\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x) (c i+d i x)} \, dx\) [88]

Optimal result

Integrand size = 42, antiderivative size = 44 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d) g i} \]

[Out]

1/3*(A+B*ln(e*(b*x+a)/(d*x+c)))^3/B/(-a*d+b*c)/g/i

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2562, 2339, 30} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g i (b c-a d)} \]

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)*(c*i + d*i*x)),x]

[Out]

(A + B*Log[(e*(a + b*x))/(c + d*x)])^3/(3*B*(b*c - a*d)*g*i)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(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] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g i} \\ & = \frac {\text {Subst}\left (\int x^2 \, dx,x,A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B (b c-a d) g i} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.80 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {3 A^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A B \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+B^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )}{3 b c g i-3 a d g i} \]

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)*(c*i + d*i*x)),x]

[Out]

(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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(105\) vs. \(2(42)=84\).

Time = 0.66 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.41

method	result	size
parallelrisch	\(-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} b^{2} d^{2}+3 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{2} d^{2}+3 A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} d^{2}}{3 b^{2} d^{2} g i \left (a d -c b \right )}\)	\(106\)
norman	\(-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a d -c b \right )}-\frac {A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a d -c b \right )}-\frac {B A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a d -c b \right )}\)	\(113\)
parts	\(\frac {A^{2} \left (\frac {\ln \left (d x +c \right )}{a d -c b}-\frac {\ln \left (b x +a \right )}{a d -c b}\right )}{g i}-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a d -c b \right )}-\frac {B A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a d -c b \right )}\)	\(123\)
risch	\(\frac {A^{2} \ln \left (d x +c \right )}{g i \left (a d -c b \right )}-\frac {A^{2} \ln \left (b x +a \right )}{g i \left (a d -c b \right )}-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a d -c b \right )}-\frac {B A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a d -c b \right )}\)	\(130\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{2} g}+\frac {d^{2} A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (a d -c b \right )^{2} g}+\frac {d^{2} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (a d -c b \right )^{2} g}\right )}{d^{2}}\)	\(182\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{2} g}+\frac {d^{2} A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (a d -c b \right )^{2} g}+\frac {d^{2} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (a d -c b \right )^{2} g}\right )}{d^{2}}\)	\(182\)

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x,method=_RETURNVERBOSE)

[Out]

-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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (42) = 84\).

Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.98 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, A B \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, A^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left (b c - a d\right )} g i} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x, algorithm="fricas")

[Out]

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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (31) = 62\).

Time = 0.38 (sec) , antiderivative size = 206, normalized size of antiderivative = 4.68 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=A^{2} \left (\frac {\log {\left (x + \frac {- \frac {a^{2} d^{2}}{a d - b c} + \frac {2 a b c d}{a d - b c} + a d - \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )} - \frac {\log {\left (x + \frac {\frac {a^{2} d^{2}}{a d - b c} - \frac {2 a b c d}{a d - b c} + a d + \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )}\right ) - \frac {A B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a d g i - b c g i} - \frac {B^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a d g i - 3 b c g i} \]

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)/(d*i*x+c*i),x)

[Out]

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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (42) = 84\).

Time = 0.22 (sec) , antiderivative size = 397, normalized size of antiderivative = 9.02 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=B^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2} + 2 \, A B {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {1}{3} \, B^{2} {\left (\frac {3 \, {\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b c g i - a d g i} - \frac {\log \left (b x + a\right )^{3} - 3 \, \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - \log \left (d x + c\right )^{3}}{b c g i - a d g i}\right )} + A^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} - \frac {{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} A B}{b c g i - a d g i} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x, algorithm="maxima")

[Out]

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 - lo
g(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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (42) = 84\).

Time = 0.47 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.00 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {{\left (B^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, A B e \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, A^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{3 \, g i} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i),x, algorithm="giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 2.58 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.18 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=-\frac {-6{}\mathrm {i}\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,A^2+3\,A\,B\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2+B^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i\,\left (a\,d-b\,c\right )} \]

[In]

int((A + B*log((e*(a + b*x))/(c + d*x)))^2/((a*g + b*g*x)*(c*i + d*i*x)),x)

[Out]

-(B^2*log((e*(a + b*x))/(c + d*x))^3 - A^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*6i + 3*A*B*log((e*(a
+ b*x))/(c + d*x))^2)/(3*g*i*(a*d - b*c))