3.188 \(\int \frac{(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx\)

Optimal. Leaf size=142 \[ \frac{2 b \text{PolyLog}\left (2,-\frac{f h-e i}{i (e+f x)}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)}+\frac{2 b^2 \text{PolyLog}\left (3,-\frac{f h-e i}{i (e+f x)}\right )}{d (f h-e i)}-\frac{\log \left (\frac{f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)} \]

[Out]

-(((a + b*Log[c*(e + f*x)])^2*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h - e*i))) + (2*b*(a + b*Log[c*(e + f*
x)])*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)) + (2*b^2*PolyLog[3, -((f*h - e*i)/(i*(e + f*x))
)])/(d*(f*h - e*i))

________________________________________________________________________________________

Rubi [A]  time = 0.383463, antiderivative size = 168, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2411, 12, 2344, 2302, 30, 2317, 2374, 6589} \[ -\frac{2 b \text{PolyLog}\left (2,-\frac{i (e+f x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)}+\frac{2 b^2 \text{PolyLog}\left (3,-\frac{i (e+f x)}{f h-e i}\right )}{d (f h-e i)}+\frac{(a+b \log (c (e+f x)))^3}{3 b d (f h-e i)}-\frac{\log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(e + f*x)])^2/((d*e + d*f*x)*(h + i*x)),x]

[Out]

(a + b*Log[c*(e + f*x)])^3/(3*b*d*(f*h - e*i)) - ((a + b*Log[c*(e + f*x)])^2*Log[(f*(h + i*x))/(f*h - e*i)])/(
d*(f*h - e*i)) - (2*b*(a + b*Log[c*(e + f*x)])*PolyLog[2, -((i*(e + f*x))/(f*h - e*i))])/(d*(f*h - e*i)) + (2*
b^2*PolyLog[3, -((i*(e + f*x))/(f*h - e*i))])/(d*(f*h - e*i))

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2344

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

Rule 2302

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 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{(a+b \log (c (e+f x)))^2}{(h+188 x) (d e+d f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{d x \left (\frac{-188 e+f h}{f}+\frac{188 x}{f}\right )} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x \left (\frac{-188 e+f h}{f}+\frac{188 x}{f}\right )} \, dx,x,e+f x\right )}{d f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d (188 e-f h)}+\frac{188 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{\frac{-188 e+f h}{f}+\frac{188 x}{f}} \, dx,x,e+f x\right )}{d f (188 e-f h)}\\ &=\frac{\log \left (-\frac{f (h+188 x)}{188 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (188 e-f h)}-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d (188 e-f h)}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(a+b \log (c x)) \log \left (1+\frac{188 x}{-188 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (188 e-f h)}\\ &=\frac{\log \left (-\frac{f (h+188 x)}{188 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (188 e-f h)}-\frac{(a+b \log (c (e+f x)))^3}{3 b d (188 e-f h)}+\frac{2 b (a+b \log (c (e+f x))) \text{Li}_2\left (\frac{188 (e+f x)}{188 e-f h}\right )}{d (188 e-f h)}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{188 x}{-188 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (188 e-f h)}\\ &=\frac{\log \left (-\frac{f (h+188 x)}{188 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (188 e-f h)}-\frac{(a+b \log (c (e+f x)))^3}{3 b d (188 e-f h)}+\frac{2 b (a+b \log (c (e+f x))) \text{Li}_2\left (\frac{188 (e+f x)}{188 e-f h}\right )}{d (188 e-f h)}-\frac{2 b^2 \text{Li}_3\left (\frac{188 (e+f x)}{188 e-f h}\right )}{d (188 e-f h)}\\ \end{align*}

Mathematica [A]  time = 0.19291, size = 189, normalized size = 1.33 \[ \frac{-6 b \text{PolyLog}\left (2,\frac{i (e+f x)}{e i-f h}\right ) (a+b \log (c (e+f x)))+6 b^2 \text{PolyLog}\left (3,\frac{i (e+f x)}{e i-f h}\right )+3 a^2 \log (e+f x)-3 a^2 \log (h+i x)-6 a b \log (c (e+f x)) \log \left (\frac{f (h+i x)}{f h-e i}\right )+3 a b \log ^2(c (e+f x))-3 b^2 \log ^2(c (e+f x)) \log \left (\frac{f (h+i x)}{f h-e i}\right )+b^2 \log ^3(c (e+f x))}{3 d (f h-e i)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(e + f*x)])^2/((d*e + d*f*x)*(h + i*x)),x]

[Out]

(3*a^2*Log[e + f*x] + 3*a*b*Log[c*(e + f*x)]^2 + b^2*Log[c*(e + f*x)]^3 - 3*a^2*Log[h + i*x] - 6*a*b*Log[c*(e
+ f*x)]*Log[(f*(h + i*x))/(f*h - e*i)] - 3*b^2*Log[c*(e + f*x)]^2*Log[(f*(h + i*x))/(f*h - e*i)] - 6*b*(a + b*
Log[c*(e + f*x)])*PolyLog[2, (i*(e + f*x))/(-(f*h) + e*i)] + 6*b^2*PolyLog[3, (i*(e + f*x))/(-(f*h) + e*i)])/(
3*d*(f*h - e*i))

