3.1.56 \(\int (a g+b g x)^2 (c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [56]

Optimal. Leaf size=450 \[ -\frac {B^2 (b c-a d)^3 g^2 i x}{3 b d^2}+\frac {B^2 (b c-a d)^2 g^2 i (c+d x)^2}{12 d^3}-\frac {B (b c-a d)^2 g^2 i (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b}+\frac {B (b c-a d)^3 g^2 i (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b^2 d^2}+\frac {B (b c-a d)^4 g^2 i \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i \log (c+d x)}{6 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3} \]

[Out]

-1/3*B^2*(-a*d+b*c)^3*g^2*i*x/b/d^2+1/12*B^2*(-a*d+b*c)^2*g^2*i*(d*x+c)^2/d^3-1/12*B*(-a*d+b*c)^2*g^2*i*(b*x+a
)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2/d-1/6*B*(-a*d+b*c)*g^2*i*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2+1/12*(-
a*d+b*c)*g^2*i*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^2+1/4*g^2*i*(b*x+a)^3*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+
c)))^2/b+1/12*B*(-a*d+b*c)^3*g^2*i*(b*x+a)*(2*A+B+2*B*ln(e*(b*x+a)/(d*x+c)))/b^2/d^2+1/12*B*(-a*d+b*c)^4*g^2*i
*ln((-a*d+b*c)/b/(d*x+c))*(2*A+3*B+2*B*ln(e*(b*x+a)/(d*x+c)))/b^2/d^3+1/6*B^2*(-a*d+b*c)^4*g^2*i*ln(d*x+c)/b^2
/d^3+1/6*B^2*(-a*d+b*c)^4*g^2*i*polylog(2,d*(b*x+a)/b/(d*x+c))/b^2/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2562, 2383, 2381, 2384, 2354, 2438, 2373, 45} \begin {gather*} \frac {B^2 g^2 i (b c-a d)^4 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3}+\frac {B g^2 i (b c-a d)^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{12 b^2 d^3}+\frac {B g^2 i (a+b x) (b c-a d)^3 \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{12 b^2 d^2}-\frac {B g^2 i (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{12 b^2 d}+\frac {g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{12 b^2}-\frac {B g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b}+\frac {B^2 g^2 i (b c-a d)^4 \log (c+d x)}{6 b^2 d^3}+\frac {B^2 g^2 i (c+d x)^2 (b c-a d)^2}{12 d^3}-\frac {B^2 g^2 i x (b c-a d)^3}{3 b d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(B^2*(b*c - a*d)^3*g^2*i*x)/(b*d^2) + (B^2*(b*c - a*d)^2*g^2*i*(c + d*x)^2)/(12*d^3) - (B*(b*c - a*d)^2*g
^2*i*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^2*d) - (B*(b*c - a*d)*g^2*i*(a + b*x)^3*(A + B*Lo
g[(e*(a + b*x))/(c + d*x)]))/(6*b^2) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/
(12*b^2) + (g^2*i*(a + b*x)^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b) + (B*(b*c - a*d)^3*g^2*i
*(a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^2*d^2) + (B*(b*c - a*d)^4*g^2*i*Log[(b*c - a*d)
/(b*(c + d*x))]*(2*A + 3*B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^2*d^3) + (B^2*(b*c - a*d)^4*g^2*i*Log[c
+ d*x])/(6*b^2*d^3) + (B^2*(b*c - a*d)^4*g^2*i*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(6*b^2*d^3)

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 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 2373

