3.2.28 \(\int (a g+b g x)^4 (A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2 \, dx\) [128]

Optimal. Leaf size=377 \[ -\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}+\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (2 A+B+2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (6 A+7 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^3}+\frac {2 B (b c-a d)^4 g^4 (a+b x) \left (6 A+13 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^4}+\frac {2 B (b c-a d)^5 g^4 \left (6 A+25 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{15 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \]

[Out]

-1/5*B*(-a*d+b*c)*g^4*(b*x+a)^4*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/d+1/5*g^4*(b*x+a)^5*(A+B*ln(e*(b*x+a)^2/(d*x
+c)^2))^2/b+2/15*B*(-a*d+b*c)^2*g^4*(b*x+a)^3*(2*A+B+2*B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/d^2-1/15*B*(-a*d+b*c)^3*
g^4*(b*x+a)^2*(6*A+7*B+6*B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/d^3+2/15*B*(-a*d+b*c)^4*g^4*(b*x+a)*(6*A+13*B+6*B*ln(e
*(b*x+a)^2/(d*x+c)^2))/b/d^4+2/15*B*(-a*d+b*c)^5*g^4*(6*A+25*B+6*B*ln(e*(b*x+a)^2/(d*x+c)^2))*ln((-a*d+b*c)/b/
(d*x+c))/b/d^5+8/5*B^2*(-a*d+b*c)^5*g^4*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^5

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2550, 2381, 2384, 2354, 2438} \begin {gather*} \frac {8 B^2 g^4 (b c-a d)^5 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}+\frac {2 B g^4 (b c-a d)^5 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+25 B\right )}{15 b d^5}+\frac {2 B g^4 (a+b x) (b c-a d)^4 \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+13 B\right )}{15 b d^4}-\frac {B g^4 (a+b x)^2 (b c-a d)^3 \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+7 B\right )}{15 b d^3}+\frac {2 B g^4 (a+b x)^3 (b c-a d)^2 \left (2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 A+B\right )}{15 b d^2}-\frac {B g^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2,x]

[Out]

-1/5*(B*(b*c - a*d)*g^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(b*d) + (g^4*(a + b*x)^5*(A + B*
Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(5*b) + (2*B*(b*c - a*d)^2*g^4*(a + b*x)^3*(2*A + B + 2*B*Log[(e*(a + b*x
)^2)/(c + d*x)^2]))/(15*b*d^2) - (B*(b*c - a*d)^3*g^4*(a + b*x)^2*(6*A + 7*B + 6*B*Log[(e*(a + b*x)^2)/(c + d*
x)^2]))/(15*b*d^3) + (2*B*(b*c - a*d)^4*g^4*(a + b*x)*(6*A + 13*B + 6*B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(15
*b*d^4) + (2*B*(b*c - a*d)^5*g^4*(6*A + 25*B + 6*B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d
*x))])/(15*b*d^5) + (8*B^2*(b*c - a*d)^5*g^4*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5)

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2381

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Dist[b*n*(p/(d*(q + 1))), Int[(
f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m
+ q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2550

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.), x_Symbol] :> Dist[(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] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

