Integrand size = 28, antiderivative size = 222 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=-\frac {b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f g-e h)^2 (g+h x)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}+\frac {b^2 f^2 p^2 q^2 \log (g+h x)}{h (f g-e h)^2}-\frac {b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (1+\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2}+\frac {b^2 f^2 p^2 q^2 \operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2} \]
-b*f*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))/(-e*h+f*g)^2/(h*x+g)-1/2*(a+b *ln(c*(d*(f*x+e)^p)^q))^2/h/(h*x+g)^2+b^2*f^2*p^2*q^2*ln(h*x+g)/h/(-e*h+f* g)^2-b*f^2*p*q*(a+b*ln(c*(d*(f*x+e)^p)^q))*ln(1+(-e*h+f*g)/h/(f*x+e))/h/(- e*h+f*g)^2+b^2*f^2*p^2*q^2*polylog(2,(e*h-f*g)/h/(f*x+e))/h/(-e*h+f*g)^2
Time = 0.31 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=-\frac {\left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+\frac {2 b p q \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (h (e+f x) (e h-f (2 g+h x)) \log (e+f x)+f (g+h x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )\right )}{(f g-e h)^2}+\frac {b^2 p^2 q^2 \left (h (e+f x) (e h-f (2 g+h x)) \log ^2(e+f x)-2 f^2 (g+h x)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 f (g+h x) \log (e+f x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )+2 f^2 (g+h x)^2 \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )}{(f g-e h)^2}}{2 h (g+h x)^2} \]
-1/2*((a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2 + (2*b*p*q*( a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])*(h*(e + f*x)*(e*h - f *(2*g + h*x))*Log[e + f*x] + f*(g + h*x)*(h*(e + f*x) + f*(g + h*x)*Log[(f *(g + h*x))/(f*g - e*h)])))/(f*g - e*h)^2 + (b^2*p^2*q^2*(h*(e + f*x)*(e*h - f*(2*g + h*x))*Log[e + f*x]^2 - 2*f^2*(g + h*x)^2*Log[(f*(g + h*x))/(f* g - e*h)] + 2*f*(g + h*x)*Log[e + f*x]*(h*(e + f*x) + f*(g + h*x)*Log[(f*( g + h*x))/(f*g - e*h)]) + 2*f^2*(g + h*x)^2*PolyLog[2, (h*(e + f*x))/(-(f* g) + e*h)]))/(f*g - e*h)^2)/(h*(g + h*x)^2)
Time = 1.18 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {2895, 2845, 2858, 27, 2789, 2751, 16, 2779, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3}dx\) |
\(\Big \downarrow \) 2845 |
\(\displaystyle \frac {b f p q \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(e+f x) (g+h x)^2}dx}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\) |
\(\Big \downarrow \) 2858 |
\(\displaystyle \frac {b p q \int \frac {f^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(e+f x) \left (f \left (g-\frac {e h}{f}\right )+h (e+f x)\right )^2}d(e+f x)}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b f^2 p q \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) (f g-e h+h (e+f x))^2}d(e+f x)}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {b f^2 p q \left (\frac {\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) (f g-e h+h (e+f x))}d(e+f x)}{f g-e h}-\frac {h \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(f g-e h+h (e+f x))^2}d(e+f x)}{f g-e h}\right )}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\) |
\(\Big \downarrow \) 2751 |
\(\displaystyle \frac {b f^2 p q \left (\frac {\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) (f g-e h+h (e+f x))}d(e+f x)}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {b p q \int \frac {1}{f g-e h+h (e+f x)}d(e+f x)}{f g-e h}\right )}{f g-e h}\right )}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {b f^2 p q \left (\frac {\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) (f g-e h+h (e+f x))}d(e+f x)}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {b p q \log (h (e+f x)-e h+f g)}{h (f g-e h)}\right )}{f g-e h}\right )}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle \frac {b f^2 p q \left (\frac {\frac {b p q \int \frac {\log \left (\frac {f g-e h}{h (e+f x)}+1\right )}{e+f x}d(e+f x)}{f g-e h}-\frac {\log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{f g-e h}}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {b p q \log (h (e+f x)-e h+f g)}{h (f g-e h)}\right )}{f g-e h}\right )}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {b f^2 p q \left (\frac {\frac {b p q \operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right )}{f g-e h}-\frac {\log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{f g-e h}}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {b p q \log (h (e+f x)-e h+f g)}{h (f g-e h)}\right )}{f g-e h}\right )}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\) |
-1/2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(h*(g + h*x)^2) + (b*f^2*p*q*(-((h *(((e + f*x)*(a + b*Log[c*d^q*(e + f*x)^(p*q)]))/((f*g - e*h)*(f*g - e*h + h*(e + f*x))) - (b*p*q*Log[f*g - e*h + h*(e + f*x)])/(h*(f*g - e*h))))/(f *g - e*h)) + (-(((a + b*Log[c*d^q*(e + f*x)^(p*q)])*Log[1 + (f*g - e*h)/(h *(e + f*x))])/(f*g - e*h)) + (b*p*q*PolyLog[2, -((f*g - e*h)/(h*(e + f*x)) )])/(f*g - e*h))/(f*g - e*h)))/h
3.5.34.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
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] - Simp[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] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(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]
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]]
\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\left (h x +g \right )^{3}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \]
integral((b^2*log(((f*x + e)^p*d)^q*c)^2 + 2*a*b*log(((f*x + e)^p*d)^q*c) + a^2)/(h^3*x^3 + 3*g*h^2*x^2 + 3*g^2*h*x + g^3), x)
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right )^{3}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \]
a*b*f*p*q*(f*log(f*x + e)/(f^2*g^2*h - 2*e*f*g*h^2 + e^2*h^3) - f*log(h*x + g)/(f^2*g^2*h - 2*e*f*g*h^2 + e^2*h^3) + 1/(f*g^2*h - e*g*h^2 + (f*g*h^2 - e*h^3)*x)) - 1/2*b^2*(log(((f*x + e)^p)^q)^2/(h^3*x^2 + 2*g*h^2*x + g^2 *h) - 2*integrate((e*h*q^2*log(d)^2 + 2*e*h*q*log(c)*log(d) + e*h*log(c)^2 + (f*h*q^2*log(d)^2 + 2*f*h*q*log(c)*log(d) + f*h*log(c)^2)*x + (f*g*p*q + 2*e*h*q*log(d) + 2*e*h*log(c) + (f*h*p*q + 2*f*h*q*log(d) + 2*f*h*log(c) )*x)*log(((f*x + e)^p)^q))/(f*h^4*x^4 + e*g^3*h + (3*f*g*h^3 + e*h^4)*x^3 + 3*(f*g^2*h^2 + e*g*h^3)*x^2 + (f*g^3*h + 3*e*g^2*h^2)*x), x)) - a*b*log( ((f*x + e)^p*d)^q*c)/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 1/2*a^2/(h^3*x^2 + 2* g*h^2*x + g^2*h)
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{{\left (g+h\,x\right )}^3} \,d x \]