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

Optimal. Leaf size=306 \[ -\frac{p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}+\frac{n p r t \text{PolyLog}\left (3,\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}+\frac{n q r t \text{PolyLog}\left (3,\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h k n t}-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{2 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{2 h k n t} \]

[Out]

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

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Rubi [A]  time = 0.287542, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109, Rules used = {2499, 2396, 2433, 2374, 6589} \[ -\frac{p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}+\frac{n p r t \text{PolyLog}\left (3,\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}+\frac{n q r t \text{PolyLog}\left (3,\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h k n t}-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{2 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{2 h k n t} \]

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]))/(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])^2)/(2*h*k*n*t) - (q*r*Log[-((h*(c + d*x))/(
d*g - c*h))]*(s + t*Log[i*(g + h*x)^n])^2)/(2*h*k*n*t) + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g
+ h*x)^n])^2)/(2*h*k*n*t) - (p*r*(s + t*Log[i*(g + h*x)^n])*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/(h*k) - (q
*r*(s + t*Log[i*(g + h*x)^n])*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/(h*k) + (n*p*r*t*PolyLog[3, (b*(g + h*x))
/(b*g - a*h)])/(h*k) + (n*q*r*t*PolyLog[3, (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 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 (52 (g+h x)^n\right )\right )}{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 (52 (g+h x)^n\right )\right )^2}{2 h k n t}-\frac{(b p r) \int \frac{\left (s+t \log \left (52 (g+h x)^n\right )\right )^2}{a+b x} \, dx}{2 h k n t}-\frac{(d q r) \int \frac{\left (s+t \log \left (52 (g+h x)^n\right )\right )^2}{c+d x} \, dx}{2 h k n t}\\ &=-\frac{p r \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (52 (g+h x)^n\right )\right )^2}{2 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (52 (g+h x)^n\right )\right )^2}{2 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 (52 (g+h x)^n\right )\right )^2}{2 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 (52 (g+h x)^n\right )\right )}{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 (52 (g+h x)^n\right )\right )}{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 (52 (g+h x)^n\right )\right )^2}{2 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (52 (g+h x)^n\right )\right )^2}{2 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 (52 (g+h x)^n\right )\right )^2}{2 h k n t}+\frac{(p r) \operatorname{Subst}\left (\int \frac{\left (s+t \log \left (52 x^n\right )\right ) \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 (52 x^n\right )\right ) \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 (52 (g+h x)^n\right )\right )^2}{2 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (52 (g+h x)^n\right )\right )^2}{2 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 (52 (g+h x)^n\right )\right )^2}{2 h k n t}-\frac{p r \left (s+t \log \left (52 (g+h x)^n\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \left (s+t \log \left (52 (g+h x)^n\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{(n p r t) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}+\frac{(n q r t) \operatorname{Subst}\left (\int \frac{\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 (52 (g+h x)^n\right )\right )^2}{2 h k n t}-\frac{q r \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (52 (g+h x)^n\right )\right )^2}{2 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 (52 (g+h x)^n\right )\right )^2}{2 h k n t}-\frac{p r \left (s+t \log \left (52 (g+h x)^n\right )\right ) \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \left (s+t \log \left (52 (g+h x)^n\right )\right ) \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{n p r t \text{Li}_3\left (\frac{b (g+h x)}{b g-a h}\right )}{h k}+\frac{n q r t \text{Li}_3\left (\frac{d (g+h x)}{d g-c h}\right )}{h k}\\ \end{align*}

Mathematica [A]  time = 2.55186, size = 436, normalized size = 1.42 \[ \frac{-2 p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )+2 n p r t \text{PolyLog}\left (3,\frac{b (g+h x)}{b g-a h}\right )-2 q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )+2 n q r t \text{PolyLog}\left (3,\frac{d (g+h x)}{d g-c h}\right )+2 t \log (g+h x) \log \left (i (g+h x)^n\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-n t \log ^2(g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 s \log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 p r t \log (g+h x) \log \left (\frac{h (a+b x)}{a h-b g}\right ) \log \left (i (g+h x)^n\right )+n p r t \log ^2(g+h x) \log \left (\frac{h (a+b x)}{a h-b g}\right )-2 p r s \log (g+h x) \log \left (\frac{h (a+b x)}{a h-b g}\right )-2 q r t \log (g+h x) \log \left (\frac{h (c+d x)}{c h-d g}\right ) \log \left (i (g+h x)^n\right )+n q r t \log ^2(g+h x) \log \left (\frac{h (c+d x)}{c h-d g}\right )-2 q r s \log (g+h x) \log \left (\frac{h (c+d x)}{c h-d g}\right )}{2 h k} \]

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]))/(g*k + h*k*x),x]

[Out]

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

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Maple [F]  time = 0.75, 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 ) }{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))/(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))/(h*k*x+g*k),x)

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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))/(h*k*x+g*k),x, algorithm="maxima")

[Out]

-1/2*((n*t*log(h*x + g)^2 - 2*t*log(h*x + g)*log((h*x + g)^n) - 2*(t*log(i) + s)*log(h*x + g))*log(((b*x + a)^
p)^r) + (n*t*log(h*x + g)^2 - 2*t*log(h*x + g)*log((h*x + g)^n) - 2*(t*log(i) + s)*log(h*x + g))*log(((d*x + c
)^q)^r))/(h*k) - integrate(-1/2*(2*((t*log(i) + s)*h*log(e) + (t*log(i) + s)*h*log(f^r))*b*d*x^2 + 2*((t*log(i
) + s)*h*log(e) + (t*log(i) + s)*h*log(f^r))*a*c + ((p*r + q*r)*b*d*h*n*t*x^2 + b*c*g*n*p*r*t + a*d*g*n*q*r*t
+ (a*d*h*n*q*r*t + (c*h*n*p*r*t + (p*r + q*r)*d*g*n*t)*b)*x)*log(h*x + g)^2 + 2*(((t*log(i) + s)*h*log(e) + (t
*log(i) + s)*h*log(f^r))*b*c + ((t*log(i) + s)*h*log(e) + (t*log(i) + s)*h*log(f^r))*a*d)*x - 2*((p*r*s + q*r*
s + (p*r + q*r)*t*log(i))*b*d*h*x^2 + (p*r*t*log(i) + p*r*s)*b*c*g + (q*r*t*log(i) + q*r*s)*a*d*g + ((q*r*t*lo
g(i) + q*r*s)*a*d*h + ((p*r*s + q*r*s + (p*r + q*r)*t*log(i))*d*g + (p*r*t*log(i) + p*r*s)*c*h)*b)*x)*log(h*x
+ g) + 2*((h*t*log(e) + h*t*log(f^r))*b*d*x^2 + (h*t*log(e) + h*t*log(f^r))*a*c + ((h*t*log(e) + h*t*log(f^r))
*b*c + (h*t*log(e) + h*t*log(f^r))*a*d)*x - ((p*r + q*r)*b*d*h*t*x^2 + b*c*g*p*r*t + a*d*g*q*r*t + (a*d*h*q*r*
t + (c*h*p*r*t + (p*r + q*r)*d*g*t)*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 \log \left ({\left (h x + g\right )}^{n} i\right ) + s\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))/(h*k*x+g*k),x, algorithm="fricas")

[Out]

integral((t*log((h*x + g)^n*i) + s)*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))/(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 )} \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))/(h*k*x+g*k),x, algorithm="giac")

[Out]

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