________________________________________________________________________________________

Maple [B]  time = 0.363, size = 383, normalized size = 2.7 \begin{align*} -{\frac{{a}^{2}\ln \left ( cfx+ce \right ) }{d \left ( ei-fh \right ) }}+{\frac{{a}^{2}\ln \left ( -cei+hcf+ \left ( cfx+ce \right ) i \right ) }{d \left ( ei-fh \right ) }}-{\frac{{b}^{2} \left ( \ln \left ( cfx+ce \right ) \right ) ^{3}}{3\,d \left ( ei-fh \right ) }}+{\frac{{b}^{2} \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{d \left ( ei-fh \right ) }\ln \left ( 1+{\frac{ \left ( cfx+ce \right ) i}{-cei+hcf}} \right ) }+2\,{\frac{{b}^{2}\ln \left ( cfx+ce \right ) }{d \left ( ei-fh \right ) }{\it polylog} \left ( 2,-{\frac{ \left ( cfx+ce \right ) i}{-cei+hcf}} \right ) }-2\,{\frac{{b}^{2}}{d \left ( ei-fh \right ) }{\it polylog} \left ( 3,-{\frac{ \left ( cfx+ce \right ) i}{-cei+hcf}} \right ) }-{\frac{ab \left ( \ln \left ( cfx+ce \right ) \right ) ^{2}}{d \left ( ei-fh \right ) }}+2\,{\frac{ab}{d \left ( ei-fh \right ) }{\it dilog} \left ({\frac{-cei+hcf+ \left ( cfx+ce \right ) i}{-cei+hcf}} \right ) }+2\,{\frac{ab\ln \left ( cfx+ce \right ) }{d \left ( ei-fh \right ) }\ln \left ({\frac{-cei+hcf+ \left ( cfx+ce \right ) i}{-cei+hcf}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(f*x+e)))^2/(d*f*x+d*e)/(i*x+h),x)

[Out]

-1/d*a^2/(e*i-f*h)*ln(c*f*x+c*e)+1/d*a^2/(e*i-f*h)*ln(-c*e*i+h*c*f+(c*f*x+c*e)*i)-1/3/d*b^2/(e*i-f*h)*ln(c*f*x
+c*e)^3+1/d*b^2/(e*i-f*h)*ln(c*f*x+c*e)^2*ln(1+i/(-c*e*i+c*f*h)*(c*f*x+c*e))+2/d*b^2/(e*i-f*h)*ln(c*f*x+c*e)*p
olylog(2,-i/(-c*e*i+c*f*h)*(c*f*x+c*e))-2/d*b^2/(e*i-f*h)*polylog(3,-i/(-c*e*i+c*f*h)*(c*f*x+c*e))-1/d*a*b/(e*
i-f*h)*ln(c*f*x+c*e)^2+2/d*a*b/(e*i-f*h)*dilog((-c*e*i+h*c*f+(c*f*x+c*e)*i)/(-c*e*i+c*f*h))+2/d*a*b/(e*i-f*h)*
ln(c*f*x+c*e)*ln((-c*e*i+h*c*f+(c*f*x+c*e)*i)/(-c*e*i+c*f*h))

________________________________________________________________________________________

Maxima [B]  time = 1.2654, size = 447, normalized size = 3.15 \begin{align*} a^{2}{\left (\frac{\log \left (f x + e\right )}{d f h - d e i} - \frac{\log \left (i x + h\right )}{d f h - d e i}\right )} - \frac{{\left (\log \left (f x + e\right )^{2} \log \left (\frac{f i x + e i}{f h - e i} + 1\right ) + 2 \,{\rm Li}_2\left (-\frac{f i x + e i}{f h - e i}\right ) \log \left (f x + e\right ) - 2 \,{\rm Li}_{3}(-\frac{f i x + e i}{f h - e i})\right )} b^{2}}{{\left (f h - e i\right )} d} - \frac{2 \,{\left (b^{2} \log \left (c\right ) + a b\right )}{\left (\log \left (f x + e\right ) \log \left (\frac{f i x + e i}{f h - e i} + 1\right ) +{\rm Li}_2\left (-\frac{f i x + e i}{f h - e i}\right )\right )}}{{\left (f h - e i\right )} d} - \frac{{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} \log \left (i x + h\right )}{{\left (f h - e i\right )} d} + \frac{b^{2} \log \left (f x + e\right )^{3} + 3 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (f x + e\right )^{2} + 3 \,{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} \log \left (f x + e\right )}{3 \,{\left (f h - e i\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(log(f*x + e)/(d*f*h - d*e*i) - log(i*x + h)/(d*f*h - d*e*i)) - (log(f*x + e)^2*log((f*i*x + e*i)/(f*h - e
*i) + 1) + 2*dilog(-(f*i*x + e*i)/(f*h - e*i))*log(f*x + e) - 2*polylog(3, -(f*i*x + e*i)/(f*h - e*i)))*b^2/((
f*h - e*i)*d) - 2*(b^2*log(c) + a*b)*(log(f*x + e)*log((f*i*x + e*i)/(f*h - e*i) + 1) + dilog(-(f*i*x + e*i)/(
f*h - e*i)))/((f*h - e*i)*d) - (b^2*log(c)^2 + 2*a*b*log(c))*log(i*x + h)/((f*h - e*i)*d) + 1/3*(b^2*log(f*x +
e)^3 + 3*(b^2*log(c) + a*b)*log(f*x + e)^2 + 3*(b^2*log(c)^2 + 2*a*b*log(c))*log(f*x + e))/((f*h - e*i)*d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log(c*f*x + c*e)^2 + 2*a*b*log(c*f*x + c*e) + a^2)/(d*f*i*x^2 + d*e*h + (d*f*h + d*e*i)*x), x)

________________________________________________________________________________________

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((a+b*ln(c*(f*x+e)))**2/(d*f*x+d*e)/(i*x+h),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(f*x+e)))^2/(d*f*x+d*e)/(i*x+h),x, algorithm="giac")

[Out]

integrate((b*log((f*x + e)*c) + a)^2/((d*f*x + d*e)*(i*x + h)), x)