Integrand size = 30, antiderivative size = 98 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^2} \, dx=\frac {A (a+b x)}{(b c-a d) i^2 (c+d x)}-\frac {B (a+b x)}{(b c-a d) i^2 (c+d x)}+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d) i^2 (c+d x)} \] Output:
A*(b*x+a)/(-a*d+b*c)/i^2/(d*x+c)-B*(b*x+a)/(-a*d+b*c)/i^2/(d*x+c)+B*(b*x+a )*ln(e*(b*x+a)/(d*x+c))/(-a*d+b*c)/i^2/(d*x+c)
Time = 0.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.06 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^2} \, dx=\frac {A b c-b B c-a A d+a B d-b B (c+d x) \log (a+b x)+B (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+b B c \log (c+d x)+b B d x \log (c+d x)}{d (-b c+a d) i^2 (c+d x)} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(c*i + d*i*x)^2,x]
Output:
(A*b*c - b*B*c - a*A*d + a*B*d - b*B*(c + d*x)*Log[a + b*x] + B*(b*c - a*d )*Log[(e*(a + b*x))/(c + d*x)] + b*B*c*Log[c + d*x] + b*B*d*x*Log[c + d*x] )/(d*(-(b*c) + a*d)*i^2*(c + d*x))
Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2952, 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.
\(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(c i+d i x)^2} \, dx\) |
\(\Big \downarrow \) 2952 |
\(\displaystyle \frac {\int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )d\frac {a+b x}{c+d x}}{i^2 (b c-a d)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}-\frac {B (a+b x)}{c+d x}}{i^2 (b c-a d)}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(c*i + d*i*x)^2,x]
Output:
((A*(a + b*x))/(c + d*x) - (B*(a + b*x))/(c + d*x) + (B*(a + b*x)*Log[(e*( a + b*x))/(c + d*x)])/(c + d*x))/((b*c - a*d)*i^2)
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 = 1.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84
method | result | size |
parts | \(-\frac {A}{i^{2} \left (d x +c \right ) d}-\frac {B \left (\frac {\ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) e \left (b x +a \right )}{d x +c}-\frac {e \left (b x +a \right )}{d x +c}\right )}{i^{2} e \left (d a -b c \right )}\) | \(82\) |
norman | \(\frac {\frac {\left (A -B \right ) x}{i c}-\frac {a B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (d a -b c \right ) i}-\frac {B b x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (d a -b c \right ) i}}{i \left (d x +c \right )}\) | \(91\) |
parallelrisch | \(-\frac {-B a b \,d^{3}+B \,b^{2} c \,d^{2}+A a b \,d^{3}-A \,b^{2} c \,d^{2}+B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} d^{3}+B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a b \,d^{3}}{i^{2} \left (d x +c \right ) b \,d^{3} \left (d a -b c \right )}\) | \(110\) |
risch | \(-\frac {B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{d \,i^{2} \left (d x +c \right )}-\frac {B \ln \left (b x +a \right ) b d x -B \ln \left (-d x -c \right ) b d x +B \ln \left (b x +a \right ) b c -B \ln \left (-d x -c \right ) b c +A d a -A b c -B a d +B b c}{i^{2} \left (d x +c \right ) d \left (d a -b c \right )}\) | \(127\) |
orering | \(-\frac {3 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) \left (b x +a \right ) \left (d x +c \right )}{\left (d i x +c i \right )^{2} \left (d a -b c \right )}-\frac {\left (b x +a \right ) \left (d x +c \right )^{2} \left (\frac {B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (d i x +c i \right )^{2}}-\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) d i}{\left (d i x +c i \right )^{3}}\right )}{\left (d a -b c \right ) d}\) | \(168\) |
derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}+\frac {d^{2} B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}\right )}{d^{2}}\) | \(170\) |
default | \(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}+\frac {d^{2} B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}\right )}{d^{2}}\) | \(170\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x,method=_RETURNVERBOSE)
Output:
-A/i^2/(d*x+c)/d-B/i^2/e/(a*d-b*c)*(ln(e*(b*x+a)/(d*x+c))*e*(b*x+a)/(d*x+c )-e*(b*x+a)/(d*x+c))
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^2} \, dx=-\frac {{\left (A - B\right )} b c - {\left (A - B\right )} a d - {\left (B b d x + B a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} i^{2} x + {\left (b c^{2} d - a c d^{2}\right )} i^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x, algorithm="fricas" )
Output:
-((A - B)*b*c - (A - B)*a*d - (B*b*d*x + B*a*d)*log((b*e*x + a*e)/(d*x + c )))/((b*c*d^2 - a*d^3)*i^2*x + (b*c^2*d - a*c*d^2)*i^2)
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (78) = 156\).
