3.2.44 \(\int \frac {(a g+b g x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(c i+d i x)^2} \, dx\) [144]

Optimal. Leaf size=275 \[ -\frac {2 B (b c-a d) g^2 n (a+b x)}{d^2 i^2 (c+d x)}+\frac {(b c-a d) g^2 (2 A+B n) (a+b x)}{d^2 i^2 (c+d x)}+\frac {2 B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^2 i^2 (c+d x)}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d i^2 (c+d x)}+\frac {b (b c-a d) g^2 \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^3 i^2}+\frac {2 b B (b c-a d) g^2 n \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2} \]

[Out]

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

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Rubi [A]
time = 0.21, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2561, 2384, 45, 2393, 2332, 2354, 2438} \begin {gather*} \frac {2 b B g^2 n (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}+\frac {b g^2 (b c-a d) \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+B n\right )}{d^3 i^2}+\frac {g^2 (a+b x) (2 A+B n) (b c-a d)}{d^2 i^2 (c+d x)}+\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 )}{d i^2 (c+d x)}+\frac {2 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^2 (c+d x)}-\frac {2 B g^2 n (a+b x) (b c-a d)}{d^2 i^2 (c+d x)} \end {gather*}

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)^2,x]

[Out]

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

Rule 45

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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2354

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

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 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 {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(144 c+144 d x)^2} \, dx &=\int \left (\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 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 )}{20736 d^2 (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 )}{10368 d^2 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{20736 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} \, dx}{10368 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)^2} \, dx}{20736 d^2}\\ &=\frac {A b^2 g^2 x}{20736 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 )}{20736 d^3 (c+d x)}-\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 ) \log (c+d x)}{10368 d^3}+\frac {\left (b^2 B g^2\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{20736 d^2}+\frac {\left (b B (b c-a d) 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}{10368 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)^2} \, dx}{20736 d^3}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 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 )}{20736 d^3 (c+d x)}-\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 ) \log (c+d x)}{10368 d^3}+\frac {\left (b B (b c-a d) 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}{10368 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {1}{c+d x} \, dx}{20736 d^2}+\frac {\left (B (b c-a d)^3 g^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{20736 d^3}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 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 )}{20736 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 n \log (c+d x)}{20736 d^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 ) \log (c+d x)}{10368 d^3}+\frac {\left (b^2 B (b c-a d) g^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{10368 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{10368 d^2}+\frac {\left (B (b c-a d)^3 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}{20736 d^3}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {B (b c-a d)^2 g^2 n}{20736 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 n \log (a+b x)}{20736 d^3}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 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 )}{20736 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 n \log (c+d x)}{10368 d^3}+\frac {b B (b c-a d) g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10368 d^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 ) \log (c+d x)}{10368 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{10368 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{10368 d^2}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {B (b c-a d)^2 g^2 n}{20736 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 n \log (a+b x)}{20736 d^3}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 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 )}{20736 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 n \log (c+d x)}{10368 d^3}+\frac {b B (b c-a d) g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10368 d^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 ) \log (c+d x)}{10368 d^3}-\frac {b B (b c-a d) g^2 n \log ^2(c+d x)}{20736 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{10368 d^3}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {B (b c-a d)^2 g^2 n}{20736 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 n \log (a+b x)}{20736 d^3}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 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 )}{20736 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 n \log (c+d x)}{10368 d^3}+\frac {b B (b c-a d) g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10368 d^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 ) \log (c+d x)}{10368 d^3}-\frac {b B (b c-a d) g^2 n \log ^2(c+d x)}{20736 d^3}+\frac {b B (b c-a d) g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{10368 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 252, normalized size = 0.92 \begin {gather*} \frac {g^2 \left (A b^2 d x+\frac {B (b c-a d)^2 n}{c+d x}+b B (b c-a d) n \log (a+b x)+b B d (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 )}{c+d x}-2 b B (b c-a d) n \log (c+d x)-2 b (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+b B (b c-a d) n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3 i^2} \end {gather*}

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)^2,x]

[Out]

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

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (d i x +c i \right )^{2}}\, dx\]

