3.51 \(\int \frac{\log (e (f (a+b x)^p (c+d x)^q)^r) (s+t \log (i (g+h x)^n))^2}{g k+h k x} \, dx\)

Optimal. Leaf size=410 \[ -\frac{p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac{2 n p r t \text{PolyLog}\left (3,\frac{b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac{2 n^2 p r t^2 \text{PolyLog}\left (4,\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac{2 n q r t \text{PolyLog}\left (3,\frac{d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac{2 n^2 q r t^2 \text{PolyLog}\left (4,\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t} \]

[Out]

-(p*r*Log[-((h*(a + b*x))/(b*g - a*h))]*(s + t*Log[i*(g + h*x)^n])^3)/(3*h*k*n*t) - (q*r*Log[-((h*(c + d*x))/(
d*g - c*h))]*(s + t*Log[i*(g + h*x)^n])^3)/(3*h*k*n*t) + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g
+ h*x)^n])^3)/(3*h*k*n*t) - (p*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/(h*k) -
(q*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/(h*k) + (2*n*p*r*t*(s + t*Log[i*(g +
h*x)^n])*PolyLog[3, (b*(g + h*x))/(b*g - a*h)])/(h*k) + (2*n*q*r*t*(s + t*Log[i*(g + h*x)^n])*PolyLog[3, (d*(g
+ h*x))/(d*g - c*h)])/(h*k) - (2*n^2*p*r*t^2*PolyLog[4, (b*(g + h*x))/(b*g - a*h)])/(h*k) - (2*n^2*q*r*t^2*Po
lyLog[4, (d*(g + h*x))/(d*g - c*h)])/(h*k)

________________________________________________________________________________________

Rubi [A]  time = 0.471232, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2499, 2396, 2433, 2374, 2383, 6589} \[ -\frac{p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac{2 n p r t \text{PolyLog}\left (3,\frac{b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac{2 n^2 p r t^2 \text{PolyLog}\left (4,\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac{2 n q r t \text{PolyLog}\left (3,\frac{d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac{2 n^2 q r t^2 \text{PolyLog}\left (4,\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t} \]

Antiderivative was successfully verified.

[In]

Int[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g + h*x)^n])^2)/(g*k + h*k*x),x]

[Out]

-(p*r*Log[-((h*(a + b*x))/(b*g - a*h))]*(s + t*Log[i*(g + h*x)^n])^3)/(3*h*k*n*t) - (q*r*Log[-((h*(c + d*x))/(
d*g - c*h))]*(s + t*Log[i*(g + h*x)^n])^3)/(3*h*k*n*t) + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g
+ h*x)^n])^3)/(3*h*k*n*t) - (p*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/(h*k) -
(q*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/(h*k) + (2*n*p*r*t*(s + t*Log[i*(g +
h*x)^n])*PolyLog[3, (b*(g + h*x))/(b*g - a*h)])/(h*k) + (2*n*q*r*t*(s + t*Log[i*(g + h*x)^n])*PolyLog[3, (d*(g
+ h*x))/(d*g - c*h)])/(h*k) - (2*n^2*p*r*t^2*PolyLog[4, (b*(g + h*x))/(b*g - a*h)])/(h*k) - (2*n^2*q*r*t^2*Po
lyLog[4, (d*(g + h*x))/(d*g - c*h)])/(h*k)

Rule 2499

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.))^(m_.))/((j_.) + (k_.)*(x_)), x_Symbol] :> Simp[((s + t*Log[i*(g + h*x)^n])^(m + 1)
*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(k*n*t*(m + 1)), x] + (-Dist[(b*p*r)/(k*n*t*(m + 1)), Int[(s + t*Log[i*
(g + h*x)^n])^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(k*n*t*(m + 1)), Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)
/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] &
& EqQ[h*j - g*k, 0] && IGtQ[m, 0]

