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

Optimal. Leaf size=211 \[ -\frac{B g^2 n (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac{g^2 (a+b x) (b c-a d) \left (2 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 A+B n\right )}{2 d^2 i}-\frac{g^2 (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 A+3 B n\right )}{2 d^3 i}+\frac{g^2 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.487975, antiderivative size = 343, normalized size of antiderivative = 1.63, 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, 2525, 12, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{B g^2 n (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^3 i}+\frac{g^2 (b c-a d)^2 \log (c i+d i x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i}+\frac{g^2 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d i}-\frac{A b g^2 x (b c-a d)}{d^2 i}-\frac{B g^2 (a+b x) (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d^2 i}-\frac{b B g^2 n x (b c-a d)}{2 d^2 i}+\frac{B g^2 n (b c-a d)^2 \log ^2(i (c+d x))}{2 d^3 i}+\frac{3 B g^2 n (b c-a d)^2 \log (c+d x)}{2 d^3 i}-\frac{B g^2 n (b c-a d)^2 \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^3 i} \]

Antiderivative was successfully verified.

[In]

Int[((a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(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*n*x)/(2*d^2*i) - (B*(b*c - a*d)*g^2*(a + b*x)*Log[e*
((a + b*x)/(c + d*x))^n])/(d^2*i) + (g^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*i) + (3*B*(b
*c - a*d)^2*g^2*n*Log[c + d*x])/(2*d^3*i) + (B*(b*c - a*d)^2*g^2*n*Log[i*(c + d*x)]^2)/(2*d^3*i) - (B*(b*c - a
*d)^2*g^2*n*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))^n])*Log[c*i + d*i*x])/(d^3*i) - (B*(b*c - a*d)^2*g^2*n*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 (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{136 c+136 d x} \, dx &=\int \left (-\frac{b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{136 d^2}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2 (136 c+136 d x)}+\frac{b g (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{136 d}\right ) \, dx\\ &=\frac{(b g) \int (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{136 d}-\frac{\left (b (b c-a d) g^2\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{136 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{136 c+136 d x} \, dx}{d^2}\\ &=-\frac{A b (b c-a d) g^2 x}{136 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}-\frac{\left (b B (b c-a d) g^2\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{136 d^2}-\frac{(B n) \int \frac{(b c-a d) g^2 (a+b x)}{c+d x} \, dx}{272 d}-\frac{\left (B (b c-a d)^2 g^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 (136 c+136 d x)}{a+b x} \, dx}{136 d^3}\\ &=-\frac{A b (b c-a d) g^2 x}{136 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}-\frac{\left (B (b c-a d) g^2 n\right ) \int \frac{a+b x}{c+d x} \, dx}{272 d}-\frac{\left (B (b c-a d)^2 g^2 n\right ) \int \left (\frac{b \log (136 c+136 d x)}{a+b x}-\frac{d \log (136 c+136 d x)}{c+d x}\right ) \, dx}{136 d^3}+\frac{\left (B (b c-a d)^2 g^2 n\right ) \int \frac{1}{c+d x} \, dx}{136 d^2}\\ &=-\frac{A b (b c-a d) g^2 x}{136 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac{B (b c-a d)^2 g^2 n \log (c+d x)}{136 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}-\frac{\left (B (b c-a d) g^2 n\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{272 d}-\frac{\left (b B (b c-a d)^2 g^2 n\right ) \int \frac{\log (136 c+136 d x)}{a+b x} \, dx}{136 d^3}+\frac{\left (B (b c-a d)^2 g^2 n\right ) \int \frac{\log (136 c+136 d x)}{c+d x} \, dx}{136 d^2}\\ &=-\frac{A b (b c-a d) g^2 x}{136 d^2}-\frac{b B (b c-a d) g^2 n x}{272 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac{3 B (b c-a d)^2 g^2 n \log (c+d x)}{272 d^3}-\frac{B (b c-a d)^2 g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (136 c+136 d x)}{136 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}+\frac{\left (B (b c-a d)^2 g^2 n\right ) \operatorname{Subst}\left (\int \frac{136 \log (x)}{x} \, dx,x,136 c+136 d x\right )}{18496 d^3}+\frac{\left (B (b c-a d)^2 g^2 n\right ) \int \frac{\log \left (\frac{136 d (a+b x)}{-136 b c+136 a d}\right )}{136 c+136 d x} \, dx}{d^2}\\ &=-\frac{A b (b c-a d) g^2 x}{136 d^2}-\frac{b B (b c-a d) g^2 n x}{272 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac{3 B (b c-a d)^2 g^2 n \log (c+d x)}{272 d^3}-\frac{B (b c-a d)^2 g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (136 c+136 d x)}{136 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}+\frac{\left (B (b c-a d)^2 g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,136 c+136 d x\right )}{136 d^3}+\frac{\left (B (b c-a d)^2 g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-136 b c+136 a d}\right )}{x} \, dx,x,136 c+136 d x\right )}{136 d^3}\\ &=-\frac{A b (b c-a d) g^2 x}{136 d^2}-\frac{b B (b c-a d) g^2 n x}{272 d^2}-\frac{B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac{3 B (b c-a d)^2 g^2 n \log (c+d x)}{272 d^3}+\frac{B (b c-a d)^2 g^2 n \log ^2(136 (c+d x))}{272 d^3}-\frac{B (b c-a d)^2 g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (136 c+136 d x)}{136 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}-\frac{B (b c-a d)^2 g^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{136 d^3}\\ \end{align*}

