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

Optimal. Leaf size=203 \[ -\frac{B^2 i (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^2 d}+\frac{B i (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 d}-\frac{B i (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2}+\frac{i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 d}+\frac{B^2 i (b c-a d)^2 \log (c+d x)}{b^2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.432791, antiderivative size = 283, normalized size of antiderivative = 1.39, number of steps used = 16, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{B^2 i (b c-a d)^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d}-\frac{B i (b c-a d)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 d}+\frac{i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 d}-\frac{A B i x (b c-a d)}{b}-\frac{B^2 i (a+b x) (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2}+\frac{B^2 i (b c-a d)^2 \log ^2(a+b x)}{2 b^2 d}+\frac{B^2 i (b c-a d)^2 \log (c+d x)}{b^2 d}-\frac{B^2 i (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 12

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

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 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]

Rubi steps

\begin{align*} \int (58 c+58 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}-\frac{B \int \frac{3364 (b c-a d) (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{58 d}\\ &=\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}-\frac{(58 B (b c-a d)) \int \frac{(c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d}\\ &=\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}-\frac{(58 B (b c-a d)) \int \left (\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b}+\frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b (a+b x)}\right ) \, dx}{d}\\ &=\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}-\frac{(58 B (b c-a d)) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b}-\frac{\left (58 B (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b d}\\ &=-\frac{58 A B (b c-a d) x}{b}-\frac{58 B (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 d}+\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}-\frac{\left (58 B^2 (b c-a d)\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{b}+\frac{\left (58 B^2 (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+b x)}{e (a+b x)} \, dx}{b^2 d}\\ &=-\frac{58 A B (b c-a d) x}{b}-\frac{58 B^2 (b c-a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2}-\frac{58 B (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 d}+\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac{\left (58 B^2 (b c-a d)^2\right ) \int \frac{1}{c+d x} \, dx}{b^2}+\frac{\left (58 B^2 (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+b x)}{a+b x} \, dx}{b^2 d e}\\ &=-\frac{58 A B (b c-a d) x}{b}-\frac{58 B^2 (b c-a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2}-\frac{58 B (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 d}+\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac{58 B^2 (b c-a d)^2 \log (c+d x)}{b^2 d}+\frac{\left (58 B^2 (b c-a d)^2\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}{b^2 d e}\\ &=-\frac{58 A B (b c-a d) x}{b}-\frac{58 B^2 (b c-a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2}-\frac{58 B (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 d}+\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac{58 B^2 (b c-a d)^2 \log (c+d x)}{b^2 d}-\frac{\left (58 B^2 (b c-a d)^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2}+\frac{\left (58 B^2 (b c-a d)^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b d}\\ &=-\frac{58 A B (b c-a d) x}{b}-\frac{58 B^2 (b c-a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2}-\frac{58 B (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 d}+\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac{58 B^2 (b c-a d)^2 \log (c+d x)}{b^2 d}-\frac{58 B^2 (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 d}+\frac{\left (58 B^2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 d}+\frac{\left (58 B^2 (b c-a d)^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b d}\\ &=-\frac{58 A B (b c-a d) x}{b}+\frac{29 B^2 (b c-a d)^2 \log ^2(a+b x)}{b^2 d}-\frac{58 B^2 (b c-a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2}-\frac{58 B (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 d}+\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac{58 B^2 (b c-a d)^2 \log (c+d x)}{b^2 d}-\frac{58 B^2 (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 d}+\frac{\left (58 B^2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 d}\\ &=-\frac{58 A B (b c-a d) x}{b}+\frac{29 B^2 (b c-a d)^2 \log ^2(a+b x)}{b^2 d}-\frac{58 B^2 (b c-a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2}-\frac{58 B (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 d}+\frac{29 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac{58 B^2 (b c-a d)^2 \log (c+d x)}{b^2 d}-\frac{58 B^2 (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 d}-\frac{58 B^2 (b c-a d)^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d}\\ \end{align*}

Mathematica [A]  time = 0.207744, size = 205, normalized size = 1.01 \[ \frac{i \left ((c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2-\frac{B (b c-a d) \left (2 B (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+2 (b c-a d) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+B \log \left (\frac{b (c+d x)}{b c-a d}\right )+A\right )+2 \left (B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )+\log (c+d x) (a B d-b B c)+A b d x\right )+\log ^2(a+b x) (a B d-b B c)\right )}{b^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.796, size = 0, normalized size = 0. \begin{align*} \int \left ( dix+ci \right ) \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.6418, size = 855, normalized size = 4.21 \begin{align*} \frac{1}{2} \, A^{2} d i x^{2} + 2 \,{\left (x \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) + \frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} A B c i +{\left (x^{2} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) - \frac{a^{2} \log \left (b x + a\right )}{b^{2}} + \frac{c^{2} \log \left (d x + c\right )}{d^{2}} - \frac{{\left (b c - a d\right )} x}{b d}\right )} A B d i + A^{2} c i x - \frac{{\left ({\left (i \log \left (e\right ) - i\right )} b c^{2} + a c d i\right )} B^{2} \log \left (d x + c\right )}{b d} - \frac{{\left (b^{2} c^{2} i - 2 \, a b c d i + a^{2} d^{2} i\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^{2}}{b^{2} d} + \frac{B^{2} b^{2} d^{2} i x^{2} \log \left (e\right )^{2} + 2 \,{\left (a b d^{2} i \log \left (e\right ) +{\left (i \log \left (e\right )^{2} - i \log \left (e\right )\right )} b^{2} c d\right )} B^{2} x +{\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x +{\left (2 \, a b c d i - a^{2} d^{2} i\right )} B^{2}\right )} \log \left (b x + a\right )^{2} +{\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left (d x + c\right )^{2} + 2 \,{\left (B^{2} b^{2} d^{2} i x^{2} \log \left (e\right ) +{\left ({\left (2 \, i \log \left (e\right ) - i\right )} b^{2} c d + a b d^{2} i\right )} B^{2} x +{\left ({\left (2 \, i \log \left (e\right ) - i\right )} a b c d -{\left (i \log \left (e\right ) - i\right )} a^{2} d^{2}\right )} B^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left (B^{2} b^{2} d^{2} i x^{2} \log \left (e\right ) +{\left ({\left (2 \, i \log \left (e\right ) - i\right )} b^{2} c d + a b d^{2} i\right )} B^{2} x +{\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x +{\left (2 \, a b c d i - a^{2} d^{2} i\right )} B^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{2 \, b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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