\(\int \frac {1}{(a g+b g x)^3 (A+B \log (\frac {e (c+d x)}{a+b x}))^2} \, dx\) [200]

Optimal result

Integrand size = 32, antiderivative size = 159 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\frac {d e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 (b c-a d)^2 e g^3}-\frac {2 b e^{-\frac {2 A}{B}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{B}\right )}{B^2 (b c-a d)^2 e^2 g^3}+\frac {c+d x}{B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \] Output:

d*Ei((A+B*ln(e*(d*x+c)/(b*x+a)))/B)/B^2/(-a*d+b*c)^2/e/exp(A/B)/g^3-2*b*Ei 
(2*(A+B*ln(e*(d*x+c)/(b*x+a)))/B)/B^2/(-a*d+b*c)^2/e^2/exp(2*A/B)/g^3+(d*x 
+c)/B/(-a*d+b*c)/g^3/(b*x+a)^2/(A+B*ln(e*(d*x+c)/(b*x+a)))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\frac {\frac {d e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A}{B}+\log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{e}-\frac {2 b e^{-\frac {2 A}{B}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{B}\right )}{e^2}+\frac {B (b c-a d) (c+d x)}{(a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}}{B^2 (b c-a d)^2 g^3} \] Input:

Integrate[1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2),x]
 

Output:

((d*ExpIntegralEi[A/B + Log[(e*(c + d*x))/(a + b*x)]])/(e*E^(A/B)) - (2*b* 
ExpIntegralEi[(2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/B])/(e^2*E^((2*A)/B 
)) + (B*(b*c - a*d)*(c + d*x))/((a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + 
b*x)])))/(B^2*(b*c - a*d)^2*g^3)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2952, 2757, 2736, 2609, 2767, 2009}

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.

\(\displaystyle \int \frac {1}{(a g+b g x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2} \, dx\)

\(\Big \downarrow \) 2952

\(\displaystyle \frac {\int \frac {d-\frac {b (c+d x)}{a+b x}}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}d\frac {c+d x}{a+b x}}{g^3 (b c-a d)^2}\)

\(\Big \downarrow \) 2757

\(\displaystyle \frac {-\frac {d \int \frac {1}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{B}+\frac {2 \int \frac {d-\frac {b (c+d x)}{a+b x}}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{B}-\frac {(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}{B (a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}}{g^3 (b c-a d)^2}\)

\(\Big \downarrow \) 2736

\(\displaystyle \frac {-\frac {d \int \frac {e (c+d x)}{(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}d\log \left (\frac {e (c+d x)}{a+b x}\right )}{B e}+\frac {2 \int \frac {d-\frac {b (c+d x)}{a+b x}}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{B}-\frac {(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}{B (a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}}{g^3 (b c-a d)^2}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\frac {2 \int \frac {d-\frac {b (c+d x)}{a+b x}}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{B}-\frac {d e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 e}-\frac {(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}{B (a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}}{g^3 (b c-a d)^2}\)

\(\Big \downarrow \) 2767

\(\displaystyle \frac {\frac {2 \int \left (\frac {d}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}-\frac {b (c+d x)}{(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}\right )d\frac {c+d x}{a+b x}}{B}-\frac {d e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 e}-\frac {(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}{B (a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}}{g^3 (b c-a d)^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {d e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 e}+\frac {2 \left (\frac {d e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B e}-\frac {b e^{-\frac {2 A}{B}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{B}\right )}{B e^2}\right )}{B}-\frac {(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}{B (a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}}{g^3 (b c-a d)^2}\)

Input:

Int[1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2),x]
 

Output:

(-((d*ExpIntegralEi[(A + B*Log[(e*(c + d*x))/(a + b*x)])/B])/(B^2*e*E^(A/B 
))) + (2*((d*ExpIntegralEi[(A + B*Log[(e*(c + d*x))/(a + b*x)])/B])/(B*e*E 
^(A/B)) - (b*ExpIntegralEi[(2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/B])/(B 
*e^2*E^((2*A)/B))))/B - ((c + d*x)*(d - (b*(c + d*x))/(a + b*x)))/(B*(a + 
b*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)])))/((b*c - a*d)^2*g^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 2736

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]
 

rule 2757

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x 
_Symbol] :> Simp[x*(d + e*x)^q*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), 
x] + (-Simp[(q + 1)/(b*n*(p + 1))   Int[(d + e*x)^q*(a + b*Log[c*x^n])^(p + 
 1), x], x] + Simp[d*(q/(b*n*(p + 1)))   Int[(d + e*x)^(q - 1)*(a + b*Log[c 
*x^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] && LtQ[p, -1] && Gt 
Q[q, 0]
 

rule 2767

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( 
q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x 
^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] 
&& IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
 

rule 2952

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])
 