Mathematica [A]  time = 0.175108, size = 266, normalized size = 1.26 \[ \frac{g^2 \left (-B n (b c-a d)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (i (c+d x)) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (i (c+d x))\right )\right )+d^2 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \log (i (c+d x)) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 A b d x (b c-a d)+2 B d (a+b x) (a d-b c) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 B n (b c-a d)^2 \log (c+d x)-B n (b c-a d) ((a d-b c) \log (c+d x)+b d x)\right )}{2 d^3 i} \]

Antiderivative was successfully verified.

[In]

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

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Maple [F]  time = 0.695, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bgx+ag \right ) ^{2}}{dix+ci} \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((b*g*x+a*g)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i),x)

[Out]

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

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Maxima [B]  time = 2.66897, size = 846, normalized size = 4.01 \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} n - 2 \, a b c d g^{2} n + a^{2} d^{2} g^{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}{d^{3} i} + \frac{{\left (2 \, a^{2} d^{2} g^{2} \log \left (e\right ) +{\left (3 \, g^{2} n + 2 \, g^{2} \log \left (e\right )\right )} b^{2} c^{2} - 4 \,{\left (g^{2} n + g^{2} \log \left (e\right )\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 ) - 2 \,{\left (b^{2} c^{2} g^{2} n - 2 \, a b c d g^{2} n + a^{2} d^{2} g^{2} n\right )} B \log \left (b x + a\right ) \log \left (d x + c\right ) +{\left (b^{2} c^{2} g^{2} n - 2 \, a b c d g^{2} n + a^{2} d^{2} g^{2} n\right )} B \log \left (d x + c\right )^{2} -{\left ({\left (g^{2} n + 2 \, g^{2} \log \left (e\right )\right )} b^{2} c d -{\left (g^{2} n + 4 \, g^{2} \log \left (e\right )\right )} a b d^{2}\right )} B x -{\left (2 \, a b c d g^{2} n - 3 \, a^{2} d^{2} g^{2} n\right )} B \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 + 2 \,{\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 )\right )} \log \left ({\left (b x + a\right )}^{n}\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 + 2 \,{\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 )\right )} \log \left ({\left (d x + c\right )}^{n}\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))^n))/(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*n - 2*a*b*c*d*g^2*n + a^2*d^2*g^2*n)*(log(b*x + a)*l
og((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) +
(3*g^2*n + 2*g^2*log(e))*b^2*c^2 - 4*(g^2*n + g^2*log(e))*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) - 2*(b^2*c^2*g^2*n - 2*a*b*c*d*g^2*n + a^2*d^2*g^2*n)*B*log(b*x + a)*log(d*x + c) + (b^2*c^2*g^2*
n - 2*a*b*c*d*g^2*n + a^2*d^2*g^2*n)*B*log(d*x + c)^2 - ((g^2*n + 2*g^2*log(e))*b^2*c*d - (g^2*n + 4*g^2*log(e
))*a*b*d^2)*B*x - (2*a*b*c*d*g^2*n - 3*a^2*d^2*g^2*n)*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 + 2*(b^2*c^2*g^2 - 2*a*b*c*d*g^2 + a^2*d^2*g^2)*B*log(d*x + c))*log((b*x + a)^n) - (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*(b^2*c^2*g^2 - 2*a*b*c*d*g^2 + a^2*d^2*g^2)*B*log(d*x + c)
)*log((d*x + c)^n))/(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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\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))^n))/(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(e*((b*x
+ a)/(d*x + c))^n))/(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))**n))/(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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\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))^n))/(d*i*x+c*i),x, algorithm="giac")

[Out]

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