Rule 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

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

Rubi steps

\begin{align*} \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^2}{g k+h k x} \, dx &=\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{(b p r) \int \frac{\left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{a+b x} \, dx}{3 h k n t}-\frac{(d q r) \int \frac{\left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{c+d x} \, dx}{3 h k n t}\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac{(p r) \int \frac{\log \left (\frac{h (a+b x)}{-b g+a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^2}{g+h x} \, dx}{k}+\frac{(q r) \int \frac{\log \left (\frac{h (c+d x)}{-d g+c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^2}{g+h x} \, dx}{k}\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac{(p r) \operatorname{Subst}\left (\int \frac{\left (s+t \log \left (51 x^n\right )\right )^2 \log \left (\frac{h \left (\frac{-b g+a h}{h}+\frac{b x}{h}\right )}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}+\frac{(q r) \operatorname{Subst}\left (\int \frac{\left (s+t \log \left (51 x^n\right )\right )^2 \log \left (\frac{h \left (\frac{-d g+c h}{h}+\frac{d x}{h}\right )}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h k}\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{p r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{(2 n p r t) \operatorname{Subst}\left (\int \frac{\left (s+t \log \left (51 x^n\right )\right ) \text{Li}_2\left (-\frac{b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}+\frac{(2 n q r t) \operatorname{Subst}\left (\int \frac{\left (s+t \log \left (51 x^n\right )\right ) \text{Li}_2\left (-\frac{d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h k}\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{p r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{2 n p r t \left (s+t \log \left (51 (g+h x)^n\right )\right ) \text{Li}_3\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}+\frac{2 n q r t \left (s+t \log \left (51 (g+h x)^n\right )\right ) \text{Li}_3\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}-\frac{\left (2 n^2 p r t^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}-\frac{\left (2 n^2 q r t^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h k}\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}+\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (51 (g+h x)^n\right )\right )^3}{3 h k n t}-\frac{p r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \left (s+t \log \left (51 (g+h x)^n\right )\right )^2 \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{2 n p r t \left (s+t \log \left (51 (g+h x)^n\right )\right ) \text{Li}_3\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}+\frac{2 n q r t \left (s+t \log \left (51 (g+h x)^n\right )\right ) \text{Li}_3\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}-\frac{2 n^2 p r t^2 \text{Li}_4\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{2 n^2 q r t^2 \text{Li}_4\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}\\ \end{align*}

Mathematica [B]  time = 7.54479, size = 22595, normalized size = 55.11 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g + h*x)^n])^2)/(g*k + h*k*x),x]

[Out]

Result too large to show

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Maple [F]  time = 0.982, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \left ( s+t\ln \left ( i \left ( hx+g \right ) ^{n} \right ) \right ) ^{2}}{hkx+gk}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*ln(i*(h*x+g)^n))^2/(h*k*x+g*k),x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*ln(i*(h*x+g)^n))^2/(h*k*x+g*k),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*log(i*(h*x+g)^n))^2/(h*k*x+g*k),x, algorithm="maxima")

[Out]

