\(\int (g+h x)^2 (a+b \log (c (d (e+f x)^p)^q))^2 \, dx\) [429]

Optimal result

Integrand size = 28, antiderivative size = 323 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {2 b^2 (f g-e h)^2 p^2 q^2 x}{f^2}+\frac {b^2 h (f g-e h) p^2 q^2 (e+f x)^2}{2 f^3}+\frac {2 b^2 h^2 p^2 q^2 (e+f x)^3}{27 f^3}+\frac {b^2 (f g-e h)^3 p^2 q^2 \log ^2(e+f x)}{3 f^3 h}-\frac {2 b (f g-e h)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {2 b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac {2 b (f g-e h)^3 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^3 h}+\frac {(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h} \]

[Out]

2*b^2*(-e*h+f*g)^2*p^2*q^2*x/f^2+1/2*b^2*h*(-e*h+f*g)*p^2*q^2*(f*x+e)^2/f^3+2/27*b^2*h^2*p^2*q^2*(f*x+e)^3/f^3
+1/3*b^2*(-e*h+f*g)^3*p^2*q^2*ln(f*x+e)^2/f^3/h-2*b*(-e*h+f*g)^2*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^3-b
*h*(-e*h+f*g)*p*q*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^3-2/9*b*h^2*p*q*(f*x+e)^3*(a+b*ln(c*(d*(f*x+e)^p)^q)
)/f^3-2/3*b*(-e*h+f*g)^3*p*q*ln(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^3/h+1/3*(h*x+g)^3*(a+b*ln(c*(d*(f*x+e)^p)
^q))^2/h

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2445, 2458, 45, 2372, 12, 14, 2338, 2495} \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=-\frac {2 b p q (f g-e h)^3 \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^3 h}-\frac {2 b p q (e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {b h p q (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {2 b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}+\frac {(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac {b^2 h p^2 q^2 (e+f x)^2 (f g-e h)}{2 f^3}+\frac {b^2 p^2 q^2 (f g-e h)^3 \log ^2(e+f x)}{3 f^3 h}+\frac {2 b^2 h^2 p^2 q^2 (e+f x)^3}{27 f^3}+\frac {2 b^2 p^2 q^2 x (f g-e h)^2}{f^2} \]

[In]

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

[Out]

(2*b^2*(f*g - e*h)^2*p^2*q^2*x)/f^2 + (b^2*h*(f*g - e*h)*p^2*q^2*(e + f*x)^2)/(2*f^3) + (2*b^2*h^2*p^2*q^2*(e
+ f*x)^3)/(27*f^3) + (b^2*(f*g - e*h)^3*p^2*q^2*Log[e + f*x]^2)/(3*f^3*h) - (2*b*(f*g - e*h)^2*p*q*(e + f*x)*(
a + b*Log[c*(d*(e + f*x)^p)^q]))/f^3 - (b*h*(f*g - e*h)*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/f^3
- (2*b*h^2*p*q*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(9*f^3) - (2*b*(f*g - e*h)^3*p*q*Log[e + f*x]*(a
+ b*Log[c*(d*(e + f*x)^p)^q]))/(3*f^3*h) + ((g + h*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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)^2 \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)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\text {Subst}\left (\frac {(2 b f p q) \int \frac {(g+h x)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x} \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\text {Subst}\left (\frac {(2 b p q) \text {Subst}\left (\int \frac {\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^3 \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, dx,x,e+f x\right )}{3 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)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {2 b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac {2 b (f g-e h)^3 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^3 h}+\frac {(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\text {Subst}\left (\frac {\left (2 b^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {h x \left (18 f^2 g^2+9 f g h (-4 e+x)+h^2 \left (18 e^2-9 e x+2 x^2\right )\right )+6 (f g-e h)^3 \log (x)}{6 f^3 x} \, dx,x,e+f x\right )}{3 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)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {2 b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac {2 b (f g-e h)^3 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^3 h}+\frac {(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\text {Subst}\left (\frac {\left (b^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {h x \left (18 f^2 g^2+9 f g h (-4 e+x)+h^2 \left (18 e^2-9 e x+2 x^2\right )\right )+6 (f g-e h)^3 \log (x)}{x} \, dx,x,e+f x\right )}{9 f^3 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)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {2 b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac {2 b (f g-e h)^3 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^3 h}+\frac {(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\text {Subst}\left (\frac {\left (b^2 p^2 q^2\right ) \text {Subst}\left (\int \left (h \left (18 (f g-e h)^2+9 h (f g-e h) x+2 h^2 x^2\right )+\frac {6 (f g-e h)^3 \log (x)}{x}\right ) \, dx,x,e+f x\right )}{9 f^3 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)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {2 b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac {2 b (f g-e h)^3 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^3 h}+\frac {(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\text {Subst}\left (\frac {\left (b^2 p^2 q^2\right ) \text {Subst}\left (\int \left (18 (f g-e h)^2+9 h (f g-e h) x+2 h^2 x^2\right ) \, dx,x,e+f x\right )}{9 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (2 b^2 (f g-e h)^3 p^2 q^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+f x\right )}{3 f^3 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)^2 p^2 q^2 x}{f^2}+\frac {b^2 h (f g-e h) p^2 q^2 (e+f x)^2}{2 f^3}+\frac {2 b^2 h^2 p^2 q^2 (e+f x)^3}{27 f^3}+\frac {b^2 (f g-e h)^3 p^2 q^2 \log ^2(e+f x)}{3 f^3 h}-\frac {2 b (f g-e h)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac {2 b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac {2 b (f g-e h)^3 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^3 h}+\frac {(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.86 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {54 (f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+54 h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+18 h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-108 b (f g-e h)^2 p q \left (f (a-b p q) x+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+27 b h (f g-e h) p q \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+4 b h^2 p q \left (b f p q x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{54 f^3} \]

