3.2.22 \(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^2} \, dx\) [122]

Optimal. Leaf size=259 \[ -\frac {B (b c-a d) i^2 n (c+d x)}{b^2 g^2 (a+b x)}+\frac {d^2 i^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac {(b c-a d) i^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}-\frac {B d (b c-a d) i^2 n \log (c+d x)}{b^3 g^2}-\frac {2 d (b c-a d) i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {2 B d (b c-a d) i^2 n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2} \]

[Out]

-B*(-a*d+b*c)*i^2*n*(d*x+c)/b^2/g^2/(b*x+a)+d^2*i^2*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/g^2-(-a*d+b*c)
*i^2*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/g^2/(b*x+a)-B*d*(-a*d+b*c)*i^2*n*ln(d*x+c)/b^3/g^2-2*d*(-a*d+
b*c)*i^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1-b*(d*x+c)/d/(b*x+a))/b^3/g^2+2*B*d*(-a*d+b*c)*i^2*n*polylog(2,b*
(d*x+c)/d/(b*x+a))/b^3/g^2

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Rubi [A]
time = 0.22, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2561, 46, 2393, 2341, 2351, 31, 2379, 2438} \begin {gather*} \frac {2 B d i^2 n (b c-a d) \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {d^2 i^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2}-\frac {2 d i^2 (b c-a d) \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^3 g^2}-\frac {i^2 (c+d 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 g^2 (a+b x)}-\frac {B d i^2 n (b c-a d) \log (c+d x)}{b^3 g^2}-\frac {B i^2 n (c+d x) (b c-a d)}{b^2 g^2 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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 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 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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 2561

