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

Optimal. Leaf size=409 \[ \frac {2 b^2 (f g-e h)^3 p^2 q^2 x}{f^3}+\frac {3 b^2 h (f g-e h)^2 p^2 q^2 (e+f x)^2}{4 f^4}+\frac {2 b^2 h^2 (f g-e h) p^2 q^2 (e+f x)^3}{9 f^4}+\frac {b^2 h^3 p^2 q^2 (e+f x)^4}{32 f^4}+\frac {b^2 (f g-e h)^4 p^2 q^2 \log ^2(e+f x)}{4 f^4 h}-\frac {2 b (f g-e h)^3 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^4}-\frac {3 b h (f g-e h)^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4}-\frac {2 b h^2 (f g-e h) p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^4}-\frac {b h^3 p q (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{8 f^4}-\frac {b (f g-e h)^4 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h} \]

[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]
time = 0.71, antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2445, 2458, 45, 2372, 12, 2338, 2495} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx &=\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 {b p q \left (\frac {48 h (f g-e h)^3 (e+f x)}{f^4}+\frac {36 h^2 (f g-e h)^2 (e+f x)^2}{f^4}+\frac {16 h^3 (f g-e h) (e+f x)^3}{f^4}+\frac {3 h^4 (e+f x)^4}{f^4}+\frac {12 (f g-e h)^4 \log (e+f x)}{f^4}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{24 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 {b p q \left (\frac {48 h (f g-e h)^3 (e+f x)}{f^4}+\frac {36 h^2 (f g-e h)^2 (e+f x)^2}{f^4}+\frac {16 h^3 (f g-e h) (e+f x)^3}{f^4}+\frac {3 h^4 (e+f x)^4}{f^4}+\frac {12 (f g-e h)^4 \log (e+f x)}{f^4}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{24 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 {b p q \left (\frac {48 h (f g-e h)^3 (e+f x)}{f^4}+\frac {36 h^2 (f g-e h)^2 (e+f x)^2}{f^4}+\frac {16 h^3 (f g-e h) (e+f x)^3}{f^4}+\frac {3 h^4 (e+f x)^4}{f^4}+\frac {12 (f g-e h)^4 \log (e+f x)}{f^4}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{24 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 {b p q \left (\frac {48 h (f g-e h)^3 (e+f x)}{f^4}+\frac {36 h^2 (f g-e h)^2 (e+f x)^2}{f^4}+\frac {16 h^3 (f g-e h) (e+f x)^3}{f^4}+\frac {3 h^4 (e+f x)^4}{f^4}+\frac {12 (f g-e h)^4 \log (e+f x)}{f^4}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{24 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]
time = 0.47, size = 634, normalized size = 1.55 \begin {gather*} \frac {72 b^2 e \left (-4 f^3 g^3+6 e f^2 g^2 h-4 e^2 f g h^2+e^3 h^3\right ) p^2 q^2 \log ^2(e+f x)-12 b e p q \log (e+f x) \left (-12 a \left (4 f^3 g^3-6 e f^2 g^2 h+4 e^2 f g h^2-e^3 h^3\right )+b \left (48 f^3 g^3-108 e f^2 g^2 h+88 e^2 f g h^2-25 e^3 h^3\right ) p q-12 b \left (4 f^3 g^3-6 e f^2 g^2 h+4 e^2 f g h^2-e^3 h^3\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+f x \left (72 a^2 f^3 \left (4 g^3+6 g^2 h x+4 g h^2 x^2+h^3 x^3\right )-12 a b p q \left (-12 e^3 h^3+6 e^2 f h^2 (8 g+h x)-4 e f^2 h \left (18 g^2+6 g h x+h^2 x^2\right )+f^3 \left (48 g^3+36 g^2 