\(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^2}{(g+h x)^3} \, dx\) [453]

Optimal result

Integrand size = 28, antiderivative size = 222 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=-\frac {b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f g-e h)^2 (g+h x)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}+\frac {b^2 f^2 p^2 q^2 \log (g+h x)}{h (f g-e h)^2}-\frac {b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (1+\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2}+\frac {b^2 f^2 p^2 q^2 \operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2} \] Output:

-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
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=-\frac {\left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+\frac {2 b p q \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (h (e+f x) (e h-f (2 g+h x)) \log (e+f x)+f (g+h x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )\right )}{(f g-e h)^2}+\frac {b^2 p^2 q^2 \left (h (e+f x) (e h-f (2 g+h x)) \log ^2(e+f x)-2 f^2 (g+h x)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 f (g+h x) \log (e+f x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )+2 f^2 (g+h x)^2 \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )}{(f g-e h)^2}}{2 h (g+h x)^2} \] Input:

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^3,x]
 

Output:

-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)
 

Rubi [A] (warning: unable to verify)

Time = 2.07 (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

\(\displaystyle \frac {b f^2 p q \left (\frac {\frac {b p q \operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right )}{f g-e h}-\frac {\log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{f g-e h}}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {b p q \log (h (e+f x)-e h+f g)}{h (f g-e h)}\right )}{f g-e h}\right )}{h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (g+h x)^2}\)

Input:

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^3,x]
 

Output:

-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
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2751

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]
 

rule 2779

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]
 

rule 2789

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]
 

rule 2838

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]
 

rule 2845

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]))
 

rule 2858

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]
 

rule 2895

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]]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\left (h x +g \right )^{3}}d x\]

Input:

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^3,x)
 

Output:

int((a+b*ln(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^3,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^3,x, algorithm="fricas")
 

Output:

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)
 

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right )^{3}}\, dx \] Input:

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**2/(h*x+g)**3,x)
 

Output:

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**2/(g + h*x)**3, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

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)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^2/(h*x+g)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{{\left (g+h\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

(4*int((log(d**q*(e + f*x)**(p*q)*c)*x)/(e*g**3 + 3*e*g**2*h*x + 3*e*g*h** 
2*x**2 + e*h**3*x**3 + f*g**3*x + 3*f*g**2*h*x**2 + 3*f*g*h**2*x**3 + f*h* 
*3*x**4),x)*b**2*e**3*f*g**4*h**4*p*q + 8*int((log(d**q*(e + f*x)**(p*q)*c 
)*x)/(e*g**3 + 3*e*g**2*h*x + 3*e*g*h**2*x**2 + e*h**3*x**3 + f*g**3*x + 3 
*f*g**2*h*x**2 + 3*f*g*h**2*x**3 + f*h**3*x**4),x)*b**2*e**3*f*g**3*h**5*p 
*q*x + 4*int((log(d**q*(e + f*x)**(p*q)*c)*x)/(e*g**3 + 3*e*g**2*h*x + 3*e 
*g*h**2*x**2 + e*h**3*x**3 + f*g**3*x + 3*f*g**2*h*x**2 + 3*f*g*h**2*x**3 
+ f*h**3*x**4),x)*b**2*e**3*f*g**2*h**6*p*q*x**2 - 12*int((log(d**q*(e + f 
*x)**(p*q)*c)*x)/(e*g**3 + 3*e*g**2*h*x + 3*e*g*h**2*x**2 + e*h**3*x**3 + 
f*g**3*x + 3*f*g**2*h*x**2 + 3*f*g*h**2*x**3 + f*h**3*x**4),x)*b**2*e**2*f 
**2*g**5*h**3*p*q - 24*int((log(d**q*(e + f*x)**(p*q)*c)*x)/(e*g**3 + 3*e* 
g**2*h*x + 3*e*g*h**2*x**2 + e*h**3*x**3 + f*g**3*x + 3*f*g**2*h*x**2 + 3* 
f*g*h**2*x**3 + f*h**3*x**4),x)*b**2*e**2*f**2*g**4*h**4*p*q*x - 12*int((l 
og(d**q*(e + f*x)**(p*q)*c)*x)/(e*g**3 + 3*e*g**2*h*x + 3*e*g*h**2*x**2 + 
e*h**3*x**3 + f*g**3*x + 3*f*g**2*h*x**2 + 3*f*g*h**2*x**3 + f*h**3*x**4), 
x)*b**2*e**2*f**2*g**3*h**5*p*q*x**2 + 12*int((log(d**q*(e + f*x)**(p*q)*c 
)*x)/(e*g**3 + 3*e*g**2*h*x + 3*e*g*h**2*x**2 + e*h**3*x**3 + f*g**3*x + 3 
*f*g**2*h*x**2 + 3*f*g*h**2*x**3 + f*h**3*x**4),x)*b**2*e*f**3*g**6*h**2*p 
*q + 24*int((log(d**q*(e + f*x)**(p*q)*c)*x)/(e*g**3 + 3*e*g**2*h*x + 3*e* 
g*h**2*x**2 + e*h**3*x**3 + f*g**3*x + 3*f*g**2*h*x**2 + 3*f*g*h**2*x**...