1/3*((n^2*t^2*log(h*x + g)^3 + 3*t^2*log(h*x + g)*log((h*x + g)^n)^2 - 3*(n*t^2*log(i) + n*s*t)*log(h*x + g)^2
+ 3*(t^2*log(i)^2 + 2*s*t*log(i) + s^2)*log(h*x + g) - 3*(n*t^2*log(h*x + g)^2 - 2*(t^2*log(i) + s*t)*log(h*x
+ g))*log((h*x + g)^n))*log(((b*x + a)^p)^r) + (n^2*t^2*log(h*x + g)^3 + 3*t^2*log(h*x + g)*log((h*x + g)^n)^
2 - 3*(n*t^2*log(i) + n*s*t)*log(h*x + g)^2 + 3*(t^2*log(i)^2 + 2*s*t*log(i) + s^2)*log(h*x + g) - 3*(n*t^2*lo
g(h*x + g)^2 - 2*(t^2*log(i) + s*t)*log(h*x + g))*log((h*x + g)^n))*log(((d*x + c)^q)^r))/(h*k) - integrate(-1
/3*(3*((t^2*log(i)^2 + 2*s*t*log(i) + s^2)*h*log(e) + (t^2*log(i)^2 + 2*s*t*log(i) + s^2)*h*log(f^r))*b*d*x^2
- ((p*r + q*r)*b*d*h*n^2*t^2*x^2 + b*c*g*n^2*p*r*t^2 + a*d*g*n^2*q*r*t^2 + (a*d*h*n^2*q*r*t^2 + (c*h*n^2*p*r*t
^2 + (p*r + q*r)*d*g*n^2*t^2)*b)*x)*log(h*x + g)^3 + 3*((t^2*log(i)^2 + 2*s*t*log(i) + s^2)*h*log(e) + (t^2*lo
g(i)^2 + 2*s*t*log(i) + s^2)*h*log(f^r))*a*c + 3*(((p*r + q*r)*n*t^2*log(i) + (p*r*s + q*r*s)*n*t)*b*d*h*x^2 +
(n*p*r*t^2*log(i) + n*p*r*s*t)*b*c*g + (n*q*r*t^2*log(i) + n*q*r*s*t)*a*d*g + ((n*q*r*t^2*log(i) + n*q*r*s*t)
*a*d*h + (((p*r + q*r)*n*t^2*log(i) + (p*r*s + q*r*s)*n*t)*d*g + (n*p*r*t^2*log(i) + n*p*r*s*t)*c*h)*b)*x)*log
(h*x + g)^2 + 3*((h*t^2*log(e) + h*t^2*log(f^r))*b*d*x^2 + (h*t^2*log(e) + h*t^2*log(f^r))*a*c + ((h*t^2*log(e
) + h*t^2*log(f^r))*b*c + (h*t^2*log(e) + h*t^2*log(f^r))*a*d)*x - ((p*r + q*r)*b*d*h*t^2*x^2 + b*c*g*p*r*t^2
+ a*d*g*q*r*t^2 + (a*d*h*q*r*t^2 + (c*h*p*r*t^2 + (p*r + q*r)*d*g*t^2)*b)*x)*log(h*x + g))*log((h*x + g)^n)^2
+ 3*(((t^2*log(i)^2 + 2*s*t*log(i) + s^2)*h*log(e) + (t^2*log(i)^2 + 2*s*t*log(i) + s^2)*h*log(f^r))*b*c + ((t
^2*log(i)^2 + 2*s*t*log(i) + s^2)*h*log(e) + (t^2*log(i)^2 + 2*s*t*log(i) + s^2)*h*log(f^r))*a*d)*x - 3*(((p*r
+ q*r)*t^2*log(i)^2 + p*r*s^2 + q*r*s^2 + 2*(p*r*s + q*r*s)*t*log(i))*b*d*h*x^2 + (p*r*t^2*log(i)^2 + 2*p*r*s
*t*log(i) + p*r*s^2)*b*c*g + (q*r*t^2*log(i)^2 + 2*q*r*s*t*log(i) + q*r*s^2)*a*d*g + ((q*r*t^2*log(i)^2 + 2*q*
r*s*t*log(i) + q*r*s^2)*a*d*h + (((p*r + q*r)*t^2*log(i)^2 + p*r*s^2 + q*r*s^2 + 2*(p*r*s + q*r*s)*t*log(i))*d
*g + (p*r*t^2*log(i)^2 + 2*p*r*s*t*log(i) + p*r*s^2)*c*h)*b)*x)*log(h*x + g) + 3*(2*((t^2*log(i) + s*t)*h*log(
e) + (t^2*log(i) + s*t)*h*log(f^r))*b*d*x^2 + 2*((t^2*log(i) + s*t)*h*log(e) + (t^2*log(i) + s*t)*h*log(f^r))*
a*c + ((p*r + q*r)*b*d*h*n*t^2*x^2 + b*c*g*n*p*r*t^2 + a*d*g*n*q*r*t^2 + (a*d*h*n*q*r*t^2 + (c*h*n*p*r*t^2 + (
p*r + q*r)*d*g*n*t^2)*b)*x)*log(h*x + g)^2 + 2*(((t^2*log(i) + s*t)*h*log(e) + (t^2*log(i) + s*t)*h*log(f^r))*
b*c + ((t^2*log(i) + s*t)*h*log(e) + (t^2*log(i) + s*t)*h*log(f^r))*a*d)*x - 2*(((p*r + q*r)*t^2*log(i) + (p*r
*s + q*r*s)*t)*b*d*h*x^2 + (p*r*t^2*log(i) + p*r*s*t)*b*c*g + (q*r*t^2*log(i) + q*r*s*t)*a*d*g + ((q*r*t^2*log
(i) + q*r*s*t)*a*d*h + (((p*r + q*r)*t^2*log(i) + (p*r*s + q*r*s)*t)*d*g + (p*r*t^2*log(i) + p*r*s*t)*c*h)*b)*
x)*log(h*x + g))*log((h*x + g)^n))/(b*d*h^2*k*x^3 + a*c*g*h*k + (a*d*h^2*k + (d*g*h*k + c*h^2*k)*b)*x^2 + (b*c
*g*h*k + (d*g*h*k + c*h^2*k)*a)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (t^{2} \log \left ({\left (h x + g\right )}^{n} i\right )^{2} + 2 \, s t \log \left ({\left (h x + g\right )}^{n} i\right ) + s^{2}\right )} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*log(i*(h*x+g)^n))^2/(h*k*x+g*k),x, algorithm="fricas")

[Out]

integral((t^2*log((h*x + g)^n*i)^2 + 2*s*t*log((h*x + g)^n*i) + s^2)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(h*k
*x + g*k), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)*(s+t*ln(i*(h*x+g)**n))**2/(h*k*x+g*k),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*log(i*(h*x+g)^n))^2/(h*k*x+g*k),x, algorithm="giac")

[Out]

integrate((t*log((h*x + g)^n*i) + s)^2*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(h*k*x + g*k), x)