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

Rule 2381

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

Rule 2383

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Dist[(m + q + 2)/(d*(q + 1)),
Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Dist[b*n*(p/(d*(q + 1))), Int[(f*x)^m*(d + e*x)^(
q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p,
0] && LtQ[q, -1] && GtQ[m, 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 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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(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] && EqQ[n + mn, 0] && IGtQ[n, 0] && 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 (56 c+56 d x) (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\int \left (\frac {56 (b c-a d) (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b}+\frac {56 d (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}\right ) \, dx\\ &=\frac {(56 (b c-a d)) \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b}+\frac {(56 d) \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b g}\\ &=\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {(28 B d) \int \frac {(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{b^2 g^2}-\frac {(112 B (b c-a d)) \int \frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{3 b^2 g}\\ &=\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {\left (28 B d (b c-a d) g^2\right ) \int \frac {(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{b^2}-\frac {\left (112 B (b c-a d)^2 g^2\right ) \int \frac {(a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{3 b^2}\\ &=\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {\left (28 B d (b c-a d) g^2\right ) \int \left (\frac {b (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {b (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {(-b c+a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b^2}-\frac {\left (112 B (b c-a d)^2 g^2\right ) \int \left (-\frac {b (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {b (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {(-b c+a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 (c+d x)}\right ) \, dx}{3 b^2}\\ &=\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {\left (28 B (b c-a d) g^2\right ) \int (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b}+\frac {\left (28 B (b c-a d)^2 g^2\right ) \int (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d}-\frac {\left (112 B (b c-a d)^2 g^2\right ) \int (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b d}-\frac {\left (28 B (b c-a d)^3 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d^2}+\frac {\left (112 B (b c-a d)^3 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b d^2}+\frac {\left (28 B (b c-a d)^4 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 d^2}-\frac {\left (112 B (b c-a d)^4 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {28 A B (b c-a d)^3 g^2 x}{3 b d^2}-\frac {14 B (b c-a d)^2 g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 d}-\frac {28 B (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {28 B (b c-a d)^4 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (28 B^2 (b c-a d) g^2\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{3 b^2}-\frac {\left (14 B^2 (b c-a d)^2 g^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{b^2 d}+\frac {\left (56 B^2 (b c-a d)^2 g^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{3 b^2 d}-\frac {\left (28 B^2 (b c-a d)^3 g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b d^2}+\frac {\left (112 B^2 (b c-a d)^3 g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{3 b d^2}-\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^2 d^3}+\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{3 b^2 d^3}\\ &=\frac {28 A B (b c-a d)^3 g^2 x}{3 b d^2}+\frac {28 B^2 (b c-a d)^3 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^2 d^2}-\frac {14 B (b c-a d)^2 g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 d}-\frac {28 B (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {28 B (b c-a d)^4 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (28 B^2 (b c-a d)^2 g^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b^2}-\frac {\left (14 B^2 (b c-a d)^3 g^2\right ) \int \frac {a+b x}{c+d x} \, dx}{b^2 d}+\frac {\left (56 B^2 (b c-a d)^3 g^2\right ) \int \frac {a+b x}{c+d x} \, dx}{3 b^2 d}+\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \int \frac {1}{c+d x} \, dx}{b^2 d^2}-\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \int \frac {1}{c+d x} \, dx}{3 b^2 d^2}-\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 d^3 e}+\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^2 d^3 e}\\ &=\frac {28 A B (b c-a d)^3 g^2 x}{3 b d^2}+\frac {28 B^2 (b c-a d)^3 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^2 d^2}-\frac {14 B (b c-a d)^2 g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 d}-\frac {28 B (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {28 B^2 (b c-a d)^4 g^2 \log (c+d x)}{3 b^2 d^3}-\frac {28 B (b c-a d)^4 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (28 