\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(f+g x)^3} \, dx\) [246]

Optimal result

Integrand size = 29, antiderivative size = 369 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^3} \, dx=\frac {B (b c-a d) g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b f-a g)^2 (d f-c g) (f+g x)}+\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (b f-a g)^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (f+g x)^2}+\frac {B^2 (b c-a d)^2 g \log \left (\frac {f+g x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2} \]

[Out]

B*(-a*d+b*c)*g*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*g+b*f)^2/(-c*g+d*f)/(g*x+f)+1/2*b^2*(A+B*ln(e*(b*x+a)/(
d*x+c)))^2/g/(-a*g+b*f)^2-1/2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/g/(g*x+f)^2+B^2*(-a*d+b*c)^2*g*ln((g*x+f)/(d*x+c))
/(-a*g+b*f)^2/(-c*g+d*f)^2+B*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(A+B*ln(e*(b*x+a)/(d*x+c)))*ln(1-(-c*g+d*f)*(b*
x+a)/(-a*g+b*f)/(d*x+c))/(-a*g+b*f)^2/(-c*g+d*f)^2+B^2*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*polylog(2,(-c*g+d*f)*
(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*g+b*f)^2/(-c*g+d*f)^2

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2554, 2398, 2404, 2338, 2351, 31, 2354, 2438} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^3} \, dx=\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g (b f-a g)^2}+\frac {B g (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(f+g x) (b f-a g)^2 (d f-c g)}+\frac {B (b c-a d) (-a d g-b c g+2 b d f) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(b f-a g)^2 (d f-c g)^2}-\frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g (f+g x)^2}+\frac {B^2 (b c-a d) (-a d g-b c g+2 b d f) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B^2 g (b c-a d)^2 \log \left (\frac {f+g x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)^2} \]

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x)^3,x]

[Out]

(B*(b*c - a*d)*g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*f - a*g)^2*(d*f - c*g)*(f + g*x)) + (b^2*
(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*g*(b*f - a*g)^2) - (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(2*g*(f +
g*x)^2) + (B^2*(b*c - a*d)^2*g*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)^2*(d*f - c*g)^2) + (B*(b*c - a*d)*(2*b*
d*f - b*c*g - a*d*g)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*
x))])/((b*f - a*g)^2*(d*f - c*g)^2) + (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*PolyLog[2, ((d*f - c*g)*(a +
b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g)^2)

Rule 31

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

Rule 2338

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 2351

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

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 2398

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] - Dist[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]

Rule 2404

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]

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 2554

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, Subst[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] && EqQ[n + mn, 0] && IGt
Q[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = (b c-a d) \text {Subst}\left (\int \frac {(b-d x) (A+B \log (e x))^2}{(b f-a g-(d f-c g) x)^3} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (f+g x)^2}+\frac {B \text {Subst}\left (\int \frac {(b-d x)^2 (A+B \log (e x))}{x (b f-a g+(-d f+c g) x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{g} \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (f+g x)^2}+\frac {B \text {Subst}\left (\int \left (\frac {b^2 (A+B \log (e x))}{(b f-a g)^2 x}+\frac {(b c-a d)^2 g^2 (A+B \log (e x))}{(b f-a g) (d f-c g) (b f-a g-(d f-c g) x)^2}+\frac {(b c-a d) g (-2 b d f+b c g+a d g) (A+B \log (e x))}{(b f-a g)^2 (d f-c g) (b f-a g-(d f-c g) x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{g} \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (f+g x)^2}+\frac {\left (b^2 B\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g (b f-a g)^2}+\frac {\left (B (b c-a d)^2 g\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{(b f-a g+(-d f+c g) x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b f-a g) (d f-c g)}-\frac {(B (b c-a d) (2 b d f-b c g-a d g)) \text {Subst}\left (\int \frac {A+B \log (e x)}{b f-a g+(-d f+c g) x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)} \\ & = \frac {B (b c-a d) g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b f-a g)^2 (d f-c g) (f+g x)}+\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (b f-a g)^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (f+g x)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2}-\frac {\left (B^2 (b c-a d)^2 g\right ) \text {Subst}\left (\int \frac {1}{b f-a g+(-d f+c g) x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)}-\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g)\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)^2} \\ & = \frac {B (b c-a d) g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b f-a g)^2 (d f-c g) (f+g x)}+\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (b f-a g)^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 g (f+g x)^2}+\frac {B^2 (b c-a d)^2 g \log \left (\frac {f+g x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.61 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^3} \, dx=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B (f+g x) \left (2 (b c-a d) g (b f-a g) (d f-c g) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 b^2 (d f-c g)^2 (f+g x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d^2 (b f-a g)^2 (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+2 (b c-a d) g (-2 b d f+b c g+a d g) (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)-2 B (b c-a d) g (f+g x) (b (d f-c g) \log (a+b x)+(-b d f+a d g) \log (c+d x)+(b c-a d) g \log (f+g x))+b^2 B (d f-c g)^2 (f+g x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-B d^2 (b f-a g)^2 (f+g x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )-2 B (b c-a d) g (-2 b d f+b c g+a d g) (f+g x) \left (\left (\log \left (\frac {g (a+b x)}{-b f+a g}\right )-\log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)+\operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )-\operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )\right )\right )}{(b f-a g)^2 (d f-c g)^2}}{2 g (f+g x)^2} \]