[In]

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

[Out]

(54*(f*g - e*h)^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + 54*h*(f*g - e*h)*(e + f*x)^2*(a + b*Log[c*(d*
(e + f*x)^p)^q])^2 + 18*h^2*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 - 108*b*(f*g - e*h)^2*p*q*(f*(a - b
*p*q)*x + b*(e + f*x)*Log[c*(d*(e + f*x)^p)^q]) + 27*b*h*(f*g - e*h)*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])) + 4*b*h^2*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])))/(54*f^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(938\) vs. \(2(311)=622\).

Time = 5.68 (sec) , antiderivative size = 939, normalized size of antiderivative = 2.91

method	result	size
parallelrisch	\(\frac {162 \ln \left (f x +e \right ) b^{2} e^{3} f g h \,p^{2} q^{2}+108 x a b \,e^{2} f^{2} g h p q -108 \ln \left (f x +e \right ) a b \,e^{3} f g h p q +108 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) a b e \,f^{3} g^{2}-108 \ln \left (f x +e \right ) b^{2} e^{2} f^{2} g^{2} p^{2} q^{2}+36 \ln \left (f x +e \right ) a b \,e^{4} h^{2} p q +4 x^{3} b^{2} e \,f^{3} h^{2} p^{2} q^{2}-15 x^{2} b^{2} e^{2} f^{2} h^{2} p^{2} q^{2}+66 x \,b^{2} e^{3} f \,h^{2} p^{2} q^{2}+108 x \,b^{2} e \,f^{3} g^{2} p^{2} q^{2}+36 x^{3} \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) a b e \,f^{3} h^{2}+54 x^{2} {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{2} e \,f^{3} g h +108 \ln \left (f x +e \right ) a b \,e^{2} f^{2} g^{2} p q -12 x^{3} \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b^{2} e \,f^{3} h^{2} p q +27 x^{2} b^{2} e \,f^{3} g h \,p^{2} q^{2}-12 x^{3} a b e \,f^{3} h^{2} p q +18 x^{2} \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b^{2} e^{2} f^{2} h^{2} p q -108 a b \,e^{3} f g h p q -54 x^{2} \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b^{2} e \,f^{3} g h p q -54 x^{2} a b e \,f^{3} g h p q +108 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b^{2} e^{2} f^{2} g h p q -162 x \,b^{2} e^{2} f^{2} g h \,p^{2} q^{2}+18 x^{2} a b \,e^{2} f^{2} h^{2} p q -36 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b^{2} e^{3} f \,h^{2} p q -108 x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) b^{2} e \,f^{3} g^{2} p q +108 x^{2} \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) a b e \,f^{3} g h -36 x a b \,e^{3} f \,h^{2} p q -108 x a b e \,f^{3} g^{2} p q -108 b^{2} e^{2} f^{2} g^{2} p^{2} q^{2}+36 a b \,e^{4} h^{2} p q +54 x \,a^{2} e \,f^{3} g^{2}+18 x^{3} a^{2} e \,f^{3} h^{2}+54 {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{2} e^{2} f^{2} g^{2}+108 a b \,e^{2} f^{2} g^{2} p q +18 x^{3} {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{2} e \,f^{3} h^{2}+54 x {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{2} e \,f^{3} g^{2}+54 x^{2} a^{2} e \,f^{3} g h -54 {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{2} e^{3} f g h -66 \ln \left (f x +e \right ) b^{2} e^{4} h^{2} p^{2} q^{2}-66 b^{2} e^{4} h^{2} p^{2} q^{2}+162 b^{2} e^{3} f g h \,p^{2} q^{2}-54 a^{2} e^{2} f^{2} g^{2}+18 {\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}^{2} b^{2} e^{4} h^{2}}{54 e \,f^{3}}\)	\(939\)

[In]

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

[Out]