B^2 (b c-a d)^2 g^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^2}-\frac {\left (14 B^2 (b c-a d)^3 g^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{b^2 d}+\frac {\left (56 B^2 (b c-a d)^3 g^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{3 b^2 d}-\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^2 d^3 e}+\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{3 b^2 d^3 e}\\ &=\frac {28 A B (b c-a d)^3 g^2 x}{3 b d^2}-\frac {14 B^2 (b c-a d)^3 g^2 x}{3 b d^2}+\frac {14 B^2 (b c-a d)^2 g^2 (a+b x)^2}{3 b^2 d}+\frac {28 B^2 (b c-a d)^3 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^2 d^2}-\frac {14 B (b c-a d)^2 g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 d}-\frac {28 B (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {14 B^2 (b c-a d)^4 g^2 \log (c+d x)}{3 b^2 d^3}-\frac {28 B (b c-a d)^4 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b d^3}+\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b d^3}+\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 d^2}-\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {28 A B (b c-a d)^3 g^2 x}{3 b d^2}-\frac {14 B^2 (b c-a d)^3 g^2 x}{3 b d^2}+\frac {14 B^2 (b c-a d)^2 g^2 (a+b x)^2}{3 b^2 d}+\frac {28 B^2 (b c-a d)^3 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^2 d^2}-\frac {14 B (b c-a d)^2 g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 d}-\frac {28 B (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {14 B^2 (b c-a d)^4 g^2 \log (c+d x)}{3 b^2 d^3}+\frac {28 B^2 (b c-a d)^4 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {28 B (b c-a d)^4 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 d^3}-\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^2 d^3}+\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 d^2}-\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {28 A B (b c-a d)^3 g^2 x}{3 b d^2}-\frac {14 B^2 (b c-a d)^3 g^2 x}{3 b d^2}+\frac {14 B^2 (b c-a d)^2 g^2 (a+b x)^2}{3 b^2 d}+\frac {28 B^2 (b c-a d)^3 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^2 d^2}-\frac {14 B (b c-a d)^2 g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 d}-\frac {28 B (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {14 B^2 (b c-a d)^4 g^2 \log (c+d x)}{3 b^2 d^3}+\frac {28 B^2 (b c-a d)^4 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {28 B (b c-a d)^4 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {14 B^2 (b c-a d)^4 g^2 \log ^2(c+d x)}{3 b^2 d^3}+\frac {\left (28 B^2 (b c-a d)^4 g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 d^3}-\frac {\left (112 B^2 (b c-a d)^4 g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^2 d^3}\\ &=\frac {28 A B (b c-a d)^3 g^2 x}{3 b d^2}-\frac {14 B^2 (b c-a d)^3 g^2 x}{3 b d^2}+\frac {14 B^2 (b c-a d)^2 g^2 (a+b x)^2}{3 b^2 d}+\frac {28 B^2 (b c-a d)^3 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^2 d^2}-\frac {14 B (b c-a d)^2 g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 d}-\frac {28 B (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2}+\frac {56 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2}+\frac {14 d g^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}-\frac {14 B^2 (b c-a d)^4 g^2 \log (c+d x)}{3 b^2 d^3}+\frac {28 B^2 (b c-a d)^4 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {28 B (b c-a d)^4 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {14 B^2 (b c-a d)^4 g^2 \log ^2(c+d x)}{3 b^2 d^3}+\frac {28 B^2 (b c-a d)^4 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 680, normalized size = 1.51 \begin {gather*} \frac {g^2 i \left (4 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+3 d (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {4 B (b c-a d)^2 \left (2 A b d (b c-a d) x+2 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 B (b c-a d)^2 \log (c+d x)-2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 \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}-\frac {B (b c-a d) \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 \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}\right )}{12 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*i*(4*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 3*d*(a + b*x)^4*(A + B*Log[(e*(a +
b*x))/(c + d*x)])^2 + (4*B*(b*c - a*d)^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)] - d^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*B*(b*c - a*d)^2*Log[c + d*x] - 2*(b*c
- a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + B*(b*c - a*d)*(b*d*x + (-(b*c) + a*d)*Log[c + d*x
]) + B*(b*c - a*d)^2*((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 - (B*(b*c - a*d)*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a
+ b*x))/(c + d*x)] + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*d^3*(a + b*x)^
3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] - 6*(b*c - a*d)^3*(A + B*Log[(e*(a + b
*x))/(c + d*x)])*Log[c + d*x] + B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c +
d*x]) + 3*B*(b*c - a*d)^2*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c - a*d)^3*((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))/(12*b^2)

