3.102 \(\int \frac {(a g+b g x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(c i+d i x)^3} \, dx\)

Optimal. Leaf size=141 \[ \frac {g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 i^3 (c+d x)^2 (b c-a d)}-\frac {B g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}+\frac {B^2 g (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.96, antiderivative size = 634, normalized size of antiderivative = 4.50, number of steps used = 58, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac {b^2 B^2 g \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac {b^2 B^2 g \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac {b^2 B g \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^3 (b c-a d)}-\frac {b^2 B g \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^3 (b c-a d)}-\frac {b g \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^2 i^3 (c+d x)}+\frac {b B g \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^3 (c+d x)}+\frac {g (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^2 i^3 (c+d x)^2}-\frac {B g (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 i^3 (c+d x)^2}-\frac {b^2 B^2 g \log ^2(a+b x)}{2 d^2 i^3 (b c-a d)}-\frac {b^2 B^2 g \log ^2(c+d x)}{2 d^2 i^3 (b c-a d)}-\frac {b^2 B^2 g \log (a+b x)}{2 d^2 i^3 (b c-a d)}+\frac {b^2 B^2 g \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac {b^2 B^2 g \log (c+d x)}{2 d^2 i^3 (b c-a d)}+\frac {b^2 B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac {B^2 g (b c-a d)}{4 d^2 i^3 (c+d x)^2}-\frac {b B^2 g}{2 d^2 i^3 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B^2*(b*c - a*d)*g)/(4*d^2*i^3*(c + d*x)^2) - (b*B^2*g)/(2*d^2*i^3*(c + d*x)) - (b^2*B^2*g*Log[a + b*x])/(2*d^
2*(b*c - a*d)*i^3) - (b^2*B^2*g*Log[a + b*x]^2)/(2*d^2*(b*c - a*d)*i^3) - (B*(b*c - a*d)*g*(A + B*Log[(e*(a +
b*x))/(c + d*x)]))/(2*d^2*i^3*(c + d*x)^2) + (b*B*g*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*i^3*(c + d*x))
+ (b^2*B*g*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*(b*c - a*d)*i^3) + ((b*c - a*d)*g*(A + B*Lo
g[(e*(a + b*x))/(c + d*x)])^2)/(2*d^2*i^3*(c + d*x)^2) - (b*g*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*i^3
*(c + d*x)) + (b^2*B^2*g*Log[c + d*x])/(2*d^2*(b*c - a*d)*i^3) + (b^2*B^2*g*Log[-((d*(a + b*x))/(b*c - a*d))]*
Log[c + d*x])/(d^2*(b*c - a*d)*i^3) - (b^2*B*g*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(d^2*(b*c -
a*d)*i^3) - (b^2*B^2*g*Log[c + d*x]^2)/(2*d^2*(b*c - a*d)*i^3) + (b^2*B^2*g*Log[a + b*x]*Log[(b*(c + d*x))/(b*
c - a*d)])/(d^2*(b*c - a*d)*i^3) + (b^2*B^2*g*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d^2*(b*c - a*d)*i^3)
+ (b^2*B^2*g*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^2*(b*c - a*d)*i^3)

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 g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(102 c+102 d x)^3} \, dx &=\int \left (\frac {(-b c+a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d (c+d x)^3}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d (c+d x)^2}\right ) \, dx\\ &=\frac {(b g) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2} \, dx}{1061208 d}-\frac {((b c-a d) g) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^3} \, dx}{1061208 d}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {(b B g) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}-\frac {(B (b c-a d) g) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^3} \, dx}{1061208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {(b B (b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}-\frac {\left (B (b c-a d)^2 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^3} \, dx}{1061208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {(b B (b c-a d) g) \int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{530604 d^2}-\frac {\left (B (b c-a d)^2 g\right ) \int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{1061208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {(b B