Time = 0.63 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.36 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^2} \, dx=\frac {B b \log {\left (x + \frac {- \frac {B a^{2} b d^{2}}{a d - b c} + \frac {2 B a b^{2} c d}{a d - b c} + B a b d - \frac {B b^{3} c^{2}}{a d - b c} + B b^{2} c}{2 B b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac {B b \log {\left (x + \frac {\frac {B a^{2} b d^{2}}{a d - b c} - \frac {2 B a b^{2} c d}{a d - b c} + B a b d + \frac {B b^{3} c^{2}}{a d - b c} + B b^{2} c}{2 B b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{c d i^{2} + d^{2} i^{2} x} + \frac {- A + B}{c d i^{2} + d^{2} i^{2} x} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)**2,x)
Output:
B*b*log(x + (-B*a**2*b*d**2/(a*d - b*c) + 2*B*a*b**2*c*d/(a*d - b*c) + B*a *b*d - B*b**3*c**2/(a*d - b*c) + B*b**2*c)/(2*B*b**2*d))/(d*i**2*(a*d - b* c)) - B*b*log(x + (B*a**2*b*d**2/(a*d - b*c) - 2*B*a*b**2*c*d/(a*d - b*c) + B*a*b*d + B*b**3*c**2/(a*d - b*c) + B*b**2*c)/(2*B*b**2*d))/(d*i**2*(a*d - b*c)) - B*log(e*(a + b*x)/(c + d*x))/(c*d*i**2 + d**2*i**2*x) + (-A + B )/(c*d*i**2 + d**2*i**2*x)
Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.37 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^2} \, dx=-B {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac {1}{d^{2} i^{2} x + c d i^{2}} - \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} + \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} - \frac {A}{d^{2} i^{2} x + c d i^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x, algorithm="maxima" )
Output:
-B*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^2*i^2*x + c*d*i^2) - 1/(d^2*i^ 2*x + c*d*i^2) - b*log(b*x + a)/((b*c*d - a*d^2)*i^2) + b*log(d*x + c)/((b *c*d - a*d^2)*i^2)) - A/(d^2*i^2*x + c*d*i^2)
Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.17 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^2} \, dx={\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} {\left (\frac {{\left (b e x + a e\right )} B \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (d x + c\right )} i^{2}} + \frac {{\left (b e x + a e\right )} {\left (A - B\right )}}{{\left (d x + c\right )} i^{2}}\right )} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x, algorithm="giac")
Output:
(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))*(( b*e*x + a*e)*B*log((b*e*x + a*e)/(d*x + c))/((d*x + c)*i^2) + (b*e*x + a*e )*(A - B)/((d*x + c)*i^2))
Time = 26.59 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^2} \, dx=-\frac {A-B}{x\,d^2\,i^2+c\,d\,i^2}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{d^2\,i^2\,\left (x+\frac {c}{d}\right )}+\frac {B\,b\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d\,i^2\,\left (a\,d-b\,c\right )} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))/(c*i + d*i*x)^2,x)
Output:
(B*b*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*2i)/(d*i^2*(a*d - b*c)) - (B*log((e*(a + b*x))/(c + d*x)))/(d^2*i^2*(x + c/d)) - (A - B)/(d^2*i^2*x + c*d*i^2)
Time = 0.24 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.47 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^2} \, dx=\frac {\mathrm {log}\left (b x +a \right ) a b c +\mathrm {log}\left (b x +a \right ) a b d x -\mathrm {log}\left (d x +c \right ) a b c -\mathrm {log}\left (d x +c \right ) a b d x -\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a b d x +\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{2} c x -a^{2} d x +a b c x +a b d x -b^{2} c x}{c \left (a \,d^{2} x -b c d x +a c d -b \,c^{2}\right )} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x)
Output:
(log(a + b*x)*a*b*c + log(a + b*x)*a*b*d*x - log(c + d*x)*a*b*c - log(c + d*x)*a*b*d*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)/(c*(a*c*d + a *d**2*x - b*c**2 - b*c*d*x))