1/54*(162*ln(f*x+e)*b^2*e^3*f*g*h*p^2*q^2+108*x*a*b*e^2*f^2*g*h*p*q-108*ln(f*x+e)*a*b*e^3*f*g*h*p*q+108*x*ln(c
*(d*(f*x+e)^p)^q)*a*b*e*f^3*g^2-108*ln(f*x+e)*b^2*e^2*f^2*g^2*p^2*q^2+36*ln(f*x+e)*a*b*e^4*h^2*p*q+4*x^3*b^2*e
*f^3*h^2*p^2*q^2-15*x^2*b^2*e^2*f^2*h^2*p^2*q^2+66*x*b^2*e^3*f*h^2*p^2*q^2+108*x*b^2*e*f^3*g^2*p^2*q^2+36*x^3*
ln(c*(d*(f*x+e)^p)^q)*a*b*e*f^3*h^2+54*x^2*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e*f^3*g*h+108*ln(f*x+e)*a*b*e^2*f^2*g^2
*p*q-12*x^3*ln(c*(d*(f*x+e)^p)^q)*b^2*e*f^3*h^2*p*q+27*x^2*b^2*e*f^3*g*h*p^2*q^2-12*x^3*a*b*e*f^3*h^2*p*q+18*x
^2*ln(c*(d*(f*x+e)^p)^q)*b^2*e^2*f^2*h^2*p*q-108*a*b*e^3*f*g*h*p*q-54*x^2*ln(c*(d*(f*x+e)^p)^q)*b^2*e*f^3*g*h*
p*q-54*x^2*a*b*e*f^3*g*h*p*q+108*x*ln(c*(d*(f*x+e)^p)^q)*b^2*e^2*f^2*g*h*p*q-162*x*b^2*e^2*f^2*g*h*p^2*q^2+18*
x^2*a*b*e^2*f^2*h^2*p*q-36*x*ln(c*(d*(f*x+e)^p)^q)*b^2*e^3*f*h^2*p*q-108*x*ln(c*(d*(f*x+e)^p)^q)*b^2*e*f^3*g^2
*p*q+108*x^2*ln(c*(d*(f*x+e)^p)^q)*a*b*e*f^3*g*h-36*x*a*b*e^3*f*h^2*p*q-108*x*a*b*e*f^3*g^2*p*q-108*b^2*e^2*f^
2*g^2*p^2*q^2+36*a*b*e^4*h^2*p*q+54*x*a^2*e*f^3*g^2+18*x^3*a^2*e*f^3*h^2+54*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e^2*f^
2*g^2+108*a*b*e^2*f^2*g^2*p*q+18*x^3*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e*f^3*h^2+54*x*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e*
f^3*g^2+54*x^2*a^2*e*f^3*g*h-54*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e^3*f*g*h-66*ln(f*x+e)*b^2*e^4*h^2*p^2*q^2-66*b^2*
e^4*h^2*p^2*q^2+162*b^2*e^3*f*g*h*p^2*q^2-54*a^2*e^2*f^2*g^2+18*ln(c*(d*(f*x+e)^p)^q)^2*b^2*e^4*h^2)/e/f^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1137 vs. \(2 (311) = 622\).

