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

Optimal. Leaf size=503 \[ -\frac {2 B g^4 (b c-a d)^5 \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 b d^5}-\frac {2 B g^4 (c+d x) (b c-a d)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 d^5}+\frac {B g^4 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 b d^3}-\frac {2 B g^4 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{15 b d^2}+\frac {B g^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b}+\frac {2 B^2 g^4 (b c-a d)^5 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}-\frac {5 B^2 g^4 (b c-a d)^5 \log (a+b x)}{6 b d^5}-\frac {13 B^2 g^4 (b c-a d)^5 \log \left (\frac {c+d x}{a+b x}\right )}{30 b d^5}+\frac {13 B^2 g^4 x (b c-a d)^4}{30 d^4}-\frac {7 B^2 g^4 (a+b x)^2 (b c-a d)^3}{60 b d^3}+\frac {B^2 g^4 (a+b x)^3 (b c-a d)^2}{30 b d^2} \]

[Out]

13/30*B^2*(-a*d+b*c)^4*g^4*x/d^4-7/60*B^2*(-a*d+b*c)^3*g^4*(b*x+a)^2/b/d^3+1/30*B^2*(-a*d+b*c)^2*g^4*(b*x+a)^3
/b/d^2-5/6*B^2*(-a*d+b*c)^5*g^4*ln(b*x+a)/b/d^5-13/30*B^2*(-a*d+b*c)^5*g^4*ln((d*x+c)/(b*x+a))/b/d^5+1/5*B*(-a
*d+b*c)^3*g^4*(b*x+a)^2*(A+B*ln(e*(d*x+c)/(b*x+a)))/b/d^3-2/15*B*(-a*d+b*c)^2*g^4*(b*x+a)^3*(A+B*ln(e*(d*x+c)/
(b*x+a)))/b/d^2+1/10*B*(-a*d+b*c)*g^4*(b*x+a)^4*(A+B*ln(e*(d*x+c)/(b*x+a)))/b/d-2/5*B*(-a*d+b*c)^4*g^4*(d*x+c)
*(A+B*ln(e*(d*x+c)/(b*x+a)))/d^5+1/5*g^4*(b*x+a)^5*(A+B*ln(e*(d*x+c)/(b*x+a)))^2/b-2/5*B*(-a*d+b*c)^5*g^4*(A+B
*ln(e*(d*x+c)/(b*x+a)))*ln(1-d*(b*x+a)/b/(d*x+c))/b/d^5+2/5*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 = 0.82, antiderivative size = 557, normalized size of antiderivative = 1.11, number of steps used = 28, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2525, 12, 2528, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {2 B^2 g^4 (b c-a d)^5 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{5 b d^5}+\frac {2 B g^4 (b c-a d)^5 \log (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 b d^5}+\frac {B g^4 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{5 b d^3}-\frac {2 B g^4 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{15 b d^2}-\frac {2 A B g^4 x (b c-a d)^4}{5 d^4}+\frac {B g^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b}-\frac {2 B^2 g^4 (a+b x) (b c-a d)^4 \log \left (\frac {e (c+d x)}{a+b x}\right )}{5 b d^4}+\frac {13 B^2 g^4 x (b c-a d)^4}{30 d^4}-\frac {7 B^2 g^4 (a+b x)^2 (b c-a d)^3}{60 b d^3}+\frac {B^2 g^4 (a+b x)^3 (b c-a d)^2}{30 b d^2}-\frac {B^2 g^4 (b c-a d)^5 \log ^2(c+d x)}{5 b d^5}-\frac {5 B^2 g^4 (b c-a d)^5 \log (c+d x)}{6 b d^5}+\frac {2 B^2 g^4 (b c-a d)^5 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{5 b d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*B*(b*c - a*d)^4*g^4*x)/(5*d^4) + (13*B^2*(b*c - a*d)^4*g^4*x)/(30*d^4) - (7*B^2*(b*c - a*d)^3*g^4*(a + b
*x)^2)/(60*b*d^3) + (B^2*(b*c - a*d)^2*g^4*(a + b*x)^3)/(30*b*d^2) - (5*B^2*(b*c - a*d)^5*g^4*Log[c + d*x])/(6
*b*d^5) + (2*B^2*(b*c - a*d)^5*g^4*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(5*b*d^5) - (B^2*(b*c - a*d
)^5*g^4*Log[c + d*x]^2)/(5*b*d^5) - (2*B^2*(b*c - a*d)^4*g^4*(a + b*x)*Log[(e*(c + d*x))/(a + b*x)])/(5*b*d^4)
+ (B*(b*c - a*d)^3*g^4*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(5*b*d^3) - (2*B*(b*c - a*d)^2*g^4*(
a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(15*b*d^2) + (B*(b*c - a*d)*g^4*(a + b*x)^4*(A + B*Log[(e*(c
+ d*x))/(a + b*x)]))/(10*b*d) + (2*B*(b*c - a*d)^5*g^4*Log[c + d*x]*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(5*b
*d^5) + (g^4*(a + b*x)^5*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(5*b) + (2*B^2*(b*c - a*d)^5*g^4*PolyLog[2, (
b*(c + d*x))/(b*c - a*d)])/(5*b*d^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