h x+16 g h^2 x^2+3 h^3 x^3\right )\right )+b^2 p^2 q^2 \left (-300 e^3 h^3+6 e^2 f h^2 (176 g+13 h x)-4 e f^2 h \left (324 g^2+60 g h x+7 h^2 x^2\right )+f^3 \left (576 g^3+216 g^2 h x+64 g h^2 x^2+9 h^3 x^3\right )\right )+12 b \left (12 a f^3 \left (4 g^3+6 g^2 h x+4 g h^2 x^2+h^3 x^3\right )-b p q \left (-12 e^3 h^3+6 e^2 f h^2 (8 g+h x)-4 e f^2 h \left (18 g^2+6 g h x+h^2 x^2\right )+f^3 \left (48 g^3+36 g^2 h x+16 g h^2 x^2+3 h^3 x^3\right )\right )\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+72 b^2 f^3 \left (4 g^3+6 g^2 h x+4 g h^2 x^2+h^3 x^3\right ) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )\right )}{288 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(72*b^2*e*(-4*f^3*g^3 + 6*e*f^2*g^2*h - 4*e^2*f*g*h^2 + e^3*h^3)*p^2*q^2*Log[e + f*x]^2 - 12*b*e*p*q*Log[e + f
*x]*(-12*a*(4*f^3*g^3 - 6*e*f^2*g^2*h + 4*e^2*f*g*h^2 - e^3*h^3) + b*(48*f^3*g^3 - 108*e*f^2*g^2*h + 88*e^2*f*
g*h^2 - 25*e^3*h^3)*p*q - 12*b*(4*f^3*g^3 - 6*e*f^2*g^2*h + 4*e^2*f*g*h^2 - e^3*h^3)*Log[c*(d*(e + f*x)^p)^q])
+ f*x*(72*a^2*f^3*(4*g^3 + 6*g^2*h*x + 4*g*h^2*x^2 + h^3*x^3) - 12*a*b*p*q*(-12*e^3*h^3 + 6*e^2*f*h^2*(8*g +
h*x) - 4*e*f^2*h*(18*g^2 + 6*g*h*x + h^2*x^2) + f^3*(48*g^3 + 36*g^2*h*x + 16*g*h^2*x^2 + 3*h^3*x^3)) + b^2*p^
2*q^2*(-300*e^3*h^3 + 6*e^2*f*h^2*(176*g + 13*h*x) - 4*e*f^2*h*(324*g^2 + 60*g*h*x + 7*h^2*x^2) + f^3*(576*g^3
+ 216*g^2*h*x + 64*g*h^2*x^2 + 9*h^3*x^3)) + 12*b*(12*a*f^3*(4*g^3 + 6*g^2*h*x + 4*g*h^2*x^2 + h^3*x^3) - b*p
*q*(-12*e^3*h^3 + 6*e^2*f*h^2*(8*g + h*x) - 4*e*f^2*h*(18*g^2 + 6*g*h*x + h^2*x^2) + f^3*(48*g^3 + 36*g^2*h*x
+ 16*g*h^2*x^2 + 3*h^3*x^3)))*Log[c*(d*(e + f*x)^p)^q] + 72*b^2*f^3*(4*g^3 + 6*g^2*h*x + 4*g*h^2*x^2 + h^3*x^3
)*Log[c*(d*(e + f*x)^p)^q]^2))/(288*f^4)

________________________________________________________________________________________

Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \left (h x +g \right )^{3} \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 914 vs. \(2 (414) = 828\).
time = 0.31, size = 914, normalized size = 2.23 \begin {gather*} \frac {1}{4} \, b^{2} h^{3} x^{4} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + \frac {1}{2} \, a b h^{3} x^{4} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + b^{2} g h^{2} x^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + \frac {1}{4} \, a^{2} h^{3} x^{4} - 2 \, a b f g^{3} p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} - \frac {3}{2} \, a b f g^{2} h p q {\left (\frac {f x^{2} - 2 \, x e}{f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{f^{3}}\right )} - \frac {1}{3} \, a b f g h^{2} p q {\left (\frac {2 \, f^{2} x^{3} - 3 \, f x^{2} e + 6 \, x e^{2}}{f^{3}} - \frac {6 \, e^{3} \log \left (f x + e\right )}{f^{4}}\right )} - \frac {1}{24} \, a b f h^{3} p q {\left (\frac {3 \, f^{3} x^{4} - 4 \, f^{2} x^{3} e + 6 \, f x^{2} e^{2} - 12 \, x e^{3}}{f^{4}} + \frac {12 \, e^{4} \log \left (f x + e\right )}{f^{5}}\right )} + 2 \, a b g h^{2} x^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac {3}{2} \, b^{2} g^{2} h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + a^{2} g h^{2} x^{3} + 3 \, a b g^{2} h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + b^{2} g^{3} x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + \frac {3}{2} \, a^{2} g^{2} h x^{2} + 2 \, a b g^{3} 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^{3} - \frac {3}{4} \, {\left (2 \, f p q {\left (\frac {f x^{2} - 2 \, x e}{f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{f^{3}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - \frac {{\left (f^{2} x^{2} - 6 \, f x e + 2 \, e^{2} \log \left (f x + e\right )^{2} + 6 \, e^{2} \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{2}}\right )} b^{2} g^{2} h - \frac {1}{18} \, {\left (6 \, f p q {\left (\frac {2 \, f^{2} x^{3} - 3 \, f x^{2} e + 6 \, x e^{2}}{f^{3}} - \frac {6 \, e^{3} \log \left (f x + e\right )}{f^{4}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - \frac {{\left (4 \, f^{3} x^{3} - 15 \, f^{2} x^{2} e + 66 \, f x e^{2} - 18 \, e^{3} \log \left (f x + e\right )^{2} - 66 \, e^{3} \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{3}}\right )} b^{2} g h^{2} - \frac {1}{288} \, {\left (12 \, f p q {\left (\frac {3 \, f^{3} x^{4} - 4 \, f^{2} x^{3} e + 6 \, f x^{2} e^{2} - 12 \, x e^{3}}{f^{4}} + \frac {12 \, e^{4} \log \left (f x + e\right )}{f^{5}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - \frac {{\left (9 \, f^{4} x^{4} - 28 \, f^{3} x^{3} e + 78 \, f^{2} x^{2} e^{2} - 300 \, f x e^{3} + 72 \, e^{4} \log \left (f x + e\right )^{2} + 300 \, e^{4} \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{4}}\right )} b^{2} h^{3} + a^{2} g^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1892 vs. \(2 (414) = 828\).
time = 0.39, size = 1892, normalized size = 4.63 \begin {gather*} \frac {9 \, {\left (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}\right )} x^{4} + 32 \, {\left (2 \, b^{2} f^{4} g h^{2} p^{2} q^{2} - 6 \, a b f^{4} g h^{2} p q + 9 \, a^{2} f^{4} g h^{2}\right )} x^{3} + 216 \, {\left (b^{2} f^{4} g^{2} h p^{2} q^{2} - 2 \, a b f^{4} g^{2} h p q + 2 \, a^{2} f^{4} g^{2} h\right )} x^{2} - 12 \, {\left (25 \, b^{2} f h^{3} p^{2} q^{2} - 12 \, a b f h^{3} p q\right )} x e^{3} + 72 \, {\left (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} f^{3} g^{3} p^{2} q^{2} e - 6 \, b^{2} f^{2} g^{2} h p^{2} q^{2} e^{2} + 4 \, b^{2} f g h^{2} p^{2} q^{2} e^{3} - b^{2} h^{3} p^{2} q^{2} e^{4}\right )} \log \left (f x + e\right )^{2} + 72 \, {\left (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\right )} \log \left (c\right )^{2} + 72 \, {\left (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\right )} \log \left (d\right )^{2} + 288 \, {\left (2 \, b^{2} f^{4} g^{3} p^{2} q^{2} - 2 \, a b f^{4} g^{3} p q + a^{2} f^{4} g^{3}\right )} x + 6 \, {\left ({\left (13 \, b^{2} f^{2} h^{3} p^{2} q^{2} - 12 \, a b f^{2} h^{3} p q\right )} x^{2} + 16 \, {\left (11 \, b^{2} f^{2} g h^{2} p^{2} q^{2} - 6 \, a b f^{2} g h^{2} p q\right )} x\right )} e^{2} - 4 \, {\left ({\left (7 \, b^{2} f^{3} h^{3} p^{2} q^{2} - 12 \, a b f^{3} h^{3} p q\right )} x^{3} + 12 \, {\left (5 \, b^{2} f^{3} g h^{2} p^{2} q^{2} - 6 \, a b f^{3} g h^{2} p q\right )} x^{2} + 108 \, {\left (3 \, b^{2} f^{3} g^{2} h p^{2} q^{2} - 2 \, a b f^{3} g^{2} h p q\right )} x\right )} e - 12 \, {\left (3 \, {\left (b^{2} f^{4} h^{3} p^{2} q^{2} - 4 \, a b f^{4} h^{3} p q\right )} x^{4} + 16 \, {\left (b^{2} f^{4} g h^{2} p^{2} q^{2} - 3 \, a b f^{4} g h^{2} p q\right )} x^{3} + 36 \, {\left (b^{2} f^{4} g^{2} h p^{2} q^{2} - 2 \, a b f^{4} g^{2} h p q\right )} x^{2} + 48 \, {\left (b^{2} f^{4} g^{3} p^{2} q^{2} - a b f^{4} g^{3} p q\right )} x - {\left (25 \, b^{2} h^{3} p^{2} q^{2} - 12 \, a b h^{3} p q\right )} e^{4} - 4 \, {\left (3 \, b^{2} f h^{3} p^{2} q^{2} x - 22 \, b^{2} f g h^{2} p^{2} q^{2} + 12 \, a b f g h^{2} p q\right )} e^{3} + 6 \, {\left (b^{2} f^{2} h^{3} p^{2} q^{2} x^{2} + 8 \, b^{2} f^{2} g h^{2} p^{2} q^{2} x - 18 \, b^{2} f^{2} g^{2} h p^{2} q^{2} + 12 \, a b f^{2} g^{2} h p q\right )} e^{2} - 4 \, {\left (b^{2} f^{3} h^{3} p^{2} q^{2} x^{3} + 6 \, b^{2} f^{3} g h^{2} p^{2} q^{2} x^{2} + 18 \, b^{2} f^{3} g^{2} h p^{2} q^{2} x - 12 \, b^{2} f^{3} g^{3} p^{2} q^{2} + 12 \, a b f^{3} g^{3} p q\right )} e - 12 \, {\left (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} f^{3} g^{3} p q e - 6 \, b^{2} f^{2} g^{2} h p q e^{2} + 4 \, b^{2} f g h^{2} p q e^{3} - b^{2} h^{3} p q e^{4}\right )} \log \left (c\right ) - 12 \, {\left (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} f^{3} g^{3} p q^{2} e - 6 \, b^{2} f^{2} g^{2} h p q^{2} e^{2} + 4 \, b^{2} f g h^{2} p q^{2} e^{3} - b^{2} h^{3} p q^{2} e^{4}\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 12 \, {\left (12 \, b^{2} f h^{3} p q x e^{3} - 3 \, {\left (b^{2} f^{4} h^{3} p q - 4 \, a b f^{4} h^{3}\right )} x^{4} - 16 \, {\left (b^{2} f^{4} g h^{2} p q - 3 \, a b f^{4} g h^{2}\right )} x^{3} - 36 \, {\left (b^{2} f^{4} g^{2} h p q - 2 \, a b f^{4} g^{2} h\right )} x^{2} - 48 \, {\left (b^{2} f^{4} g^{3} p q - a b f^{4} g^{3}\right )} x - 6 \, {\left (b^{2} f^{2} h^{3} p q x^{2} + 8 \, b^{2} f^{2} g h^{2} p q x\right )} e^{2} + 4 \, {\left (b^{2} f^{3} h^{3} p q x^{3} + 6 \, b^{2} f^{3} g h^{2} p q x^{2} + 18 \, b^{2} f^{3} g^{2} h p q x\right )} e\right )} \log \left (c\right ) + 12 \, {\left (12 \, b^{2} f h^{3} p q^{2} x e^{3} - 3 \, {\left (b^{2} f^{4} h^{3} p q^{2} - 4 \, a b f^{4} h^{3} q\right )} x^{4} - 16 \, {\left (b^{2} f^{4} g h^{2} p q^{2} - 3 \, a b f^{4} g h^{2} q\right )} x^{3} - 36 \, {\left (b^{2} f^{4} g^{2} h p q^{2} - 2 \, a b f^{4} g^{2} h q\right )} x^{2} - 48 \, {\left (b^{2} f^{4} g^{3} p q^{2} - a b f^{4} g^{3} q\right )} x - 6 \, {\left (b^{2} f^{2} h^{3} p q^{2} x^{2} + 8 \, b^{2} f^{2} g h^{2} p q^{2} x\right )} e^{2} + 4 \, {\left (b^{2} f^{3} h^{3} p q^{2} x^{3} + 6 \, b^{2} f^{3} g h^{2} p q^{2} x^{2} + 18 \, b^{2} f^{3} g^{2} h p q^{2} x\right )} e + 12 \, {\left (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\right )} \log \left (c\right )\right )} \log \left (d\right )}{288 \, f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + 32*(2*b^2*f^4*g*h^2*p^2*q^2 - 6*a*b*f
^4*g*h^2*p*q + 9*a^2*f^4*g*h^2)*x^3 + 216*(b^2*f^4*g^2*h*p^2*q^2 - 2*a*b*f^4*g^2*h*p*q + 2*a^2*f^4*g^2*h)*x^2
- 12*(25*b^2*f*h^3*p^2*q^2 - 12*a*b*f*h^3*p*q)*x*e^3 + 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*f^3*g^3*p^2*q^2*e - 6*b^2*f^2*g^2*h*p^2*q^2
*e^2 + 4*b^2*f*g*h^2*p^2*q^2*e^3 - b^2*h^3*p^2*q^2*e^4)*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 + 288*(2*b^2*f^4*g^3*p^2*q^2 - 2*a*b*f^4*g^3*p*q + a^2*
f^4*g^3)*x + 6*((13*b^2*f^2*h^3*p^2*q^2 - 12*a*b*f^2*h^3*p*q)*x^2 + 16*(11*b^2*f^2*g*h^2*p^2*q^2 - 6*a*b*f^2*g
*h^2*p*q)*x)*e^2 - 4*((7*b^2*f^3*h^3*p^2*q^2 - 12*a*b*f^3*h^3*p*q)*x^3 + 12*(5*b^2*f^3*g*h^2*p^2*q^2 - 6*a*b*f
^3*g*h^2*p*q)*x^2 + 108*(3*b^2*f^3*g^2*h*p^2*q^2 - 2*a*b*f^3*g^2*h*p*q)*x)*e - 12*(3*(b^2*f^4*h^3*p^2*q^2 - 4*
a*b*f^4*h^3*p*q)*x^4 + 16*(b^2*f^4*g*h^2*p^2*q^2 - 3*a*b*f^4*g*h^2*p*q)*x^3 + 36*(b^2*f^4*g^2*h*p^2*q^2 - 2*a*
b*f^4*g^2*h*p*q)*x^2 + 48*(b^2*f^4*g^3*p^2*q^2 - a*b*f^4*g^3*p*q)*x - (25*b^2*h^3*p^2*q^2 - 12*a*b*h^3*p*q)*e^
4 - 4*(3*b^2*f*h^3*p^2*q^2*x - 22*b^2*f*g*h^2*p^2*q^2 + 12*a*b*f*g*h^2*p*q)*e^3 + 6*(b^2*f^2*h^3*p^2*q^2*x^2 +
8*b^2*f^2*g*h^2*p^2*q^2*x - 18*b^2*f^2*g^2*h*p^2*q^2 + 12*a*b*f^2*g^2*h*p*q)*e^2 - 4*(b^2*f^3*h^3*p^2*q^2*x^3
+ 6*b^2*f^3*g*h^2*p^2*q^2*x^2 + 18*b^2*f^3*g^2*h*p^2*q^2*x - 12*b^2*f^3*g^3*p^2*q^2 + 12*a*b*f^3*g^3*p*q)*e -
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*f^3
*g^3*p*q*e - 6*b^2*f^2*g^2*h*p*q*e^2 + 4*b^2*f*g*h^2*p*q*e^3 - b^2*h^3*p*q*e^4)*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*f^3*g^3*p*q^2*e -
6*b^2*f^2*g^2*h*p*q^2*e^2 + 4*b^2*f*g*h^2*p*q^2*e^3 - b^2*h^3*p*q^2*e^4)*log(d))*log(f*x + e) + 12*(12*b^2*f*
h^3*p*q*x*e^3 - 3*(b^2*f^4*h^3*p*q - 4*a*b*f^4*h^3)*x^4 - 16*(b^2*f^4*g*h^2*p*q - 3*a*b*f^4*g*h^2)*x^3 - 36*(b
^2*f^4*g^2*h*p*q - 2*a*b*f^4*g^2*h)*x^2 - 48*(b^2*f^4*g^3*p*q - a*b*f^4*g^3)*x - 6*(b^2*f^2*h^3*p*q*x^2 + 8*b^
2*f^2*g*h^2*p*q*x)*e^2 + 4*(b^2*f^3*h^3*p*q*x^3 + 6*b^2*f^3*g*h^2*p*q*x^2 + 18*b^2*f^3*g^2*h*p*q*x)*e)*log(c)
+ 12*(12*b^2*f*h^3*p*q^2*x*e^3 - 3*(b^2*f^4*h^3*p*q^2 - 4*a*b*f^4*h^3*q)*x^4 - 16*(b^2*f^4*g*h^2*p*q^2 - 3*a*b
*f^4*g*h^2*q)*x^3 - 36*(b^2*f^4*g^2*h*p*q^2 - 2*a*b*f^4*g^2*h*q)*x^2 - 48*(b^2*f^4*g^3*p*q^2 - a*b*f^4*g^3*q)*
x - 6*(b^2*f^2*h^3*p*q^2*x^2 + 8*b^2*f^2*g*h^2*p*q^2*x)*e^2 + 4*(b^2*f^3*h^3*p*q^2*x^3 + 6*b^2*f^3*g*h^2*p*q^2
*x^2 + 18*b^2*f^3*g^2*h*p*q^2*x)*e + 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] Leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (394) = 788\).