Maple [A] (verified)

Time = 5.39 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.69

method	result	size
derivativedivides	\(-\frac {\frac {b \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-2 \,{\mathrm e}^{-\frac {2 A}{B}} \operatorname {expIntegral}_{1}\left (-2 \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )\right )}{B^{2}}-\frac {d e \left (-\frac {\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}}{\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )\right )}{B^{2}}}{e^{2} \left (d a -b c \right )^{2} g^{3}}\)	\(268\)
default	\(-\frac {\frac {b \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-2 \,{\mathrm e}^{-\frac {2 A}{B}} \operatorname {expIntegral}_{1}\left (-2 \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )\right )}{B^{2}}-\frac {d e \left (-\frac {\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}}{\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )\right )}{B^{2}}}{e^{2} \left (d a -b c \right )^{2} g^{3}}\)	\(268\)
risch	\(-\frac {d x +c}{\left (d a -b c \right ) B \left (b x +a \right )^{2} g^{3} \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )}-\frac {a \,d^{2} {\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \,g^{3} B^{2} \left (d a -b c \right )^{3}}+\frac {b c d \,{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \,g^{3} B^{2} \left (d a -b c \right )^{3}}+\frac {2 d b a \,{\mathrm e}^{-\frac {2 A}{B}} \operatorname {expIntegral}_{1}\left (-2 \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )}{e^{2} g^{3} B^{2} \left (d a -b c \right )^{3}}-\frac {2 c \,b^{2} {\mathrm e}^{-\frac {2 A}{B}} \operatorname {expIntegral}_{1}\left (-2 \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )}{e^{2} g^{3} B^{2} \left (d a -b c \right )^{3}}\)	\(338\)

Input:

int(1/(b*g*x+a*g)^3/(A+B*ln(e*(d*x+c)/(b*x+a)))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/e^2/(a*d-b*c)^2/g^3*(b/B^2*(-(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2/(ln(d*e/b- 
e*(a*d-b*c)/b/(b*x+a))+A/B)-2*exp(-2*A/B)*Ei(1,-2*ln(d*e/b-e*(a*d-b*c)/b/( 
b*x+a))-2*A/B))-d*e/B^2*(-(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(ln(d*e/b-e*(a*d-b 
*c)/b/(b*x+a))+A/B)-exp(-A/B)*Ei(1,-ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))-A/B)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (157) = 314\).

Time = 0.09 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.67 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\frac {{\left ({\left (B b c d - B a d^{2}\right )} e^{2} x + {\left (B b c^{2} - B a c d\right )} e^{2}\right )} e^{\left (\frac {2 \, A}{B}\right )} - 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b + {\left (B b^{3} x^{2} + 2 \, B a b^{2} x + B a^{2} b\right )} \log \left (\frac {d e x + c e}{b x + a}\right )\right )} \operatorname {log\_integral}\left (\frac {{\left (d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}\right )} e^{\left (\frac {2 \, A}{B}\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + {\left ({\left (B b^{2} d e x^{2} + 2 \, B a b d e x + B a^{2} d e\right )} e^{\frac {A}{B}} \log \left (\frac {d e x + c e}{b x + a}\right ) + {\left (A b^{2} d e x^{2} + 2 \, A a b d e x + A a^{2} d e\right )} e^{\frac {A}{B}}\right )} \operatorname {log\_integral}\left (\frac {{\left (d e x + c e\right )} e^{\frac {A}{B}}}{b x + a}\right )}{{\left ({\left (B^{3} b^{4} c^{2} - 2 \, B^{3} a b^{3} c d + B^{3} a^{2} b^{2} d^{2}\right )} e^{2} g^{3} x^{2} + 2 \, {\left (B^{3} a b^{3} c^{2} - 2 \, B^{3} a^{2} b^{2} c d + B^{3} a^{3} b d^{2}\right )} e^{2} g^{3} x + {\left (B^{3} a^{2} b^{2} c^{2} - 2 \, B^{3} a^{3} b c d + B^{3} a^{4} d^{2}\right )} e^{2} g^{3}\right )} e^{\left (\frac {2 \, A}{B}\right )} \log \left (\frac {d e x + c e}{b x + a}\right ) + {\left ({\left (A B^{2} b^{4} c^{2} - 2 \, A B^{2} a b^{3} c d + A B^{2} a^{2} b^{2} d^{2}\right )} e^{2} g^{3} x^{2} + 2 \, {\left (A B^{2} a b^{3} c^{2} - 2 \, A B^{2} a^{2} b^{2} c d + A B^{2} a^{3} b d^{2}\right )} e^{2} g^{3} x + {\left (A B^{2} a^{2} b^{2} c^{2} - 2 \, A B^{2} a^{3} b c d + A B^{2} a^{4} d^{2}\right )} e^{2} g^{3}\right )} e^{\left (\frac {2 \, A}{B}\right )}} \] Input:

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(d*x+c)/(b*x+a)))^2,x, algorithm="fri 
cas")
 

Output:

(((B*b*c*d - B*a*d^2)*e^2*x + (B*b*c^2 - B*a*c*d)*e^2)*e^(2*A/B) - 2*(A*b^ 
3*x^2 + 2*A*a*b^2*x + A*a^2*b + (B*b^3*x^2 + 2*B*a*b^2*x + B*a^2*b)*log((d 
*e*x + c*e)/(b*x + a)))*log_integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2) 
*e^(2*A/B)/(b^2*x^2 + 2*a*b*x + a^2)) + ((B*b^2*d*e*x^2 + 2*B*a*b*d*e*x + 
B*a^2*d*e)*e^(A/B)*log((d*e*x + c*e)/(b*x + a)) + (A*b^2*d*e*x^2 + 2*A*a*b 
*d*e*x + A*a^2*d*e)*e^(A/B))*log_integral((d*e*x + c*e)*e^(A/B)/(b*x + a)) 
)/(((B^3*b^4*c^2 - 2*B^3*a*b^3*c*d + B^3*a^2*b^2*d^2)*e^2*g^3*x^2 + 2*(B^3 
*a*b^3*c^2 - 2*B^3*a^2*b^2*c*d + B^3*a^3*b*d^2)*e^2*g^3*x + (B^3*a^2*b^2*c 
^2 - 2*B^3*a^3*b*c*d + B^3*a^4*d^2)*e^2*g^3)*e^(2*A/B)*log((d*e*x + c*e)/( 
b*x + a)) + ((A*B^2*b^4*c^2 - 2*A*B^2*a*b^3*c*d + A*B^2*a^2*b^2*d^2)*e^2*g 
^3*x^2 + 2*(A*B^2*a*b^3*c^2 - 2*A*B^2*a^2*b^2*c*d + A*B^2*a^3*b*d^2)*e^2*g 
^3*x + (A*B^2*a^2*b^2*c^2 - 2*A*B^2*a^3*b*c*d + A*B^2*a^4*d^2)*e^2*g^3)*e^ 
(2*A/B))
 

Sympy [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\frac {- c - d x}{A B a^{3} d g^{3} - A B a^{2} b c g^{3} + 2 A B a^{2} b d g^{3} x - 2 A B a b^{2} c g^{3} x + A B a b^{2} d g^{3} x^{2} - A B b^{3} c g^{3} x^{2} + \left (B^{2} a^{3} d g^{3} - B^{2} a^{2} b c g^{3} + 2 B^{2} a^{2} b d g^{3} x - 2 B^{2} a b^{2} c g^{3} x + B^{2} a b^{2} d g^{3} x^{2} - B^{2} b^{3} c g^{3} x^{2}\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}} - \frac {\int \left (- \frac {a d}{A a^{3} + 3 A a^{2} b x + 3 A a b^{2} x^{2} + A b^{3} x^{3} + B a^{3} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 3 B a^{2} b x \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 3 B a b^{2} x^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + B b^{3} x^{3} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\right )\, dx + \int \frac {2 b c}{A a^{3} + 3 A a^{2} b x + 3 A a b^{2} x^{2} + A b^{3} x^{3} + B a^{3} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 3 B a^{2} b x \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 3 B a b^{2} x^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + B b^{3} x^{3} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {b d x}{A a^{3} + 3 A a^{2} b x + 3 A a b^{2} x^{2} + A b^{3} x^{3} + B a^{3} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 3 B a^{2} b x \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 3 B a b^{2} x^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + B b^{3} x^{3} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx}{B g^{3} \left (a d - b c\right )} \] Input:

integrate(1/(b*g*x+a*g)**3/(A+B*ln(e*(d*x+c)/(b*x+a)))**2,x)
 

Output:

(-c - d*x)/(A*B*a**3*d*g**3 - A*B*a**2*b*c*g**3 + 2*A*B*a**2*b*d*g**3*x - 
2*A*B*a*b**2*c*g**3*x + A*B*a*b**2*d*g**3*x**2 - A*B*b**3*c*g**3*x**2 + (B 
**2*a**3*d*g**3 - B**2*a**2*b*c*g**3 + 2*B**2*a**2*b*d*g**3*x - 2*B**2*a*b 
**2*c*g**3*x + B**2*a*b**2*d*g**3*x**2 - B**2*b**3*c*g**3*x**2)*log(e*(c + 
 d*x)/(a + b*x))) - (Integral(-a*d/(A*a**3 + 3*A*a**2*b*x + 3*A*a*b**2*x** 
2 + A*b**3*x**3 + B*a**3*log(c*e/(a + b*x) + d*e*x/(a + b*x)) + 3*B*a**2*b 
*x*log(c*e/(a + b*x) + d*e*x/(a + b*x)) + 3*B*a*b**2*x**2*log(c*e/(a + b*x 
) + d*e*x/(a + b*x)) + B*b**3*x**3*log(c*e/(a + b*x) + d*e*x/(a + b*x))), 
x) + Integral(2*b*c/(A*a**3 + 3*A*a**2*b*x + 3*A*a*b**2*x**2 + A*b**3*x**3 
 + B*a**3*log(c*e/(a + b*x) + d*e*x/(a + b*x)) + 3*B*a**2*b*x*log(c*e/(a + 
 b*x) + d*e*x/(a + b*x)) + 3*B*a*b**2*x**2*log(c*e/(a + b*x) + d*e*x/(a + 
b*x)) + B*b**3*x**3*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x) + Integral(b 
*d*x/(A*a**3 + 3*A*a**2*b*x + 3*A*a*b**2*x**2 + A*b**3*x**3 + B*a**3*log(c 
*e/(a + b*x) + d*e*x/(a + b*x)) + 3*B*a**2*b*x*log(c*e/(a + b*x) + d*e*x/( 
a + b*x)) + 3*B*a*b**2*x**2*log(c*e/(a + b*x) + d*e*x/(a + b*x)) + B*b**3* 
x**3*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x))/(B*g**3*(a*d - b*c))
 

Maxima [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2}} \,d x } \] Input:

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(d*x+c)/(b*x+a)))^2,x, algorithm="max 
ima")
 

Output:

(d*x + c)/((a^2*b*c*g^3 - a^3*d*g^3)*A*B + (a^2*b*c*g^3*log(e) - a^3*d*g^3 
*log(e))*B^2 + ((b^3*c*g^3 - a*b^2*d*g^3)*A*B + (b^3*c*g^3*log(e) - a*b^2* 
d*g^3*log(e))*B^2)*x^2 + 2*((a*b^2*c*g^3 - a^2*b*d*g^3)*A*B + (a*b^2*c*g^3 
*log(e) - a^2*b*d*g^3*log(e))*B^2)*x - ((b^3*c*g^3 - a*b^2*d*g^3)*B^2*x^2 
+ 2*(a*b^2*c*g^3 - a^2*b*d*g^3)*B^2*x + (a^2*b*c*g^3 - a^3*d*g^3)*B^2)*log 
(b*x + a) + ((b^3*c*g^3 - a*b^2*d*g^3)*B^2*x^2 + 2*(a*b^2*c*g^3 - a^2*b*d* 
g^3)*B^2*x + (a^2*b*c*g^3 - a^3*d*g^3)*B^2)*log(d*x + c)) - integrate(-(b* 
d*x + 2*b*c - a*d)/(((b^4*c*g^3 - a*b^3*d*g^3)*A*B + (b^4*c*g^3*log(e) - a 
*b^3*d*g^3*log(e))*B^2)*x^3 + (a^3*b*c*g^3 - a^4*d*g^3)*A*B + (a^3*b*c*g^3 
*log(e) - a^4*d*g^3*log(e))*B^2 + 3*((a*b^3*c*g^3 - a^2*b^2*d*g^3)*A*B + ( 
a*b^3*c*g^3*log(e) - a^2*b^2*d*g^3*log(e))*B^2)*x^2 + 3*((a^2*b^2*c*g^3 - 
a^3*b*d*g^3)*A*B + (a^2*b^2*c*g^3*log(e) - a^3*b*d*g^3*log(e))*B^2)*x - (( 
b^4*c*g^3 - a*b^3*d*g^3)*B^2*x^3 + 3*(a*b^3*c*g^3 - a^2*b^2*d*g^3)*B^2*x^2 
 + 3*(a^2*b^2*c*g^3 - a^3*b*d*g^3)*B^2*x + (a^3*b*c*g^3 - a^4*d*g^3)*B^2)* 
log(b*x + a) + ((b^4*c*g^3 - a*b^3*d*g^3)*B^2*x^3 + 3*(a*b^3*c*g^3 - a^2*b 
^2*d*g^3)*B^2*x^2 + 3*(a^2*b^2*c*g^3 - a^3*b*d*g^3)*B^2*x + (a^3*b*c*g^3 - 
 a^4*d*g^3)*B^2)*log(d*x + c)), x)
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx={\left (\frac {d e {\rm Ei}\left (\frac {A}{B} + \log \left (\frac {d e x + c e}{b x + a}\right )\right ) e^{\left (-\frac {A}{B}\right )}}{B^{2} b c e g^{3} - B^{2} a d e g^{3}} - \frac {2 \, b {\rm Ei}\left (\frac {2 \, A}{B} + 2 \, \log \left (\frac {d e x + c e}{b x + a}\right )\right ) e^{\left (-\frac {2 \, A}{B}\right )}}{B^{2} b c e g^{3} - B^{2} a d e g^{3}} - \frac {\frac {{\left (d e x + c e\right )} d e}{b x + a} - \frac {{\left (d e x + c e\right )}^{2} b}{{\left (b x + a\right )}^{2}}}{B^{2} b c e g^{3} \log \left (\frac {d e x + c e}{b x + a}\right ) - B^{2} a d e g^{3} \log \left (\frac {d e x + c e}{b x + a}\right ) + A B b c e g^{3} - A B a d e g^{3}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(d*x+c)/(b*x+a)))^2,x, algorithm="gia 
c")
 