Time = 0.34 (sec) , antiderivative size = 1137, normalized size of antiderivative = 3.52 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/54*(2*(2*b^2*f^3*h^2*p^2*q^2 - 6*a*b*f^3*h^2*p*q + 9*a^2*f^3*h^2)*x^3 + 3*(18*a^2*f^3*g*h + (9*b^2*f^3*g*h -
5*b^2*e*f^2*h^2)*p^2*q^2 - 6*(3*a*b*f^3*g*h - a*b*e*f^2*h^2)*p*q)*x^2 + 18*(b^2*f^3*h^2*p^2*q^2*x^3 + 3*b^2*f
^3*g*h*p^2*q^2*x^2 + 3*b^2*f^3*g^2*p^2*q^2*x + (3*b^2*e*f^2*g^2 - 3*b^2*e^2*f*g*h + b^2*e^3*h^2)*p^2*q^2)*log(
f*x + e)^2 + 18*(b^2*f^3*h^2*x^3 + 3*b^2*f^3*g*h*x^2 + 3*b^2*f^3*g^2*x)*log(c)^2 + 18*(b^2*f^3*h^2*q^2*x^3 + 3
*b^2*f^3*g*h*q^2*x^2 + 3*b^2*f^3*g^2*q^2*x)*log(d)^2 + 6*(9*a^2*f^3*g^2 + (18*b^2*f^3*g^2 - 27*b^2*e*f^2*g*h +
11*b^2*e^2*f*h^2)*p^2*q^2 - 6*(3*a*b*f^3*g^2 - 3*a*b*e*f^2*g*h + a*b*e^2*f*h^2)*p*q)*x - 6*((18*b^2*e*f^2*g^2
- 27*b^2*e^2*f*g*h + 11*b^2*e^3*h^2)*p^2*q^2 + 2*(b^2*f^3*h^2*p^2*q^2 - 3*a*b*f^3*h^2*p*q)*x^3 - 6*(3*a*b*e*f
^2*g^2 - 3*a*b*e^2*f*g*h + a*b*e^3*h^2)*p*q - 3*(6*a*b*f^3*g*h*p*q - (3*b^2*f^3*g*h - b^2*e*f^2*h^2)*p^2*q^2)*
x^2 - 6*(3*a*b*f^3*g^2*p*q - (3*b^2*f^3*g^2 - 3*b^2*e*f^2*g*h + b^2*e^2*f*h^2)*p^2*q^2)*x - 6*(b^2*f^3*h^2*p*q
*x^3 + 3*b^2*f^3*g*h*p*q*x^2 + 3*b^2*f^3*g^2*p*q*x + (3*b^2*e*f^2*g^2 - 3*b^2*e^2*f*g*h + b^2*e^3*h^2)*p*q)*lo
g(c) - 6*(b^2*f^3*h^2*p*q^2*x^3 + 3*b^2*f^3*g*h*p*q^2*x^2 + 3*b^2*f^3*g^2*p*q^2*x + (3*b^2*e*f^2*g^2 - 3*b^2*e
^2*f*g*h + b^2*e^3*h^2)*p*q^2)*log(d))*log(f*x + e) - 6*(2*(b^2*f^3*h^2*p*q - 3*a*b*f^3*h^2)*x^3 - 3*(6*a*b*f^
3*g*h - (3*b^2*f^3*g*h - b^2*e*f^2*h^2)*p*q)*x^2 - 6*(3*a*b*f^3*g^2 - (3*b^2*f^3*g^2 - 3*b^2*e*f^2*g*h + b^2*e
^2*f*h^2)*p*q)*x)*log(c) - 6*(2*(b^2*f^3*h^2*p*q^2 - 3*a*b*f^3*h^2*q)*x^3 - 3*(6*a*b*f^3*g*h*q - (3*b^2*f^3*g*
h - b^2*e*f^2*h^2)*p*q^2)*x^2 - 6*(3*a*b*f^3*g^2*q - (3*b^2*f^3*g^2 - 3*b^2*e*f^2*g*h + b^2*e^2*f*h^2)*p*q^2)*
x - 6*(b^2*f^3*h^2*q*x^3 + 3*b^2*f^3*g*h*q*x^2 + 3*b^2*f^3*g^2*q*x)*log(c))*log(d))/f^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (311) = 622\).

