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

Optimal. Leaf size=211 \[ \frac{(e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}-\frac{b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2}-\frac{2 a b p q x (f g-e h)}{f}-\frac{2 b^2 p q (e+f x) (f g-e h) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2}+\frac{b^2 h p^2 q^2 (e+f x)^2}{4 f^2}+\frac{2 b^2 p^2 q^2 x (f g-e h)}{f} \]

[Out]

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

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Rubi [A]  time = 0.389334, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2401, 2389, 2296, 2295, 2390, 2305, 2304, 2445} \[ \frac{(e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}-\frac{b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2}-\frac{2 a b p q x (f g-e h)}{f}-\frac{2 b^2 p q (e+f x) (f g-e h) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2}+\frac{b^2 h p^2 q^2 (e+f x)^2}{4 f^2}+\frac{2 b^2 p^2 q^2 x (f g-e h)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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

Mathematica [A]  time = 0.0880622, size = 164, normalized size = 0.78 \[ \frac{4 (e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-8 b p q (f g-e h) \left (f x (a-b p q)+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+2 h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+b h p q \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{4 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.285, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) \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)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2,x)

[Out]

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

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Maxima [A]  time = 1.16868, size = 470, normalized size = 2.23 \begin{align*} -2 \, a b f g p q{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} - \frac{1}{2} \, a b f 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{1}{2} \, b^{2} h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + a b h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + b^{2} g x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + \frac{1}{2} \, a^{2} h x^{2} + 2 \, a b g 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 - \frac{1}{4} \,{\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} h + a^{2} g x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.1887, size = 1320, normalized size = 6.26 \begin{align*} \frac{{\left (b^{2} f^{2} h p^{2} q^{2} - 2 \, a b f^{2} h p q + 2 \, a^{2} f^{2} h\right )} x^{2} + 2 \,{\left (b^{2} f^{2} h p^{2} q^{2} x^{2} + 2 \, b^{2} f^{2} g p^{2} q^{2} x +{\left (2 \, b^{2} e f g - b^{2} e^{2} h\right )} p^{2} q^{2}\right )} \log \left (f x + e\right )^{2} + 2 \,{\left (b^{2} f^{2} h x^{2} + 2 \, b^{2} f^{2} g x\right )} \log \left (c\right )^{2} + 2 \,{\left (b^{2} f^{2} h q^{2} x^{2} + 2 \, b^{2} f^{2} g q^{2} x\right )} \log \left (d\right )^{2} + 2 \,{\left (2 \, a^{2} f^{2} g +{\left (4 \, b^{2} f^{2} g - 3 \, b^{2} e f h\right )} p^{2} q^{2} - 2 \,{\left (2 \, a b f^{2} g - a b e f h\right )} p q\right )} x - 2 \,{\left ({\left (4 \, b^{2} e f g - 3 \, b^{2} e^{2} h\right )} p^{2} q^{2} - 2 \,{\left (2 \, a b e f g - a b e^{2} h\right )} p q +{\left (b^{2} f^{2} h p^{2} q^{2} - 2 \, a b f^{2} h p q\right )} x^{2} - 2 \,{\left (2 \, a b f^{2} g p q -{\left (2 \, b^{2} f^{2} g - b^{2} e f h\right )} p^{2} q^{2}\right )} x - 2 \,{\left (b^{2} f^{2} h p q x^{2} + 2 \, b^{2} f^{2} g p q x +{\left (2 \, b^{2} e f g - b^{2} e^{2} h\right )} p q\right )} \log \left (c\right ) - 2 \,{\left (b^{2} f^{2} h p q^{2} x^{2} + 2 \, b^{2} f^{2} g p q^{2} x +{\left (2 \, b^{2} e f g - b^{2} e^{2} h\right )} p q^{2}\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) - 2 \,{\left ({\left (b^{2} f^{2} h p q - 2 \, a b f^{2} h\right )} x^{2} - 2 \,{\left (2 \, a b f^{2} g -{\left (2 \, b^{2} f^{2} g - b^{2} e f h\right )} p q\right )} x\right )} \log \left (c\right ) - 2 \,{\left ({\left (b^{2} f^{2} h p q^{2} - 2 \, a b f^{2} h q\right )} x^{2} - 2 \,{\left (2 \, a b f^{2} g q -{\left (2 \, b^{2} f^{2} g - b^{2} e f h\right )} p q^{2}\right )} x - 2 \,{\left (b^{2} f^{2} h q x^{2} + 2 \, b^{2} f^{2} g q x\right )} \log \left (c\right )\right )} \log \left (d\right )}{4 \, f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 11.3038, size = 879, normalized size = 4.17 \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)*(a+b*ln(c*(d*(f*x+e)**p)**q))**2,x)

