Integrand size = 42, antiderivative size = 44 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d) g i} \] Output:
1/3*(A+B*ln(e*(b*x+a)/(d*x+c)))^3/B/(-a*d+b*c)/g/i
Time = 0.40 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.80 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {3 A^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A B \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+B^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )}{3 b c g i-3 a d g i} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)*(c*i + d*i *x)),x]
Output:
(3*A^2*Log[(e*(a + b*x))/(c + d*x)] + 3*A*B*Log[(e*(a + b*x))/(c + d*x)]^2 + B^2*Log[(e*(a + b*x))/(c + d*x)]^3)/(3*b*c*g*i - 3*a*d*g*i)
Time = 0.36 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 2739, 15}
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.
\(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x) (c i+d i x)} \, dx\) |
\(\Big \downarrow \) 2962 |
\(\displaystyle \frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}}{g i (b c-a d)}\) |
\(\Big \downarrow \) 2739 |
\(\displaystyle \frac {\int \frac {(a+b x)^2}{(c+d x)^2}d\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B g i (b c-a d)}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {(a+b x)^3}{3 B g i (c+d x)^3 (b c-a d)}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)*(c*i + d*i*x)),x ]
Output:
(a + b*x)^3/(3*B*(b*c - a*d)*g*i*(c + d*x)^3)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
Leaf count of result is larger than twice the leaf count of optimal. \(105\) vs. \(2(42)=84\).
Time = 1.50 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.41
method | result | size |
parallelrisch | \(-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} b^{2} d^{2}+3 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{2} d^{2}+3 A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} d^{2}}{3 b^{2} d^{2} g i \left (d a -b c \right )}\) | \(106\) |
norman | \(-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (d a -b c \right )}-\frac {A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (d a -b c \right )}-\frac {A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (d a -b c \right )}\) | \(113\) |
parts | \(\frac {A^{2} \left (\frac {\ln \left (d x +c \right )}{d a -b c}-\frac {\ln \left (b x +a \right )}{d a -b c}\right )}{g i}-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (d a -b c \right )}-\frac {A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (d a -b c \right )}\) | \(123\) |
risch | \(\frac {A^{2} \ln \left (d x +c \right )}{g i \left (d a -b c \right )}-\frac {A^{2} \ln \left (b x +a \right )}{g i \left (d a -b c \right )}-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (d a -b c \right )}-\frac {A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (d a -b c \right )}\) | \(130\) |
derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{2} g}+\frac {d^{2} A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (d a -b c \right )^{2} g}+\frac {d^{2} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (d a -b c \right )^{2} g}\right )}{d^{2}}\) | \(182\) |
default | \(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{2} g}+\frac {d^{2} A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (d a -b c \right )^{2} g}+\frac {d^{2} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (d a -b c \right )^{2} g}\right )}{d^{2}}\) | \(182\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x,method=_RETURN VERBOSE)
Output:
-1/3*(B^2*ln(e*(b*x+a)/(d*x+c))^3*b^2*d^2+3*A*B*ln(e*(b*x+a)/(d*x+c))^2*b^ 2*d^2+3*A^2*ln(e*(b*x+a)/(d*x+c))*b^2*d^2)/b^2/d^2/g/i/(a*d-b*c)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (42) = 84\).
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.98 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, A B \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, A^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left (b c - a d\right )} g i} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x, algori thm="fricas")
Output:
1/3*(B^2*log((b*e*x + a*e)/(d*x + c))^3 + 3*A*B*log((b*e*x + a*e)/(d*x + c ))^2 + 3*A^2*log((b*e*x + a*e)/(d*x + c)))/((b*c - a*d)*g*i)
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (31) = 62\).
