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

Optimal. Leaf size=156 \[ \frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 B g i^2 (b c-a d)^2}-\frac {A d (a+b x)}{g i^2 (c+d x) (b c-a d)^2}-\frac {B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{g i^2 (c+d x) (b c-a d)^2}+\frac {B d (a+b x)}{g i^2 (c+d x) (b c-a d)^2} \]

[Out]

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

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Rubi [C]  time = 0.71, antiderivative size = 432, normalized size of antiderivative = 2.77, number of steps used = 24, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44} \[ \frac {b B \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac {b B \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac {b \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^2 (b c-a d)^2}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{g i^2 (c+d x) (b c-a d)}-\frac {b \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^2 (b c-a d)^2}-\frac {B}{g i^2 (c+d x) (b c-a d)}-\frac {b B \log ^2(a+b x)}{2 g i^2 (b c-a d)^2}-\frac {b B \log ^2(c+d x)}{2 g i^2 (b c-a d)^2}+\frac {b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}-\frac {b B \log (a+b x)}{g i^2 (b c-a d)^2}+\frac {b B \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac {b B \log (c+d x)}{g i^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 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 \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(43 c+43 d x)^2 (a g+b g x)} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d) g (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g (c+d x)}\right ) \, dx\\ &=\frac {b^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1849 (b c-a d)^2 g}-\frac {(b d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1849 (b c-a d)^2 g}-\frac {d \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1849 (b c-a d) g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {(b B) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{1849 (b c-a d)^2 g}+\frac {(b B) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{1849 (b c-a d)^2 g}-\frac {B \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1849 (b c-a d) g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {B \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1849 g}-\frac {(b B) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{1849 (b c-a d)^2 e g}+\frac {(b B) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{1849 (b c-a d)^2 e g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {B \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1849 g}-\frac {(b B) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{1849 (b c-a d)^2 e g}+\frac {(b B) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{1849 (b c-a d)^2 e g}\\ &=-\frac {B}{1849 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x)}{1849 (b c-a d)^2 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac {b B \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {\left (b^2 B\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{1849 (b c-a d)^2 g}+\frac {\left (b^2 B\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1849 (b c-a d)^2 g}+\frac {(b B d) \int \frac {\log (a+b x)}{c+d x} \, dx}{1849 (b c-a d)^2 g}-\frac {(b B d) \int \frac {\log (c+d x)}{c+d x} \, dx}{1849 (b c-a d)^2 g}\\ &=-\frac {B}{1849 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x)}{1849 (b c-a d)^2 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac {b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac {b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}+\frac {b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}-\frac {(b B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{1849 (b c-a d)^2 g}-\frac {(b B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1849 (b c-a d)^2 g}-\frac {\left (b^2 B\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1849 (b c-a d)^2 g}-\frac {(b B d) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1849 (b c-a d)^2 g}\\ &=-\frac {B}{1849 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x)}{1849 (b c-a d)^2 g}-\frac {b B \log ^2(a+b x)}{3698 (b c-a d)^2 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac {b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac {b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b B \log ^2(c+d x)}{3698 (b c-a d)^2 g}+\frac {b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}-\frac {(b B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1849 (b c-a d)^2 g}-\frac {(b B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1849 (b c-a d)^2 g}\\ &=-\frac {B}{1849 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x)}{1849 (b c-a d)^2 g}-\frac {b B \log ^2(a+b x)}{3698 (b c-a d)^2 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac {b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac {b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac {b B \log ^2(c+d x)}{3698 (b c-a d)^2 g}+\frac {b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}+\frac {b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}+\frac {b B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}\\ \end {align*}

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Mathematica [C]  time = 0.28, size = 292, normalized size = 1.87 \[ \frac {2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b B (c+d x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)}{2 g i^2 (c+d x) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.89, size = 151, normalized size = 0.97 \[ \frac {2 \, {\left (A - B\right )} b c - 2 \, {\left (A - B\right )} a d + {\left (B b d x + B b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left ({\left (A - B\right )} b d x + A b c - B a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.00, size = 204, normalized size = 1.31 \[ -\frac {{\left (B b e \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 2 \, A b e \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {2 \, {\left (b x e + a e\right )} B d \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {2 \, {\left (b x e + a e\right )} A d}{d x + c} + \frac {2 \, {\left (b x e + a e\right )} B d}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{2 \, {\left (b c g - a d g\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(B*b*e*log((b*x*e + a*e)/(d*x + c))^2 + 2*A*b*e*log((b*x*e + a*e)/(d*x + c)) - 2*(b*x*e + a*e)*B*d*log((b
*x*e + a*e)/(d*x + c))/(d*x + c) - 2*(b*x*e + a*e)*A*d/(d*x + c) + 2*(b*x*e + a*e)*B*d/(d*x + c))*(b*c/((b*c*e
 - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/(b*c*g - a*d*g)

