\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x) (c i+d i x)} \, dx\) [190]

Optimal result

Integrand size = 45, antiderivative size = 50 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d) g i n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2561, 2339, 30} \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g i n (b c-a d)} \]

[In]

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

[Out]

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

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 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(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] && 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 {\left (A+B \log \left (e x^n\right )\right )^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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B (b c-a d) g i n} \\ & = \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d) g i n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.80 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {3 A^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A B \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B^2 \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b c g i n-3 a d g i n} \]

[In]

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

[Out]

(3*A^2*Log[e*((a + b*x)/(c + d*x))^n] + 3*A*B*Log[e*((a + b*x)/(c + d*x))^n]^2 + B^2*Log[e*((a + b*x)/(c + d*x
))^n]^3)/(3*b*c*g*i*n - 3*a*d*g*i*n)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs. \(2(48)=96\).

Time = 1.44 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.36

method	result	size
parallelrisch	\(-\frac {B^{2} a^{2} c^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{3}+3 A B \,a^{2} c^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}+3 A^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} c^{2}}{3 i g \,c^{2} a^{2} n \left (a d -c b \right )}\)	\(118\)
default	\(\frac {A^{2} \left (\frac {\ln \left (d x +c \right )}{a d -c b}-\frac {\ln \left (b x +a \right )}{a d -c b}\right )}{g i}-\frac {B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{3}}{3 g i n \left (a d -c b \right )}-\frac {A B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}{g i n \left (a d -c b \right )}\)	\(135\)
parts	\(\frac {A^{2} \left (\frac {\ln \left (d x +c \right )}{a d -c b}-\frac {\ln \left (b x +a \right )}{a d -c b}\right )}{g i}-\frac {B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{3}}{3 g i n \left (a d -c b \right )}-\frac {A B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}{g i n \left (a d -c b \right )}\)	\(135\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (48) = 96\).

Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.98 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 3 \, B^{2} \log \left (e\right )^{2} \log \left (\frac {b x + a}{d x + c}\right ) + 3 \, A B n \log \left (\frac {b x + a}{d x + c}\right )^{2} + 3 \, A^{2} \log \left (\frac {b x + a}{d x + c}\right ) + 3 \, {\left (B^{2} n \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, A B \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right )}{3 \, {\left (b c - a d\right )} g i} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (48) = 96\).

Time = 0.22 (sec) , antiderivative size = 407, normalized size of antiderivative = 8.14 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=B^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )^{2} + 2 \, A B {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{3} \, {\left (\frac {{\left (\log \left (b x + a\right )^{3} - 3 \, \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - \log \left (d x + c\right )^{3}\right )} n^{2}}{b c g i - a d g i} - \frac {3 \, {\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} n \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{b c g i - a d g i}\right )} B^{2} - \frac {{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} A B n}{b c g i - a d g i} + A^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (48) = 96\).

Time = 0.55 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.34 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=\frac {{\left (B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 3 \, B^{2} n \log \left (e\right ) \log \left (\frac {b x + a}{d x + c}\right )^{2} + 3 \, B^{2} \log \left (e\right )^{2} \log \left (\frac {b x + a}{d x + c}\right ) + 3 \, A B n \log \left (\frac {b x + a}{d x + c}\right )^{2} + 6 \, A B \log \left (e\right ) \log \left (\frac {b x + a}{d x + c}\right ) + 3 \, A^{2} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{3 \, g i} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.60 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.44 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)} \, dx=-\frac {\frac {B^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3}+A\,B\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{g\,i\,n\,\left (a\,d-b\,c\right )}+\frac {A^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{g\,i\,\left (a\,d-b\,c\right )} \]

[In]

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

[Out]

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