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

Optimal. Leaf size=289 \[ \frac{B i^2 n (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac{d i^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}-\frac{i^2 (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac{i^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b g}-\frac{B d i^2 n x (b c-a d)}{2 b^2 g}-\frac{B i^2 n (b c-a d)^2 \log \left (\frac{a+b x}{c+d x}\right )}{2 b^3 g}-\frac{3 B i^2 n (b c-a d)^2 \log (c+d x)}{2 b^3 g} \]

[Out]

-(B*d*(b*c - a*d)*i^2*n*x)/(2*b^2*g) + (d*(b*c - a*d)*i^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b
^3*g) + (i^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*g) - (B*(b*c - a*d)^2*i^2*n*Log[(a + b*x
)/(c + d*x)])/(2*b^3*g) - (3*B*(b*c - a*d)^2*i^2*n*Log[c + d*x])/(2*b^3*g) - ((b*c - a*d)^2*i^2*(A + B*Log[e*(
(a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (B*(b*c - a*d)^2*i^2*n*PolyLog[2, (b*
(c + d*x))/(d*(a + b*x))])/(b^3*g)

________________________________________________________________________________________

Rubi [A]  time = 0.489496, antiderivative size = 369, normalized size of antiderivative = 1.28, number of steps used = 18, number of rules used = 13, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.302, Rules used = {2528, 2486, 31, 2524, 2418, 2390, 12, 2301, 2394, 2393, 2391, 2525, 43} \[ \frac{B i^2 n (b c-a d)^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{i^2 (b c-a d)^2 \log (a g+b g x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac{A d i^2 x (b c-a d)}{b^2 g}+\frac{i^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b g}+\frac{B d i^2 (a+b x) (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}-\frac{B d i^2 n x (b c-a d)}{2 b^2 g}-\frac{B i^2 n (b c-a d)^2 \log ^2(g (a+b x))}{2 b^3 g}-\frac{B i^2 n (b c-a d)^2 \log (a+b x)}{2 b^3 g}-\frac{B i^2 n (b c-a d)^2 \log (c+d x)}{b^3 g}+\frac{B i^2 n (b c-a d)^2 \log (a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g} \]

Antiderivative was successfully verified.

[In]

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

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]

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 31

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

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 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 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 12

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

Rule 2301

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

Rule 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 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 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 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 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])

Rubi steps

\begin{align*} \int \frac{(121 c+121 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx &=\int \left (\frac{14641 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac{121 d (121 c+121 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g}+\frac{14641 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 (a g+b g x)}\right ) \, dx\\ &=\frac{\left (14641 (b c-a d)^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a g+b g x} \, dx}{b^2}+\frac{(121 d) \int (121 c+121 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b g}+\frac{(14641 d (b c-a d)) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g}\\ &=\frac{14641 A d (b c-a d) x}{b^2 g}+\frac{14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b g}+\frac{14641 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^3 g}+\frac{(14641 B d (b c-a d)) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 g}-\frac{(B n) \int \frac{14641 (b c-a d) (c+d x)}{a+b x} \, dx}{2 b g}-\frac{\left (14641 B (b c-a d)^2 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a g+b g x)}{a+b x} \, dx}{b^3 g}\\ &=\frac{14641 A d (b c-a d) x}{b^2 g}+\frac{14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b g}+\frac{14641 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^3 g}-\frac{(14641 B (b c-a d) n) \int \frac{c+d x}{a+b x} \, dx}{2 b g}-\frac{\left (14641 B (b c-a d)^2 n\right ) \int \left (\frac{b \log (a g+b g x)}{a+b x}-\frac{d \log (a g+b g x)}{c+d x}\right ) \, dx}{b^3 g}-\frac{\left (14641 B d (b c-a d)^2 n\right ) \int \frac{1}{c+d x} \, dx}{b^3 g}\\ &=\frac{14641 A d (b c-a d) x}{b^2 g}+\frac{14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac{14641 B (b c-a d)^2 n \log (c+d x)}{b^3 g}+\frac{14641 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^3 g}-\frac{(14641 B (b c-a d) n) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{2 b g}-\frac{\left (14641 B (b c-a d)^2 n\right ) \int \frac{\log (a g+b g x)}{a+b x} \, dx}{b^2 g}+\frac{\left (14641 B d (b c-a d)^2 n\right ) \int \frac{\log (a g+b g x)}{c+d x} \, dx}{b^3 g}\\ &=\frac{14641 A d (b c-a d) x}{b^2 g}-\frac{14641 B d (b c-a d) n x}{2 b^2 g}-\frac{14641 B (b c-a d)^2 n \log (a+b x)}{2 b^3 g}+\frac{14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac{14641 B (b c-a d)^2 n \log (c+d x)}{b^3 g}+\frac{14641 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^3 g}+\frac{14641 B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{\left (14641 B (b c-a d)^2 n\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 (14641 B (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{g \log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g^2}\\ &=\frac{14641 A d (b c-a d) x}{b^2 g}-\frac{14641 B d (b c-a d) n x}{2 b^2 g}-\frac{14641 B (b c-a d)^2 n \log (a+b x)}{2 b^3 g}+\frac{14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac{14641 B (b c-a d)^2 n \log (c+d x)}{b^3 g}+\frac{14641 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^3 g}+\frac{14641 B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{\left (14641 B (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g}-\frac{\left (14641 B (b c-a d)^2 n\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{14641 A d (b c-a d) x}{b^2 g}-\frac{14641 B d (b c-a d) n x}{2 b^2 g}-\frac{14641 B (b c-a d)^2 n \log (a+b x)}{2 b^3 g}-\frac{14641 B (b c-a d)^2 n \log ^2(g (a+b x))}{2 b^3 g}+\frac{14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac{14641 B (b c-a d)^2 n \log (c+d x)}{b^3 g}+\frac{14641 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^3 g}+\frac{14641 B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}+\frac{14641 B (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}\\ \end{align*}

