Integrand size = 30, antiderivative size = 63 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx=-\frac {B}{b g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) g^2 (a+b x)} \] Output:
-B/b/g^2/(b*x+a)-(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)/g^2/(b*x+a )
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.67 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx=\frac {-A b c-b B c+a A d+a B d-B d (a+b x) \log (a+b x)+(-b B c+a B d) \log \left (\frac {e (a+b x)}{c+d x}\right )+a B d \log (c+d x)+b B d x \log (c+d x)}{b (b c-a d) g^2 (a+b x)} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^2,x]
Output:
(-(A*b*c) - b*B*c + a*A*d + a*B*d - B*d*(a + b*x)*Log[a + b*x] + (-(b*B*c) + a*B*d)*Log[(e*(a + b*x))/(c + d*x)] + a*B*d*Log[c + d*x] + b*B*d*x*Log[ c + d*x])/(b*(b*c - a*d)*g^2*(a + b*x))
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2950, 2741}
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.
\(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(a g+b g x)^2} \, dx\) |
\(\Big \downarrow \) 2950 |
\(\displaystyle \frac {\int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 (b c-a d)}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle \frac {-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {B (c+d x)}{a+b x}}{g^2 (b c-a d)}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^2,x]
Output:
(-((B*(c + d*x))/(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x )]))/(a + b*x))/((b*c - a*d)*g^2)
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
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/b)^m Subst[Int[x^m*((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] && E qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
Time = 0.87 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.38
method | result | size |
norman | \(\frac {\frac {\left (A +B \right ) x}{g a}+\frac {c B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (d a -b c \right ) g}+\frac {B d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (d a -b c \right ) g}}{g \left (b x +a \right )}\) | \(87\) |
parallelrisch | \(-\frac {A a \,b^{2} d^{2}-A \,b^{3} c d +B a \,b^{2} d^{2}-B \,b^{3} c d -B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{2}-B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c d}{g^{2} \left (b x +a \right ) b^{3} d \left (d a -b c \right )}\) | \(112\) |
parts | \(-\frac {A}{g^{2} \left (b x +a \right ) b}-\frac {B e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{g^{2} \left (d a -b c \right )}\) | \(126\) |
risch | \(-\frac {B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{b \,g^{2} \left (b x +a \right )}-\frac {-B \ln \left (-b x -a \right ) b d x +B \ln \left (d x +c \right ) b d x -B \ln \left (-b x -a \right ) a d +B \ln \left (d x +c \right ) a d +A d a -A b c +B a d -B b c}{\left (b x +a \right ) g^{2} b \left (d a -b c \right )}\) | \(127\) |
orering | \(\frac {3 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) \left (b x +a \right ) \left (d x +c \right )}{\left (b g x +a g \right )^{2} \left (d a -b c \right )}+\frac {\left (b x +a \right )^{2} \left (d x +c \right ) \left (\frac {B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{2}}-\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{3}}\right )}{b \left (d a -b c \right )}\) | \(167\) |
derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A}{\left (d a -b c \right )^{2} g^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{2} g^{2}}\right )}{d^{2}}\) | \(173\) |
default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A}{\left (d a -b c \right )^{2} g^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{2} g^{2}}\right )}{d^{2}}\) | \(173\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x,method=_RETURNVERBOSE)
Output:
((A+B)/g/a*x+c*B/(a*d-b*c)/g*ln(e*(b*x+a)/(d*x+c))+B*d/(a*d-b*c)/g*x*ln(e* (b*x+a)/(d*x+c)))/g/(b*x+a)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.32 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx=-\frac {{\left (A + B\right )} b c - {\left (A + B\right )} a d + {\left (B b d x + B b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="fricas" )
Output:
-((A + B)*b*c - (A + B)*a*d + (B*b*d*x + B*b*c)*log((b*e*x + a*e)/(d*x + c )))/((b^3*c - a*b^2*d)*g^2*x + (a*b^2*c - a^2*b*d)*g^2)
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (49) = 98\).
