Integrand size = 33, antiderivative size = 102 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx=\frac {A (a+b x)}{(b c-a d) g^2 (c+d x)}-\frac {B n (a+b x)}{(b c-a d) g^2 (c+d x)}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) g^2 (c+d x)} \] Output:
A*(b*x+a)/(-a*d+b*c)/g^2/(d*x+c)-B*n*(b*x+a)/(-a*d+b*c)/g^2/(d*x+c)+B*(b*x +a)*ln(e*((b*x+a)/(d*x+c))^n)/(-a*d+b*c)/g^2/(d*x+c)
Time = 0.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d g (c g+d g x)}+\frac {B (b c-a d) n \left (\frac {1}{(b c-a d) (c+d x)}+\frac {b \log (a+b x)}{(b c-a d)^2}-\frac {b \log (c+d x)}{(b c-a d)^2}\right )}{d g^2} \] Input:
Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^2,x]
Output:
-((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(d*g*(c*g + d*g*x))) + (B*(b*c - a*d)*n*(1/((b*c - a*d)*(c + d*x)) + (b*Log[a + b*x])/(b*c - a*d)^2 - (b*Lo g[c + d*x])/(b*c - a*d)^2))/(d*g^2)
Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2951, 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.
\(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(c g+d g x)^2} \, dx\) |
\(\Big \downarrow \) 2951 |
\(\displaystyle \frac {\int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )d\frac {a+b x}{c+d x}}{g^2 (b c-a d)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}-\frac {B n (a+b x)}{c+d x}}{g^2 (b c-a d)}\) |
Input:
Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^2,x]
Output:
((A*(a + b*x))/(c + d*x) - (B*n*(a + b*x))/(c + d*x) + (B*(a + b*x)*Log[e* ((a + b*x)/(c + d*x))^n])/(c + d*x))/((b*c - a*d)*g^2)
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 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] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, - 1])
Time = 2.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {A}{g^{2} \left (d x +c \right ) d}-\frac {B \left (\frac {\left (b x +a \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{d x +c}-\frac {n \left (b x +a \right )}{d x +c}\right )}{g^{2} \left (d a -b c \right )}\) | \(81\) |
parts | \(-\frac {A}{g^{2} \left (d x +c \right ) d}-\frac {B \left (\frac {\left (b x +a \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{d x +c}-\frac {n \left (b x +a \right )}{d x +c}\right )}{g^{2} \left (d a -b c \right )}\) | \(81\) |
parallelrisch | \(-\frac {-B a b \,d^{3} n^{2}+B \,b^{2} c \,d^{2} n^{2}+A a b \,d^{3} n -A \,b^{2} c \,d^{2} n +B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{2} d^{3} n +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a b \,d^{3} n}{g^{2} \left (d x +c \right ) b \,d^{3} n \left (d a -b c \right )}\) | \(129\) |
Input:
int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x,method=_RETURNVERBOSE)
Output:
-1/g^2*A/(d*x+c)/d-1/g^2*B/(a*d-b*c)*((b*x+a)/(d*x+c)*ln(e*((b*x+a)/(d*x+c ))^n)-n*(b*x+a)/(d*x+c))
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.03 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx=-\frac {A b c - A a d - {\left (B b c - B a d\right )} n + {\left (B b c - B a d\right )} \log \left (e\right ) - {\left (B b d n x + B a d n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} g^{2} x + {\left (b c^{2} d - a c d^{2}\right )} g^{2}} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x, algorithm="fri cas")
Output:
-(A*b*c - A*a*d - (B*b*c - B*a*d)*n + (B*b*c - B*a*d)*log(e) - (B*b*d*n*x + B*a*d*n)*log((b*x + a)/(d*x + c)))/((b*c*d^2 - a*d^3)*g^2*x + (b*c^2*d - a*c*d^2)*g^2)
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (82) = 164\).