Mathematica [A]  time = 0.185614, size = 264, normalized size = 0.91 \[ \frac{i^2 \left (B n (b c-a d)^2 \left (2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-\log (g (a+b x)) \left (\log (g (a+b x))-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )\right )+b^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \log (g (a+b x)) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 A b d x (b c-a d)+2 B d (a+b x) (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 B n (b c-a d)^2 \log (c+d x)-B n (b c-a d) ((b c-a d) \log (a+b x)+b d x)\right )}{2 b^3 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(i^2*(2*A*b*d*(b*c - a*d)*x - B*(b*c - a*d)*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) + 2*B*d*(b*c - a*d)*(a + b*x)
*Log[e*((a + b*x)/(c + d*x))^n] + b^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*(b*c - a*d)^2*Log
[g*(a + b*x)]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 2*B*(b*c - a*d)^2*n*Log[c + d*x] + B*(b*c - a*d)^2*n*(-
(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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Maple [F]  time = 0.686, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{bgx+ag} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 3.19895, size = 783, normalized size = 2.71 \begin{align*} 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} n - 2 \, a c d i^{2} n\right )} B \log \left (d x + c\right )}{2 \, b^{2} g} + \frac{{\left (b^{2} c^{2} i^{2} n - 2 \, a b c d i^{2} n + a^{2} d^{2} i^{2} n\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 \left (e\right ) -{\left (b^{2} c^{2} i^{2} n - 2 \, a b c d i^{2} n + a^{2} d^{2} i^{2} n\right )} B \log \left (b x + a\right )^{2} -{\left ({\left (i^{2} n - 4 \, i^{2} \log \left (e\right )\right )} b^{2} c d -{\left (i^{2} n - 2 \, i^{2} \log \left (e\right )\right )} a b d^{2}\right )} B x +{\left (2 \, b^{2} c^{2} i^{2} \log \left (e\right ) + 4 \,{\left (i^{2} n - i^{2} \log \left (e\right )\right )} a b c d -{\left (3 \, i^{2} n - 2 \, i^{2} \log \left (e\right )\right )} a^{2} d^{2}\right )} B \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 ({\left (b x + a\right )}^{n}\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 ({\left (d x + c\right )}^{n}\right )}{2 \, b^{3} g} \end{align*}

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b g x + a g}, x\right ) \end{align*}

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))^n))/(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(e*((b*x
+ a)/(d*x + c))^n))/(b*g*x + a*g), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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))**n))/(b*g*x+a*g),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{b g x + a g}\,{d x} \end{align*}

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))^n))/(b*g*x+a*g),x, algorithm="giac")

[Out]

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