Integrand size = 30, antiderivative size = 290 \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {B (b c-a d) g n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}-\frac {(b f-a g)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g}+\frac {(f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 g}+\frac {B (b c-a d) (2 b d f-b c g-a d g) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{b^2 d^2}+\frac {B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b^2 d^2}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2} \]
-B*(-a*d+b*c)*g*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/d-1/2*(-a*g+ b*f)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^2/g+1/2*(g*x+f)^2*(A+B*ln(e*((b *x+a)/(d*x+c))^n))^2/g+B*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*n*(A+B*ln(e*((b *x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x+c))/b^2/d^2+B^2*(-a*d+b*c)^2*g*n^2 *ln(d*x+c)/b^2/d^2+B^2*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*n^2*polylog(2,d*( b*x+a)/b/(d*x+c))/b^2/d^2
Time = 0.20 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.25 \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {(f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {B n \left (2 A b d (b c-a d) g^2 x+2 B d (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 d^2 (b f-a g)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 g^2 n \log (c+d x)-2 b^2 (d f-c g)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-B d^2 (b f-a g)^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+b^2 B (d f-c g)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b^2 d^2}}{2 g} \]
((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (B*n*(2*A*b*d*(b*c - a*d)*g^2*x + 2*B*d*(b*c - a*d)*g^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x) )^n] + 2*d^2*(b*f - a*g)^2*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x)) ^n]) - 2*B*(b*c - a*d)^2*g^2*n*Log[c + d*x] - 2*b^2*(d*f - c*g)^2*(A + B*L og[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - B*d^2*(b*f - a*g)^2*n*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, ( d*(a + b*x))/(-(b*c) + a*d)]) + b^2*B*(d*f - c*g)^2*n*((2*Log[(d*(a + b*x) )/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x) )/(b*c - a*d)])))/(b^2*d^2))/(2*g)
Time = 0.78 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2953, 2798, 2804, 2009}
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.
| \(\displaystyle \int (f+g x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2 \, dx\) |
| \(\Big \downarrow \) 2953 |
| \(\displaystyle (b c-a d) \int \frac {\left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2798 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d f-c g)}{c+d x}-a g+b f\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {(c+d x) \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{g (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d f-c g)}{c+d x}-a g+b f\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \left (\frac {(b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g^2}{b d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {(b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g}{b^2 d \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(b f-a g)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{g (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d f-c g)}{c+d x}-a g+b f\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (-\frac {g (b c-a d) (-a d g-b c g+2 b d f) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d^2}+\frac {(b f-a g)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^2 B n}+\frac {g^2 (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {B g n (b c-a d) (-a d g-b c g+2 b d f) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}+\frac {B g^2 n (b c-a d)^2 \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2 d^2}\right )}{g (b c-a d)}\right )\) |
(b*c - a*d)*(((b*f - a*g - ((d*f - c*g)*(a + b*x))/(c + d*x))^2*(A + B*Log [e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)*g*(b - (d*(a + b*x))/(c + d *x))^2) - (B*n*(((b*c - a*d)^2*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^2*d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + ((b*f - a*g)^ 2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*B*n) + (B*(b*c - a*d)^2 *g^2*n*Log[b - (d*(a + b*x))/(c + d*x)])/(b^2*d^2) - ((b*c - a*d)*g*(2*b*d *f - b*c*g - a*d*g)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2) - (B*(b*c - a*d)*g*(2*b*d*f - b*c*g - a*d *g)*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2)))/((b*c - a*d)*g) )
3.1.69.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*(( f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Simp[b*n*(p/((q + 1) *(e*f - d*g))) Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d) Sub st[Int[(b*f - a*g - (d*f - c*g)*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] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
\[\int \left (g x +f \right ) {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]
\[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (g x + f\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]
integral(A^2*g*x + A^2*f + (B^2*g*x + B^2*f)*log(e*((b*x + a)/(d*x + c))^n )^2 + 2*(A*B*g*x + A*B*f)*log(e*((b*x + a)/(d*x + c))^n), x)
Timed out. \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (285) = 570\).
