Integrand size = 32, antiderivative size = 65 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^2} \, dx=-\frac {2 B}{b g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) g^2 (a+b x)} \] Output:
-2*B/b/g^2/(b*x+a)-(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)/g^2/ (b*x+a)
Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.71 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^2} \, dx=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b g^2 (a+b x)}+\frac {2 B (b c-a d) \left (-\frac {1}{(b c-a d) (a+b x)}-\frac {d \log (a+b x)}{(b c-a d)^2}+\frac {d \log (c+d x)}{(b c-a d)^2}\right )}{b g^2} \] Input:
Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^2,x]
Output:
-((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(b*g^2*(a + b*x))) + (2*B*(b*c - a*d)*(-(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 = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.
\(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{(a g+b g x)^2} \, dx\) |
\(\Big \downarrow \) 2950 |
\(\displaystyle \frac {\int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 (b c-a d)}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle \frac {-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{a+b x}-\frac {2 B (c+d x)}{a+b x}}{g^2 (b c-a d)}\) |
Input:
Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^2,x]
Output:
((-2*B*(c + d*x))/(a + b*x) - ((c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d *x)^2]))/(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 = 1.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.43
method | result | size |
norman | \(\frac {\frac {\left (A +2 B \right ) x}{g a}+\frac {c B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (d a -b c \right ) g}+\frac {B d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (d a -b c \right ) g}}{g \left (b x +a \right )}\) | \(93\) |
parts | \(-\frac {A}{g^{2} \left (b x +a \right ) b}+\frac {\frac {2 B x}{a g}+\frac {c B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (d a -b c \right ) g}+\frac {B d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (d a -b c \right ) g}}{\left (b x +a \right ) g}\) | \(107\) |
parallelrisch | \(-\frac {-2 B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{2}-2 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c d +2 A a \,b^{2} d^{2}-2 A \,b^{3} c d +4 B a \,b^{2} d^{2}-4 B \,b^{3} c d}{2 g^{2} \left (b x +a \right ) b^{3} d \left (d a -b c \right )}\) | \(118\) |
risch | \(-\frac {B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{b \,g^{2} \left (b x +a \right )}-\frac {-2 B \ln \left (-b x -a \right ) b d x +2 B \ln \left (d x +c \right ) b d x -2 B \ln \left (-b x -a \right ) a d +2 B \ln \left (d x +c \right ) a d +A d a -A b c +2 B a d -2 B b c}{\left (b x +a \right ) g^{2} b \left (d a -b c \right )}\) | \(132\) |
derivativedivides | \(-\frac {-\frac {d^{2} A}{g^{2} \left (\frac {d a -b c}{d x +c}+b \right ) \left (d a -b c \right )}+\frac {\frac {2 d^{2} B}{b g \left (d x +c \right )}-\frac {d^{2} B \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (d a -b c \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}}{d}\) | \(145\) |
default | \(-\frac {-\frac {d^{2} A}{g^{2} \left (\frac {d a -b c}{d x +c}+b \right ) \left (d a -b c \right )}+\frac {\frac {2 d^{2} B}{b g \left (d x +c \right )}-\frac {d^{2} B \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (d a -b c \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}}{d}\) | \(145\) |
orering | \(\frac {3 \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right ) \left (b x +a \right ) \left (d x +c \right )}{\left (b g x +a g \right )^{2} \left (d a -b c \right )}+\frac {\left (b x +a \right )^{2} \left (d x +c \right ) \left (\frac {B \left (\frac {2 e \left (b x +a \right ) b}{\left (d x +c \right )^{2}}-\frac {2 e \left (b x +a \right )^{2} d}{\left (d x +c \right )^{3}}\right ) \left (d x +c \right )^{2}}{e \left (b x +a \right )^{2} \left (b g x +a g \right )^{2}}-\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right ) b g}{\left (b g x +a g \right )^{3}}\right )}{b \left (d a -b c \right )}\) | \(181\) |
Input:
int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^2,x,method=_RETURNVERBOSE)
Output:
((A+2*B)/g/a*x+c*B/(a*d-b*c)/g*ln(e*(b*x+a)^2/(d*x+c)^2)+B*d/(a*d-b*c)/g*x *ln(e*(b*x+a)^2/(d*x+c)^2))/g/(b*x+a)
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.69 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^2} \, dx=-\frac {{\left (A + 2 \, B\right )} b c - {\left (A + 2 \, B\right )} a d + {\left (B b d x + B b c\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^2,x, algorithm="fri cas")
Output:
-((A + 2*B)*b*c - (A + 2*B)*a*d + (B*b*d*x + B*b*c)*log((b^2*e*x^2 + 2*a*b *e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/((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. 255 vs. \(2 (54) = 108\).