Time = 2.59 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.77 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\begin {cases} a^{2} g^{2} x + a^{2} g h x^{2} + \frac {a^{2} h^{2} x^{3}}{3} + \frac {2 a b e^{3} h^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{3 f^{3}} - \frac {2 a b e^{2} g h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f^{2}} - \frac {2 a b e^{2} h^{2} p q x}{3 f^{2}} + \frac {2 a b e g^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {2 a b e g h p q x}{f} + \frac {a b e h^{2} p q x^{2}}{3 f} - 2 a b g^{2} p q x + 2 a b g^{2} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - a b g h p q x^{2} + 2 a b g h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {2 a b h^{2} p q x^{3}}{9} + \frac {2 a b h^{2} x^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{3} - \frac {11 b^{2} e^{3} h^{2} p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{9 f^{3}} + \frac {b^{2} e^{3} h^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{3 f^{3}} + \frac {3 b^{2} e^{2} g h p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f^{2}} - \frac {b^{2} e^{2} g h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f^{2}} + \frac {11 b^{2} e^{2} h^{2} p^{2} q^{2} x}{9 f^{2}} - \frac {2 b^{2} e^{2} h^{2} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{3 f^{2}} - \frac {2 b^{2} e g^{2} p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {b^{2} e g^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} - \frac {3 b^{2} e g h p^{2} q^{2} x}{f} + \frac {2 b^{2} e g h p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - \frac {5 b^{2} e h^{2} p^{2} q^{2} x^{2}}{18 f} + \frac {b^{2} e h^{2} p q x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{3 f} + 2 b^{2} g^{2} p^{2} q^{2} x - 2 b^{2} g^{2} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + b^{2} g^{2} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + \frac {b^{2} g h p^{2} q^{2} x^{2}}{2} - b^{2} g h p q x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + b^{2} g h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + \frac {2 b^{2} h^{2} p^{2} q^{2} x^{3}}{27} - \frac {2 b^{2} h^{2} p q x^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{9} + \frac {b^{2} h^{2} x^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{3} & \text {for}\: f \neq 0 \\\left (a + b \log {\left (c \left (d e^{p}\right )^{q} \right )}\right )^{2} \left (g^{2} x + g h x^{2} + \frac {h^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*g**2*x + a**2*g*h*x**2 + a**2*h**2*x**3/3 + 2*a*b*e**3*h**2*log(c*(d*(e + f*x)**p)**q)/(3*f**3
) - 2*a*b*e**2*g*h*log(c*(d*(e + f*x)**p)**q)/f**2 - 2*a*b*e**2*h**2*p*q*x/(3*f**2) + 2*a*b*e*g**2*log(c*(d*(e
+ f*x)**p)**q)/f + 2*a*b*e*g*h*p*q*x/f + a*b*e*h**2*p*q*x**2/(3*f) - 2*a*b*g**2*p*q*x + 2*a*b*g**2*x*log(c*(d
*(e + f*x)**p)**q) - a*b*g*h*p*q*x**2 + 2*a*b*g*h*x**2*log(c*(d*(e + f*x)**p)**q) - 2*a*b*h**2*p*q*x**3/9 + 2*
a*b*h**2*x**3*log(c*(d*(e + f*x)**p)**q)/3 - 11*b**2*e**3*h**2*p*q*log(c*(d*(e + f*x)**p)**q)/(9*f**3) + b**2*
e**3*h**2*log(c*(d*(e + f*x)**p)**q)**2/(3*f**3) + 3*b**2*e**2*g*h*p*q*log(c*(d*(e + f*x)**p)**q)/f**2 - b**2*
e**2*g*h*log(c*(d*(e + f*x)**p)**q)**2/f**2 + 11*b**2*e**2*h**2*p**2*q**2*x/(9*f**2) - 2*b**2*e**2*h**2*p*q*x*
log(c*(d*(e + f*x)**p)**q)/(3*f**2) - 2*b**2*e*g**2*p*q*log(c*(d*(e + f*x)**p)**q)/f + b**2*e*g**2*log(c*(d*(e
+ f*x)**p)**q)**2/f - 3*b**2*e*g*h*p**2*q**2*x/f + 2*b**2*e*g*h*p*q*x*log(c*(d*(e + f*x)**p)**q)/f - 5*b**2*e
*h**2*p**2*q**2*x**2/(18*f) + b**2*e*h**2*p*q*x**2*log(c*(d*(e + f*x)**p)**q)/(3*f) + 2*b**2*g**2*p**2*q**2*x
- 2*b**2*g**2*p*q*x*log(c*(d*(e + f*x)**p)**q) + b**2*g**2*x*log(c*(d*(e + f*x)**p)**q)**2 + b**2*g*h*p**2*q**
2*x**2/2 - b**2*g*h*p*q*x**2*log(c*(d*(e + f*x)**p)**q) + b**2*g*h*x**2*log(c*(d*(e + f*x)**p)**q)**2 + 2*b**2
*h**2*p**2*q**2*x**3/27 - 2*b**2*h**2*p*q*x**3*log(c*(d*(e + f*x)**p)**q)/9 + b**2*h**2*x**3*log(c*(d*(e + f*x
)**p)**q)**2/3, Ne(f, 0)), ((a + b*log(c*(d*e**p)**q))**2*(g**2*x + g*h*x**2 + h**2*x**3/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.87 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} h^{2} x^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} - 2 \, a b f g^{2} p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + \frac {1}{9} \, a b f h^{2} p q {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{f^{3}}\right )} - a b f g h p q {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{f^{3}} + \frac {f x^{2} - 2 \, e x}{f^{2}}\right )} + \frac {2}{3} \, a b h^{2} x^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + b^{2} g h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + \frac {1}{3} \, a^{2} h^{2} x^{3} + 2 \, a b g h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + b^{2} g^{2} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + a^{2} g h x^{2} + 2 \, a b g^{2} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - {\left (2 \, f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} p^{2} q^{2}}{f}\right )} b^{2} g^{2} - \frac {1}{2} \, {\left (2 \, f p q {\left (\frac {2 \, e^{2} \log \left (f x + e\right )}{f^{3}} + \frac {f x^{2} - 2 \, e x}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - \frac {{\left (f^{2} x^{2} + 2 \, e^{2} \log \left (f x + e\right )^{2} - 6 \, e f x + 6 \, e^{2} \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{2}}\right )} b^{2} g h + \frac {1}{54} \, {\left (6 \, f p q {\left (\frac {6 \, e^{3} \log \left (f x + e\right )}{f^{4}} - \frac {2 \, f^{2} x^{3} - 3 \, e f x^{2} + 6 \, e^{2} x}{f^{3}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac {{\left (4 \, f^{3} x^{3} - 15 \, e f^{2} x^{2} - 18 \, e^{3} \log \left (f x + e\right )^{2} + 66 \, e^{2} f x - 66 \, e^{3} \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{3}}\right )} b^{2} h^{2} + a^{2} g^{2} x \]

[In]

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

[Out]

1/3*b^2*h^2*x^3*log(((f*x + e)^p*d)^q*c)^2 - 2*a*b*f*g^2*p*q*(x/f - e*log(f*x + e)/f^2) + 1/9*a*b*f*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) - a*b*f*g*h*p*q*(2*e^2*log(f*x + e)/f^3 + (f*x^
2 - 2*e*x)/f^2) + 2/3*a*b*h^2*x^3*log(((f*x + e)^p*d)^q*c) + b^2*g*h*x^2*log(((f*x + e)^p*d)^q*c)^2 + 1/3*a^2*
h^2*x^3 + 2*a*b*g*h*x^2*log(((f*x + e)^p*d)^q*c) + b^2*g^2*x*log(((f*x + e)^p*d)^q*c)^2 + a^2*g*h*x^2 + 2*a*b*
g^2*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^2 - 1/2*(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*h + 1/54*(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*h^2
+ a^2*g^2*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2106 vs. \(2 (311) = 622\).

