3.152 \(\int \frac{(a g+b g x)^2 (A+B \log (e (\frac{a+b x}{c+d x})^n))}{(c i+d i x)^3} \, dx\)

Optimal. Leaf size=263 \[ -\frac{b^2 B g^2 n \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 i^3}-\frac{b^2 g^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^3}-\frac{g^2 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d i^3 (c+d x)^2}-\frac{A b g^2 (a+b x)}{d^2 i^3 (c+d x)}-\frac{b B g^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d^2 i^3 (c+d x)}+\frac{b B g^2 n (a+b x)}{d^2 i^3 (c+d x)}+\frac{B g^2 n (a+b x)^2}{4 d i^3 (c+d x)^2} \]

[Out]

(B*g^2*n*(a + b*x)^2)/(4*d*i^3*(c + d*x)^2) - (A*b*g^2*(a + b*x))/(d^2*i^3*(c + d*x)) + (b*B*g^2*n*(a + b*x))/
(d^2*i^3*(c + d*x)) - (b*B*g^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^2*i^3*(c + d*x)) - (g^2*(a + b*x)^
2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*i^3*(c + d*x)^2) - (b^2*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^
n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^3*i^3) - (b^2*B*g^2*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^3
)

________________________________________________________________________________________

Rubi [A]  time = 0.597234, antiderivative size = 356, normalized size of antiderivative = 1.35, number of steps used = 18, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{b^2 B g^2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^3 i^3}+\frac{b^2 g^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^3}+\frac{2 b g^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^3 (c+d x)}-\frac{g^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 i^3 (c+d x)^2}-\frac{b^2 B g^2 n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^3 i^3}-\frac{3 b^2 B g^2 n \log (a+b x)}{2 d^3 i^3}-\frac{3 b B g^2 n (b c-a d)}{2 d^3 i^3 (c+d x)}+\frac{B g^2 n (b c-a d)^2}{4 d^3 i^3 (c+d x)^2}+\frac{b^2 B g^2 n \log ^2(c+d x)}{2 d^3 i^3}+\frac{3 b^2 B g^2 n \log (c+d x)}{2 d^3 i^3} \]

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)^3,x]

[Out]

(B*(b*c - a*d)^2*g^2*n)/(4*d^3*i^3*(c + d*x)^2) - (3*b*B*(b*c - a*d)*g^2*n)/(2*d^3*i^3*(c + d*x)) - (3*b^2*B*g
^2*n*Log[a + b*x])/(2*d^3*i^3) - ((b*c - a*d)^2*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d^3*i^3*(c + d*
x)^2) + (2*b*(b*c - a*d)*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d^3*i^3*(c + d*x)) + (3*b^2*B*g^2*n*Log[
c + d*x])/(2*d^3*i^3) - (b^2*B*g^2*n*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^3*i^3) + (b^2*g^2*(A +
B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(d^3*i^3) + (b^2*B*g^2*n*Log[c + d*x]^2)/(2*d^3*i^3) - (b^2*B
*g^2*n*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^3*i^3)

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 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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 )}{(152 c+152 d x)^3} \, dx &=\int \left (\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 )}{3511808 d^2 (c+d x)^3}-\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 )}{1755904 d^2 (c+d x)^2}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3511808 d^2 (c+d x)}\right ) \, dx\\ &=\frac{\left (b^2 g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3511808 d^2}-\frac{\left (b (b c-a d) g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{1755904 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 )}{(c+d x)^3} \, dx}{3511808 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 )}{7023616 d^3 (c+d x)^2}+\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 )}{1755904 d^3 (c+d x)}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}-\frac{\left (b^2 B 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 (c+d x)}{a+b x} \, dx}{3511808 d^3}-\frac{\left (b B (b c-a d) g^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1755904 d^3}+\frac{\left (B (b c-a d)^2 g^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{7023616 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 )}{7023616 d^3 (c+d x)^2}+\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 )}{1755904 d^3 (c+d x)}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}-\frac{\left (b^2 B g^2 n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{3511808 d^3}-\frac{\left (b B (b c-a d)^2 g^2 n\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1755904 d^3}+\frac{\left (B (b c-a d)^3 g^2 n\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{7023616 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 )}{7023616 d^3 (c+d x)^2}+\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 )}{1755904 d^3 (c+d x)}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}-\frac{\left (b^3 B g^2 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3511808 d^3}+\frac{\left (b^2 B g^2 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3511808 d^2}-\frac{\left (b B (b c-a d)^2 g^2 n\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1755904 d^3}+\frac{\left (B (b c-a d)^3 g^2 n\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7023616 d^3}\\ &=\frac{B (b c-a d)^2 g^2 n}{14047232 d^3 (c+d x)^2}-\frac{3 b B (b c-a d) g^2 n}{7023616 d^3 (c+d x)}-\frac{3 b^2 B g^2 n \log (a+b x)}{7023616 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 )}{7023616 d^3 (c+d x)^2}+\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 )}{1755904 d^3 (c+d x)}+\frac{3 b^2 B g^2 n \log (c+d x)}{7023616 d^3}-\frac{b^2 B g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}+\frac{\left (b^2 B g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3511808 d^3}+\frac{\left (b^2 B g^2 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3511808 d^2}\\ &=\frac{B (b c-a d)^2 g^2 n}{14047232 d^3 (c+d x)^2}-\frac{3 b B (b c-a d) g^2 n}{7023616 d^3 (c+d x)}-\frac{3 b^2 B g^2 n \log (a+b x)}{7023616 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 )}{7023616 d^3 (c+d x)^2}+\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 )}{1755904 d^3 (c+d x)}+\frac{3 b^2 B g^2 n \log (c+d x)}{7023616 d^3}-\frac{b^2 B g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 B g^2 n \log ^2(c+d x)}{7023616 d^3}+\frac{\left (b^2 B g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3511808 d^3}\\ &=\frac{B (b c-a d)^2 g^2 n}{14047232 d^3 (c+d x)^2}-\frac{3 b B (b c-a d) g^2 n}{7023616 d^3 (c+d x)}-\frac{3 b^2 B g^2 n \log (a+b x)}{7023616 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 )}{7023616 d^3 (c+d x)^2}+\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 )}{1755904 d^3 (c+d x)}+\frac{3 b^2 B g^2 n \log (c+d x)}{7023616 d^3}-\frac{b^2 B g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 B g^2 n \log ^2(c+d x)}{7023616 d^3}-\frac{b^2 B g^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3511808 d^3}\\ \end{align*}