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

Rule 2391

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2486

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

Rule 2524

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

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {(2 B) \int \frac {(b c-a d) g^5 (a+b x)^4 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{c+d x} \, dx}{5 b g}\\ &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {\left (2 B (b c-a d) g^4\right ) \int \frac {(a+b x)^4 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{c+d x} \, dx}{5 b}\\ &=\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {\left (2 B (b c-a d) g^4\right ) \int \left (-\frac {b (b c-a d)^3 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{d^4}+\frac {b (b c-a d)^2 (a+b x) \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x)^2 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{d^2}+\frac {b (a+b x)^3 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{d}+\frac {(-b c+a d)^4 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\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 (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {\left (2 B (b c-a d) g^4\right ) \int (a+b x)^3 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx}{5 d}+\frac {\left (2 B (b c-a d)^2 g^4\right ) \int (a+b x)^2 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx}{5 d^2}-\frac {\left (2 B (b c-a d)^3 g^4\right ) \int (a+b x) \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx}{5 d^3}+\frac {\left (2 B (b c-a d)^4 g^4\right ) \int \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx}{5 d^4}-\frac {\left (2 B (b c-a d)^5 g^4\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{c+d x} \, dx}{5 b d^4}\\ &=-\frac {2 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\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 (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {2 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {\left (B^2 (b c-a d) g^4\right ) \int \frac {(-b c+a d) (a+b x)^3}{c+d x} \, dx}{10 b d}+\frac {\left (2 B^2 (b c-a d)^2 g^4\right ) \int \frac {(-b c+a d) (a+b x)^2}{c+d x} \, dx}{15 b d^2}-\frac {\left (B^2 (b c-a d)^3 g^4\right ) \int \frac {(b c-a d) (-a-b x)}{c+d x} \, dx}{5 b d^3}-\frac {\left (2 B^2 (b c-a d)^4 g^4\right ) \int \log \left (\frac {e (c+d x)}{a+b x}\right ) \, dx}{5 d^4}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \int \frac {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{e (c+d x)} \, dx}{5 b d^5}\\ &=-\frac {2 A B (b c-a d)^4 g^4 x}{5 d^4}-\frac {2 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )}{5 b d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\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 (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {2 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {\left (B^2 (b c-a d)^2 g^4\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{10 b d}-\frac {\left (2 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 (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 (2 B^2 (b c-a d)^5 g^4\right ) \int \frac {1}{c+d x} \, dx}{5 b d^4}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \int \frac {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{c+d x} \, dx}{5 b d^5 e}\\ &=-\frac {2 A B (b c-a d)^4 g^4 x}{5 d^4}-\frac {2 B^2 (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5}-\frac {2 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )}{5 b d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\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 (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {2 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {\left (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}{10 b d}-\frac {\left (2 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 (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 (2 B^2 (b c-a d)^5 g^4\right ) \int \left (-\frac {b e \log (c+d x)}{a+b x}+\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{5 