Time = 39.99 (sec) , antiderivative size = 444, normalized size of antiderivative = 4.35 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx=\begin {cases} - \frac {A}{c d g^{2} + d^{2} g^{2} x} - \frac {B \log {\left (e \left (\frac {b c}{c d + d^{2} x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c d g^{2} + d^{2} g^{2} x} & \text {for}\: a = \frac {b c}{d} \\\frac {A x + \frac {B a \log {\left (e \left (\frac {a}{c} + \frac {b x}{c}\right )^{n} \right )}}{b} - B n x + B x \log {\left (e \left (\frac {a}{c} + \frac {b x}{c}\right )^{n} \right )}}{c^{2} g^{2}} & \text {for}\: d = 0 \\- \frac {A a d}{a c d^{2} g^{2} + a d^{3} g^{2} x - b c^{2} d g^{2} - b c d^{2} g^{2} x} + \frac {A b c}{a c d^{2} g^{2} + a d^{3} g^{2} x - b c^{2} d g^{2} - b c d^{2} g^{2} x} + \frac {B a d n}{a c d^{2} g^{2} + a d^{3} g^{2} x - b c^{2} d g^{2} - b c d^{2} g^{2} x} - \frac {B a d \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a c d^{2} g^{2} + a d^{3} g^{2} x - b c^{2} d g^{2} - b c d^{2} g^{2} x} - \frac {B b c n}{a c d^{2} g^{2} + a d^{3} g^{2} x - b c^{2} d g^{2} - b c d^{2} g^{2} x} - \frac {B b d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a c d^{2} g^{2} + a d^{3} g^{2} x - b c^{2} d g^{2} - b c d^{2} g^{2} x} & \text {otherwise} \end {cases} \] Input:
integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(d*g*x+c*g)**2,x)
Output:
Piecewise((-A/(c*d*g**2 + d**2*g**2*x) - B*log(e*(b*c/(c*d + d**2*x) + b*x /(c + d*x))**n)/(c*d*g**2 + d**2*g**2*x), Eq(a, b*c/d)), ((A*x + B*a*log(e *(a/c + b*x/c)**n)/b - B*n*x + B*x*log(e*(a/c + b*x/c)**n))/(c**2*g**2), E q(d, 0)), (-A*a*d/(a*c*d**2*g**2 + a*d**3*g**2*x - b*c**2*d*g**2 - b*c*d** 2*g**2*x) + A*b*c/(a*c*d**2*g**2 + a*d**3*g**2*x - b*c**2*d*g**2 - b*c*d** 2*g**2*x) + B*a*d*n/(a*c*d**2*g**2 + a*d**3*g**2*x - b*c**2*d*g**2 - b*c*d **2*g**2*x) - B*a*d*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a*c*d**2*g**2 + a*d**3*g**2*x - b*c**2*d*g**2 - b*c*d**2*g**2*x) - B*b*c*n/(a*c*d**2*g* *2 + a*d**3*g**2*x - b*c**2*d*g**2 - b*c*d**2*g**2*x) - B*b*d*x*log(e*(a/( c + d*x) + b*x/(c + d*x))**n)/(a*c*d**2*g**2 + a*d**3*g**2*x - b*c**2*d*g* *2 - b*c*d**2*g**2*x), True))
Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.33 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx=B n {\left (\frac {1}{d^{2} g^{2} x + c d g^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} g^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} g^{2}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{d^{2} g^{2} x + c d g^{2}} - \frac {A}{d^{2} g^{2} x + c d g^{2}} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x, algorithm="max ima")
Output:
B*n*(1/(d^2*g^2*x + c*d*g^2) + b*log(b*x + a)/((b*c*d - a*d^2)*g^2) - b*lo g(d*x + c)/((b*c*d - a*d^2)*g^2)) - B*log(e*(b*x/(d*x + c) + a/(d*x + c))^ n)/(d^2*g^2*x + c*d*g^2) - A/(d^2*g^2*x + c*d*g^2)
Time = 0.39 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx={\left (\frac {{\left (b x + a\right )} B n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (d x + c\right )} g^{2}} - \frac {{\left (B n - B \log \left (e\right ) - A\right )} {\left (b x + a\right )}}{{\left (d x + c\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x, algorithm="gia c")
Output:
((b*x + a)*B*n*log((b*x + a)/(d*x + c))/((d*x + c)*g^2) - (B*n - B*log(e) - A)*(b*x + a)/((d*x + c)*g^2))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
Time = 25.69 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx=-\frac {A-B\,n}{x\,d^2\,g^2+c\,d\,g^2}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{d\,\left (c\,g^2+d\,g^2\,x\right )}+\frac {B\,b\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d\,g^2\,\left (a\,d-b\,c\right )} \] Input:
int((A + B*log(e*((a + b*x)/(c + d*x))^n))/(c*g + d*g*x)^2,x)
Output:
(B*b*n*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*2i)/(d*g^2*(a*d - b*c)) - (B*log(e*((a + b*x)/(c + d*x))^n))/(d*(c*g^2 + d*g^2*x)) - (A - B*n)/(d^ 2*g^2*x + c*d*g^2)
Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.54 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx=\frac {-\mathrm {log}\left (b x +a \right ) a b c n -\mathrm {log}\left (b x +a \right ) a b d n x +\mathrm {log}\left (d x +c \right ) a b c n +\mathrm {log}\left (d x +c \right ) a b d n x +\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a b d x -\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{2} c x +a^{2} d x -a b c x -a b d n x +b^{2} c n x}{c \,g^{2} \left (a \,d^{2} x -b c d x +a c d -b \,c^{2}\right )} \] Input:
int((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x)
Output:
( - log(a + b*x)*a*b*c*n - log(a + b*x)*a*b*d*n*x + log(c + d*x)*a*b*c*n + log(c + d*x)*a*b*d*n*x + log(((a + b*x)**n*e)/(c + d*x)**n)*a*b*d*x - log (((a + b*x)**n*e)/(c + d*x)**n)*b**2*c*x + a**2*d*x - a*b*c*x - a*b*d*n*x + b**2*c*n*x)/(c*g**2*(a*c*d + a*d**2*x - b*c**2 - b*c*d*x))