e\,f^2\,g^2\,h+24\,a^2\,f^3\,g^3-48\,a\,b\,f^3\,g^3\,p\,q-12\,b^2\,e^3\,h^3\,p^2\,q^2+48\,b^2\,e^2\,f\,g\,h^2\,p^2\,q^2-72\,b^2\,e\,f^2\,g^2\,h\,p^2\,q^2+48\,b^2\,f^3\,g^3\,p^2\,q^2}{24\,f^3}+\frac {e\,\left (\frac {e\,\left (\frac {h^2\,\left (6\,a^2\,e\,h+18\,a^2\,f\,g-b^2\,e\,h\,p^2\,q^2+4\,b^2\,f\,g\,p^2\,q^2-12\,a\,b\,f\,g\,p\,q\right )}{6\,f}-\frac {e\,h^3\,\left (8\,a^2-4\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{8\,f}\right )}{f}-\frac {h\,\left (12\,a^2\,e\,f\,g\,h+12\,a^2\,f^2\,g^2-12\,a\,b\,f^2\,g^2\,p\,q+b^2\,e^2\,h^2\,p^2\,q^2-4\,b^2\,e\,f\,g\,h\,p^2\,q^2+6\,b^2\,f^2\,g^2\,p^2\,q^2\right )}{4\,f^2}\right )}{f}\right )-x^2\,\left (\frac {e\,\left (\frac {h^2\,\left (6\,a^2\,e\,h+18\,a^2\,f\,g-b^2\,e\,h\,p^2\,q^2+4\,b^2\,f\,g\,p^2\,q^2-12\,a\,b\,f\,g\,p\,q\right )}{6\,f}-\frac {e\,h^3\,\left (8\,a^2-4\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{8\,f}\right )}{2\,f}-\frac {h\,\left (12\,a^2\,e\,f\,g\,h+12\,a^2\,f^2\,g^2-12\,a\,b\,f^2\,g^2\,p\,q+b^2\,e^2\,h^2\,p^2\,q^2-4\,b^2\,e\,f\,g\,h\,p^2\,q^2+6\,b^2\,f^2\,g^2\,p^2\,q^2\right )}{8\,f^2}\right )+\frac {\ln \left (e+f\,x\right )\,\left (25\,b^2\,e^4\,h^3\,p^2\,q^2-88\,b^2\,e^3\,f\,g\,h^2\,p^2\,q^2+108\,b^2\,e^2\,f^2\,g^2\,h\,p^2\,q^2-48\,b^2\,e\,f^3\,g^3\,p^2\,q^2-12\,a\,b\,e^4\,h^3\,p\,q+48\,a\,b\,e^3\,f\,g\,h^2\,p\,q-72\,a\,b\,e^2\,f^2\,g^2\,h\,p\,q+48\,a\,b\,e\,f^3\,g^3\,p\,q\right )}{24\,f^4}+\frac {h^3\,x^4\,\left (8\,a^2-4\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{32} \]

[In]

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

[Out]

x^3*((h^2*(6*a^2*e*h + 18*a^2*f*g - b^2*e*h*p^2*q^2 + 4*b^2*f*g*p^2*q^2 - 12*a*b*f*g*p*q))/(18*f) - (e*h^3*(8*
a^2 + b^2*p^2*q^2 - 4*a*b*p*q))/(24*f)) + log(c*(d*(e + f*x)^p)^q)*((x*((e*((e*((4*b*h^2*(a*e*h + 3*a*f*g - b*
f*g*p*q))/f - (b*e*h^3*(4*a - b*p*q))/f))/f - (6*b*g*h*(2*a*e*h + 2*a*f*g - b*f*g*p*q))/f))/f + (4*b*g^2*(3*a*
e*h + a*f*g - b*f*g*p*q))/f))/2 + (x^3*((4*b*h^2*(a*e*h + 3*a*f*g - b*f*g*p*q))/(3*f) - (b*e*h^3*(4*a - b*p*q)
)/(3*f)))/2 - (x^2*((e*((4*b*h^2*(a*e*h + 3*a*f*g - b*f*g*p*q))/f - (b*e*h^3*(4*a - b*p*q))/f))/(2*f) - (3*b*g
*h*(2*a*e*h + 2*a*f*g - b*f*g*p*q))/f))/2 + (b*h^3*x^4*(4*a - b*p*q))/8) + log(c*(d*(e + f*x)^p)^q)^2*(b^2*g^3
*x - (e*(b^2*e^3*h^3 - 4*b^2*f^3*g^3 + 6*b^2*e*f^2*g^2*h - 4*b^2*e^2*f*g*h^2))/(4*f^4) + (b^2*h^3*x^4)/4 + (3*
b^2*g^2*h*x^2)/2 + b^2*g*h^2*x^3) + x*((24*a^2*f^3*g^3 - 12*b^2*e^3*h^3*p^2*q^2 + 48*b^2*f^3*g^3*p^2*q^2 + 72*
a^2*e*f^2*g^2*h - 48*a*b*f^3*g^3*p*q - 72*b^2*e*f^2*g^2*h*p^2*q^2 + 48*b^2*e^2*f*g*h^2*p^2*q^2)/(24*f^3) + (e*
((e*((h^2*(6*a^2*e*h + 18*a^2*f*g - b^2*e*h*p^2*q^2 + 4*b^2*f*g*p^2*q^2 - 12*a*b*f*g*p*q))/(6*f) - (e*h^3*(8*a
^2 + b^2*p^2*q^2 - 4*a*b*p*q))/(8*f)))/f - (h*(12*a^2*f^2*g^2 + b^2*e^2*h^2*p^2*q^2 + 6*b^2*f^2*g^2*p^2*q^2 +
12*a^2*e*f*g*h - 12*a*b*f^2*g^2*p*q - 4*b^2*e*f*g*h*p^2*q^2))/(4*f^2)))/f) - x^2*((e*((h^2*(6*a^2*e*h + 18*a^2
*f*g - b^2*e*h*p^2*q^2 + 4*b^2*f*g*p^2*q^2 - 12*a*b*f*g*p*q))/(6*f) - (e*h^3*(8*a^2 + b^2*p^2*q^2 - 4*a*b*p*q)
)/(8*f)))/(2*f) - (h*(12*a^2*f^2*g^2 + b^2*e^2*h^2*p^2*q^2 + 6*b^2*f^2*g^2*p^2*q^2 + 12*a^2*e*f*g*h - 12*a*b*f
^2*g^2*p*q - 4*b^2*e*f*g*h*p^2*q^2))/(8*f^2)) + (log(e + f*x)*(25*b^2*e^4*h^3*p^2*q^2 - 12*a*b*e^4*h^3*p*q - 4
8*b^2*e*f^3*g^3*p^2*q^2 - 88*b^2*e^3*f*g*h^2*p^2*q^2 + 108*b^2*e^2*f^2*g^2*h*p^2*q^2 + 48*a*b*e*f^3*g^3*p*q +
48*a*b*e^3*f*g*h^2*p*q - 72*a*b*e^2*f^2*g^2*h*p*q))/(24*f^4) + (h^3*x^4*(8*a^2 + b^2*p^2*q^2 - 4*a*b*p*q))/32