Time = 0.34 (sec) , antiderivative size = 2106, normalized size of antiderivative = 6.52 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

(f*x + e)*b^2*g^2*p^2*q^2*log(f*x + e)^2/f + (f*x + e)^2*b^2*g*h*p^2*q^2*log(f*x + e)^2/f^2 - 2*(f*x + e)*b^2*
e*g*h*p^2*q^2*log(f*x + e)^2/f^2 + 1/3*(f*x + e)^3*b^2*h^2*p^2*q^2*log(f*x + e)^2/f^3 - (f*x + e)^2*b^2*e*h^2*
p^2*q^2*log(f*x + e)^2/f^3 + (f*x + e)*b^2*e^2*h^2*p^2*q^2*log(f*x + e)^2/f^3 - 2*(f*x + e)*b^2*g^2*p^2*q^2*lo
g(f*x + e)/f - (f*x + e)^2*b^2*g*h*p^2*q^2*log(f*x + e)/f^2 + 4*(f*x + e)*b^2*e*g*h*p^2*q^2*log(f*x + e)/f^2 -
2/9*(f*x + e)^3*b^2*h^2*p^2*q^2*log(f*x + e)/f^3 + (f*x + e)^2*b^2*e*h^2*p^2*q^2*log(f*x + e)/f^3 - 2*(f*x +
e)*b^2*e^2*h^2*p^2*q^2*log(f*x + e)/f^3 + 2*(f*x + e)*b^2*g^2*p*q^2*log(f*x + e)*log(d)/f + 2*(f*x + e)^2*b^2*
g*h*p*q^2*log(f*x + e)*log(d)/f^2 - 4*(f*x + e)*b^2*e*g*h*p*q^2*log(f*x + e)*log(d)/f^2 + 2/3*(f*x + e)^3*b^2*
h^2*p*q^2*log(f*x + e)*log(d)/f^3 - 2*(f*x + e)^2*b^2*e*h^2*p*q^2*log(f*x + e)*log(d)/f^3 + 2*(f*x + e)*b^2*e^
2*h^2*p*q^2*log(f*x + e)*log(d)/f^3 + 2*(f*x + e)*b^2*g^2*p^2*q^2/f + 1/2*(f*x + e)^2*b^2*g*h*p^2*q^2/f^2 - 4*
(f*x + e)*b^2*e*g*h*p^2*q^2/f^2 + 2/27*(f*x + e)^3*b^2*h^2*p^2*q^2/f^3 - 1/2*(f*x + e)^2*b^2*e*h^2*p^2*q^2/f^3
+ 2*(f*x + e)*b^2*e^2*h^2*p^2*q^2/f^3 + 2*(f*x + e)*b^2*g^2*p*q*log(f*x + e)*log(c)/f + 2*(f*x + e)^2*b^2*g*h
*p*q*log(f*x + e)*log(c)/f^2 - 4*(f*x + e)*b^2*e*g*h*p*q*log(f*x + e)*log(c)/f^2 + 2/3*(f*x + e)^3*b^2*h^2*p*q
*log(f*x + e)*log(c)/f^3 - 2*(f*x + e)^2*b^2*e*h^2*p*q*log(f*x + e)*log(c)/f^3 + 2*(f*x + e)*b^2*e^2*h^2*p*q*l
og(f*x + e)*log(c)/f^3 - 2*(f*x + e)*b^2*g^2*p*q^2*log(d)/f - (f*x + e)^2*b^2*g*h*p*q^2*log(d)/f^2 + 4*(f*x +
e)*b^2*e*g*h*p*q^2*log(d)/f^2 - 2/9*(f*x + e)^3*b^2*h^2*p*q^2*log(d)/f^3 + (f*x + e)^2*b^2*e*h^2*p*q^2*log(d)/
f^3 - 2*(f*x + e)*b^2*e^2*h^2*p*q^2*log(d)/f^3 + (f*x + e)*b^2*g^2*q^2*log(d)^2/f + (f*x + e)^2*b^2*g*h*q^2*lo