Time = 0.66 (sec) , antiderivative size = 255, normalized size of antiderivative = 3.92 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^2} \, dx=- \frac {B \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{a b g^{2} + b^{2} g^{2} x} - \frac {2 B d \log {\left (x + \frac {- \frac {2 B a^{2} d^{3}}{a d - b c} + \frac {4 B a b c d^{2}}{a d - b c} + 2 B a d^{2} - \frac {2 B b^{2} c^{2} d}{a d - b c} + 2 B b c d}{4 B b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {2 B d \log {\left (x + \frac {\frac {2 B a^{2} d^{3}}{a d - b c} - \frac {4 B a b c d^{2}}{a d - b c} + 2 B a d^{2} + \frac {2 B b^{2} c^{2} d}{a d - b c} + 2 B b c d}{4 B b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {- A - 2 B}{a b g^{2} + b^{2} g^{2} x} \] Input:
integrate((A+B*ln(e*(b*x+a)**2/(d*x+c)**2))/(b*g*x+a*g)**2,x)
Output:
-B*log(e*(a + b*x)**2/(c + d*x)**2)/(a*b*g**2 + b**2*g**2*x) - 2*B*d*log(x + (-2*B*a**2*d**3/(a*d - b*c) + 4*B*a*b*c*d**2/(a*d - b*c) + 2*B*a*d**2 - 2*B*b**2*c**2*d/(a*d - b*c) + 2*B*b*c*d)/(4*B*b*d**2))/(b*g**2*(a*d - b*c )) + 2*B*d*log(x + (2*B*a**2*d**3/(a*d - b*c) - 4*B*a*b*c*d**2/(a*d - b*c) + 2*B*a*d**2 + 2*B*b**2*c**2*d/(a*d - b*c) + 2*B*b*c*d)/(4*B*b*d**2))/(b* g**2*(a*d - b*c)) + (-A - 2*B)/(a*b*g**2 + b**2*g**2*x)
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (65) = 130\).
Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.88 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^2} \, dx=-B {\left (\frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {2}{b^{2} g^{2} x + a b g^{2}} + \frac {2 \, d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {2 \, d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {A}{b^{2} g^{2} x + a b g^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^2,x, algorithm="max ima")
Output:
-B*(log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(b^2*g^2*x + a*b*g^2) + 2/(b^2* g^2*x + a*b*g^2) + 2*d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - 2*d*log(d*x + c)/((b^2*c - a*b*d)*g^2)) - A/(b^2*g^2*x + a*b*g^2)
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (65) = 130\).
Time = 0.16 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.88 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^2} \, dx={\left (2 \, {\left (b^{2} c g^{2} - a b d g^{2}\right )} {\left (\frac {d \log \left ({\left | \frac {b c g}{b g x + a g} - \frac {a d g}{b g x + a g} + d \right |}\right )}{b^{4} c^{2} g^{4} - 2 \, a b^{3} c d g^{4} + a^{2} b^{2} d^{2} g^{4}} - \frac {1}{{\left (b^{2} c g^{2} - a b d g^{2}\right )} {\left (b g x + a g\right )} b g}\right )} - \frac {\log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right )}{{\left (b g x + a g\right )} b g}\right )} B - \frac {A}{{\left (b g x + a g\right )} b g} \] Input:
integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^2,x, algorithm="gia c")
Output:
(2*(b^2*c*g^2 - a*b*d*g^2)*(d*log(abs(b*c*g/(b*g*x + a*g) - a*d*g/(b*g*x + a*g) + d))/(b^4*c^2*g^4 - 2*a*b^3*c*d*g^4 + a^2*b^2*d^2*g^4) - 1/((b^2*c* g^2 - a*b*d*g^2)*(b*g*x + a*g)*b*g)) - log((b*x + a)^2*e/(d*x + c)^2)/((b* g*x + a*g)*b*g))*B - A/((b*g*x + a*g)*b*g)
Time = 26.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.66 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^2} \, dx=-\frac {A+2\,B}{x\,b^2\,g^2+a\,b\,g^2}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B\,d\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \] Input:
int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))/(a*g + b*g*x)^2,x)
Output:
- (A + 2*B)/(b^2*g^2*x + a*b*g^2) - (B*log((e*(a + b*x)^2)/(c + d*x)^2))/( b^2*g^2*(x + a/b)) - (B*d*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*4i)/( b*g^2*(a*d - b*c))
Time = 0.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 3.05 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^2} \, dx=\frac {2 \,\mathrm {log}\left (b x +a \right ) a b c +2 \,\mathrm {log}\left (b x +a \right ) b^{2} c x -2 \,\mathrm {log}\left (d x +c \right ) a b c -2 \,\mathrm {log}\left (d x +c \right ) b^{2} c x +\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a b d x -\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) b^{2} c x +a^{2} d x -a b c x +2 a b d x -2 b^{2} c x}{a \,g^{2} \left (a b d x -b^{2} c x +a^{2} d -a b c \right )} \] Input:
int((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^2,x)
Output:
(2*log(a + b*x)*a*b*c + 2*log(a + b*x)*b**2*c*x - 2*log(c + d*x)*a*b*c - 2 *log(c + d*x)*b**2*c*x + log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2* c*d*x + d**2*x**2))*a*b*d*x - log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*b**2*c*x + a**2*d*x - a*b*c*x + 2*a*b*d*x - 2*b** 2*c*x)/(a*g**2*(a**2*d - a*b*c + a*b*d*x - b**2*c*x))