Integrand size = 33, antiderivative size = 75 \[ \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {(a+b x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)^2} \] Output:
(b*x+a)^2*Ei(2*ln(e*((b*x+a)/(d*x+c))^n)/n)/(-a*d+b*c)/n/((e*((b*x+a)/(d*x +c))^n)^(2/n))/(d*x+c)^2
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {(a+b x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)^2} \] Input:
Integrate[(a + b*x)/((c + d*x)^3*Log[e*((a + b*x)/(c + d*x))^n]),x]
Output:
((a + b*x)^2*ExpIntegralEi[(2*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - a*d)*n*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2)
Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx\) |
\(\Big \downarrow \) 2961 |
\(\displaystyle \frac {\int \frac {a+b x}{(c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}d\frac {a+b x}{c+d x}}{b c-a d}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {(a+b x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \int \frac {\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} (c+d x)}{a+b x}d\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (c+d x)^2 (b c-a d)}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {(a+b x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x)^2 (b c-a d)}\) |
Input:
Int[(a + b*x)/((c + d*x)^3*Log[e*((a + b*x)/(c + d*x))^n]),x]
Output:
((a + b*x)^2*ExpIntegralEi[(2*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - a*d)*n*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2)
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]
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]
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]
\[\int \frac {b x +a}{\left (d x +c \right )^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}d x\]
Input:
int((b*x+a)/(d*x+c)^3/ln(e*((b*x+a)/(d*x+c))^n),x)
Output:
int((b*x+a)/(d*x+c)^3/ln(e*((b*x+a)/(d*x+c))^n),x)
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {\operatorname {log\_integral}\left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e^{\frac {2}{n}}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (b c - a d\right )} e^{\frac {2}{n}} n} \] Input:
integrate((b*x+a)/(d*x+c)^3/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="frica s")
Output:
log_integral((b^2*x^2 + 2*a*b*x + a^2)*e^(2/n)/(d^2*x^2 + 2*c*d*x + c^2))/ ((b*c - a*d)*e^(2/n)*n)
Timed out. \[ \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)/(d*x+c)**3/ln(e*((b*x+a)/(d*x+c))**n),x)
Output:
Timed out
\[ \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {b x + a}{{\left (d x + c\right )}^{3} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \] Input:
integrate((b*x+a)/(d*x+c)^3/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="maxim a")
Output:
integrate((b*x + a)/((d*x + c)^3*log(e*((b*x + a)/(d*x + c))^n)), x)
\[ \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {b x + a}{{\left (d x + c\right )}^{3} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \] Input:
integrate((b*x+a)/(d*x+c)^3/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="giac" )
Output:
integrate((b*x + a)/((d*x + c)^3*log(e*((b*x + a)/(d*x + c))^n)), x)
Timed out. \[ \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {a+b\,x}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,{\left (c+d\,x\right )}^3} \,d x \] Input:
int((a + b*x)/(log(e*((a + b*x)/(c + d*x))^n)*(c + d*x)^3),x)
Output:
int((a + b*x)/(log(e*((a + b*x)/(c + d*x))^n)*(c + d*x)^3), x)
\[ \int \frac {a+b x}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {too large to display} \] Input:
int((b*x+a)/(d*x+c)^3/log(e*((b*x+a)/(d*x+c))^n),x)
Output:
(2*int(x/(log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**3 + 3*log(((a + b*x)**n* e)/(c + d*x)**n)*a*c**2*d*x + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c*d** 2*x**2 + log(((a + b*x)**n*e)/(c + d*x)**n)*a*d**3*x**3 + log(((a + b*x)** n*e)/(c + d*x)**n)*b*c**3*x + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*b*c**2* d*x**2 + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*b*c*d**2*x**3 + log(((a + b* x)**n*e)/(c + d*x)**n)*b*d**3*x**4),x)*a**2*b*d**3*n - 4*int(x/(log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**3 + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c **2*d*x + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c*d**2*x**2 + log(((a + b *x)**n*e)/(c + d*x)**n)*a*d**3*x**3 + log(((a + b*x)**n*e)/(c + d*x)**n)*b *c**3*x + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*b*c**2*d*x**2 + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*b*c*d**2*x**3 + log(((a + b*x)**n*e)/(c + d*x)** n)*b*d**3*x**4),x)*a*b**2*c*d**2*n + 2*int(x/(log(((a + b*x)**n*e)/(c + d* x)**n)*a*c**3 + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**2*d*x + 3*log((( a + b*x)**n*e)/(c + d*x)**n)*a*c*d**2*x**2 + log(((a + b*x)**n*e)/(c + d*x )**n)*a*d**3*x**3 + log(((a + b*x)**n*e)/(c + d*x)**n)*b*c**3*x + 3*log((( a + b*x)**n*e)/(c + d*x)**n)*b*c**2*d*x**2 + 3*log(((a + b*x)**n*e)/(c + d *x)**n)*b*c*d**2*x**3 + log(((a + b*x)**n*e)/(c + d*x)**n)*b*d**3*x**4),x) *b**3*c**2*d*n + int(1/(log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**3 + 3*log( ((a + b*x)**n*e)/(c + d*x)**n)*a*c**2*d*x + 3*log(((a + b*x)**n*e)/(c + d* x)**n)*a*c*d**2*x**2 + log(((a + b*x)**n*e)/(c + d*x)**n)*a*d**3*x**3 +...