b d^5 e}\\ &=-\frac {2 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {13 B^2 (b c-a d)^4 g^4 x}{30 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {5 B^2 (b c-a d)^5 g^4 \log (c+d x)}{6 b d^5}-\frac {2 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )}{5 b d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\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 (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {2 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {\left (2 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 (2 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 {2 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {13 B^2 (b c-a d)^4 g^4 x}{30 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {5 B^2 (b c-a d)^5 g^4 \log (c+d x)}{6 b d^5}+\frac {2 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 {2 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )}{5 b d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\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 (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {2 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{5 b d^5}-\frac {\left (2 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 {2 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {13 B^2 (b c-a d)^4 g^4 x}{30 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {5 B^2 (b c-a d)^5 g^4 \log (c+d x)}{6 b d^5}+\frac {2 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 {B^2 (b c-a d)^5 g^4 \log ^2(c+d x)}{5 b d^5}-\frac {2 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )}{5 b d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\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 (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {2 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \operatorname {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 {2 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac {13 B^2 (b c-a d)^4 g^4 x}{30 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {5 B^2 (b c-a d)^5 g^4 \log (c+d x)}{6 b d^5}+\frac {2 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 {B^2 (b c-a d)^5 g^4 \log ^2(c+d x)}{5 b d^5}-\frac {2 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )}{5 b d^4}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\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 (c+d x)}{a+b x}\right )\right )}{10 b d}+\frac {2 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}+\frac {2 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 = 0.51, size = 512, normalized size = 1.02 \[ \frac {g^4 \left ((a+b x)^5 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2-\frac {B (b c-a d) \left (-6 d^4 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+8 d^3 (a+b x)^3 (b c-a d) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )-12 d^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )-24 (b c-a d)^4 \log (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+24 A b d x (b c-a d)^3-4 B (b c-a d)^2 \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )-B (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+24 b B (c+d x) (b c-a d)^3 \log \left (\frac {e (c+d x)}{a+b x}\right )-12 B (b c-a d)^4 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+24 B (b c-a d)^4 \log (a+b x)-12 B (b c-a d)^3 ((a d-b c) \log (c+d x)+b d x)\right )}{12 d^5}\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^4*((a + b*x)^5*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2 - (B*(b*c - a*d)*(24*A*b*d*(b*c - a*d)^3*x + 24*B*(b*
c - a*d)^4*Log[a + b*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]) + 24*b*B*(b*c - a*d)^3*(c + d*
x)*Log[(e*(c + d*x))/(a + b*x)] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]) + 8*d^
3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]) - 6*d^4*(a + b*x)^4*(A + B*Log[(e*(c + d*x))/(a
+ b*x)]) - 24*(b*c - a*d)^4*Log[c + d*x]*(A + B*Log[(e*(c + d*x))/(a + b*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)])))/(12*d^5)
))/(5*b)