g(d)^2/f^2 - 2*(f*x + e)*b^2*e*g*h*q^2*log(d)^2/f^2 + 1/3*(f*x + e)^3*b^2*h^2*q^2*log(d)^2/f^3 - (f*x + e)^2*b
^2*e*h^2*q^2*log(d)^2/f^3 + (f*x + e)*b^2*e^2*h^2*q^2*log(d)^2/f^3 + 2*(f*x + e)*a*b*g^2*p*q*log(f*x + e)/f +
2*(f*x + e)^2*a*b*g*h*p*q*log(f*x + e)/f^2 - 4*(f*x + e)*a*b*e*g*h*p*q*log(f*x + e)/f^2 + 2/3*(f*x + e)^3*a*b*
h^2*p*q*log(f*x + e)/f^3 - 2*(f*x + e)^2*a*b*e*h^2*p*q*log(f*x + e)/f^3 + 2*(f*x + e)*a*b*e^2*h^2*p*q*log(f*x
+ e)/f^3 - 2*(f*x + e)*b^2*g^2*p*q*log(c)/f - (f*x + e)^2*b^2*g*h*p*q*log(c)/f^2 + 4*(f*x + e)*b^2*e*g*h*p*q*l
og(c)/f^2 - 2/9*(f*x + e)^3*b^2*h^2*p*q*log(c)/f^3 + (f*x + e)^2*b^2*e*h^2*p*q*log(c)/f^3 - 2*(f*x + e)*b^2*e^
2*h^2*p*q*log(c)/f^3 + 2*(f*x + e)*b^2*g^2*q*log(c)*log(d)/f + 2*(f*x + e)^2*b^2*g*h*q*log(c)*log(d)/f^2 - 4*(
f*x + e)*b^2*e*g*h*q*log(c)*log(d)/f^2 + 2/3*(f*x + e)^3*b^2*h^2*q*log(c)*log(d)/f^3 - 2*(f*x + e)^2*b^2*e*h^2
*q*log(c)*log(d)/f^3 + 2*(f*x + e)*b^2*e^2*h^2*q*log(c)*log(d)/f^3 - 2*(f*x + e)*a*b*g^2*p*q/f - (f*x + e)^2*a
*b*g*h*p*q/f^2 + 4*(f*x + e)*a*b*e*g*h*p*q/f^2 - 2/9*(f*x + e)^3*a*b*h^2*p*q/f^3 + (f*x + e)^2*a*b*e*h^2*p*q/f
^3 - 2*(f*x + e)*a*b*e^2*h^2*p*q/f^3 + (f*x + e)*b^2*g^2*log(c)^2/f + (f*x + e)^2*b^2*g*h*log(c)^2/f^2 - 2*(f*
x + e)*b^2*e*g*h*log(c)^2/f^2 + 1/3*(f*x + e)^3*b^2*h^2*log(c)^2/f^3 - (f*x + e)^2*b^2*e*h^2*log(c)^2/f^3 + (f
*x + e)*b^2*e^2*h^2*log(c)^2/f^3 + 2*(f*x + e)*a*b*g^2*q*log(d)/f + 2*(f*x + e)^2*a*b*g*h*q*log(d)/f^2 - 4*(f*
x + e)*a*b*e*g*h*q*log(d)/f^2 + 2/3*(f*x + e)^3*a*b*h^2*q*log(d)/f^3 - 2*(f*x + e)^2*a*b*e*h^2*q*log(d)/f^3 +
2*(f*x + e)*a*b*e^2*h^2*q*log(d)/f^3 + 2*(f*x + e)*a*b*g^2*log(c)/f + 2*(f*x + e)^2*a*b*g*h*log(c)/f^2 - 4*(f*
x + e)*a*b*e*g*h*log(c)/f^2 + 2/3*(f*x + e)^3*a*b*h^2*log(c)/f^3 - 2*(f*x + e)^2*a*b*e*h^2*log(c)/f^3 + 2*(f*x
+ e)*a*b*e^2*h^2*log(c)/f^3 + (f*x + e)*a^2*g^2/f + (f*x + e)^2*a^2*g*h/f^2 - 2*(f*x + e)*a^2*e*g*h/f^2 + 1/3
*(f*x + e)^3*a^2*h^2/f^3 - (f*x + e)^2*a^2*e*h^2/f^3 + (f*x + e)*a^2*e^2*h^2/f^3

Mupad [B] (verification not implemented)