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maple [B]  time = 0.05, size = 759, normalized size = 4.87 \[ -\frac {B \,a^{2} d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {2 B a b c d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {B a b d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{3} g \,i^{2}}-\frac {B \,b^{2} c^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}-\frac {B \,b^{2} c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{3} g \,i^{2}}-\frac {A \,a^{2} d^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {2 A a b c d}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {A a b d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} g \,i^{2}}-\frac {A \,b^{2} c^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}-\frac {A \,b^{2} c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} g \,i^{2}}+\frac {B \,a^{2} d^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}-\frac {2 B a b c d}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}-\frac {B a b d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} g \,i^{2}}+\frac {B \,b^{2} c^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) g \,i^{2}}+\frac {B \,b^{2} c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{3} g \,i^{2}}-\frac {A a b d}{\left (a d -b c \right )^{3} g \,i^{2}}+\frac {A \,b^{2} c}{\left (a d -b c \right )^{3} g \,i^{2}}+\frac {B a b d}{\left (a d -b c \right )^{3} g \,i^{2}}-\frac {B \,b^{2} c}{\left (a d -b c \right )^{3} g \,i^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d/i^2/(a*d-b*c)^3/g*A*b*a+1/i^2/(a*d-b*c)^3/g*A*b^2*c-d^2/i^2/(a*d-b*c)^3/g*A/(d*x+c)*a^2+2*d/i^2/(a*d-b*c)^3
/g*A/(d*x+c)*a*b*c-1/i^2/(a*d-b*c)^3/g*A/(d*x+c)*b^2*c^2+d/i^2/(a*d-b*c)^3/g*A*b*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*
e)*a-1/i^2/(a*d-b*c)^3/g*A*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c-d/i^2/(a*d-b*c)^3/g*B*ln(b/d*e+(a*d-b*c)/(d*x
+c)/d*e)*b*a+1/i^2/(a*d-b*c)^3/g*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^2*c-d^2/i^2/(a*d-b*c)^3/g*B*ln(b/d*e+(a*d
-b*c)/(d*x+c)/d*e)/(d*x+c)*a^2+2*d/i^2/(a*d-b*c)^3/g*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)*a*b*c-1/i^2/(a*
d-b*c)^3/g*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)*b^2*c^2+d^2/i^2/(a*d-b*c)^3/g*B/(d*x+c)*a^2-2*d/i^2/(a*d-
b*c)^3/g*B/(d*x+c)*a*b*c+1/i^2/(a*d-b*c)^3/g*B/(d*x+c)*b^2*c^2+d/i^2/(a*d-b*c)^3/g*B*b*a-1/i^2/(a*d-b*c)^3/g*B
*b^2*c+1/2*d/i^2/(a*d-b*c)^3/g*B*b*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*a-1/2/i^2/(a*d-b*c)^3/g*B*b^2*ln(b/d*e+(a
*d-b*c)/(d*x+c)/d*e)^2*c

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maxima [B]  time = 1.32, size = 421, normalized size = 2.70 \[ B {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + A {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} - \frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{2 \, {\left (b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.81, size = 247, normalized size = 1.58 \[ \frac {B\,b\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {A-B}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {b\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a^2\,d^2\,g\,i^2-b^2\,c^2\,g\,i^2}{g\,i^2\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-B\right )\,2{}\mathrm {i}}{g\,i^2\,{\left (a\,d-b\,c\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*log((e*(a + b*x))/(c + d*x))^2)/(2*g*i^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (b*atan(((2*b*d*x + (a^2*d^2*
g*i^2 - b^2*c^2*g*i^2)/(g*i^2*(a*d - b*c)))*1i)/(a*d - b*c))*(A - B)*2i)/(g*i^2*(a*d - b*c)^2) - (A - B)/((a*d
 - b*c)*(c*g*i^2 + d*g*i^2*x)) - (B*log((e*(a + b*x))/(c + d*x))*(a*d - b*c))/(b*d*g*i^2*(x/b + c/(b*d))*(a^2*
d^2 + b^2*c^2 - 2*a*b*c*d))

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sympy [B]  time = 2.50, size = 386, normalized size = 2.47 \[ \frac {B b \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{2} d^{2} g i^{2} - 4 a b c d g i^{2} + 2 b^{2} c^{2} g i^{2}} - \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a c d g i^{2} + a d^{2} g i^{2} x - b c^{2} g i^{2} - b c d g i^{2} x} + \left (A - B\right ) \left (- \frac {b \log {\left (x + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} + \frac {b \log {\left (x + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} - \frac {1}{a c d g i^{2} - b c^{2} g i^{2} + x \left (a d^{2} g i^{2} - b c d g i^{2}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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