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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{2} \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1888 vs. \(2 (425) = 850\).
time = 0.39, size = 1888, normalized size = 4.20 \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*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="maxima")

[Out]

1/4*I*A^2*b^2*d*g^2*x^4 + 1/3*I*A^2*b^2*c*g^2*x^3 + 2/3*I*A^2*a*b*d*g^2*x^3 + I*A^2*a*b*c*g^2*x^2 + 1/2*I*A^2*
a^2*d*g^2*x^2 + 2*I*(x*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*A*B*a^2*c*g
^2 + 2*I*(x^2*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)
*x/(b*d))*A*B*a*b*c*g^2 + 1/3*I*(2*x^3*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*l
og(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*b^2*c*g^2 + I*(x^2*log(b*
x*e/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*a^2*d*
g^2 + 2/3*I*(2*x^3*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((
b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*a*b*d*g^2 + 1/12*I*(6*x^4*log(b*x*e/(d*x + c)
+ a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*
c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*b^2*d*g^2 + I*A^2*a^2*c*g^2*x - 1/12*(3*I*b^3
*c^4*g^2 - 10*I*a*b^2*c^3*d*g^2 + 11*I*a^2*b*c^2*d^2*g^2 + 2*I*a^3*c*d^3*g^2)*B^2*log(d*x + c)/(b*d^3) - 1/6*(
I*b^4*c^4*g^2 - 4*I*a*b^3*c^3*d*g^2 + 6*I*a^2*b^2*c^2*d^2*g^2 - 4*I*a^3*b*c*d^3*g^2 + I*a^4*d^4*g^2)*(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^2/(b^2*d^3) + 1/12*(3*I*B^2*b^4
*d^4*g^2*x^4 - 2*(-I*b^4*c*d^3*g^2 - 5*I*a*b^3*d^4*g^2)*B^2*x^3 - 6*(-I*a*b^3*c*d^3*g^2 - 2*I*a^2*b^2*d^4*g^2)
*B^2*x^2 + (I*b^4*c^3*d*g^2 - 3*I*a*b^3*c^2*d^2*g^2 + 9*I*a^2*b^2*c*d^3*g^2 + 5*I*a^3*b*d^4*g^2)*B^2*x + (3*I*
B^2*b^4*d^4*g^2*x^4 + 12*I*B^2*a^2*b^2*c*d^3*g^2*x - 4*(-I*b^4*c*d^3*g^2 - 2*I*a*b^3*d^4*g^2)*B^2*x^3 - 6*(-2*
I*a*b^3*c*d^3*g^2 - I*a^2*b^2*d^4*g^2)*B^2*x^2 + (4*I*a^3*b*c*d^3*g^2 - I*a^4*d^4*g^2)*B^2)*log(b*x + a)^2 + (
3*I*B^2*b^4*d^4*g^2*x^4 + 12*I*B^2*a^2*b^2*c*d^3*g^2*x - 4*(-I*b^4*c*d^3*g^2 - 2*I*a*b^3*d^4*g^2)*B^2*x^3 - 6*
(-2*I*a*b^3*c*d^3*g^2 - I*a^2*b^2*d^4*g^2)*B^2*x^2 + (I*b^4*c^4*g^2 - 4*I*a*b^3*c^3*d*g^2 + 6*I*a^2*b^2*c^2*d^
2*g^2)*B^2)*log(d*x + c)^2 + (6*I*B^2*b^4*d^4*g^2*x^4 - 6*(-I*b^4*c*d^3*g^2 - 3*I*a*b^3*d^4*g^2)*B^2*x^3 + (-I
*b^4*c^2*d^2*g^2 + 20*I*a*b^3*c*d^3*g^2 + 17*I*a^2*b^2*d^4*g^2)*B^2*x^2 - 2*(-I*b^4*c^3*d*g^2 + 4*I*a*b^3*c^2*
d^2*g^2 - 14*I*a^2*b^2*c*d^3*g^2 - I*a^3*b*d^4*g^2)*B^2*x + (2*I*a*b^3*c^3*d*g^2 - 7*I*a^2*b^2*c^2*d^2*g^2 + 1
4*I*a^3*b*c*d^3*g^2 - 3*I*a^4*d^4*g^2)*B^2)*log(b*x + a) + (-6*I*B^2*b^4*d^4*g^2*x^4 - 6*(I*b^4*c*d^3*g^2 + 3*
I*a*b^3*d^4*g^2)*B^2*x^3 + (I*b^4*c^2*d^2*g^2 - 20*I*a*b^3*c*d^3*g^2 - 17*I*a^2*b^2*d^4*g^2)*B^2*x^2 - 2*(I*b^
4*c^3*d*g^2 - 4*I*a*b^3*c^2*d^2*g^2 + 14*I*a^2*b^2*c*d^3*g^2 + I*a^3*b*d^4*g^2)*B^2*x - 2*(3*I*B^2*b^4*d^4*g^2
*x^4 + 12*I*B^2*a^2*b^2*c*d^3*g^2*x + 4*(I*b^4*c*d^3*g^2 + 2*I*a*b^3*d^4*g^2)*B^2*x^3 + 6*(2*I*a*b^3*c*d^3*g^2
+ I*a^2*b^2*d^4*g^2)*B^2*x^2 + (4*I*a^3*b*c*d^3*g^2 - I*a^4*d^4*g^2)*B^2)*log(b*x + a))*log(d*x + c))/(b^2*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*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

