Integrand size = 32, antiderivative size = 53 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \, dx=-\frac {e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B (b c-a d) e g^2} \] Output:
-Ei((A+B*ln(e*(d*x+c)/(b*x+a)))/B)/B/(-a*d+b*c)/e/exp(A/B)/g^2
Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \, dx=\frac {e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A}{B}+\log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{B (-b c+a d) e g^2} \] Input:
Integrate[1/((a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)])),x]
Output:
ExpIntegralEi[A/B + Log[(e*(c + d*x))/(a + b*x)]]/(B*(-(b*c) + a*d)*e*E^(A /B)*g^2)
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2952, 2736, 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.
| \(\displaystyle \int \frac {1}{(a g+b g x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )} \, dx\) |
| \(\Big \downarrow \) 2952 |
| \(\displaystyle -\frac {\int \frac {1}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{g^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2736 |
| \(\displaystyle -\frac {\int \frac {e (c+d x)}{(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}d\log \left (\frac {e (c+d x)}{a+b x}\right )}{e g^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle -\frac {e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B e g^2 (b c-a d)}\) |
Input:
Int[1/((a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)])),x]
Output:
-(ExpIntegralEi[(A + B*Log[(e*(c + d*x))/(a + b*x)])/B]/(B*(b*c - a*d)*e*E ^(A/B)*g^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_), x_Symbol] :> Simp[1/(n*c^(1 /n)) Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b , c, p}, x] && IntegerQ[1/n]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( m + 1)*(g/d)^m Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, ( a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[ n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
Time = 4.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )-\frac {A}{B}\right )}{g^{2} e \left (d a -b c \right ) B}\) | \(55\) |
| derivativedivides | \(-\frac {{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \left (d a -b c \right ) g^{2} B}\) | \(69\) |
| default | \(-\frac {{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \left (d a -b c \right ) g^{2} B}\) | \(69\) |
Input:
int(1/(b*g*x+a*g)^2/(A+B*ln(e*(d*x+c)/(b*x+a))),x,method=_RETURNVERBOSE)
Output:
-1/g^2/e/(a*d-b*c)/B*exp(-A/B)*Ei(1,-ln(e*(d*x+c)/(b*x+a))-A/B)
Time = 0.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \, dx=-\frac {e^{\left (-\frac {A}{B}\right )} \operatorname {log\_integral}\left (\frac {{\left (d e x + c e\right )} e^{\frac {A}{B}}}{b x + a}\right )}{{\left (B b c - B a d\right )} e g^{2}} \] Input:
integrate(1/(b*g*x+a*g)^2/(A+B*log(e*(d*x+c)/(b*x+a))),x, algorithm="frica s")
Output:
-e^(-A/B)*log_integral((d*e*x + c*e)*e^(A/B)/(b*x + a))/((B*b*c - B*a*d)*e *g^2)
\[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \, dx=\frac {\int \frac {1}{A a^{2} + 2 A a b x + A b^{2} x^{2} + B a^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 2 B a b x \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + B b^{2} x^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx}{g^{2}} \] Input:
integrate(1/(b*g*x+a*g)**2/(A+B*ln(e*(d*x+c)/(b*x+a))),x)
Output:
Integral(1/(A*a**2 + 2*A*a*b*x + A*b**2*x**2 + B*a**2*log(c*e/(a + b*x) + d*e*x/(a + b*x)) + 2*B*a*b*x*log(c*e/(a + b*x) + d*e*x/(a + b*x)) + B*b**2 *x**2*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x)/g**2
\[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}} \,d x } \] Input:
integrate(1/(b*g*x+a*g)^2/(A+B*log(e*(d*x+c)/(b*x+a))),x, algorithm="maxim a")
Output:
integrate(1/((b*g*x + a*g)^2*(B*log((d*x + c)*e/(b*x + a)) + A)), x)
\[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{2} {\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}} \,d x } \] Input:
integrate(1/(b*g*x+a*g)^2/(A+B*log(e*(d*x+c)/(b*x+a))),x, algorithm="giac" )
Output:
integrate(1/((b*g*x + a*g)^2*(B*log((d*x + c)*e/(b*x + a)) + A)), x)
Timed out. \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \, dx=\int \frac {1}{{\left (a\,g+b\,g\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )} \,d x \] Input:
int(1/((a*g + b*g*x)^2*(A + B*log((e*(c + d*x))/(a + b*x)))),x)
Output:
int(1/((a*g + b*g*x)^2*(A + B*log((e*(c + d*x))/(a + b*x)))), x)
\[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \, dx=\frac {-\left (\int \frac {1}{\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b c +\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b d x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{2} c x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{2} d \,x^{2}+\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b^{3} c \,x^{2}+\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b^{3} d \,x^{3}+a^{3} c +a^{3} d x +2 a^{2} b c x +2 a^{2} b d \,x^{2}+a \,b^{2} c \,x^{2}+a \,b^{2} d \,x^{3}}d x \right ) a^{2} b \,d^{2}+2 \left (\int \frac {1}{\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b c +\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b d x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{2} c x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{2} d \,x^{2}+\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b^{3} c \,x^{2}+\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b^{3} d \,x^{3}+a^{3} c +a^{3} d x +2 a^{2} b c x +2 a^{2} b d \,x^{2}+a \,b^{2} c \,x^{2}+a \,b^{2} d \,x^{3}}d x \right ) a \,b^{2} c d -\left (\int \frac {1}{\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b c +\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b d x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{2} c x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{2} d \,x^{2}+\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b^{3} c \,x^{2}+\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b^{3} d \,x^{3}+a^{3} c +a^{3} d x +2 a^{2} b c x +2 a^{2} b d \,x^{2}+a \,b^{2} c \,x^{2}+a \,b^{2} d \,x^{3}}d x \right ) b^{3} c^{2}+\mathrm {log}\left (\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b +a \right ) d}{b^{2} g^{2} \left (a d -b c \right )} \] Input:
int(1/(b*g*x+a*g)^2/(A+B*log(e*(d*x+c)/(b*x+a))),x)
Output:
( - int(1/(log((c*e + d*e*x)/(a + b*x))*a**2*b*c + log((c*e + d*e*x)/(a + b*x))*a**2*b*d*x + 2*log((c*e + d*e*x)/(a + b*x))*a*b**2*c*x + 2*log((c*e + d*e*x)/(a + b*x))*a*b**2*d*x**2 + log((c*e + d*e*x)/(a + b*x))*b**3*c*x* *2 + log((c*e + d*e*x)/(a + b*x))*b**3*d*x**3 + a**3*c + a**3*d*x + 2*a**2 *b*c*x + 2*a**2*b*d*x**2 + a*b**2*c*x**2 + a*b**2*d*x**3),x)*a**2*b*d**2 + 2*int(1/(log((c*e + d*e*x)/(a + b*x))*a**2*b*c + log((c*e + d*e*x)/(a + b *x))*a**2*b*d*x + 2*log((c*e + d*e*x)/(a + b*x))*a*b**2*c*x + 2*log((c*e + d*e*x)/(a + b*x))*a*b**2*d*x**2 + log((c*e + d*e*x)/(a + b*x))*b**3*c*x** 2 + log((c*e + d*e*x)/(a + b*x))*b**3*d*x**3 + a**3*c + a**3*d*x + 2*a**2* b*c*x + 2*a**2*b*d*x**2 + a*b**2*c*x**2 + a*b**2*d*x**3),x)*a*b**2*c*d - i nt(1/(log((c*e + d*e*x)/(a + b*x))*a**2*b*c + log((c*e + d*e*x)/(a + b*x)) *a**2*b*d*x + 2*log((c*e + d*e*x)/(a + b*x))*a*b**2*c*x + 2*log((c*e + d*e *x)/(a + b*x))*a*b**2*d*x**2 + log((c*e + d*e*x)/(a + b*x))*b**3*c*x**2 + log((c*e + d*e*x)/(a + b*x))*b**3*d*x**3 + a**3*c + a**3*d*x + 2*a**2*b*c* x + 2*a**2*b*d*x**2 + a*b**2*c*x**2 + a*b**2*d*x**3),x)*b**3*c**2 + log(lo g((c*e + d*e*x)/(a + b*x))*b + a)*d)/(b**2*g**2*(a*d - b*c))