\(\int (a g+b g x)^2 (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [128]

Optimal result

Integrand size = 43, antiderivative size = 387 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^5 g^2 i^3 n x}{60 b^3 d^2}-\frac {B (b c-a d)^4 g^2 i^3 n (c+d x)^2}{120 b^2 d^3}-\frac {B (b c-a d)^3 g^2 i^3 n (c+d x)^3}{180 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 n (c+d x)^4}{120 d^3}-\frac {b B (b c-a d) g^2 i^3 n (c+d x)^5}{30 d^3}+\frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log (c+d x)}{60 b^4 d^3} \]

[Out]

-1/60*B*(-a*d+b*c)^5*g^2*i^3*n*x/b^3/d^2-1/120*B*(-a*d+b*c)^4*g^2*i^3*n*(d*x+c)^2/b^2/d^3-1/180*B*(-a*d+b*c)^3
*g^2*i^3*n*(d*x+c)^3/b/d^3+7/120*B*(-a*d+b*c)^2*g^2*i^3*n*(d*x+c)^4/d^3-1/30*b*B*(-a*d+b*c)*g^2*i^3*n*(d*x+c)^
5/d^3+1/4*(-a*d+b*c)^2*g^2*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-2/5*b*(-a*d+b*c)*g^2*i^3*(d*x+c)^
5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3+1/6*b^2*g^2*i^3*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-1/60*B*(-a
*d+b*c)^6*g^2*i^3*n*ln((b*x+a)/(d*x+c))/b^4/d^3-1/60*B*(-a*d+b*c)^6*g^2*i^3*n*ln(d*x+c)/b^4/d^3

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2561, 45, 2382, 12, 907} \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {b^2 g^2 i^3 (c+d x)^6 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3}+\frac {g^2 i^3 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3}-\frac {2 b g^2 i^3 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3}-\frac {B g^2 i^3 n (b c-a d)^6 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac {B g^2 i^3 n (b c-a d)^6 \log (c+d x)}{60 b^4 d^3}-\frac {B g^2 i^3 n x (b c-a d)^5}{60 b^3 d^2}-\frac {B g^2 i^3 n (c+d x)^2 (b c-a d)^4}{120 b^2 d^3}-\frac {B g^2 i^3 n (c+d x)^3 (b c-a d)^3}{180 b d^3}+\frac {7 B g^2 i^3 n (c+d x)^4 (b c-a d)^2}{120 d^3}-\frac {b B g^2 i^3 n (c+d x)^5 (b c-a d)}{30 d^3} \]

[In]

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

[Out]

-1/60*(B*(b*c - a*d)^5*g^2*i^3*n*x)/(b^3*d^2) - (B*(b*c - a*d)^4*g^2*i^3*n*(c + d*x)^2)/(120*b^2*d^3) - (B*(b*
c - a*d)^3*g^2*i^3*n*(c + d*x)^3)/(180*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*n*(c + d*x)^4)/(120*d^3) - (b*B*(b*
c - a*d)*g^2*i^3*n*(c + d*x)^5)/(30*d^3) + ((b*c - a*d)^2*g^2*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]))/(4*d^3) - (2*b*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^3) + (b^2*
g^2*i^3*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^3) - (B*(b*c - a*d)^6*g^2*i^3*n*Log[(a + b*x)
/(c + d*x)])/(60*b^4*d^3) - (B*(b*c - a*d)^6*g^2*i^3*n*Log[c + d*x])/(60*b^4*d^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2382

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[
x^m*(d + e*x)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ
[{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]

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*} \text {integral}& = \left ((b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 \left (A+B \log \left (e x^n\right )\right )}{(b-d x)^7} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3}-\left (B (b c-a d)^6 g^2 i^3 n\right ) \text {Subst}\left (\int \frac {b^2-6 b d x+15 d^2 x^2}{60 d^3 x (b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3}-\frac {\left (B (b c-a d)^6 g^2 i^3 n\right ) \text {Subst}\left (\int \frac {b^2-6 b d x+15 d^2 x^2}{x (b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{60 d^3} \\ & = \frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3}-\frac {\left (B (b c-a d)^6 g^2 i^3 n\right ) \text {Subst}\left (\int \left (\frac {1}{b^4 x}+\frac {10 b d}{(b-d x)^6}-\frac {14 d}{(b-d x)^5}+\frac {d}{b (b-d x)^4}+\frac {d}{b^2 (b-d x)^3}+\frac {d}{b^3 (b-d x)^2}+\frac {d}{b^4 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{60 d^3} \\ & = -\frac {B (b c-a d)^5 g^2 i^3 n x}{60 b^3 d^2}-\frac {B (b c-a d)^4 g^2 i^3 n (c+d x)^2}{120 b^2 d^3}-\frac {B (b c-a d)^3 g^2 i^3 n (c+d x)^3}{180 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 n (c+d x)^4}{120 d^3}-\frac {b B (b c-a d) g^2 i^3 n (c+d x)^5}{30 d^3}+\frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log (c+d x)}{60 b^4 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.14 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^2 i^3 \left (-15 B (b c-a d)^3 n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+12 B (b c-a d)^2 n \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )-B (b c-a d) n \left (60 b d (b c-a d)^4 x+30 b^2 (b c-a d)^3 (c+d x)^2+20 b^3 (b c-a d)^2 (c+d x)^3+15 b^4 (b c-a d) (c+d x)^4+12 b^5 (c+d x)^5+60 (b c-a d)^5 \log (a+b x)\right )+90 b^4 (b c-a d)^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-144 b^5 (b c-a d) (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+60 b^6 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{360 b^4 d^3} \]