Time = 0.67 (sec) , antiderivative size = 233, normalized size of antiderivative = 3.70 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx=- \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a b g^{2} + b^{2} g^{2} x} - \frac {B d \log {\left (x + \frac {- \frac {B a^{2} d^{3}}{a d - b c} + \frac {2 B a b c d^{2}}{a d - b c} + B a d^{2} - \frac {B b^{2} c^{2} d}{a d - b c} + B b c d}{2 B b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {B d \log {\left (x + \frac {\frac {B a^{2} d^{3}}{a d - b c} - \frac {2 B a b c d^{2}}{a d - b c} + B a d^{2} + \frac {B b^{2} c^{2} d}{a d - b c} + B b c d}{2 B b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {- A - B}{a b g^{2} + b^{2} g^{2} x} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**2,x)
Output:
-B*log(e*(a + b*x)/(c + d*x))/(a*b*g**2 + b**2*g**2*x) - B*d*log(x + (-B*a **2*d**3/(a*d - b*c) + 2*B*a*b*c*d**2/(a*d - b*c) + B*a*d**2 - B*b**2*c**2 *d/(a*d - b*c) + B*b*c*d)/(2*B*b*d**2))/(b*g**2*(a*d - b*c)) + B*d*log(x + (B*a**2*d**3/(a*d - b*c) - 2*B*a*b*c*d**2/(a*d - b*c) + B*a*d**2 + B*b**2 *c**2*d/(a*d - b*c) + B*b*c*d)/(2*B*b*d**2))/(b*g**2*(a*d - b*c)) + (-A - B)/(a*b*g**2 + b**2*g**2*x)
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (63) = 126\).
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.10 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx=-B {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {A}{b^{2} g^{2} x + a b g^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="maxima" )
Output:
-B*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^2*g^2*x + a*b*g^2) + 1/(b^2*g^ 2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b ^2*c - a*b*d)*g^2)) - A/(b^2*g^2*x + a*b*g^2)
Time = 0.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.98 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx=-{\left (\frac {{\left (d x + c\right )} B e^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )} g^{2}} + \frac {{\left (A e^{2} + B e^{2}\right )} {\left (d x + c\right )}}{{\left (b e x + a e\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="giac")
Output:
-((d*x + c)*B*e^2*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)*g^2) + (A*e^ 2 + B*e^2)*(d*x + c)/((b*e*x + a*e)*g^2))*(b*c/((b*c*e - a*d*e)*(b*c - a*d )) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
Time = 27.49 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.65 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx=-\frac {A+B}{x\,b^2\,g^2+a\,b\,g^2}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B\,d\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))/(a*g + b*g*x)^2,x)
Output:
- (A + B)/(b^2*g^2*x + a*b*g^2) - (B*log((e*(a + b*x))/(c + d*x)))/(b^2*g^ 2*(x + a/b)) - (B*d*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*2i)/(b*g^2* (a*d - b*c))
Time = 0.15 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.37 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx=\frac {\mathrm {log}\left (b x +a \right ) a b c +\mathrm {log}\left (b x +a \right ) b^{2} c x -\mathrm {log}\left (d x +c \right ) a b c -\mathrm {log}\left (d x +c \right ) b^{2} c x +\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a b d x -\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{2} c x +a^{2} d x -a b c x +a b d x -b^{2} c x}{a \,g^{2} \left (a b d x -b^{2} c x +a^{2} d -a b c \right )} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x)
Output:
(log(a + b*x)*a*b*c + log(a + b*x)*b**2*c*x - log(c + d*x)*a*b*c - log(c + d*x)*b**2*c*x + log((a*e + b*e*x)/(c + d*x))*a*b*d*x - log((a*e + b*e*x)/ (c + d*x))*b**2*c*x + a**2*d*x - a*b*c*x + a*b*d*x - b**2*c*x)/(a*g**2*(a* *2*d - a*b*c + a*b*d*x - b**2*c*x))