1/12*(3*I*B^2*b^2*d*g^2*x^4 + 12*I*B^2*a^2*c*g^2*x - 4*(-I*B^2*b^2*c - 2*I*B^2*a*b*d)*g^2*x^3 - 6*(-2*I*B^2*a*
b*c - I*B^2*a^2*d)*g^2*x^2)*log((b*x + a)*e/(d*x + c))^2 + integral(1/6*(6*I*A^2*b^3*d^2*g^2*x^5 + 6*I*A^2*a^3
*c^2*g^2 - 6*(-2*I*A^2*b^3*c*d - 3*I*A^2*a*b^2*d^2)*g^2*x^4 - 6*(-I*A^2*b^3*c^2 - 6*I*A^2*a*b^2*c*d - 3*I*A^2*
a^2*b*d^2)*g^2*x^3 - 6*(-3*I*A^2*a*b^2*c^2 - 6*I*A^2*a^2*b*c*d - I*A^2*a^3*d^2)*g^2*x^2 - 6*(-3*I*A^2*a^2*b*c^
2 - 2*I*A^2*a^3*c*d)*g^2*x + (12*I*A*B*b^3*d^2*g^2*x^5 + 12*I*A*B*a^3*c^2*g^2 - 3*((-8*I*A*B + I*B^2)*b^3*c*d
+ (-12*I*A*B - I*B^2)*a*b^2*d^2)*g^2*x^4 - 4*((-3*I*A*B + I*B^2)*b^3*c^2 + (-18*I*A*B + I*B^2)*a*b^2*c*d + (-9
*I*A*B - 2*I*B^2)*a^2*b*d^2)*g^2*x^3 - 6*(2*(-3*I*A*B + I*B^2)*a*b^2*c^2 + (-12*I*A*B - I*B^2)*a^2*b*c*d + (-2
*I*A*B - I*B^2)*a^3*d^2)*g^2*x^2 - 12*((-3*I*A*B + I*B^2)*a^2*b*c^2 + (-2*I*A*B - I*B^2)*a^3*c*d)*g^2*x)*log((
b*x + a)*e/(d*x + c)))/(b*d*x^2 + a*c + (b*c + a*d)*x), 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*(d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))**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((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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