Rubi steps

\begin {align*} \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {(2 B) \int \frac {2 (b c-a d) g^5 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{c+d x} \, dx}{5 b g}\\ &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {\left (4 B (b c-a d) g^4\right ) \int \frac {(a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{c+d x} \, dx}{5 b}\\ &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {\left (4 B (b c-a d) g^4\right ) \int \left (-\frac {b (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{d^4}+\frac {b (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{d^2}+\frac {b (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{d}+\frac {(-b c+a d)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{d^4 (c+d x)}\right ) \, dx}{5 b}\\ &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {\left (4 B (b c-a d) g^4\right ) \int (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{5 d}+\frac {\left (4 B (b c-a d)^2 g^4\right ) \int (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{5 d^2}-\frac {\left (4 B (b c-a d)^3 g^4\right ) \int (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{5 d^3}+\frac {\left (4 B (b c-a d)^4 g^4\right ) \int \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{5 d^4}-\frac {\left (4 B (b c-a d)^5 g^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{5 b d^4}\\ &=\frac {4 A B (b c-a d)^4 g^4 x}{5 d^4}-\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{5 b d^5}+\frac {\left (B^2 (b c-a d) g^4\right ) \int \frac {2 (b c-a d) (a+b x)^3}{c+d x} \, dx}{5 b d}-\frac {\left (4 B^2 (b c-a d)^2 g^4\right ) \int \frac {2 (b c-a d) (a+b x)^2}{c+d x} \, dx}{15 b d^2}+\frac {\left (2 B^2 (b c-a d)^3 g^4\right ) \int \frac {2 (b c-a d) (a+b x)}{c+d x} \, dx}{5 b d^3}+\frac {\left (4 B^2 (b c-a d)^4 g^4\right ) \int \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \, dx}{5 d^4}+\frac {\left (4 B^2 (b c-a d)^5 g^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{5 b d^5}\\ &=\frac {4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{5 b d^4}-\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{5 b d^5}+\frac {\left (2 B^2 (b c-a d)^2 g^4\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{5 b d}-\frac {\left (8 B^2 (b c-a d)^3 g^4\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{15 b d^2}+\frac {\left (4 B^2 (b c-a d)^4 g^4\right ) \int \frac {a+b x}{c+d x} \, dx}{5 b d^3}-\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \int \frac {1}{c+d x} \, dx}{5 b d^4}+\frac {\left (4 B^2 (b c-a d)^5 g^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{5 b d^5 e}\\ &=\frac {4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{5 b d^4}-\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {8 B^2 (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{5 b d^5}+\frac {\left (2 B^2 (b c-a d)^2 g^4\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{5 b d}-\frac {\left (8 B^2 (b c-a d)^3 g^4\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{15 b d^2}+\frac {\left (4 B^2 (b c-a d)^4 g^4\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{5 b d^3}+\frac {\left (4 B^2 (b c-a d)^5 g^4\right ) \int \left (\frac {2 b e \log (c+d x)}{a+b x}-\frac {2 d e \log (c+d x)}{c+d x}\right ) \, dx}{5 b d^5 e}\\ &=\frac {4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac {2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}+\frac {4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{5 b d^4}-\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {10 B^2 (b c-a d)^5 g^4 \log (c+d x)}{3 b d^5}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{5 b d^5}+\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{5 d^5}-\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{5 b d^4}\\ &=\frac {4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac {2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}+\frac {4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{5 b d^4}-\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {10 B^2 (b c-a d)^5 g^4 \log (c+d x)}{3 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{5 b d^5}-\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{5 b d^5}-\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b d^4}\\ &=\frac {4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac {2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}+\frac {4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{5 b d^4}-\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {10 B^2 (b c-a d)^5 g^4 \log (c+d x)}{3 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{5 b d^5}-\frac {4 B^2 (b c-a d)^5 g^4 \log ^2(c+d x)}{5 b d^5}-\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{5 b d^5}\\ &=\frac {4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac {2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}+\frac {4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{5 b d^4}-\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}-\frac {10 B^2 (b c-a d)^5 g^4 \log (c+d x)}{3 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{5 b d^5}-\frac {4 B^2 (b c-a d)^5 g^4 \log ^2(c+d x)}{5 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{5 b d^5}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 523, normalized size = 1.39 \begin {gather*} \frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {B (b c-a d) \left (12 A b d (b c-a d)^3 x+12 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-6 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+4 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-3 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-24 B (b c-a d)^4 \log (c+d x)-12 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)+4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+12 B (b c-a d)^4 \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 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 d^5}\right )}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2,x]

[Out]

(g^4*((a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (B*(b*c - a*d)*(12*A*b*d*(b*c - a*d)^3*x + 12*B
*d*(b*c - a*d)^3*(a + b*x)*Log[(e*(a + b*x)^2)/(c + d*x)^2] - 6*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a
+ b*x)^2)/(c + d*x)^2]) + 4*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) - 3*d^4*(a +
b*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 12*(b*c - a*d)^4*(A + B*L
og[(e*(a + b*x)^2)/(c + d*x)^2])*Log[c + d*x] + 4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(
b*c - a*d)^2*Log[c + d*x]) + B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(
a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + 12*B*(b*c - a*d)^3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 12*B*(
b*c - a*d)^4*((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)])))/(3*d^5)))/(5*b)