[Out]

Piecewise((a**2*g*x + a**2*h*x**2/2 - a*b*e**2*h*p*q*log(e + f*x)/f**2 + 2*a*b*e*g*p*q*log(e + f*x)/f + a*b*e*
h*p*q*x/f + 2*a*b*g*p*q*x*log(e + f*x) - 2*a*b*g*p*q*x + 2*a*b*g*q*x*log(d) + 2*a*b*g*x*log(c) + a*b*h*p*q*x**
2*log(e + f*x) - a*b*h*p*q*x**2/2 + a*b*h*q*x**2*log(d) + a*b*h*x**2*log(c) - b**2*e**2*h*p**2*q**2*log(e + f*
x)**2/(2*f**2) + 3*b**2*e**2*h*p**2*q**2*log(e + f*x)/(2*f**2) - b**2*e**2*h*p*q**2*log(d)*log(e + f*x)/f**2 -
b**2*e**2*h*p*q*log(c)*log(e + f*x)/f**2 + b**2*e*g*p**2*q**2*log(e + f*x)**2/f - 2*b**2*e*g*p**2*q**2*log(e
+ f*x)/f + 2*b**2*e*g*p*q**2*log(d)*log(e + f*x)/f + 2*b**2*e*g*p*q*log(c)*log(e + f*x)/f + b**2*e*h*p**2*q**2
*x*log(e + f*x)/f - 3*b**2*e*h*p**2*q**2*x/(2*f) + b**2*e*h*p*q**2*x*log(d)/f + b**2*e*h*p*q*x*log(c)/f + b**2
*g*p**2*q**2*x*log(e + f*x)**2 - 2*b**2*g*p**2*q**2*x*log(e + f*x) + 2*b**2*g*p**2*q**2*x + 2*b**2*g*p*q**2*x*
log(d)*log(e + f*x) - 2*b**2*g*p*q**2*x*log(d) + 2*b**2*g*p*q*x*log(c)*log(e + f*x) - 2*b**2*g*p*q*x*log(c) +
b**2*g*q**2*x*log(d)**2 + 2*b**2*g*q*x*log(c)*log(d) + b**2*g*x*log(c)**2 + b**2*h*p**2*q**2*x**2*log(e + f*x)
**2/2 - b**2*h*p**2*q**2*x**2*log(e + f*x)/2 + b**2*h*p**2*q**2*x**2/4 + b**2*h*p*q**2*x**2*log(d)*log(e + f*x
) - b**2*h*p*q**2*x**2*log(d)/2 + b**2*h*p*q*x**2*log(c)*log(e + f*x) - b**2*h*p*q*x**2*log(c)/2 + b**2*h*q**2
*x**2*log(d)**2/2 + b**2*h*q*x**2*log(c)*log(d) + b**2*h*x**2*log(c)**2/2, Ne(f, 0)), ((a + b*log(c*(d*e**p)**
q))**2*(g*x + h*x**2/2), True))

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Giac [B]  time = 1.29273, size = 1369, normalized size = 6.49 \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)*(a+b*log(c*(d*(f*x+e)^p)^q))^2,x, algorithm="giac")

[Out]

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