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)^2,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)^2,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1163 vs. \(2 (262) = 524\).
time = 0.55, size = 1163, normalized size = 4.23 \begin {gather*} -B a^{2} g^{2} n {\left (\frac {b \log \left (b x + a\right )}{b c d - a d^{2}} - \frac {b \log \left (d x + c\right )}{b c d - a d^{2}} + \frac {1}{d^{2} x + c d}\right )} + A b^{2} {\left (\frac {c^{2}}{d^{4} x + c d^{3}} - \frac {x}{d^{2}} + \frac {2 \, c \log \left (d x + c\right )}{d^{3}}\right )} g^{2} - 2 \, A a b g^{2} {\left (\frac {c}{d^{3} x + c d^{2}} + \frac {\log \left (d x + c\right )}{d^{2}}\right )} + \frac {B a^{2} g^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{d^{2} x + c d} + \frac {A a^{2} g^{2}}{d^{2} x + c d} - \frac {{\left (a b^{2} c d g^{2} {\left (3 \, n + 4\right )} - 2 \, b^{3} c^{2} g^{2} {\left (n + 1\right )} - 2 \, a^{2} b d^{2} g^{2}\right )} B \log \left (d x + c\right )}{b c d^{3} - a d^{4}} - \frac {{\left (b^{3} c d^{2} g^{2} - a b^{2} d^{3} g^{2}\right )} B x^{2} + {\left (b^{3} c^{2} d g^{2} - a b^{2} c d^{2} g^{2}\right )} B x + 2 \, {\left ({\left (b^{3} c^{2} d g^{2} n - 2 \, a b^{2} c d^{2} g^{2} n + a^{2} b d^{3} g^{2} n\right )} B x + {\left (b^{3} c^{3} g^{2} n - 2 \, a b^{2} c^{2} d g^{2} n + a^{2} b c d^{2} g^{2} n\right )} B\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left ({\left (b^{3} c^{2} d g^{2} n - 2 \, a b^{2} c d^{2} g^{2} n + a^{2} b d^{3} g^{2} n\right )} B x + {\left (b^{3} c^{3} g^{2} n - 2 \, a b^{2} c^{2} d g^{2} n + a^{2} b c d^{2} g^{2} n\right )} B\right )} \log \left (d x + c\right )^{2} + {\left (b^{3} c^{3} g^{2} {\left (n - 1\right )} - 3 \, a b^{2} c^{2} d g^{2} {\left (n - 1\right )} + 2 \, a^{2} b c d^{2} g^{2} {\left (n - 1\right )}\right )} B + {\left ({\left (b^{3} c^{2} d g^{2} n - a b^{2} c d^{2} g^{2} n - a^{2} b d^{3} g^{2} n\right )} B x + {\left (b^{3} c^{3} g^{2} n - a b^{2} c^{2} d g^{2} n - a^{2} b c d^{2} g^{2} n\right )} B\right )} \log \left (b x + a\right ) + {\left ({\left (b^{3} c d^{2} g^{2} - a b^{2} d^{3} g^{2}\right )} B x^{2} + {\left (b^{3} c^{2} d g^{2} - a b^{2} c d^{2} g^{2}\right )} B x - {\left (b^{3} c^{3} g^{2} - 3 \, a b^{2} c^{2} d g^{2} + 2 \, a^{2} b c d^{2} g^{2}\right )} B - 2 \, {\left ({\left (b^{3} c^{2} d g^{2} - 2 \, a b^{2} c d^{2} g^{2} + a^{2} b d^{3} g^{2}\right )} B x + {\left (b^{3} c^{3} g^{2} - 2 \, a b^{2} c^{2} d g^{2} + a^{2} b c d^{2} g^{2}\right )} B\right )} \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b^{3} c d^{2} g^{2} - a b^{2} d^{3} g^{2}\right )} B x^{2} + {\left (b^{3} c^{2} d g^{2} - a b^{2} c d^{2} g^{2}\right )} B x - {\left (b^{3} c^{3} g^{2} - 3 \, a b^{2} c^{2} d g^{2} + 2 \, a^{2} b c d^{2} g^{2}\right )} B - 2 \, {\left ({\left (b^{3} c^{2} d g^{2} - 2 \, a b^{2} c d^{2} g^{2} + a^{2} b d^{3} g^{2}\right )} B x + {\left (b^{3} c^{3} g^{2} - 2 \, a b^{2} c^{2} d g^{2} + a^{2} b c d^{2} g^{2}\right )} B\right )} \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b c^{2} d^{3} - a c d^{4} + {\left (b c d^{4} - a d^{5}\right )} x} + \frac {2 \, {\left (b^{2} c g^{2} n - a b d 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}} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

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

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

[Out]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1838 vs. \(2 (262) = 524\).
time = 196.84, size = 1838, normalized size = 6.68 \begin {gather*} \text {Too large to display} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