Mathematica [A]  time = 0.355202, size = 259, normalized size = 0.98 \[ \frac{g^2 \left (-2 b^2 B n \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+4 b^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{8 b (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-\frac{2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}-6 b^2 B n \log (a+b x)-\frac{6 b B n (b c-a d)}{c+d x}+\frac{B n (b c-a d)^2}{(c+d x)^2}+6 b^2 B n \log (c+d x)\right )}{4 d^3 i^3} \]

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)^3,x]

[Out]

(g^2*((B*(b*c - a*d)^2*n)/(c + d*x)^2 - (6*b*B*(b*c - a*d)*n)/(c + d*x) - 6*b^2*B*n*Log[a + b*x] - (2*(b*c - a
*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x)^2 + (8*b*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x)
)^n]))/(c + d*x) + 6*b^2*B*n*Log[c + d*x] + 4*b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 2*b^2*
B*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d
)])))/(4*d^3*i^3)

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Maple [F]  time = 0.68, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bgx+ag \right ) ^{2}}{ \left ( dix+ci \right ) ^{3}} \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)^3,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)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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)^3,x, algorithm="maxima")

[Out]

1/2*B*a*b*g^2*n*((b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/((b*c*d^4 - a*d^5)*i^3*x^2 + 2*(b*c^2*d^3 - a*c*d^4
)*i^3*x + (b*c^3*d^2 - a*c^2*d^3)*i^3) + 2*(b^2*c - 2*a*b*d)*log(b*x + a)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^
4)*i^3) - 2*(b^2*c - 2*a*b*d)*log(d*x + c)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*i^3)) + 1/4*B*a^2*g^2*n*((2*
b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)*i^3*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*i^3*x + (b*c^3*d - a*c^2*d^2)*i^3) +
2*b^2*log(b*x + a)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x + c)/((b^2*c^2*d - 2*a*b*c*d^2 +
a^2*d^3)*i^3)) + 1/2*A*b^2*g^2*((4*c*d*x + 3*c^2)/(d^5*i^3*x^2 + 2*c*d^4*i^3*x + c^2*d^3*i^3) + 2*log(d*x + c
)/(d^3*i^3)) - 1/2*B*b^2*g^2*((2*(d^2*n*x^2 + 2*c*d*n*x + c^2*n)*log(b*x + a)*log(d*x + c) - (d^2*n*x^2 + 2*c*
d*n*x + c^2*n)*log(d*x + c)^2 - (4*c*d*x + 3*c^2 + 2*(d^2*x^2 + 2*c*d*x + c^2)*log(d*x + c))*log((b*x + a)^n)
+ (4*c*d*x + 3*c^2 + 2*(d^2*x^2 + 2*c*d*x + c^2)*log(d*x + c))*log((d*x + c)^n))/(d^5*i^3*x^2 + 2*c*d^4*i^3*x
+ c^2*d^3*i^3) - 2*integrate(1/2*(2*b*d^3*x^3*log(e) + 2*a*d^3*x^2*log(e) - 3*b*c^3*n + 3*a*c^2*d*n - 4*(b*c^2
*d*n - a*c*d^2*n)*x + 2*(b*d^3*n*x^3 + a*c^2*d*n + (2*b*c*d^2*n + a*d^3*n)*x^2 + (b*c^2*d*n + 2*a*c*d^2*n)*x)*
log(b*x + a))/(b*d^6*i^3*x^4 + a*c^3*d^3*i^3 + (3*b*c*d^5*i^3 + a*d^6*i^3)*x^3 + 3*(b*c^2*d^4*i^3 + a*c*d^5*i^
3)*x^2 + (b*c^3*d^3*i^3 + 3*a*c^2*d^4*i^3)*x), x)) - (2*d*x + c)*B*a*b*g^2*log(e*(b*x/(d*x + c) + a/(d*x + c))
^n)/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3) - (2*d*x + c)*A*a*b*g^2/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2
*i^3) - 1/2*B*a^2*g^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^2*d*i^3) - 1/2*A
*a^2*g^2/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^2*d*i^3)

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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^{3} i^{3} x^{3} + 3 \, c d^{2} i^{3} x^{2} + 3 \, c^{2} d i^{3} x + c^{3} i^{3}}, 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)^3,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^3*i^3*x^3 + 3*c*d^2*i^3*x^2 + 3*c^2*d*i^3*x + c^3*i^3), 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)**3,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 )}}{{\left (d i x + c i\right )}^{3}}\,{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)^3,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)^3, x)