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x)^3,x]

[Out]

-1/2*((A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(f + g*x)*(2*(b*c - a*d)*g*(b*f - a*g)*(d*f - c*g)*(A + B*Lo
g[(e*(a + b*x))/(c + d*x)]) - 2*b^2*(d*f - c*g)^2*(f + g*x)*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)])
+ 2*d^2*(b*f - a*g)^2*(f + g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 2*(b*c - a*d)*g*(-2*b*d*f
+ b*c*g + a*d*g)*(f + g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[f + g*x] - 2*B*(b*c - a*d)*g*(f + g*x)*(b*
(d*f - c*g)*Log[a + b*x] + (-(b*d*f) + a*d*g)*Log[c + d*x] + (b*c - a*d)*g*Log[f + g*x]) + b^2*B*(d*f - c*g)^2
*(f + g*x)*(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*d^2*(b*f - a*g)^2*(f + g*x)*((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)]) - 2*B*(b*c - a*d)*g*(-2*b*d*f + b*c*g + a*d*g)*(f + g*x)*((Log[(g*(
a + b*x))/(-(b*f) + a*g)] - Log[(g*(c + d*x))/(-(d*f) + c*g)])*Log[f + g*x] + PolyLog[2, (b*(f + g*x))/(b*f -
a*g)] - PolyLog[2, (d*(f + g*x))/(d*f - c*g)])))/((b*f - a*g)^2*(d*f - c*g)^2))/(g*(f + g*x)^2)

Maple [F]

\[\int \frac {\left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (g x +f \right )^{3}}d x\]

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(g*x+f)^3,x)

[Out]

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(g*x+f)^3,x)

Fricas [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{3}} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f)^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^3} \, dx=\text {Timed out} \]

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(g*x+f)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{3}} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f)^3,x, algorithm="maxima")

[Out]

(b^2*log(b*x + a)/(b^2*f^2*g - 2*a*b*f*g^2 + a^2*g^3) - d^2*log(d*x + c)/(d^2*f^2*g - 2*c*d*f*g^2 + c^2*g^3) +
(2*(b^2*c*d - a*b*d^2)*f - (b^2*c^2 - a^2*d^2)*g)*log(g*x + f)/(b^2*d^2*f^4 + a^2*c^2*g^4 - 2*(b^2*c*d + a*b*
d^2)*f^3*g + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^2*g^2 - 2*(a*b*c^2 + a^2*c*d)*f*g^3) - (b*c - a*d)/(b*d*f^3 + a
*c*f*g^2 - (b*c + a*d)*f^2*g + (b*d*f^2*g + a*c*g^3 - (b*c + a*d)*f*g^2)*x) - log(b*e*x/(d*x + c) + a*e/(d*x +
c))/(g^3*x^2 + 2*f*g^2*x + f^2*g))*A*B - 1/2*B^2*(log(d*x + c)^2/(g^3*x^2 + 2*f*g^2*x + f^2*g) + 2*integrate(
-(d*g*x*log(e)^2 + c*g*log(e)^2 + (d*g*x + c*g)*log(b*x + a)^2 + 2*(d*g*x*log(e) + c*g*log(e))*log(b*x + a) -
((2*g*log(e) - g)*d*x + 2*c*g*log(e) - d*f + 2*(d*g*x + c*g)*log(b*x + a))*log(d*x + c))/(d*g^4*x^4 + c*f^3*g
+ (3*d*f*g^3 + c*g^4)*x^3 + 3*(d*f^2*g^2 + c*f*g^3)*x^2 + (d*f^3*g + 3*c*f^2*g^2)*x), x)) - 1/2*A^2/(g^3*x^2 +
2*f*g^2*x + f^2*g)

Giac [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{3}} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f)^3,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)*e/(d*x + c)) + A)^2/(g*x + f)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(f+g x)^3} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (f+g\,x\right )}^3} \,d x \]

[In]

int((A + B*log((e*(a + b*x))/(c + d*x)))^2/(f + g*x)^3,x)

[Out]

int((A + B*log((e*(a + b*x))/(c + d*x)))^2/(f + g*x)^3, x)