g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1061208 d}-\frac {(b B g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{530604 d}-\frac {\left (b^3 B g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1061208 d^2 (b c-a d)}+\frac {\left (b^3 B g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac {\left (b^2 B g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1061208 d (b c-a d)}-\frac {\left (b^2 B g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{530604 d (b c-a d)}+\frac {(B (b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{1061208 d}\\ &=-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {\left (b B^2 g\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1061208 d^2}-\frac {\left (b B^2 g\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}+\frac {\left (b^2 B^2 g\right ) \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}{1061208 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \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}{1061208 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \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}{530604 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \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}{530604 d^2 (b c-a d)}+\frac {\left (B^2 (b c-a d) g\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{2122416 d^2}\\ &=-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1061208 d^2}-\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}+\frac {\left (B^2 (b c-a d)^2 g\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{2122416 d^2}+\frac {\left (b^2 B^2 g\right ) \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}{1061208 d^2 (b c-a d) e}-\frac {\left (b^2 B^2 g\right ) \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}{1061208 d^2 (b c-a d) e}-\frac {\left (b^2 B^2 g\right ) \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}{530604 d^2 (b c-a d) e}+\frac {\left (b^2 B^2 g\right ) \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}{530604 d^2 (b c-a d) e}\\ &=-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {\left (b B^2 (b c-a d) g\right ) \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}{1061208 d^2}-\frac {\left (b B^2 (b c-a d) g\right ) \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}{530604 d^2}+\frac {\left (B^2 (b c-a d)^2 g\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2122416 d^2}+\frac {\left (b^2 B^2 g\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{1061208 d^2 (b c-a d) e}-\frac {\left (b^2 B^2 g\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}{1061208 d^2 (b c-a d) e}-\frac {\left (b^2 B^2 g\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{530604 d^2 (b c-a d) e}+\frac {\left (b^2 B^2 g\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}{530604 d^2 (b c-a d) e}\\ &=\frac {B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac {b B^2 g}{2122416 d^2 (c+d x)}-\frac {b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{530604 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{1061208 d (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{1061208 d (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{530604 d (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{530604 d (b c-a d)}\\ &=\frac {B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac {b B^2 g}{2122416 d^2 (c+d x)}-\frac {b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {b^2 B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{1061208 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1061208 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{530604 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{530604 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1061208 d (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{530604 d (b c-a d)}\\ &=\frac {B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac {b B^2 g}{2122416 d^2 (c+d x)}-\frac {b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac {b^2 B^2 g \log ^2(a+b