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

Optimal. Leaf size=323 \[ -\frac{2 b p q (f g-e h)^3 \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^3 h}-\frac{2 b p q (e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac{b h p q (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^3}-\frac{2 b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}+\frac{(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac{2 b^2 p^2 q^2 x (f g-e h)^2}{f^2}+\frac{b^2 h p^2 q^2 (e+f x)^2 (f g-e h)}{2 f^3}+\frac{b^2 p^2 q^2 (f g-e h)^3 \log ^2(e+f x)}{3 f^3 h}+\frac{2 b^2 h^2 p^2 q^2 (e+f x)^3}{27 f^3} \]

[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)

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Rubi [A]  time = 0.834571, antiderivative size = 264, normalized size of antiderivative = 0.82, number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2398, 2411, 43, 2334, 12, 14, 2301, 2445} \[ -\frac{b p q \left (\frac{9 h^2 (e+f x)^2 (f g-e h)}{f^3}+\frac{18 h (e+f x) (f g-e h)^2}{f^3}+\frac{6 (f g-e h)^3 \log (e+f x)}{f^3}+\frac{2 h^3 (e+f x)^3}{f^3}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac{(g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac{2 b^2 p^2 q^2 x (f g-e h)^2}{f^2}+\frac{b^2 h p^2 q^2 (e+f x)^2 (f g-e h)}{2 f^3}+\frac{b^2 p^2 q^2 (f g-e h)^3 \log ^2(e+f x)}{3 f^3 h}+\frac{2 b^2 h^2 p^2 q^2 (e+f x)^3}{27 f^3} \]

Antiderivative was successfully verified.

[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) - (b*p*q*((18*h*(f*g - e*h)^2*(e + f
*x))/f^3 + (9*h^2*(f*g - e*h)*(e + f*x)^2)/f^3 + (2*h^3*(e + f*x)^3)/f^3 + (6*(f*g - e*h)^3*Log[e + f*x])/f^3)
*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(9*h) + ((g + h*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*h)

Rule 2398

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 2411

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 43

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 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 2301

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 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx &=\operatorname{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}-\operatorname{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}-\operatorname{Subst}\left (\frac{(2 b p q) \operatorname{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{b p q \left (\frac{18 h (f g-e h)^2 (e+f x)}{f^3}+\frac{9 h^2 (f g-e h) (e+f x)^2}{f^3}+\frac{2 h^3 (e+f x)^3}{f^3}+\frac{6 (f g-e h)^3 \log (e+f x)}{f^3}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 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}+\operatorname{Subst}\left (\frac{\left (2 b^2 p^2 q^2\right ) \operatorname{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{b p q \left (\frac{18 h (f g-e h)^2 (e+f x)}{f^3}+\frac{9 h^2 (f g-e h) (e+f x)^2}{f^3}+\frac{2 h^3 (e+f x)^3}{f^3}+\frac{6 (f g-e h)^3 \log (e+f x)}{f^3}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 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}+\operatorname{Subst}\left (\frac{\left (b^2 p^2 q^2\right ) \operatorname{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{b p q \left (\frac{18 h (f g-e h)^2 (e+f x)}{f^3}+\frac{9 h^2 (f g-e h) (e+f x)^2}{f^3}+\frac{2 h^3 (e+f x)^3}{f^3}+\frac{6 (f g-e h)^3 \log (e+f x)}{f^3}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 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}+\operatorname{Subst}\left (\frac{\left (b^2 p^2 q^2\right ) \operatorname{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{b p q \left (\frac{18 h (f g-e h)^2 (e+f x)}{f^3}+\frac{9 h^2 (f g-e h) (e+f x)^2}{f^3}+\frac{2 h^3 (e+f x)^3}{f^3}+\frac{6 (f g-e h)^3 \log (e+f x)}{f^3}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 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}+\operatorname{Subst}\left (\frac{\left (b^2 p^2 q^2\right ) \operatorname{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 )+\operatorname{Subst}\left (\frac{\left (2 b^2 (f g-e h)^3 p^2 q^2\right ) \operatorname{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{b p q \left (\frac{18 h (f g-e h)^2 (e+f x)}{f^3}+\frac{9 h^2 (f g-e h) (e+f x)^2}{f^3}+\frac{2 h^3 (e+f x)^3}{f^3}+\frac{6 (f g-e h)^3 \log (e+f x)}{f^3}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 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]  time = 0.171716, size = 277, normalized size = 0.86 \[ \frac{4 b h^2 p q \left (b f p q x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+54 h (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+54 (e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-108 b p q (f g-e h)^2 \left (f x (a-b p q)+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+27 b h p q (f g-e h) \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+18 h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{54 f^3} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.5, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{2} \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11124, size = 817, normalized size = 2.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 2.24703, size = 2348, normalized size = 7.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 31.5494, size = 1692, normalized size = 5.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Giac [B]  time = 1.40241, size = 3025, normalized size = 9.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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