Time = 0.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 4.68 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=A^{2} \left (\frac {\log {\left (x + \frac {- \frac {a^{2} d^{2}}{a d - b c} + \frac {2 a b c d}{a d - b c} + a d - \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )} - \frac {\log {\left (x + \frac {\frac {a^{2} d^{2}}{a d - b c} - \frac {2 a b c d}{a d - b c} + a d + \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )}\right ) - \frac {A B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a d g i - b c g i} - \frac {B^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a d g i - 3 b c g i} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)/(d*i*x+c*i),x)
Output:
A**2*(log(x + (-a**2*d**2/(a*d - b*c) + 2*a*b*c*d/(a*d - b*c) + a*d - b**2 *c**2/(a*d - b*c) + b*c)/(2*b*d))/(g*i*(a*d - b*c)) - log(x + (a**2*d**2/( a*d - b*c) - 2*a*b*c*d/(a*d - b*c) + a*d + b**2*c**2/(a*d - b*c) + b*c)/(2 *b*d))/(g*i*(a*d - b*c))) - A*B*log(e*(a + b*x)/(c + d*x))**2/(a*d*g*i - b *c*g*i) - B**2*log(e*(a + b*x)/(c + d*x))**3/(3*a*d*g*i - 3*b*c*g*i)
Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (42) = 84\).
Time = 0.06 (sec) , antiderivative size = 397, normalized size of antiderivative = 9.02 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=B^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2} + 2 \, A B {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {1}{3} \, B^{2} {\left (\frac {3 \, {\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b c g i - a d g i} - \frac {\log \left (b x + a\right )^{3} - 3 \, \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - \log \left (d x + c\right )^{3}}{b c g i - a d g i}\right )} + A^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} - \frac {{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} A B}{b c g i - a d g i} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x, algori thm="maxima")
Output:
B^2*(log(b*x + a)/((b*c - a*d)*g*i) - log(d*x + c)/((b*c - a*d)*g*i))*log( b*e*x/(d*x + c) + a*e/(d*x + c))^2 + 2*A*B*(log(b*x + a)/((b*c - a*d)*g*i) - log(d*x + c)/((b*c - a*d)*g*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1/3*B^2*(3*(log(b*x + a)^2 - 2*log(b*x + a)*log(d*x + c) + log(d*x + c)^2) *log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b*c*g*i - a*d*g*i) - (log(b*x + a)^ 3 - 3*log(b*x + a)^2*log(d*x + c) + 3*log(b*x + a)*log(d*x + c)^2 - log(d* x + c)^3)/(b*c*g*i - a*d*g*i)) + A^2*(log(b*x + a)/((b*c - a*d)*g*i) - log (d*x + c)/((b*c - a*d)*g*i)) - (log(b*x + a)^2 - 2*log(b*x + a)*log(d*x + c) + log(d*x + c)^2)*A*B/(b*c*g*i - a*d*g*i)
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (42) = 84\).
Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.00 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {{\left (B^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, A B e \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, A^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{3 \, g i} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x, algori thm="giac")
Output:
1/3*(B^2*e*log((b*e*x + a*e)/(d*x + c))^3 + 3*A*B*e*log((b*e*x + a*e)/(d*x + c))^2 + 3*A^2*e*log((b*e*x + a*e)/(d*x + c)))*(b*c/((b*c*e - a*d*e)*(b* c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/(g*i)
Time = 27.62 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.18 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=-\frac {-6{}\mathrm {i}\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,A^2+3\,A\,B\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2+B^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i\,\left (a\,d-b\,c\right )} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))^2/((a*g + b*g*x)*(c*i + d*i*x)),x )
Output:
-(B^2*log((e*(a + b*x))/(c + d*x))^3 - A^2*atan((a*d*1i + b*c*1i + b*d*x*2 i)/(a*d - b*c))*6i + 3*A*B*log((e*(a + b*x))/(c + d*x))^2)/(3*g*i*(a*d - b *c))
Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.93 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {i \left (3 \,\mathrm {log}\left (b x +a \right ) a^{2}-3 \,\mathrm {log}\left (d x +c \right ) a^{2}+\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{3} b^{2}+3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a b \right )}{3 g \left (a d -b c \right )} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x)
Output:
(i*(3*log(a + b*x)*a**2 - 3*log(c + d*x)*a**2 + log((a*e + b*e*x)/(c + d*x ))**3*b**2 + 3*log((a*e + b*e*x)/(c + d*x))**2*a*b))/(3*g*(a*d - b*c))