time = 6.70, size = 1421, normalized size = 3.47 \begin {gather*} \begin {cases} a^{2} g^{3} x + \frac {3 a^{2} g^{2} h x^{2}}{2} + a^{2} g h^{2} x^{3} + \frac {a^{2} h^{3} x^{4}}{4} - \frac {a b e^{4} h^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2 f^{4}} + \frac {2 a b e^{3} g h^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f^{3}} + \frac {a b e^{3} h^{3} p q x}{2 f^{3}} - \frac {3 a b e^{2} g^{2} h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f^{2}} - \frac {2 a b e^{2} g h^{2} p q x}{f^{2}} - \frac {a b e^{2} h^{3} p q x^{2}}{4 f^{2}} + \frac {2 a b e g^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {3 a b e g^{2} h p q x}{f} + \frac {a b e g h^{2} p q x^{2}}{f} + \frac {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 {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {3 a b g^{2} h p q x^{2}}{2} + 3 a b g^{2} h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {2 a b g h^{2} p q x^{3}}{3} + 2 a b g h^{2} x^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {a b h^{3} p q x^{4}}{8} + \frac {a b h^{3} x^{4} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2} + \frac {25 b^{2} e^{4} h^{3} p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{24 f^{4}} - \frac {b^{2} e^{4} h^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{4 f^{4}} - \frac {11 b^{2} e^{3} g h^{2} p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{3 f^{3}} + \frac {b^{2} e^{3} g h^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f^{3}} - \frac {25 b^{2} e^{3} h^{3} p^{2} q^{2} x}{24 f^{3}} + \frac {b^{2} e^{3} h^{3} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2 f^{3}} + \frac {9 b^{2} e^{2} g^{2} h p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2 f^{2}} - \frac {3 b^{2} e^{2} g^{2} h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{2 f^{2}} + \frac {11 b^{2} e^{2} g h^{2} p^{2} q^{2} x}{3 f^{2}} - \frac {2 b^{2} e^{2} g h^{2} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f^{2}} + \frac {13 b^{2} e^{2} h^{3} p^{2} q^{2} x^{2}}{48 f^{2}} - \frac {b^{2} e^{2} h^{3} p q x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{4 f^{2}} - \frac {2 b^{2} e g^{3} p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {b^{2} e g^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} - \frac {9 b^{2} e g^{2} h p^{2} q^{2} x}{2 f} + \frac {3 b^{2} e g^{2} h p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - \frac {5 b^{2} e g h^{2} p^{2} q^{2} x^{2}}{6 f} + \frac {b^{2} e g h^{2} p q x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - \frac {7 b^{2} e h^{3} p^{2} q^{2} x^{3}}{72 f} + \frac {b^{2} e h^{3} p q x^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{6 f} + 2 b^{2} g^{3} p^{2} q^{2} x - 2 b^{2} g^{3} p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + b^{2} g^{3} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + \frac {3 b^{2} g^{2} h p^{2} q^{2} x^{2}}{4} - \frac {3 b^{2} g^{2} h p q x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2} + \frac {3 b^{2} g^{2} h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{2} + \frac {2 b^{2} g h^{2} p^{2} q^{2} x^{3}}{9} - \frac {2 b^{2} g h^{2} p q x^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{3} + b^{2} g h^{2} x^{3} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + \frac {b^{2} h^{3} p^{2} q^{2} x^{4}}{32} - \frac {b^{2} h^{3} p q x^{4} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{8} + \frac {b^{2} h^{3} x^{4} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{4} & \text {for}\: f \neq 0 \\\left (a + b \log {\left (c \left (d e^{p}\right )^{q} \right )}\right )^{2} \left (g^{3} x + \frac {3 g^{2} h x^{2}}{2} + g h^{2} x^{3} + \frac {h^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3938 vs. \(2 (414) = 828\).
time = 2.69, size = 3938, normalized size = 9.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Mupad [B]
time = 0.92, size = 1154, normalized size = 2.82 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________