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

Optimal. Leaf size=276 \[ \frac {d i^2 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g}-\frac {i^2 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g}+\frac {i^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b g}+\frac {B i^2 (b c-a d)^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}-\frac {B i^2 (b c-a d)^2 \log \left (\frac {a+b x}{c+d x}\right )}{2 b^3 g}-\frac {3 B i^2 (b c-a d)^2 \log (c+d x)}{2 b^3 g}-\frac {B d i^2 x (b c-a d)}{2 b^2 g} \]

[Out]

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

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Rubi [A]  time = 0.49, antiderivative size = 354, normalized size of antiderivative = 1.28, number of steps used = 19, number of rules used = 13, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2528, 2486, 31, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 43} \[ \frac {B i^2 (b c-a d)^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {i^2 (b c-a d)^2 \log (a g+b g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g}+\frac {A d i^2 x (b c-a d)}{b^2 g}+\frac {i^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b g}+\frac {B d i^2 (a+b x) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g}-\frac {B d i^2 x (b c-a d)}{2 b^2 g}-\frac {B i^2 (b c-a d)^2 \log ^2(g (a+b x))}{2 b^3 g}-\frac {B i^2 (b c-a d)^2 \log (a+b x)}{2 b^3 g}-\frac {B i^2 (b c-a d)^2 \log (c+d x)}{b^3 g}+\frac {B i^2 (b c-a d)^2 \log (a g+b g x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*d*(b*c - a*d)*i^2*x)/(b^2*g) - (B*d*(b*c - a*d)*i^2*x)/(2*b^2*g) - (B*(b*c - a*d)^2*i^2*Log[a + b*x])/(2*b^
3*g) - (B*(b*c - a*d)^2*i^2*Log[g*(a + b*x)]^2)/(2*b^3*g) + (B*d*(b*c - a*d)*i^2*(a + b*x)*Log[(e*(a + b*x))/(
c + d*x)])/(b^3*g) + (i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b*g) - (B*(b*c - a*d)^2*i^2*Log
[c + d*x])/(b^3*g) + ((b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[a*g + b*g*x])/(b^3*g) + (B*(b
*c - a*d)^2*i^2*Log[(b*(c + d*x))/(b*c - a*d)]*Log[a*g + b*g*x])/(b^3*g) + (B*(b*c - a*d)^2*i^2*PolyLog[2, -((
d*(a + b*x))/(b*c - a*d))])/(b^3*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 43