Time = 1.74 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.02 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx={\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (b^2\,g^2\,x+\frac {b^2\,h^2\,x^3}{3}+\frac {e\,\left (b^2\,e^2\,h^2-3\,b^2\,e\,f\,g\,h+3\,b^2\,f^2\,g^2\right )}{3\,f^3}+b^2\,g\,h\,x^2\right )+\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (\frac {x^2\,\left (\frac {3\,b\,h\,\left (a\,e\,h+2\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {b\,e\,h^2\,\left (3\,a-b\,p\,q\right )}{f}\right )}{3}-\frac {x\,\left (\frac {e\,\left (\frac {6\,b\,h\,\left (a\,e\,h+2\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {2\,b\,e\,h^2\,\left (3\,a-b\,p\,q\right )}{f}\right )}{f}-\frac {6\,b\,g\,\left (2\,a\,e\,h+a\,f\,g-b\,f\,g\,p\,q\right )}{f}\right )}{3}+\frac {2\,b\,h^2\,x^3\,\left (3\,a-b\,p\,q\right )}{9}\right )+x\,\left (\frac {18\,a^2\,e\,f\,g\,h+9\,a^2\,f^2\,g^2-18\,a\,b\,f^2\,g^2\,p\,q+6\,b^2\,e^2\,h^2\,p^2\,q^2-18\,b^2\,e\,f\,g\,h\,p^2\,q^2+18\,b^2\,f^2\,g^2\,p^2\,q^2}{9\,f^2}-\frac {e\,\left (\frac {h\,\left (3\,a^2\,e\,h+6\,a^2\,f\,g-b^2\,e\,h\,p^2\,q^2+3\,b^2\,f\,g\,p^2\,q^2-6\,a\,b\,f\,g\,p\,q\right )}{3\,f}-\frac {e\,h^2\,\left (9\,a^2-6\,a\,b\,p\,q+2\,b^2\,p^2\,q^2\right )}{9\,f}\right )}{f}\right )+x^2\,\left (\frac {h\,\left (3\,a^2\,e\,h+6\,a^2\,f\,g-b^2\,e\,h\,p^2\,q^2+3\,b^2\,f\,g\,p^2\,q^2-6\,a\,b\,f\,g\,p\,q\right )}{6\,f}-\frac {e\,h^2\,\left (9\,a^2-6\,a\,b\,p\,q+2\,b^2\,p^2\,q^2\right )}{18\,f}\right )-\frac {\ln \left (e+f\,x\right )\,\left (11\,b^2\,e^3\,h^2\,p^2\,q^2-27\,b^2\,e^2\,f\,g\,h\,p^2\,q^2+18\,b^2\,e\,f^2\,g^2\,p^2\,q^2-6\,a\,b\,e^3\,h^2\,p\,q+18\,a\,b\,e^2\,f\,g\,h\,p\,q-18\,a\,b\,e\,f^2\,g^2\,p\,q\right )}{9\,f^3}+\frac {h^2\,x^3\,\left (9\,a^2-6\,a\,b\,p\,q+2\,b^2\,p^2\,q^2\right )}{27} \]

[In]

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

[Out]

log(c*(d*(e + f*x)^p)^q)^2*(b^2*g^2*x + (b^2*h^2*x^3)/3 + (e*(b^2*e^2*h^2 + 3*b^2*f^2*g^2 - 3*b^2*e*f*g*h))/(3
*f^3) + b^2*g*h*x^2) + log(c*(d*(e + f*x)^p)^q)*((x^2*((3*b*h*(a*e*h + 2*a*f*g - b*f*g*p*q))/f - (b*e*h^2*(3*a
- b*p*q))/f))/3 - (x*((e*((6*b*h*(a*e*h + 2*a*f*g - b*f*g*p*q))/f - (2*b*e*h^2*(3*a - b*p*q))/f))/f - (6*b*g*
(2*a*e*h + a*f*g - b*f*g*p*q))/f))/3 + (2*b*h^2*x^3*(3*a - b*p*q))/9) + x*((9*a^2*f^2*g^2 + 6*b^2*e^2*h^2*p^2*
q^2 + 18*b^2*f^2*g^2*p^2*q^2 + 18*a^2*e*f*g*h - 18*a*b*f^2*g^2*p*q - 18*b^2*e*f*g*h*p^2*q^2)/(9*f^2) - (e*((h*
(3*a^2*e*h + 6*a^2*f*g - b^2*e*h*p^2*q^2 + 3*b^2*f*g*p^2*q^2 - 6*a*b*f*g*p*q))/(3*f) - (e*h^2*(9*a^2 + 2*b^2*p
^2*q^2 - 6*a*b*p*q))/(9*f)))/f) + x^2*((h*(3*a^2*e*h + 6*a^2*f*g - b^2*e*h*p^2*q^2 + 3*b^2*f*g*p^2*q^2 - 6*a*b
*f*g*p*q))/(6*f) - (e*h^2*(9*a^2 + 2*b^2*p^2*q^2 - 6*a*b*p*q))/(18*f)) - (log(e + f*x)*(11*b^2*e^3*h^2*p^2*q^2
- 6*a*b*e^3*h^2*p*q + 18*b^2*e*f^2*g^2*p^2*q^2 - 27*b^2*e^2*f*g*h*p^2*q^2 - 18*a*b*e*f^2*g^2*p*q + 18*a*b*e^2
*f*g*h*p*q))/(9*f^3) + (h^2*x^3*(9*a^2 + 2*b^2*p^2*q^2 - 6*a*b*p*q))/27