x)}{2122416 d^2 (b c-a d)}-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B^2 g \log ^2(c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1061208 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1061208 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{530604 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{530604 d^2 (b c-a d)}\\ &=\frac {B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac {b B^2 g}{2122416 d^2 (c+d x)}-\frac {b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac {b^2 B^2 g \log ^2(a+b x)}{2122416 d^2 (b c-a d)}-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B^2 g \log ^2(c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac {b^2 B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac {b^2 B^2 g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}\\ \end {align*}

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Mathematica [C]  time = 0.91, size = 767, normalized size = 5.44 \[ \frac {g \left (-B \left (4 b^2 (c+d x)^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-4 b^2 (c+d x)^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 b (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 b^2 B (c+d x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-4 b B (c+d x) (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )+4 b B (c+d x) \left (2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b B (c+d x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )+2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2-4 b (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2\right )}{4 d^2 i^3 (c+d x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 4*b*(b*c - a*d)*(c + d*x)*(A + B*Log[(e*(a + b*x)
)/(c + d*x)])^2 + 4*b*B*(c + d*x)*(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)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) - B*(2*
(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*b*(b*c - a*d)*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d
*x)]) + 4*b^2*(c + d*x)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 4*b^2*(c + d*x)^2*(A + B*Log[(e*
(a + b*x))/(c + d*x)])*Log[c + d*x] - 4*b*B*(c + d*x)*(b*c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*Log[
c + d*x]) - B*((b*c - a*d)^2 + 2*b*(b*c - a*d)*(c + d*x) + 2*b^2*(c + d*x)^2*Log[a + b*x] - 2*b^2*(c + d*x)^2*
Log[c + d*x]) - 2*b^2*B*(c + d*x)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLo
g[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 2*b^2*B*(c + d*x)^2*((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)]))))/(4*d^2*(b*c - a*d)*i^3*(c + d*x)^2)

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fricas [B]  time = 0.89, size = 295, normalized size = 2.09 \[ -\frac {2 \, {\left ({\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a b d^{2}\right )} g x - 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x + B^{2} a^{2} d^{2} g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + {\left ({\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} b^{2} c^{2} - {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} d^{2}\right )} g - 2 \, {\left ({\left (2 \, A B - B^{2}\right )} b^{2} d^{2} g x^{2} + 2 \, {\left (2 \, A B - B^{2}\right )} a b d^{2} g x + {\left (2 \, A B - B^{2}\right )} a^{2} d^{2} g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*((2*A^2 - 2*A*B + B^2)*b^2*c*d - (2*A^2 - 2*A*B + B^2)*a*b*d^2)*g*x - 2*(B^2*b^2*d^2*g*x^2 + 2*B^2*a*b
*d^2*g*x + B^2*a^2*d^2*g)*log((b*e*x + a*e)/(d*x + c))^2 + ((2*A^2 - 2*A*B + B^2)*b^2*c^2 - (2*A^2 - 2*A*B + B
^2)*a^2*d^2)*g - 2*((2*A*B - B^2)*b^2*d^2*g*x^2 + 2*(2*A*B - B^2)*a*b*d^2*g*x + (2*A*B - B^2)*a^2*d^2*g)*log((
b*e*x + a*e)/(d*x + c)))/((b*c*d^4 - a*d^5)*i^3*x^2 + 2*(b*c^2*d^3 - a*c*d^4)*i^3*x + (b*c^3*d^2 - a*c^2*d^3)*
i^3)

