Integrand size = 33, antiderivative size = 67 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx=-\frac {B n}{b g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) g^2 (a+b x)} \] Output:
-B*n/b/g^2/(b*x+a)-(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)/g^2/ (b*x+a)
Time = 0.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.72 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b g (a g+b g x)}+\frac {B (b c-a d) n \left (-\frac {1}{(b c-a d) (a+b x)}-\frac {d \log (a+b x)}{(b c-a d)^2}+\frac {d \log (c+d x)}{(b c-a d)^2}\right )}{b g^2} \] Input:
Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^2,x]
Output:
-((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(b*g*(a*g + b*g*x))) + (B*(b*c - a*d)*n*(-(1/((b*c - a*d)*(a + b*x))) - (d*Log[a + b*x])/(b*c - a*d)^2 + (d *Log[c + d*x])/(b*c - a*d)^2))/(b*g^2)
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2949, 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.
\(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(a g+b g x)^2} \, dx\) |
\(\Big \downarrow \) 2949 |
\(\displaystyle \frac {\int \frac {(c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 (b c-a d)}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle \frac {-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {B n (c+d x)}{a+b x}}{g^2 (b c-a d)}\) |
Input:
Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^2,x]
Output:
(-((B*n*(c + d*x))/(a + b*x)) - ((c + d*x)*(A + B*Log[e*((a + b*x)/(c + d* x))^n]))/(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_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 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] && Ne Q[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || Lt Q[m, -1])
Time = 2.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.96
method | result | size |
parallelrisch | \(-\frac {B a \,b^{2} d^{2} n^{2}-B \,b^{3} c d \,n^{2}+A a \,b^{2} d^{2} n -A \,b^{3} c d n -B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{3} d^{2} n -B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{3} c d n}{g^{2} \left (b x +a \right ) b^{3} d n \left (d a -b c \right )}\) | \(131\) |
Input:
int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2,x,method=_RETURNVERBOSE)
Output:
-(B*a*b^2*d^2*n^2-B*b^3*c*d*n^2+A*a*b^2*d^2*n-A*b^3*c*d*n-B*x*ln(e*((b*x+a )/(d*x+c))^n)*b^3*d^2*n-B*ln(e*((b*x+a)/(d*x+c))^n)*b^3*c*d*n)/g^2/(b*x+a) /b^3/d/n/(a*d-b*c)
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.54 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx=-\frac {A b c - A a d + {\left (B b c - B a d\right )} n + {\left (B b c - B a d\right )} \log \left (e\right ) + {\left (B b d n x + B b c n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*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*b*c*n)*log((b*x + a)/(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. 468 vs. \(2 (53) = 106\).
Time = 32.56 (sec) , antiderivative size = 468, normalized size of antiderivative = 6.99 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx=\begin {cases} \frac {\tilde {\infty } \left (A + B \log {\left (0^{n} e \right )}\right )}{g^{2} x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {A d}{b^{2} c g^{2} + b^{2} d g^{2} x} - \frac {B d \log {\left (e \left (\frac {b c}{c d + d^{2} x} + \frac {b x}{c + d x}\right )^{n} \right )}}{b^{2} c g^{2} + b^{2} d g^{2} x} & \text {for}\: a = \frac {b c}{d} \\\frac {A x + \frac {B c \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )}}{d} + B n x + B x \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )}}{a^{2} g^{2}} & \text {for}\: b = 0 \\- \frac {A a d}{a^{2} b d g^{2} - a b^{2} c g^{2} + a b^{2} d g^{2} x - b^{3} c g^{2} x} + \frac {A b c}{a^{2} b d g^{2} - a b^{2} c g^{2} + a b^{2} d g^{2} x - b^{3} c g^{2} x} - \frac {B a d n}{a^{2} b d g^{2} - a b^{2} c g^{2} + a b^{2} d g^{2} x - b^{3} c g^{2} x} + \frac {B b c n}{a^{2} b d g^{2} - a b^{2} c g^{2} + a b^{2} d g^{2} x - b^{3} c g^{2} x} + \frac {B b c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{2} b d g^{2} - a b^{2} c g^{2} + a b^{2} d g^{2} x - b^{3} c g^{2} x} + \frac {B b d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{2} b d g^{2} - a b^{2} c g^{2} + a b^{2} d g^{2} x - b^{3} c g^{2} x} & \text {otherwise} \end {cases} \] Input:
integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**2,x)
Output:
Piecewise((zoo*(A + B*log(0**n*e))/(g**2*x), Eq(a, 0) & Eq(b, 0)), (-A*d/( b**2*c*g**2 + b**2*d*g**2*x) - B*d*log(e*(b*c/(c*d + d**2*x) + b*x/(c + d* x))**n)/(b**2*c*g**2 + b**2*d*g**2*x), Eq(a, b*c/d)), ((A*x + B*c*log(e*(a /(c + d*x))**n)/d + B*n*x + B*x*log(e*(a/(c + d*x))**n))/(a**2*g**2), Eq(b , 0)), (-A*a*d/(a**2*b*d*g**2 - a*b**2*c*g**2 + a*b**2*d*g**2*x - b**3*c*g **2*x) + A*b*c/(a**2*b*d*g**2 - a*b**2*c*g**2 + a*b**2*d*g**2*x - b**3*c*g **2*x) - B*a*d*n/(a**2*b*d*g**2 - a*b**2*c*g**2 + a*b**2*d*g**2*x - b**3*c *g**2*x) + B*b*c*n/(a**2*b*d*g**2 - a*b**2*c*g**2 + a*b**2*d*g**2*x - b**3 *c*g**2*x) + B*b*c*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a**2*b*d*g**2 - a*b**2*c*g**2 + a*b**2*d*g**2*x - b**3*c*g**2*x) + B*b*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a**2*b*d*g**2 - a*b**2*c*g**2 + a*b**2*d*g**2* x - b**3*c*g**2*x), True))
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (67) = 134\).
Time = 0.04 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.04 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx=-B n {\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac {A}{b^{2} g^{2} x + a b g^{2}} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2,x, algorithm="max ima")
Output:
-B*n*(1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*l og(d*x + c)/((b^2*c - a*b*d)*g^2)) - B*log(e*(b*x/(d*x + c) + a/(d*x + c)) ^n)/(b^2*g^2*x + a*b*g^2) - A/(b^2*g^2*x + a*b*g^2)
Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx=-{\left (\frac {{\left (d x + c\right )} B n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )} g^{2}} + \frac {{\left (B n + B \log \left (e\right ) + A\right )} {\left (d x + c\right )}}{{\left (b x + a\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2,x, algorithm="gia c")
Output:
-((d*x + c)*B*n*log((b*x + a)/(d*x + c))/((b*x + a)*g^2) + (B*n + B*log(e) + A)*(d*x + c)/((b*x + a)*g^2))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
Time = 27.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.67 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx=-\frac {A+B\,n}{x\,b^2\,g^2+a\,b\,g^2}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{b\,\left (a\,g^2+b\,g^2\,x\right )}-\frac {B\,d\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \] Input:
int((A + B*log(e*((a + b*x)/(c + d*x))^n))/(a*g + b*g*x)^2,x)
Output:
- (A + B*n)/(b^2*g^2*x + a*b*g^2) - (B*log(e*((a + b*x)/(c + d*x))^n))/(b* (a*g^2 + b*g^2*x)) - (B*d*n*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 = 159, normalized size of antiderivative = 2.37 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2} \, dx=\frac {\mathrm {log}\left (b x +a \right ) a b c n +\mathrm {log}\left (b x +a \right ) b^{2} c n x -\mathrm {log}\left (d x +c \right ) a b c n -\mathrm {log}\left (d x +c \right ) b^{2} c n x +\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a b d x -\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{2} c x +a^{2} d x -a b c x +a b d n x -b^{2} c n x}{a \,g^{2} \left (a b d x -b^{2} c x +a^{2} d -a b c \right )} \] Input:
int((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2,x)
Output:
(log(a + b*x)*a*b*c*n + log(a + b*x)*b**2*c*n*x - log(c + d*x)*a*b*c*n - l og(c + d*x)*b**2*c*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)/(a*g**2*(a**2*d - a*b*c + a*b*d*x - b**2*c*x))