\(\int \frac {(c+d x)^2}{(a+b x)^4 \log (e (\frac {a+b x}{c+d x})^n)} \, dx\) [248]

Optimal result

Integrand size = 35, antiderivative size = 75 \[ \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{3/n} (c+d x)^3 \operatorname {ExpIntegralEi}\left (-\frac {3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (a+b x)^3} \] Output:

(e*((b*x+a)/(d*x+c))^n)^(3/n)*(d*x+c)^3*Ei(-3*ln(e*((b*x+a)/(d*x+c))^n)/n) 
/(-a*d+b*c)/n/(b*x+a)^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{3/n} (c+d x)^3 \operatorname {ExpIntegralEi}\left (-\frac {3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (a+b x)^3} \] Input:

Integrate[(c + d*x)^2/((a + b*x)^4*Log[e*((a + b*x)/(c + d*x))^n]),x]
 

Output:

((e*((a + b*x)/(c + d*x))^n)^(3/n)*(c + d*x)^3*ExpIntegralEi[(-3*Log[e*((a 
 + b*x)/(c + d*x))^n])/n])/((b*c - a*d)*n*(a + b*x)^3)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2961, 2747, 2609}

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.

\(\displaystyle \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx\)

\(\Big \downarrow \) 2961

\(\displaystyle \frac {\int \frac {(c+d x)^4}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}d\frac {a+b x}{c+d x}}{b c-a d}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {(c+d x)^3 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{3/n} \int \frac {\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-3/n} (c+d x)}{a+b x}d\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {(c+d x)^3 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{3/n} \operatorname {ExpIntegralEi}\left (-\frac {3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^3 (b c-a d)}\)

Input:

Int[(c + d*x)^2/((a + b*x)^4*Log[e*((a + b*x)/(c + d*x))^n]),x]
 

Output:

((e*((a + b*x)/(c + d*x))^n)^(3/n)*(c + d*x)^3*ExpIntegralEi[(-3*Log[e*((a 
 + b*x)/(c + d*x))^n])/n])/((b*c - a*d)*n*(a + b*x)^3)
 

Defintions of rubi rules used

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2747

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n))   Subst[Int[E^(((m + 1)/n 
)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
 

rule 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [F]

\[\int \frac {\left (d x +c \right )^{2}}{\left (b x +a \right )^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}d x\]

Input:

int((d*x+c)^2/(b*x+a)^4/ln(e*((b*x+a)/(d*x+c))^n),x)
 

Output:

int((d*x+c)^2/(b*x+a)^4/ln(e*((b*x+a)/(d*x+c))^n),x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.17 \[ \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {e^{\frac {3}{n}} \operatorname {log\_integral}\left (\frac {d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}{{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} e^{\frac {3}{n}}}\right )}{{\left (b c - a d\right )} n} \] Input:

integrate((d*x+c)^2/(b*x+a)^4/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="fri 
cas")
 

Output:

e^(3/n)*log_integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)/((b^3*x^3 + 
 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*e^(3/n)))/((b*c - a*d)*n)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**2/(b*x+a)**4/ln(e*((b*x+a)/(d*x+c))**n),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{4} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \] Input:

integrate((d*x+c)^2/(b*x+a)^4/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="max 
ima")
 

Output:

integrate((d*x + c)^2/((b*x + a)^4*log(e*((b*x + a)/(d*x + c))^n)), x)
 

Giac [F]

\[ \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{4} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \] Input:

integrate((d*x+c)^2/(b*x+a)^4/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="gia 
c")
 

Output:

integrate((d*x + c)^2/((b*x + a)^4*log(e*((b*x + a)/(d*x + c))^n)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,{\left (a+b\,x\right )}^4} \,d x \] Input:

int((c + d*x)^2/(log(e*((a + b*x)/(c + d*x))^n)*(a + b*x)^4),x)
 

Output:

int((c + d*x)^2/(log(e*((a + b*x)/(c + d*x))^n)*(a + b*x)^4), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {too large to display} \] Input:

int((d*x+c)^2/(b*x+a)^4/log(e*((b*x+a)/(d*x+c))^n),x)
 

Output:

( - 3*int(x**2/(log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*c + log(((a + b*x) 
**n*e)/(c + d*x)**n)*a**4*d*x + 4*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3* 
b*c*x + 4*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b*d*x**2 + 6*log(((a + b 
*x)**n*e)/(c + d*x)**n)*a**2*b**2*c*x**2 + 6*log(((a + b*x)**n*e)/(c + d*x 
)**n)*a**2*b**2*d*x**3 + 4*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**3*c*x** 
3 + 4*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**3*d*x**4 + log(((a + b*x)**n 
*e)/(c + d*x)**n)*b**4*c*x**4 + log(((a + b*x)**n*e)/(c + d*x)**n)*b**4*d* 
x**5),x)*a**2*b**2*d**4*n + 6*int(x**2/(log(((a + b*x)**n*e)/(c + d*x)**n) 
*a**4*c + log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*d*x + 4*log(((a + b*x)** 
n*e)/(c + d*x)**n)*a**3*b*c*x + 4*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3* 
b*d*x**2 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**2*c*x**2 + 6*log(( 
(a + b*x)**n*e)/(c + d*x)**n)*a**2*b**2*d*x**3 + 4*log(((a + b*x)**n*e)/(c 
 + d*x)**n)*a*b**3*c*x**3 + 4*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**3*d* 
x**4 + log(((a + b*x)**n*e)/(c + d*x)**n)*b**4*c*x**4 + log(((a + b*x)**n* 
e)/(c + d*x)**n)*b**4*d*x**5),x)*a*b**3*c*d**3*n - 3*int(x**2/(log(((a + b 
*x)**n*e)/(c + d*x)**n)*a**4*c + log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*d 
*x + 4*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b*c*x + 4*log(((a + b*x)**n 
*e)/(c + d*x)**n)*a**3*b*d*x**2 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a** 
2*b**2*c*x**2 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**2*d*x**3 + 4* 
log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**3*c*x**3 + 4*log(((a + b*x)**n*...