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giac [B]  time = 1.50, size = 273, normalized size = 1.94 \[ \frac {1}{4} \, {\left (\frac {2 \, {\left (b x e + a e\right )}^{2} B^{2} g i \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2}} + \frac {4 \, {\left (b x e + a e\right )}^{2} A B g i \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {2 \, {\left (b x e + a e\right )}^{2} B^{2} g i \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {2 \, {\left (b x e + a e\right )}^{2} A^{2} g i}{{\left (d x + c\right )}^{2}} - \frac {2 \, {\left (b x e + a e\right )}^{2} A B g i}{{\left (d x + c\right )}^{2}} + \frac {{\left (b x e + a e\right )}^{2} B^{2} g i}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 2449, normalized size = 17.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.52, size = 1966, normalized size = 13.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*d*x + c)*B^2*b*g*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3) +
1/4*(2*((2*b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)*i^3*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*i^3*x + (b*c^3*d - a*c^2*
d^2)*i^3) + 2*b^2*log(b*x + a)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x + c)/((b^2*c^2*d - 2*
a*b*c*d^2 + a^2*d^3)*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (7*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(b^2*d^
2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(d*x + c)^2 + 6*(b^
2*c*d - a*b*d^2)*x + 6*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a) - 2*(3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3
*b^2*c^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a))*log(d*x + c))/(b^2*c^4*d*i^3 - 2*a*b*c^3*d^2*
i^3 + a^2*c^2*d^3*i^3 + (b^2*c^2*d^3*i^3 - 2*a*b*c*d^4*i^3 + a^2*d^5*i^3)*x^2 + 2*(b^2*c^3*d^2*i^3 - 2*a*b*c^2
*d^3*i^3 + a^2*c*d^4*i^3)*x))*B^2*a*g + 1/4*(2*((b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/((b*c*d^4 - a*d^5)*i
^3*x^2 + 2*(b*c^2*d^3 - a*c*d^4)*i^3*x + (b*c^3*d^2 - a*c^2*d^3)*i^3) + 2*(b^2*c - 2*a*b*d)*log(b*x + a)/((b^2
*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*i^3) - 2*(b^2*c - 2*a*b*d)*log(d*x + c)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^
4)*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2 + 2*(b^2*c^3 - 2*a*b*c^2*
d + (b^2*c*d^2 - 2*a*b*d^3)*x^2 + 2*(b^2*c^2*d - 2*a*b*c*d^2)*x)*log(b*x + a)^2 + 2*(b^2*c^3 - 2*a*b*c^2*d + (
b^2*c*d^2 - 2*a*b*d^3)*x^2 + 2*(b^2*c^2*d - 2*a*b*c*d^2)*x)*log(d*x + c)^2 + 2*(b^2*c^2*d - 5*a*b*c*d^2 + 4*a^
2*d^3)*x + 2*(b^2*c^3 - 4*a*b*c^2*d + (b^2*c*d^2 - 4*a*b*d^3)*x^2 + 2*(b^2*c^2*d - 4*a*b*c*d^2)*x)*log(b*x + a
) - 2*(b^2*c^3 - 4*a*b*c^2*d + (b^2*c*d^2 - 4*a*b*d^3)*x^2 + 2*(b^2*c^2*d - 4*a*b*c*d^2)*x + 2*(b^2*c^3 - 2*a*
b*c^2*d + (b^2*c*d^2 - 2*a*b*d^3)*x^2 + 2*(b^2*c^2*d - 2*a*b*c*d^2)*x)*log(b*x + a))*log(d*x + c))/(b^2*c^4*d^
2*i^3 - 2*a*b*c^3*d^3*i^3 + a^2*c^2*d^4*i^3 + (b^2*c^2*d^4*i^3 - 2*a*b*c*d^5*i^3 + a^2*d^6*i^3)*x^2 + 2*(b^2*c
^3*d^3*i^3 - 2*a*b*c^2*d^4*i^3 + a^2*c*d^5*i^3)*x))*B^2*b*g - 1/2*A*B*b*g*(2*(2*d*x + c)*log(b*e*x/(d*x + c) +
a*e/(d*x + c))/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3) - (b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/((b*c*
d^4 - a*d^5)*i^3*x^2 + 2*(b*c^2*d^3 - a*c*d^4)*i^3*x + (b*c^3*d^2 - a*c^2*d^3)*i^3) - 2*(b^2*c - 2*a*b*d)*log(
b*x + a)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*i^3) + 2*(b^2*c - 2*a*b*d)*log(d*x + c)/((b^2*c^2*d^2 - 2*a*b*
c*d^3 + a^2*d^4)*i^3)) + 1/2*A*B*a*g*((2*b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)*i^3*x^2 + 2*(b*c^2*d^2 - a*c*
d^3)*i^3*x + (b*c^3*d - a*c^2*d^2)*i^3) - 2*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^3*i^3*x^2 + 2*c*d^2*i^3*x
+ c^2*d*i^3) + 2*b^2*log(b*x + a)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x + c)/((b^2*c^2*d -
2*a*b*c*d^2 + a^2*d^3)*i^3)) - 1/2*B^2*a*g*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(d^3*i^3*x^2 + 2*c*d^2*i^3*
x + c^2*d*i^3) - 1/2*(2*d*x + c)*A^2*b*g/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3) - 1/2*A^2*a*g/(d^3*i^3*x^
2 + 2*c*d^2*i^3*x + c^2*d*i^3)