Time = 0.68 (sec) , antiderivative size = 899, normalized size of antiderivative = 3.10 \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=A B g x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{2} \, A^{2} g x^{2} - A B g n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + 2 \, A B f n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + 2 \, A B f x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A^{2} f x - \frac {{\left (a c d g n^{2} + {\left (2 \, c d f n \log \left (e\right ) - {\left (g n^{2} + g n \log \left (e\right )\right )} c^{2}\right )} b\right )} B^{2} \log \left (d x + c\right )}{b d^{2}} + \frac {{\left (2 \, a b d^{2} f n^{2} - a^{2} d^{2} g n^{2} - {\left (2 \, c d f n^{2} - c^{2} g n^{2}\right )} b^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B^{2}}{b^{2} d^{2}} + \frac {B^{2} b^{2} d^{2} g x^{2} \log \left (e\right )^{2} + 2 \, {\left (2 \, c d f n^{2} - c^{2} g n^{2}\right )} B^{2} b^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (2 \, c d f n^{2} - c^{2} g n^{2}\right )} B^{2} b^{2} \log \left (d x + c\right )^{2} - {\left (2 \, a b d^{2} f n^{2} - a^{2} d^{2} g n^{2}\right )} B^{2} \log \left (b x + a\right )^{2} + 2 \, {\left (a b d^{2} g n \log \left (e\right ) - {\left (c d g n \log \left (e\right ) - d^{2} f \log \left (e\right )^{2}\right )} b^{2}\right )} B^{2} x + 2 \, {\left ({\left (g n^{2} - g n \log \left (e\right )\right )} a^{2} d^{2} - {\left (c d g n^{2} - 2 \, d^{2} f n \log \left (e\right )\right )} a b\right )} B^{2} \log \left (b x + a\right ) + {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} b^{2} d^{2} f x\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} b^{2} d^{2} f x\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} \log \left (e\right ) - {\left (2 \, c d f n - c^{2} g n\right )} B^{2} b^{2} \log \left (d x + c\right ) + {\left (a b d^{2} g n - {\left (c d g n - 2 \, d^{2} f \log \left (e\right )\right )} b^{2}\right )} B^{2} x + {\left (2 \, a b d^{2} f n - a^{2} d^{2} g n\right )} B^{2} \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} \log \left (e\right ) - {\left (2 \, c d f n - c^{2} g n\right )} B^{2} b^{2} \log \left (d x + c\right ) + {\left (a b d^{2} g n - {\left (c d g n - 2 \, d^{2} f \log \left (e\right )\right )} b^{2}\right )} B^{2} x + {\left (2 \, a b d^{2} f n - a^{2} d^{2} g n\right )} B^{2} \log \left (b x + a\right ) + {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} b^{2} d^{2} f x\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{2} d^{2}} \]
A*B*g*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A^2*g*x^2 - A*B*g*n *(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) + 2*A *B*f*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*f*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*f*x - (a*c*d*g*n^2 + (2*c*d*f*n*log(e) - (g*n^ 2 + g*n*log(e))*c^2)*b)*B^2*log(d*x + c)/(b*d^2) + (2*a*b*d^2*f*n^2 - a^2* d^2*g*n^2 - (2*c*d*f*n^2 - c^2*g*n^2)*b^2)*(log(b*x + a)*log((b*d*x + a*d) /(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B^2/(b^2*d^2) + 1/2 *(B^2*b^2*d^2*g*x^2*log(e)^2 + 2*(2*c*d*f*n^2 - c^2*g*n^2)*B^2*b^2*log(b*x + a)*log(d*x + c) - (2*c*d*f*n^2 - c^2*g*n^2)*B^2*b^2*log(d*x + c)^2 - (2 *a*b*d^2*f*n^2 - a^2*d^2*g*n^2)*B^2*log(b*x + a)^2 + 2*(a*b*d^2*g*n*log(e) - (c*d*g*n*log(e) - d^2*f*log(e)^2)*b^2)*B^2*x + 2*((g*n^2 - g*n*log(e))* a^2*d^2 - (c*d*g*n^2 - 2*d^2*f*n*log(e))*a*b)*B^2*log(b*x + a) + (B^2*b^2* d^2*g*x^2 + 2*B^2*b^2*d^2*f*x)*log((b*x + a)^n)^2 + (B^2*b^2*d^2*g*x^2 + 2 *B^2*b^2*d^2*f*x)*log((d*x + c)^n)^2 + 2*(B^2*b^2*d^2*g*x^2*log(e) - (2*c* d*f*n - c^2*g*n)*B^2*b^2*log(d*x + c) + (a*b*d^2*g*n - (c*d*g*n - 2*d^2*f* log(e))*b^2)*B^2*x + (2*a*b*d^2*f*n - a^2*d^2*g*n)*B^2*log(b*x + a))*log(( b*x + a)^n) - 2*(B^2*b^2*d^2*g*x^2*log(e) - (2*c*d*f*n - c^2*g*n)*B^2*b^2* log(d*x + c) + (a*b*d^2*g*n - (c*d*g*n - 2*d^2*f*log(e))*b^2)*B^2*x + (2*a *b*d^2*f*n - a^2*d^2*g*n)*B^2*log(b*x + a) + (B^2*b^2*d^2*g*x^2 + 2*B^2*b^ 2*d^2*f*x)*log((b*x + a)^n))*log((d*x + c)^n))/(b^2*d^2)
\[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (g x + f\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]
Timed out. \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int \left (f+g\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]