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

Optimal. Leaf size=198 \[ -\frac{B g^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac{g^2 (a+b x) (b c-a d) \left (2 B \log \left (\frac{e (a+b x)}{c+d x}\right )+2 A+B\right )}{2 d^2 i}-\frac{g^2 (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (\frac{e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{2 d^3 i}+\frac{g^2 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.487327, antiderivative size = 329, normalized size of antiderivative = 1.66, 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, 2525, 12, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{B g^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^3 i}+\frac{g^2 (b c-a d)^2 \log (c i+d i x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^3 i}+\frac{g^2 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d i}-\frac{A b g^2 x (b c-a d)}{d^2 i}-\frac{B g^2 (a+b x) (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right )}{d^2 i}-\frac{b B g^2 x (b c-a d)}{2 d^2 i}+\frac{B g^2 (b c-a d)^2 \log ^2(i (c+d x))}{2 d^3 i}+\frac{3 B g^2 (b c-a d)^2 \log (c+d x)}{2 d^3 i}-\frac{B g^2 (b c-a d)^2 \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^3 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{32 c+32 d x} \, dx &=\int \left (-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{32 d^2}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2 (32 c+32 d x)}+\frac{b g (a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{32 d}\right ) \, dx\\ &=\frac{(b g) \int (a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{32 d}-\frac{\left (b (b c-a d) g^2\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{32 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{32 c+32 d x} \, dx}{d^2}\\ &=-\frac{A b (b c-a d) g^2 x}{32 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac{B \int \frac{(b c-a d) g^2 (a+b x)}{c+d x} \, dx}{64 d}-\frac{\left (b B (b c-a d) g^2\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{32 d^2}-\frac{\left (B (b c-a d)^2 g^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 (32 c+32 d x)}{e (a+b x)} \, dx}{32 d^3}\\ &=-\frac{A b (b c-a d) g^2 x}{32 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{32 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac{\left (B (b c-a d) g^2\right ) \int \frac{a+b x}{c+d x} \, dx}{64 d}+\frac{\left (B (b c-a d)^2 g^2\right ) \int \frac{1}{c+d x} \, dx}{32 d^2}-\frac{\left (B (b c-a d)^2 g^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 (32 c+32 d x)}{a+b x} \, dx}{32 d^3 e}\\ &=-\frac{A b (b c-a d) g^2 x}{32 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{32 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac{B (b c-a d)^2 g^2 \log (c+d x)}{32 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac{\left (B (b c-a d) g^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{64 d}-\frac{\left (B (b c-a d)^2 g^2\right ) \int \left (\frac{b e \log (32 c+32 d x)}{a+b x}-\frac{d e \log (32 c+32 d x)}{c+d x}\right ) \, dx}{32 d^3 e}\\ &=-\frac{A b (b c-a d) g^2 x}{32 d^2}-\frac{b B (b c-a d) g^2 x}{64 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{32 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac{3 B (b c-a d)^2 g^2 \log (c+d x)}{64 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac{\left (b B (b c-a d)^2 g^2\right ) \int \frac{\log (32 c+32 d x)}{a+b x} \, dx}{32 d^3}+\frac{\left (B (b c-a d)^2 g^2\right ) \int \frac{\log (32 c+32 d x)}{c+d x} \, dx}{32 d^2}\\ &=-\frac{A b (b c-a d) g^2 x}{32 d^2}-\frac{b B (b c-a d) g^2 x}{64 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{32 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac{3 B (b c-a d)^2 g^2 \log (c+d x)}{64 d^3}-\frac{B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (32 c+32 d x)}{32 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}+\frac{\left (B (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{32 \log (x)}{x} \, dx,x,32 c+32 d x\right )}{1024 d^3}+\frac{\left (B (b c-a d)^2 g^2\right ) \int \frac{\log \left (\frac{32 d (a+b x)}{-32 b c+32 a d}\right )}{32 c+32 d x} \, dx}{d^2}\\ &=-\frac{A b (b c-a d) g^2 x}{32 d^2}-\frac{b B (b c-a d) g^2 x}{64 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{32 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac{3 B (b c-a d)^2 g^2 \log (c+d x)}{64 d^3}-\frac{B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (32 c+32 d x)}{32 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}+\frac{\left (B (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,32 c+32 d x\right )}{32 d^3}+\frac{\left (B (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-32 b c+32 a d}\right )}{x} \, dx,x,32 c+32 d x\right )}{32 d^3}\\ &=-\frac{A b (b c-a d) g^2 x}{32 d^2}-\frac{b B (b c-a d) g^2 x}{64 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{32 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{64 d}+\frac{3 B (b c-a d)^2 g^2 \log (c+d x)}{64 d^3}+\frac{B (b c-a d)^2 g^2 \log ^2(32 (c+d x))}{64 d^3}-\frac{B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (32 c+32 d x)}{32 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (32 c+32 d x)}{32 d^3}-\frac{B (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{32 d^3}\\ \end{align*}