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mupad [B]  time = 6.40, size = 474, normalized size = 3.36 \[ -\frac {x\,\left (2\,b\,d\,g\,A^2-2\,b\,d\,g\,A\,B+b\,d\,g\,B^2\right )+A^2\,a\,d\,g+A^2\,b\,c\,g+\frac {B^2\,a\,d\,g}{2}+\frac {B^2\,b\,c\,g}{2}-A\,B\,a\,d\,g-A\,B\,b\,c\,g}{2\,c^2\,d^2\,i^3+4\,c\,d^3\,i^3\,x+2\,d^4\,i^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,a\,g}{2\,d^2\,i^3}+\frac {B^2\,b\,c\,g}{2\,d^3\,i^3}+\frac {B^2\,b\,g\,x}{d^2\,i^3}}{2\,c\,x+d\,x^2+\frac {c^2}{d}}+\frac {B^2\,b^2\,g}{2\,d^2\,i^3\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {A\,B\,c\,g}{d^3\,i^3}-x\,\left (\frac {B^2\,g}{d^2\,i^3}-\frac {2\,A\,B\,g}{d^2\,i^3}\right )+\frac {B\,g\,\left (A\,a\,d-B\,a\,d+B\,b\,c\right )}{b\,d^3\,i^3}+\frac {B^2\,b^2\,g\,\left (\frac {a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2}{2\,b^3\,d}-\frac {c\,\left (a\,d-b\,c\right )}{2\,b^2\,d}\right )}{d^2\,i^3\,\left (a\,d-b\,c\right )}\right )}{\frac {d\,x^2}{b}+\frac {c^2}{b\,d}+\frac {2\,c\,x}{b}}+\frac {B\,b^2\,g\,\mathrm {atan}\left (\frac {\left (\frac {2\,a\,d^3\,i^3+2\,b\,c\,d^2\,i^3}{2\,d^2\,i^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A-B\right )\,1{}\mathrm {i}}{d^2\,i^3\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 13.80, size = 712, normalized size = 5.05 \[ \frac {B b^{2} g \left (2 A - B\right ) \log {\left (x + \frac {2 A B a b^{2} d g + 2 A B b^{3} c g - B^{2} a b^{2} d g - B^{2} b^{3} c g - \frac {B a^{2} b^{2} d^{2} g \left (2 A - B\right )}{a d - b c} + \frac {2 B a b^{3} c d g \left (2 A - B\right )}{a d - b c} - \frac {B b^{4} c^{2} g \left (2 A - B\right )}{a d - b c}}{4 A B b^{3} d g - 2 B^{2} b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} - \frac {B b^{2} g \left (2 A - B\right ) \log {\left (x + \frac {2 A B a b^{2} d g + 2 A B b^{3} c g - B^{2} a b^{2} d g - B^{2} b^{3} c g + \frac {B a^{2} b^{2} d^{2} g \left (2 A - B\right )}{a d - b c} - \frac {2 B a b^{3} c d g \left (2 A - B\right )}{a d - b c} + \frac {B b^{4} c^{2} g \left (2 A - B\right )}{a d - b c}}{4 A B b^{3} d g - 2 B^{2} b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} + \frac {\left (- B^{2} a^{2} g - 2 B^{2} a b g x - B^{2} b^{2} g x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a c^{2} d i^{3} + 4 a c d^{2} i^{3} x + 2 a d^{3} i^{3} x^{2} - 2 b c^{3} i^{3} - 4 b c^{2} d i^{3} x - 2 b c d^{2} i^{3} x^{2}} + \frac {- 2 A^{2} a d g - 2 A^{2} b c g + 2 A B a d g + 2 A B b c g - B^{2} a d g - B^{2} b c g + x \left (- 4 A^{2} b d g + 4 A B b d g - 2 B^{2} b d g\right )}{4 c^{2} d^{2} i^{3} + 8 c d^{3} i^{3} x + 4 d^{4} i^{3} x^{2}} + \frac {\left (- 2 A B a d g - 2 A B b c g - 4 A B b d g x + B^{2} a d g + B^{2} b c g + 2 B^{2} b d g x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d^{2} i^{3} + 4 c d^{3} i^{3} x + 2 d^{4} i^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b**2*g*(2*A - B)*log(x + (2*A*B*a*b**2*d*g + 2*A*B*b**3*c*g - B**2*a*b**2*d*g - B**2*b**3*c*g - B*a**2*b**2*
d**2*g*(2*A - B)/(a*d - b*c) + 2*B*a*b**3*c*d*g*(2*A - B)/(a*d - b*c) - B*b**4*c**2*g*(2*A - B)/(a*d - b*c))/(
4*A*B*b**3*d*g - 2*B**2*b**3*d*g))/(2*d**2*i**3*(a*d - b*c)) - B*b**2*g*(2*A - B)*log(x + (2*A*B*a*b**2*d*g +
2*A*B*b**3*c*g - B**2*a*b**2*d*g - B**2*b**3*c*g + B*a**2*b**2*d**2*g*(2*A - B)/(a*d - b*c) - 2*B*a*b**3*c*d*g
*(2*A - B)/(a*d - b*c) + B*b**4*c**2*g*(2*A - B)/(a*d - b*c))/(4*A*B*b**3*d*g - 2*B**2*b**3*d*g))/(2*d**2*i**3
*(a*d - b*c)) + (-B**2*a**2*g - 2*B**2*a*b*g*x - B**2*b**2*g*x**2)*log(e*(a + b*x)/(c + d*x))**2/(2*a*c**2*d*i
**3 + 4*a*c*d**2*i**3*x + 2*a*d**3*i**3*x**2 - 2*b*c**3*i**3 - 4*b*c**2*d*i**3*x - 2*b*c*d**2*i**3*x**2) + (-2
*A**2*a*d*g - 2*A**2*b*c*g + 2*A*B*a*d*g + 2*A*B*b*c*g - B**2*a*d*g - B**2*b*c*g + x*(-4*A**2*b*d*g + 4*A*B*b*
d*g - 2*B**2*b*d*g))/(4*c**2*d**2*i**3 + 8*c*d**3*i**3*x + 4*d**4*i**3*x**2) + (-2*A*B*a*d*g - 2*A*B*b*c*g - 4
*A*B*b*d*g*x + B**2*a*d*g + B**2*b*c*g + 2*B**2*b*d*g*x)*log(e*(a + b*x)/(c + d*x))/(2*c**2*d**2*i**3 + 4*c*d*
*3*i**3*x + 2*d**4*i**3*x**2)

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