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

Optimal. Leaf size=290 \[ \frac {B n (b c-a d) (-a d g-b c g+2 b d f) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d^2}-\frac {(b f-a g)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^2 g}-\frac {B g n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d}+\frac {(f+g x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g}+\frac {B^2 n^2 (b c-a d) (-a d g-b c g+2 b d f) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}+\frac {B^2 g n^2 (b c-a d)^2 \log (c+d x)}{b^2 d^2} \]

[Out]

-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

________________________________________________________________________________________

Rubi [A]  time = 0.83, antiderivative size = 481, normalized size of antiderivative = 1.66, number of steps used = 23, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {B^2 n^2 (b f-a g)^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}-\frac {B^2 n^2 (d f-c g)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 g}-\frac {B n (b f-a g)^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g}+\frac {B n (d f-c g)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 g}+\frac {(f+g x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g}-\frac {A B g n x (b c-a d)}{b d}+\frac {B^2 g n^2 (b c-a d)^2 \log (c+d x)}{b^2 d^2}-\frac {B^2 g n (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 d}-\frac {B^2 n^2 (b f-a g)^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {B^2 n^2 (b f-a g)^2 \log ^2(a+b x)}{2 b^2 g}-\frac {B^2 n^2 (d f-c g)^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^2 g}+\frac {B^2 n^2 (d f-c g)^2 \log ^2(c+d x)}{2 d^2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*B*(b*c - a*d)*g*n*x)/(b*d)) + (B^2*(b*f - a*g)^2*n^2*Log[a + b*x]^2)/(2*b^2*g) - (B^2*(b*c - a*d)*g*n*(a
+ b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(b^2*d) - (B*(b*f - a*g)^2*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]))/(b^2*g) + ((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*g) + (B^2*(b*c - a*d)^2*g*n^2*L
og[c + d*x])/(b^2*d^2) - (B^2*(d*f - c*g)^2*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^2*g) + (B*(
d*f - c*g)^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(d^2*g) + (B^2*(d*f - c*g)^2*n^2*Log[c + d
*x]^2)/(2*d^2*g) - (B^2*(b*f - a*g)^2*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b^2*g) - (B^2*(b*f - a
*g)^2*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^2*g) - (B^2*(d*f - c*g)^2*n^2*PolyLog[2, (b*(c + d*x))/
(b*c - a*d)])/(d^2*g)

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 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 (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}{2 g}-\frac {(B n) \int \frac {(b c-a d) (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{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) n) \int \frac {(f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{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) n) \int \left (\frac {g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b 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 )}{b (b c-a d) (a+b x)}+\frac {(d f-c g)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{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) g n) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}-\frac {\left (B (b f-a g)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g}+\frac {\left (B (d f-c g)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{d g}\\ &=-\frac {A B (b c-a d) g n x}{b d}-\frac {B (b f-a g)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{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 (d f-c g)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 g}-\frac {\left (B^2 (b c-a d) g n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b d}+\frac {\left (B^2 (b f-a g)^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 g}-\frac {\left (B^2 (d f-c g)^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{d^2 g}\\ &=-\frac {A B (b c-a d) g n x}{b d}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 d}-\frac {B (b f-a g)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{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 (d f-c g)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 g}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {1}{c+d x} \, dx}{b^2 d}+\frac {\left (B^2 (b f-a g)^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^2 g}-\frac {\left (B^2 (d f-c g)^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{d^2 g}\\ &=-\frac {A B (b c-a d) g n x}{b d}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 d}-\frac {B (b f-a g)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{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^2 (b c-a d)^2 g n^2 \log (c+d x)}{b^2 d^2}+\frac {B (d f-c g)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 g}+\frac {\left (B^2 (b f-a g)^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b g}-\frac {\left (B^2 d (b f-a g)^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 g}-\frac {\left (b B^2 (d f-c g)^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{d^2 g}+\frac {\left (B^2 (d f-c g)^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{d g}\\ &=-\frac {A B (b c-a d) g n x}{b d}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 d}-\frac {B (b f-a g)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{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^2 (b c-a d)^2 g n^2 \log (c+d x)}{b^2 d^2}-\frac {B^2 (d f-c g)^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 g}+\frac {B (d f-c g)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 g}-\frac {B^2 (b f-a g)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {\left (B^2 (b f-a g)^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}+\frac {\left (B^2 (b f-a g)^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g}+\frac {\left (B^2 (d f-c g)^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{d^2 g}+\frac {\left (B^2 (d f-c g)^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{d g}\\ &=-\frac {A B (b c-a d) g n x}{b d}+\frac {B^2 (b f-a g)^2 n^2 \log ^2(a+b x)}{2 b^2 g}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 d}-\frac {B (b f-a g)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{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^2 (b c-a d)^2 g n^2 \log (c+d x)}{b^2 d^2}-\frac {B^2 (d f-c g)^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 g}+\frac {B (d f-c g)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 g}+\frac {B^2 (d f-c g)^2 n^2 \log ^2(c+d x)}{2 d^2 g}-\frac {B^2 (b f-a g)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {\left (B^2 (b f-a g)^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}+\frac {\left (B^2 (d f-c g)^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d^2 g}\\ &=-\frac {A B (b c-a d) g n x}{b d}+\frac {B^2 (b f-a g)^2 n^2 \log ^2(a+b x)}{2 b^2 g}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 d}-\frac {B (b f-a g)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{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^2 (b c-a d)^2 g n^2 \log (c+d x)}{b^2 d^2}-\frac {B^2 (d f-c g)^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 g}+\frac {B (d f-c g)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 g}+\frac {B^2 (d f-c g)^2 n^2 \log ^2(c+d x)}{2 d^2 g}-\frac {B^2 (b f-a g)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac {B^2 (b f-a g)^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}-\frac {B^2 (d f-c g)^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 g}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 362, normalized size = 1.25 \[ \frac {(f+g x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2-\frac {B n \left (-2 b^2 (d f-c g)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 d^2 (b f-a g)^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 A b d g^2 x (b c-a d)+b^2 B n (d f-c g)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B d^2 n (b f-a g)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 B d g^2 (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B g^2 n (b c-a d)^2 \log (c+d x)\right )}{b^2 d^2}}{2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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*Log[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)

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fricas [F]  time = 1.37, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} g x + A^{2} f + {\left (B^{2} g x + B^{2} f\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B g x + A B f\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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

[Out]

Timed out

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (g x +f \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 5.19, size = 899, normalized size = 3.10 \[ 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 \relax (e) - {\left (g n^{2} + g n \log \relax (e)\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 \relax (e)^{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 \relax (e) - {\left (c d g n \log \relax (e) - d^{2} f \log \relax (e)^{2}\right )} b^{2}\right )} B^{2} x + 2 \, {\left ({\left (g n^{2} - g n \log \relax (e)\right )} a^{2} d^{2} - {\left (c d g n^{2} - 2 \, d^{2} f n \log \relax (e)\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 \relax (e) - {\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 \relax (e)\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 \relax (e) - {\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 \relax (e)\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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (f+g\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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