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

Rule 2301

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

Rule 2390

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

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

Rule 2394

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

Rule 2418

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

Rule 2486

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

Rule 2524

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

Rule 2525

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

Rule 2528

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

Rubi steps

\begin {align*} \int \frac {(14 c+14 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx &=\int \left (\frac {196 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {14 d (14 c+14 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}+\frac {196 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (a g+b g x)}\right ) \, dx\\ &=\frac {\left (196 (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a g+b g x} \, dx}{b^2}+\frac {(14 d) \int (14 c+14 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b g}+\frac {(196 d (b c-a d)) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 g}\\ &=\frac {196 A d (b c-a d) x}{b^2 g}+\frac {98 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}+\frac {196 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {B \int \frac {196 (b c-a d) (c+d x)}{a+b x} \, dx}{2 b g}+\frac {(196 B d (b c-a d)) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^2 g}-\frac {\left (196 B (b c-a d)^2\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 g+b g x)}{e (a+b x)} \, dx}{b^3 g}\\ &=\frac {196 A d (b c-a d) x}{b^2 g}+\frac {196 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g}+\frac {98 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}+\frac {196 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {(98 B (b c-a d)) \int \frac {c+d x}{a+b x} \, dx}{b g}-\frac {\left (196 B d (b c-a d)^2\right ) \int \frac {1}{c+d x} \, dx}{b^3 g}-\frac {\left (196 B (b c-a d)^2\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 g+b g x)}{a+b x} \, dx}{b^3 e g}\\ &=\frac {196 A d (b c-a d) x}{b^2 g}+\frac {196 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g}+\frac {98 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}-\frac {196 B (b c-a d)^2 \log (c+d x)}{b^3 g}+\frac {196 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {(98 B (b c-a d)) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b g}-\frac {\left (196 B (b c-a d)^2\right ) \int \left (\frac {b e \log (a g+b g x)}{a+b x}-\frac {d e \log (a g+b g x)}{c+d x}\right ) \, dx}{b^3 e g}\\ &=\frac {196 A d (b c-a d) x}{b^2 g}-\frac {98 B d (b c-a d) x}{b^2 g}-\frac {98 B (b c-a d)^2 \log (a+b x)}{b^3 g}+\frac {196 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g}+\frac {98 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}-\frac {196 B (b c-a d)^2 \log (c+d x)}{b^3 g}+\frac {196 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {\left (196 B (b c-a d)^2\right ) \int \frac {\log (a g+b g x)}{a+b x} \, dx}{b^2 g}+\frac {\left (196 B d (b c-a d)^2\right ) \int \frac {\log (a g+b g x)}{c+d x} \, dx}{b^3 g}\\ &=\frac {196 A d (b c-a d) x}{b^2 g}-\frac {98 B d (b c-a d) x}{b^2 g}-\frac {98 B (b c-a d)^2 \log (a+b x)}{b^3 g}+\frac {196 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g}+\frac {98 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}-\frac {196 B (b c-a d)^2 \log (c+d x)}{b^3 g}+\frac {196 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{b^3 g}+\frac {196 B (b c-a d)^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {\left (196 B (b c-a d)^2\right ) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right )}{a g+b g x} \, dx}{b^2}-\frac {\left (196 B (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g^2}\\ &=\frac {196 A d (b c-a d) x}{b^2 g}-\frac {98 B d (b c-a d) x}{b^2 g}-\frac {98 B (b c-a d)^2 \log (a+b x)}{b^3 g}+\frac {196 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g}+\frac {98 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}-\frac {196 B (b c-a d)^2 \log (c+d x)}{b^3 g}+\frac {196 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{b^3 g}+\frac {196 B (b c-a d)^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {\left (196 B (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g}-\frac {\left (196 B (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=\frac {196 A d (b c-a d) x}{b^2 g}-\frac {98 B d (b c-a d) x}{b^2 g}-\frac {98 B (b c-a d)^2 \log (a+b x)}{b^3 g}-\frac {98 B (b c-a d)^2 \log ^2(g (a+b x))}{b^3 g}+\frac {196 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g}+\frac {98 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}-\frac {196 B (b c-a d)^2 \log (c+d x)}{b^3 g}+\frac {196 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{b^3 g}+\frac {196 B (b c-a d)^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}+\frac {196 B (b c-a d)^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 252, normalized size = 0.91 \[ \frac {i^2 \left (b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \log (g (a+b x)) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 A b d x (b c-a d)+2 B d (a+b x) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+B (b c-a d)^2 \left (2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )-\log (g (a+b x)) \left (\log (g (a+b x))-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-2 B (b c-a d)^2 \log (c+d x)-B (b c-a d) ((b c-a d) \log (a+b x)+b d x)\right )}{2 b^3 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d^{2} i^{2} x^{2} + 2 \, A c d i^{2} x + A c^{2} i^{2} + {\left (B d^{2} i^{2} x^{2} + 2 \, B c d i^{2} x + B c^{2} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b g x + a g}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.15, size = 2538, normalized size = 9.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.80, size = 518, normalized size = 1.88 \[ 2 \, A c d i^{2} {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} + \frac {1}{2} \, A d^{2} i^{2} {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{3} g} + \frac {b x^{2} - 2 \, a x}{b^{2} g}\right )} + \frac {A c^{2} i^{2} \log \left (b g x + a g\right )}{b g} - \frac {{\left (3 \, b c^{2} i^{2} - 2 \, a c d i^{2}\right )} B \log \left (d x + c\right )}{2 \, b^{2} g} + \frac {{\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{3} g} + \frac {B b^{2} d^{2} i^{2} x^{2} \log \relax (e) + {\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} B \log \left (b x + a\right )^{2} + {\left ({\left (4 \, i^{2} \log \relax (e) - i^{2}\right )} b^{2} c d - {\left (2 \, i^{2} \log \relax (e) - i^{2}\right )} a b d^{2}\right )} B x + {\left (B b^{2} d^{2} i^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d i^{2} - a b d^{2} i^{2}\right )} B x + {\left (2 \, b^{2} c^{2} i^{2} \log \relax (e) - 4 \, {\left (i^{2} \log \relax (e) - i^{2}\right )} a b c d + {\left (2 \, i^{2} \log \relax (e) - 3 \, i^{2}\right )} a^{2} d^{2}\right )} B\right )} \log \left (b x + a\right ) - {\left (B b^{2} d^{2} i^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d i^{2} - a b d^{2} i^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} B \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{2 \, b^{3} g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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