Output:

(d*e*Ei(A/B + log((d*e*x + c*e)/(b*x + a)))*e^(-A/B)/(B^2*b*c*e*g^3 - B^2* 
a*d*e*g^3) - 2*b*Ei(2*A/B + 2*log((d*e*x + c*e)/(b*x + a)))*e^(-2*A/B)/(B^ 
2*b*c*e*g^3 - B^2*a*d*e*g^3) - ((d*e*x + c*e)*d*e/(b*x + a) - (d*e*x + c*e 
)^2*b/(b*x + a)^2)/(B^2*b*c*e*g^3*log((d*e*x + c*e)/(b*x + a)) - B^2*a*d*e 
*g^3*log((d*e*x + c*e)/(b*x + a)) + A*B*b*c*e*g^3 - A*B*a*d*e*g^3))*(b*c/( 
(b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\int \frac {1}{{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2} \,d x \] Input:

int(1/((a*g + b*g*x)^3*(A + B*log((e*(c + d*x))/(a + b*x)))^2),x)
 

Output:

int(1/((a*g + b*g*x)^3*(A + B*log((e*(c + d*x))/(a + b*x)))^2), x)
 

Reduce [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\frac {\int \frac {1}{\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right )^{2} a^{3} b^{2}+3 \mathrm {log}\left (\frac {d e x +c e}{b x +a}\right )^{2} a^{2} b^{3} x +3 \mathrm {log}\left (\frac {d e x +c e}{b x +a}\right )^{2} a \,b^{4} x^{2}+\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right )^{2} b^{5} x^{3}+2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{4} b +6 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{3} b^{2} x +6 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b^{3} x^{2}+2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{4} x^{3}+a^{5}+3 a^{4} b x +3 a^{3} b^{2} x^{2}+a^{2} b^{3} x^{3}}d x}{g^{3}} \] Input:

int(1/(b*g*x+a*g)^3/(A+B*log(e*(d*x+c)/(b*x+a)))^2,x)
 

Output:

int(1/(log((c*e + d*e*x)/(a + b*x))**2*a**3*b**2 + 3*log((c*e + d*e*x)/(a 
+ b*x))**2*a**2*b**3*x + 3*log((c*e + d*e*x)/(a + b*x))**2*a*b**4*x**2 + l 
og((c*e + d*e*x)/(a + b*x))**2*b**5*x**3 + 2*log((c*e + d*e*x)/(a + b*x))* 
a**4*b + 6*log((c*e + d*e*x)/(a + b*x))*a**3*b**2*x + 6*log((c*e + d*e*x)/ 
(a + b*x))*a**2*b**3*x**2 + 2*log((c*e + d*e*x)/(a + b*x))*a*b**4*x**3 + a 
**5 + 3*a**4*b*x + 3*a**3*b**2*x**2 + a**2*b**3*x**3),x)/g**3