\(\int (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [162]

Optimal result

Integrand size = 33, antiderivative size = 220 \[ \int (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {B (b c-a d) i n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac {i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d}+\frac {B^2 (b c-a d)^2 i n^2 \log (c+d x)}{b^2 d}+\frac {B (b c-a d)^2 i n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 d}-\frac {B^2 (b c-a d)^2 i n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 d} \]

[Out]

-B*(-a*d+b*c)*i*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2+1/2*i*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^
2/d+B^2*(-a*d+b*c)^2*i*n^2*ln(d*x+c)/b^2/d+B*(-a*d+b*c)^2*i*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1-b*(d*x+c)/d
/(b*x+a))/b^2/d-B^2*(-a*d+b*c)^2*i*n^2*polylog(2,b*(d*x+c)/d/(b*x+a))/b^2/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2551, 2356, 2389, 2379, 2438, 2351, 31} \[ \int (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B i n (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d}-\frac {B i n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2}+\frac {i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d}-\frac {B^2 i n^2 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 d}+\frac {B^2 i n^2 (b c-a d)^2 \log (c+d x)}{b^2 d} \]

[In]

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

[Out]

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

Rule 31

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

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

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 2551

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/d)^m, Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, (
a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] &&
EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.98 \[ \int (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {i \left ((c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B (b c-a d) n \left (B (b c-a d) n \log ^2(a+b x)-2 \left (A b d x+B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B (-b c+a d) n \log (c+d x)\right )-2 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 B (-b c+a d) n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )}{b^2}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (d i x +c i \right ) {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (d i x + c i\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 825 vs. \(2 (217) = 434\).

Time = 0.69 (sec) , antiderivative size = 825, normalized size of antiderivative = 3.75 \[ \int (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=A B d i x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{2} \, A^{2} d i x^{2} - A B d i n {\left (\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 )} + 2 \, A B c i n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + 2 \, A B c i x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A^{2} c i x - \frac {{\left (a c d i n^{2} - {\left (i n^{2} - i n \log \left (e\right )\right )} b c^{2}\right )} B^{2} \log \left (d x + c\right )}{b d} - \frac {{\left (b^{2} c^{2} i n^{2} - 2 \, a b c d i n^{2} + a^{2} d^{2} i n^{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^{2}}{b^{2} d} + \frac {2 \, B^{2} b^{2} c^{2} i n^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) - B^{2} b^{2} c^{2} i n^{2} \log \left (d x + c\right )^{2} + B^{2} b^{2} d^{2} i x^{2} \log \left (e\right )^{2} - {\left (2 \, a b c d i n^{2} - a^{2} d^{2} i n^{2}\right )} B^{2} \log \left (b x + a\right )^{2} + 2 \, {\left (a b d^{2} i n \log \left (e\right ) - {\left (i n \log \left (e\right ) - i \log \left (e\right )^{2}\right )} b^{2} c d\right )} B^{2} x - 2 \, {\left ({\left (i n^{2} - 2 \, i n \log \left (e\right )\right )} a b c d - {\left (i n^{2} - i n \log \left (e\right )\right )} a^{2} d^{2}\right )} B^{2} \log \left (b x + a\right ) + {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + 2 \, {\left (B^{2} b^{2} d^{2} i x^{2} \log \left (e\right ) - B^{2} b^{2} c^{2} i n \log \left (d x + c\right ) + {\left (a b d^{2} i n - {\left (i n - 2 \, i \log \left (e\right )\right )} b^{2} c d\right )} B^{2} x + {\left (2 \, a b c d i n - a^{2} d^{2} i n\right )} B^{2} \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} b^{2} d^{2} i x^{2} \log \left (e\right ) - B^{2} b^{2} c^{2} i n \log \left (d x + c\right ) + {\left (a b d^{2} i n - {\left (i n - 2 \, i \log \left (e\right )\right )} b^{2} c d\right )} B^{2} x + {\left (2 \, a b c d i n - a^{2} d^{2} i n\right )} B^{2} \log \left (b x + a\right ) + {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{2} d} \]

[In]

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

[Out]

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

Giac [F]

\[ \int (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (d i x + c i\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int \left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]

[In]

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

[Out]

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