-1/6*(2*(B*b^6*c^4*g^2*n - 4*B*a*b^5*c^3*d*g^2*n - 3*(b*x + a)*B*b^5*c^4*d*g^2*n/(d*x + c) + 6*B*a^2*b^4*c^2*d
^2*g^2*n + 12*(b*x + a)*B*a*b^4*c^3*d^2*g^2*n/(d*x + c) + 3*(b*x + a)^2*B*b^4*c^4*d^2*g^2*n/(d*x + c)^2 - 4*B*
a^3*b^3*c*d^3*g^2*n - 18*(b*x + a)*B*a^2*b^3*c^2*d^3*g^2*n/(d*x + c) - 12*(b*x + a)^2*B*a*b^3*c^3*d^3*g^2*n/(d
*x + c)^2 + B*a^4*b^2*d^4*g^2*n + 12*(b*x + a)*B*a^3*b^2*c*d^4*g^2*n/(d*x + c) + 18*(b*x + a)^2*B*a^2*b^2*c^2*
d^4*g^2*n/(d*x + c)^2 - 3*(b*x + a)*B*a^4*b*d^5*g^2*n/(d*x + c) - 12*(b*x + a)^2*B*a^3*b*c*d^5*g^2*n/(d*x + c)
^2 + 3*(b*x + a)^2*B*a^4*d^6*g^2*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/(b^3*d^3 - 3*(b*x + a)*b^2*d^4/(d*x +
c) + 3*(b*x + a)^2*b*d^5/(d*x + c)^2 - (b*x + a)^3*d^6/(d*x + c)^3) + (3*B*b^6*c^4*g^2*n - 12*B*a*b^5*c^3*d*g
^2*n - 7*(b*x + a)*B*b^5*c^4*d*g^2*n/(d*x + c) + 18*B*a^2*b^4*c^2*d^2*g^2*n + 28*(b*x + a)*B*a*b^4*c^3*d^2*g^2
*n/(d*x + c) + 4*(b*x + a)^2*B*b^4*c^4*d^2*g^2*n/(d*x + c)^2 - 12*B*a^3*b^3*c*d^3*g^2*n - 42*(b*x + a)*B*a^2*b
^3*c^2*d^3*g^2*n/(d*x + c) - 16*(b*x + a)^2*B*a*b^3*c^3*d^3*g^2*n/(d*x + c)^2 + 3*B*a^4*b^2*d^4*g^2*n + 28*(b*
x + a)*B*a^3*b^2*c*d^4*g^2*n/(d*x + c) + 24*(b*x + a)^2*B*a^2*b^2*c^2*d^4*g^2*n/(d*x + c)^2 - 7*(b*x + a)*B*a^
4*b*d^5*g^2*n/(d*x + c) - 16*(b*x + a)^2*B*a^3*b*c*d^5*g^2*n/(d*x + c)^2 + 4*(b*x + a)^2*B*a^4*d^6*g^2*n/(d*x
+ c)^2 + 2*A*b^6*c^4*g^2 + 2*B*b^6*c^4*g^2 - 8*A*a*b^5*c^3*d*g^2 - 8*B*a*b^5*c^3*d*g^2 - 6*(b*x + a)*A*b^5*c^4
*d*g^2/(d*x + c) - 6*(b*x + a)*B*b^5*c^4*d*g^2/(d*x + c) + 12*A*a^2*b^4*c^2*d^2*g^2 + 12*B*a^2*b^4*c^2*d^2*g^2
+ 24*(b*x + a)*A*a*b^4*c^3*d^2*g^2/(d*x + c) + 24*(b*x + a)*B*a*b^4*c^3*d^2*g^2/(d*x + c) + 6*(b*x + a)^2*A*b
^4*c^4*d^2*g^2/(d*x + c)^2 + 6*(b*x + a)^2*B*b^4*c^4*d^2*g^2/(d*x + c)^2 - 8*A*a^3*b^3*c*d^3*g^2 - 8*B*a^3*b^3
*c*d^3*g^2 - 36*(b*x + a)*A*a^2*b^3*c^2*d^3*g^2/(d*x + c) - 36*(b*x + a)*B*a^2*b^3*c^2*d^3*g^2/(d*x + c) - 24*
(b*x + a)^2*A*a*b^3*c^3*d^3*g^2/(d*x + c)^2 - 24*(b*x + a)^2*B*a*b^3*c^3*d^3*g^2/(d*x + c)^2 + 2*A*a^4*b^2*d^4
*g^2 + 2*B*a^4*b^2*d^4*g^2 + 24*(b*x + a)*A*a^3*b^2*c*d^4*g^2/(d*x + c) + 24*(b*x + a)*B*a^3*b^2*c*d^4*g^2/(d*
x + c) + 36*(b*x + a)^2*A*a^2*b^2*c^2*d^4*g^2/(d*x + c)^2 + 36*(b*x + a)^2*B*a^2*b^2*c^2*d^4*g^2/(d*x + c)^2 -
6*(b*x + a)*A*a^4*b*d^5*g^2/(d*x + c) - 6*(b*x + a)*B*a^4*b*d^5*g^2/(d*x + c) - 24*(b*x + a)^2*A*a^3*b*c*d^5*
g^2/(d*x + c)^2 - 24*(b*x + a)^2*B*a^3*b*c*d^5*g^2/(d*x + c)^2 + 6*(b*x + a)^2*A*a^4*d^6*g^2/(d*x + c)^2 + 6*(
b*x + a)^2*B*a^4*d^6*g^2/(d*x + c)^2)/(b^3*d^3 - 3*(b*x + a)*b^2*d^4/(d*x + c) + 3*(b*x + a)^2*b*d^5/(d*x + c)
^2 - (b*x + a)^3*d^6/(d*x + c)^3) + 2*(B*b^4*c^4*g^2*n - 4*B*a*b^3*c^3*d*g^2*n + 6*B*a^2*b^2*c^2*d^2*g^2*n - 4
*B*a^3*b*c*d^3*g^2*n + B*a^4*d^4*g^2*n)*log(b - (b*x + a)*d/(d*x + c))/(b*d^3) - 2*(B*b^4*c^4*g^2*n - 4*B*a*b^
3*c^3*d*g^2*n + 6*B*a^2*b^2*c^2*d^2*g^2*n - 4*B*a^3*b*c*d^3*g^2*n + B*a^4*d^4*g^2*n)*log((b*x + a)/(d*x + c))/
(b*d^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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