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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{4} \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^4*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2,x)

[Out]

int((b*g*x+a*g)^4*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2374 vs. \(2 (368) = 736\).
time = 0.45, size = 2374, normalized size = 6.30 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="maxima")

[Out]

1/5*A^2*b^4*g^4*x^5 + A^2*a*b^3*g^4*x^4 + 2*A^2*a^2*b^2*g^4*x^3 + 2*A^2*a^3*b*g^4*x^2 + 2*(x*log(b^2*x^2*e/(d^
2*x^2 + 2*c*d*x + c^2) + 2*a*b*x*e/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 2*a*log(b*x
+ a)/b - 2*c*log(d*x + c)/d)*A*B*a^4*g^4 + 4*(x^2*log(b^2*x^2*e/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*x*e/(d^2*x^2
+ 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - 2*a^2*log(b*x + a)/b^2 + 2*c^2*log(d*x + c)/d^2 - 2*(b*
c - a*d)*x/(b*d))*A*B*a^3*b*g^4 + 4*(x^3*log(b^2*x^2*e/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*x*e/(d^2*x^2 + 2*c*d*
x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*
b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*a^2*b^2*g^4 + 2/3*(3*x^4*log(b^2*x^2*e/(d^2*x^2 + 2*c*d*x
+ c^2) + 2*a*b*x*e/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - 6*a^4*log(b*x + a)/b^4 + 6*
c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*
x)/(b^3*d^3))*A*B*a*b^3*g^4 + 1/15*(6*x^5*log(b^2*x^2*e/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*x*e/(d^2*x^2 + 2*c*d
*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d
^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^
4)*x)/(b^4*d^4))*A*B*b^4*g^4 + A^2*a^4*g^4*x - 2/15*(31*b^4*c^5*g^4 - 143*a*b^3*c^4*d*g^4 + 256*a^2*b^2*c^3*d^
2*g^4 - 216*a^3*b*c^2*d^3*g^4 + 78*a^4*c*d^4*g^4)*B^2*log(d*x + c)/d^5 - 8/5*(b^5*c^5*g^4 - 5*a*b^4*c^4*d*g^4
+ 10*a^2*b^3*c^3*d^2*g^4 - 10*a^3*b^2*c^2*d^3*g^4 + 5*a^4*b*c*d^4*g^4 - a^5*d^5*g^4)*(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*d^5) + 1/15*(3*B^2*b^5*d^5*g^4*x^5 - 3*(b^
5*c*d^4*g^4 - 6*a*b^4*d^5*g^4)*B^2*x^4 + 6*(b^5*c^2*d^3*g^4 - 4*a*b^4*c*d^4*g^4 + 8*a^2*b^3*d^5*g^4)*B^2*x^3 -
(13*b^5*c^3*d^2*g^4 - 57*a*b^4*c^2*d^3*g^4 + 93*a^2*b^3*c*d^4*g^4 - 79*a^3*b^2*d^5*g^4)*B^2*x^2 + (38*b^5*c^4
*d*g^4 - 178*a*b^4*c^3*d^2*g^4 + 324*a^2*b^3*c^2*d^3*g^4 - 278*a^3*b^2*c*d^4*g^4 + 109*a^4*b*d^5*g^4)*B^2*x +
12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*a*b^4*d^5*g^4*x^4 + 10*B^2*a^2*b^3*d^5*g^4*x^3 + 10*B^2*a^3*b^2*d^5*g^4*x^2 +
5*B^2*a^4*b*d^5*g^4*x + B^2*a^5*d^5*g^4)*log(b*x + a)^2 + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*a*b^4*d^5*g^4*x^4 +
10*B^2*a^2*b^3*d^5*g^4*x^3 + 10*B^2*a^3*b^2*d^5*g^4*x^2 + 5*B^2*a^4*b*d^5*g^4*x + (b^5*c^5*g^4 - 5*a*b^4*c^4*d
*g^4 + 10*a^2*b^3*c^3*d^2*g^4 - 10*a^3*b^2*c^2*d^3*g^4 + 5*a^4*b*c*d^4*g^4)*B^2)*log(d*x + c)^2 + 2*(6*B^2*b^5
*d^5*g^4*x^5 - 3*(b^5*c*d^4*g^4 - 11*a*b^4*d^5*g^4)*B^2*x^4 + 4*(b^5*c^2*d^3*g^4 - 5*a*b^4*c*d^4*g^4 + 19*a^2*
b^3*d^5*g^4)*B^2*x^3 - 6*(b^5*c^3*d^2*g^4 - 5*a*b^4*c^2*d^3*g^4 + 10*a^2*b^3*c*d^4*g^4 - 16*a^3*b^2*d^5*g^4)*B
^2*x^2 + 6*(2*b^5*c^4*d*g^4 - 10*a*b^4*c^3*d^2*g^4 + 20*a^2*b^3*c^2*d^3*g^4 - 20*a^3*b^2*c*d^4*g^4 + 13*a^4*b*
d^5*g^4)*B^2*x + (12*a*b^4*c^4*d*g^4 - 54*a^2*b^3*c^3*d^2*g^4 + 94*a^3*b^2*c^2*d^3*g^4 - 77*a^4*b*c*d^4*g^4 +
31*a^5*d^5*g^4)*B^2)*log(b*x + a) - 2*(6*B^2*b^5*d^5*g^4*x^5 - 3*(b^5*c*d^4*g^4 - 11*a*b^4*d^5*g^4)*B^2*x^4 +
4*(b^5*c^2*d^3*g^4 - 5*a*b^4*c*d^4*g^4 + 19*a^2*b^3*d^5*g^4)*B^2*x^3 - 6*(b^5*c^3*d^2*g^4 - 5*a*b^4*c^2*d^3*g^
4 + 10*a^2*b^3*c*d^4*g^4 - 16*a^3*b^2*d^5*g^4)*B^2*x^2 + 6*(2*b^5*c^4*d*g^4 - 10*a*b^4*c^3*d^2*g^4 + 20*a^2*b^
3*c^2*d^3*g^4 - 20*a^3*b^2*c*d^4*g^4 + 13*a^4*b*d^5*g^4)*B^2*x + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*a*b^4*d^5*g^4
*x^4 + 10*B^2*a^2*b^3*d^5*g^4*x^3 + 10*B^2*a^3*b^2*d^5*g^4*x^2 + 5*B^2*a^4*b*d^5*g^4*x + B^2*a^5*d^5*g^4)*log(
b*x + a))*log(d*x + c))/(b*d^5)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="fricas")