Mathematica [A]  time = 0.170747, size = 254, normalized size = 1.28 \[ \frac{g^2 \left (-B (b c-a d)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (i (c+d x)) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (i (c+d x))\right )\right )+d^2 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \log (i (c+d x)) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 A b d x (b c-a d)+2 B d (a+b x) (a d-b c) \log \left (\frac{e (a+b x)}{c+d x}\right )+2 B (b c-a d)^2 \log (c+d x)-B (b c-a d) ((a d-b c) \log (c+d x)+b d x)\right )}{2 d^3 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.189, size = 2309, normalized size = 11.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*B*g^2/i*dilog(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*a^2-1/d*A*g^2/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+
c))-b*e)*a^2-3/2/d*B*g^2/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^2-3*e^2/d*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/
(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2/(d*x+c)^2*b^2*c^2+6*e/d^2*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))
/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a/(d*x+c)*b^2*c^2+2*e^2/d^2*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c
)*a-e/(d*x+c)*b*c)^2*a/(d*x+c)^2*b^3*c^3-6*e/d*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c
)*b*c)*a^2/(d*x+c)*b*c-3/2/d^3*B*g^2/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*b^2*c^2-e^2/d^2*A*g^2/i*b^3/(d*
e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c*a-e/d^2*B*g^2/i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a*c-4*e/d^2*A*g^2/i*b^2/(d*e/
(d*x+c)*a-e/(d*x+c)*b*c)*a*c+1/2*e^2/d^3*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)
^2*b^4*c^2+2*e/d*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*b+2*e/d^3*B*g^2/i*l
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*b^3*c^2+2/d^2*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x
+c))*ln(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*a*b*c+1/2*e^2/d*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d
*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*b^2-1/2*e^2*d*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+
c)*b*c)^2*a^4/(d*x+c)^2+2*e^2*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3/(d*x
+c)^2*b*c-1/d^3*A*g^2/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*b^2*c^2-1/d*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*
x+c))*ln(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*a^2-1/d^3*B*g^2/i*dilog(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))
-b*e)/b/e)*b^2*c^2+2/d^2*B*g^2/i*dilog(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*a*b*c+2/d^2*A*g^2/i*ln(d*(b
*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a*b*c-1/d^3*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-(d*(b*e/d+(a*d-b*c)*e
/d/(d*x+c))-b*e)/b/e)*b^2*c^2+2*e/d*A*g^2/i*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2+1/2*e/d^3*B*g^2/i*b^3/(d*e/(d*
x+c)*a-e/(d*x+c)*b*c)*c^2+1/2*e^2/d*A*g^2/i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2+1/2*e^2/d^3*A*g^2/i*b^4/(d
*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^2+1/2*e/d*B*g^2/i*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2+2*e/d^3*A*g^2/i*b^3/(d*e
/(d*x+c)*a-e/(d*x+c)*b*c)*c^2+2*e*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3/(d
*x+c)+3/d^2*B*g^2/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a*b*c-4*e/d^2*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+
c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a*b^2*c-2*e/d^3*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*
x+c)*b*c)*b^3*c^3/(d*x+c)-e^2/d^2*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^3*
c*a-1/2*e^2/d^3*B*g^2/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^4*c^4/(d*x+c)^2

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Maxima [B]  time = 1.45319, size = 644, normalized size = 3.25 \begin{align*} 2 \, A a b g^{2}{\left (\frac{x}{d i} - \frac{c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac{1}{2} \, A b^{2} g^{2}{\left (\frac{2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac{d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac{A a^{2} g^{2} \log \left (d i x + c i\right )}{d i} + \frac{{\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{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}{d^{3} i} + \frac{{\left (2 \, a^{2} d^{2} g^{2} \log \left (e\right ) +{\left (2 \, g^{2} \log \left (e\right ) + 3 \, g^{2}\right )} b^{2} c^{2} - 4 \,{\left (g^{2} \log \left (e\right ) + g^{2}\right )} a b c d\right )} B \log \left (d x + c\right )}{2 \, d^{3} i} + \frac{B b^{2} d^{2} g^{2} x^{2} \log \left (e\right ) -{\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{2}\right )} B \log \left (d x + c\right )^{2} -{\left ({\left (2 \, g^{2} \log \left (e\right ) + g^{2}\right )} b^{2} c d -{\left (4 \, g^{2} \log \left (e\right ) + g^{2}\right )} a b d^{2}\right )} B x +{\left (B b^{2} d^{2} g^{2} x^{2} - 2 \,{\left (b^{2} c d g^{2} - 2 \, a b d^{2} g^{2}\right )} B x -{\left (2 \, a b c d g^{2} - 3 \, a^{2} d^{2} g^{2}\right )} B\right )} \log \left (b x + a\right ) -{\left (B b^{2} d^{2} g^{2} x^{2} - 2 \,{\left (b^{2} c d g^{2} - 2 \, a b d^{2} g^{2}\right )} B x\right )} \log \left (d x + c\right )}{2 \, d^{3} i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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