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

Rubi steps

\begin {align*} \int \frac {(122 c+122 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^2} \, dx &=\int \left (\frac {14884 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac {14884 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)^2}+\frac {29768 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}\right ) \, dx\\ &=\frac {\left (14884 d^2\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g^2}+\frac {(29768 d (b c-a d)) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g^2}+\frac {\left (14884 (b c-a d)^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 g^2}\\ &=\frac {14884 A d^2 x}{b^2 g^2}-\frac {14884 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {29768 d (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 )\right )}{b^3 g^2}+\frac {\left (14884 B d^2\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 g^2}-\frac {(29768 B d (b c-a d) n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 g^2}+\frac {\left (14884 B (b c-a d)^2 n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac {14884 A d^2 x}{b^2 g^2}+\frac {14884 B d^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {14884 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {29768 d (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 )\right )}{b^3 g^2}-\frac {(29768 B d (b c-a d) n) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 g^2}-\frac {\left (14884 B d^2 (b c-a d) n\right ) \int \frac {1}{c+d x} \, dx}{b^3 g^2}+\frac {\left (14884 B (b c-a d)^3 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac {14884 A d^2 x}{b^2 g^2}+\frac {14884 B d^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {14884 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {29768 d (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 )\right )}{b^3 g^2}-\frac {14884 B d (b c-a d) n \log (c+d x)}{b^3 g^2}-\frac {(29768 B d (b c-a d) n) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^2}+\frac {\left (29768 B d^2 (b c-a d) n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^2}+\frac {\left (14884 B (b c-a d)^3 n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^2}\\ &=\frac {14884 A d^2 x}{b^2 g^2}-\frac {14884 B (b c-a d)^2 n}{b^3 g^2 (a+b x)}-\frac {14884 B d (b c-a d) n \log (a+b x)}{b^3 g^2}+\frac {14884 B d^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {14884 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {29768 d (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 )\right )}{b^3 g^2}+\frac {29768 B d (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {(29768 B d (b c-a d) n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}-\frac {(29768 B d (b c-a d) n) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}\\ &=\frac {14884 A d^2 x}{b^2 g^2}-\frac {14884 B (b c-a d)^2 n}{b^3 g^2 (a+b x)}-\frac {14884 B d (b c-a d) n \log (a+b x)}{b^3 g^2}-\frac {14884 B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}+\frac {14884 B d^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {14884 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {29768 d (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 )\right )}{b^3 g^2}+\frac {29768 B d (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {(29768 B d (b c-a d) n) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}\\ &=\frac {14884 A d^2 x}{b^2 g^2}-\frac {14884 B (b c-a d)^2 n}{b^3 g^2 (a+b x)}-\frac {14884 B d (b c-a d) n \log (a+b x)}{b^3 g^2}-\frac {14884 B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}+\frac {14884 B d^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {14884 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {29768 d (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 )\right )}{b^3 g^2}+\frac {29768 B d (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {29768 B d (b c-a d) n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 233, normalized size = 0.90 \begin {gather*} \frac {i^2 \left (A b d^2 x-\frac {B (b c-a d)^2 n}{a+b x}+B d (-b c+a d) n \log (a+b x)+B d^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\frac {(b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+2 d (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 )\right )+B d (-b c+a d) n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3 g^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (b g x +a g \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 966 vs. \(2 (241) = 482\).
time = 0.54, size = 966, normalized size = 3.73 \begin {gather*} B c^{2} n {\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} + A {\left (\frac {a^{2}}{b^{4} g^{2} x + a b^{3} g^{2}} - \frac {x}{b^{2} g^{2}} + \frac {2 \, a \log \left (b x + a\right )}{b^{3} g^{2}}\right )} d^{2} - 2 \, A c d {\left (\frac {a}{b^{3} g^{2} x + a b^{2} g^{2}} + \frac {\log \left (b x + a\right )}{b^{2} g^{2}}\right )} + \frac {B c^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {A c^{2}}{b^{2} g^{2} x + a b g^{2}} + \frac {{\left (b^{2} c^{2} d n + a b c d^{2} n - a^{2} d^{3} n\right )} B \log \left (d x + c\right )}{b^{4} c g^{2} - a b^{3} d g^{2}} - \frac {{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} B x^{2} + {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} B x - {\left ({\left (b^{3} c^{2} d n - 2 \, a b^{2} c d^{2} n + a^{2} b d^{3} n\right )} B x + {\left (a b^{2} c^{2} d n - 2 \, a^{2} b c d^{2} n + a^{3} d^{3} n\right )} B\right )} \log \left (b x + a\right )^{2} + {\left (2 \, a b^{2} c^{2} d {\left (n + 1\right )} - 3 \, a^{2} b c d^{2} {\left (n + 1\right )} + a^{3} d^{3} {\left (n + 1\right )}\right )} B + {\left ({\left (a b^{2} c d^{2} {\left (3 \, n - 4\right )} - 2 \, a^{2} b d^{3} {\left (n - 1\right )} + 2 \, b^{3} c^{2} d\right )} B x + {\left (a^{2} b c d^{2} {\left (3 \, n - 4\right )} - 2 \, a^{3} d^{3} {\left (n - 1\right )} + 2 \, a b^{2} c^{2} d\right )} B\right )} \log \left (b x + a\right ) + {\left ({\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} B x^{2} + {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} B x + {\left (2 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} B + 2 \, {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} B x + {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} B\right )} \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} B x^{2} + {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} B x + {\left (2 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} B + 2 \, {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} B x + {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} B\right )} \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{a b^{4} c g^{2} - a^{2} b^{3} d g^{2} + {\left (b^{5} c g^{2} - a b^{4} d g^{2}\right )} x} - \frac {2 \, {\left (b c d n - a d^{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}{b^{3} g^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-((A + B)*d^2*x^2 + 2*(A + B)*c*d*x + (A + B)*c^2 + (B*d^2*n*x^2 + 2*B*c*d*n*x + B*c^2*n)*log((b*x +
a)/(d*x + c)))/(b^2*g^2*x^2 + 2*a*b*g^2*x + a^2*g^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,i+d\,i\,x\right )}^2\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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