[Out]

integral(A^2*b^4*g^4*x^4 + 4*A^2*a*b^3*g^4*x^3 + 6*A^2*a^2*b^2*g^4*x^2 + 4*A^2*a^3*b*g^4*x + A^2*a^4*g^4 + (B^
2*b^4*g^4*x^4 + 4*B^2*a*b^3*g^4*x^3 + 6*B^2*a^2*b^2*g^4*x^2 + 4*B^2*a^3*b*g^4*x + B^2*a^4*g^4)*log((b^2*x^2 +
2*a*b*x + a^2)*e/(d^2*x^2 + 2*c*d*x + c^2))^2 + 2*(A*B*b^4*g^4*x^4 + 4*A*B*a*b^3*g^4*x^3 + 6*A*B*a^2*b^2*g^4*x
^2 + 4*A*B*a^3*b*g^4*x + A*B*a^4*g^4)*log((b^2*x^2 + 2*a*b*x + a^2)*e/(d^2*x^2 + 2*c*d*x + c^2)), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**4*(A+B*ln(e*(b*x+a)**2/(d*x+c)**2))**2,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="giac")

[Out]

integrate((b*g*x + a*g)^4*(B*log((b*x + a)^2*e/(d*x + c)^2) + A)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a\,g+b\,g\,x\right )}^4\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^4*(A + B*log((e*(a + b*x)^2)/(c + d*x)^2))^2,x)

[Out]

int((a*g + b*g*x)^4*(A + B*log((e*(a + b*x)^2)/(c + d*x)^2))^2, x)

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