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b^{4} g^{4} x^{4} + 4 \, A^{2} a b^{3} g^{4} x^{3} + 6 \, A^{2} a^{2} b^{2} g^{4} x^{2} + 4 \, A^{2} a^{3} b g^{4} x + A^{2} a^{4} g^{4} + {\left (B^{2} b^{4} g^{4} x^{4} + 4 \, B^{2} a b^{3} g^{4} x^{3} + 6 \, B^{2} a^{2} b^{2} g^{4} x^{2} + 4 \, B^{2} a^{3} b g^{4} x + B^{2} a^{4} g^{4}\right )} \log \left (\frac {d e x + c e}{b x + a}\right )^{2} + 2 \, {\left (A B b^{4} g^{4} x^{4} + 4 \, A B a b^{3} g^{4} x^{3} + 6 \, A B a^{2} b^{2} g^{4} x^{2} + 4 \, A B a^{3} b g^{4} x + A B a^{4} g^{4}\right )} \log \left (\frac {d e x + c e}{b x + a}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 2.20, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{4} \left (B \ln \left (\frac {\left (d x +c \right ) e}{b x +a}\right )+A \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.53, size = 2395, normalized size = 4.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)/(b*x+a)))^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(d*e*x/(b*x +
a) + c*e/(b*x + a)) - a*log(b*x + a)/b + c*log(d*x + c)/d)*A*B*a^4*g^4 + 4*(x^2*log(d*e*x/(b*x + a) + c*e/(b*x
+ a)) + a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d))*A*B*a^3*b*g^4 + 2*(2*x^3*log(d*e*x
/(b*x + a) + c*e/(b*x + a)) - 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 + 1/3*(6*x^4*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + 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 + 1/30*(12*x^5*log(d*e*x/(b*x + a) + c*e/(b*x + a)) - 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 + 1/30*((12*
g^4*log(e) - 25*g^4)*b^4*c^5 - (60*g^4*log(e) - 113*g^4)*a*b^3*c^4*d + 4*(30*g^4*log(e) - 49*g^4)*a^2*b^2*c^3*
d^2 - 12*(10*g^4*log(e) - 13*g^4)*a^3*b*c^2*d^3 + 12*(5*g^4*log(e) - 4*g^4)*a^4*c*d^4)*B^2*log(d*x + c)/d^5 -
2/5*(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)
+ 1/60*(12*B^2*b^5*d^5*g^4*x^5*log(e)^2 + 6*(b^5*c*d^4*g^4*log(e) + (10*g^4*log(e)^2 - g^4*log(e))*a*b^4*d^5)*
B^2*x^4 - 2*((4*g^4*log(e) - g^4)*b^5*c^2*d^3 - 2*(10*g^4*log(e) - g^4)*a*b^4*c*d^4 - (60*g^4*log(e)^2 - 16*g^
4*log(e) + g^4)*a^2*b^3*d^5)*B^2*x^3 + ((12*g^4*log(e) - 7*g^4)*b^5*c^3*d^2 - 3*(20*g^4*log(e) - 9*g^4)*a*b^4*
c^2*d^3 + 3*(40*g^4*log(e) - 11*g^4)*a^2*b^3*c*d^4 + (120*g^4*log(e)^2 - 72*g^4*log(e) + 13*g^4)*a^3*b^2*d^5)*
B^2*x^2 - 2*((12*g^4*log(e) - 13*g^4)*b^5*c^4*d - (60*g^4*log(e) - 59*g^4)*a*b^4*c^3*d^2 + 6*(20*g^4*log(e) -
17*g^4)*a^2*b^3*c^2*d^3 - (120*g^4*log(e) - 79*g^4)*a^3*b^2*c*d^4 - (30*g^4*log(e)^2 - 48*g^4*log(e) + 23*g^4)
*a^4*b*d^5)*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*(12*B^2*b^5*d^5*g^4*x^5*log(e) + 3*(b^5*c*d^4*g^4 + (20*g^4*log(e) - g^4)*a*b^4*d^5)*B^2*x^4 - 4*(b^
5*c^2*d^3*g^4 - 5*a*b^4*c*d^4*g^4 - 2*(15*g^4*log(e) - 2*g^4)*a^2*b^3*d^5)*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 + 2*(10*g^4*log(e) - 3*g^4)*a^3*b^2*d^5)*B^2*x^2 - 12*(b^5*c^4*d*g^4 -
5*a*b^4*c^3*d^2*g^4 + 10*a^2*b^3*c^2*d^3*g^4 - 10*a^3*b^2*c*d^4*g^4 - (5*g^4*log(e) - 4*g^4)*a^4*b*d^5)*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 - (12*g^4*log(e)
- 25*g^4)*a^5*d^5)*B^2)*log(b*x + a) + 2*(12*B^2*b^5*d^5*g^4*x^5*log(e) + 3*(b^5*c*d^4*g^4 + (20*g^4*log(e) -
g^4)*a*b^4*d^5)*B^2*x^4 - 4*(b^5*c^2*d^3*g^4 - 5*a*b^4*c*d^4*g^4 - 2*(15*g^4*log(e) - 2*g^4)*a^2*b^3*d^5)*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 + 2*(10*g^4*log(e) - 3*g^4)*a^3*b^2*d^5)
*B^2*x^2 - 12*(b^5*c^4*d*g^4 - 5*a*b^4*c^3*d^2*g^4 + 10*a^2*b^3*c^2*d^3*g^4 - 10*a^3*b^2*c*d^4*g^4 - (5*g^4*lo
g(e) - 4*g^4)*a^4*b*d^5)*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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a\,g+b\,g\,x\right )}^4\,{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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