[In]

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

[Out]

(g^2*i^3*(-15*B*(b*c - a*d)^3*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6
*(b*c - a*d)^3*Log[a + b*x]) + 12*B*(b*c - a*d)^2*n*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2
+ 4*b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]) - B*(b*c - a*d)*n*(60*b*d
*(b*c - a*d)^4*x + 30*b^2*(b*c - a*d)^3*(c + d*x)^2 + 20*b^3*(b*c - a*d)^2*(c + d*x)^3 + 15*b^4*(b*c - a*d)*(c
+ d*x)^4 + 12*b^5*(c + d*x)^5 + 60*(b*c - a*d)^5*Log[a + b*x]) + 90*b^4*(b*c - a*d)^2*(c + d*x)^4*(A + B*Log[
e*((a + b*x)/(c + d*x))^n]) - 144*b^5*(b*c - a*d)*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 60*b^6*
(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(360*b^4*d^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1720\) vs. \(2(367)=734\).

Time = 25.41 (sec) , antiderivative size = 1721, normalized size of antiderivative = 4.45

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1721\)

[In]

int((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x,method=_RETURNVERBOSE)

[Out]

1/360*(12*B*x^5*a*b^5*d^6*g^2*i^3*n^2+540*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c*d^5*g^2*i^3*n+360*B*x^3*ln(e
*((b*x+a)/(d*x+c))^n)*a^2*b^4*c*d^5*g^2*i^3*n+720*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c^2*d^4*g^2*i^3*n+540*
B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c^2*d^4*g^2*i^3*n+360*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c^3*d^3*g^
2*i^3*n+360*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c^3*d^3*g^2*i^3*n+33*B*a^5*b*c*d^5*g^2*i^3*n^2-72*B*a^4*b^2*
c^2*d^4*g^2*i^3*n^2-150*B*a^3*b^3*c^3*d^3*g^2*i^3*n^2+168*B*a^2*b^4*c^4*d^2*g^2*i^3*n^2+33*B*a*b^5*c^5*d*g^2*i
^3*n^2-900*A*a^3*b^3*c^3*d^3*g^2*i^3*n-720*A*a^2*b^4*c^4*d^2*g^2*i^3*n+90*B*x^2*a^2*b^4*c^2*d^4*g^2*i^3*n^2-10
2*B*x^2*a*b^5*c^3*d^3*g^2*i^3*n^2+540*A*x^2*a^2*b^4*c^2*d^4*g^2*i^3*n+360*A*x^2*a*b^5*c^3*d^3*g^2*i^3*n-36*B*x
*a^4*b^2*c*d^5*g^2*i^3*n^2+90*B*x*a^3*b^3*c^2*d^4*g^2*i^3*n^2-30*B*x*a^2*b^4*c^3*d^3*g^2*i^3*n^2-36*B*x*a*b^5*
c^4*d^2*g^2*i^3*n^2+360*A*x*a^2*b^4*c^3*d^3*g^2*i^3*n+90*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c^4*d^2*g^2*i^3*n
+144*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^6*g^2*i^3*n+216*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c*d^5*g^2*i^3
*n+90*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*d^6*g^2*i^3*n+270*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^2*d^4*g^
2*i^3*n+18*B*x^4*a*b^5*c*d^5*g^2*i^3*n^2+540*A*x^4*a*b^5*c*d^5*g^2*i^3*n+120*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b
^6*c^3*d^3*g^2*i^3*n+78*B*x^3*a^2*b^4*c*d^5*g^2*i^3*n^2-42*B*x^3*a*b^5*c^2*d^4*g^2*i^3*n^2+360*A*x^3*a^2*b^4*c
*d^5*g^2*i^3*n+720*A*x^3*a*b^5*c^2*d^4*g^2*i^3*n+18*B*x^2*a^3*b^3*c*d^5*g^2*i^3*n^2-36*B*ln(e*((b*x+a)/(d*x+c)
)^n)*a*b^5*c^5*d*g^2*i^3*n+36*B*ln(b*x+a)*a^5*b*c*d^5*g^2*i^3*n^2-90*B*ln(b*x+a)*a^4*b^2*c^2*d^4*g^2*i^3*n^2+1
20*B*ln(b*x+a)*a^3*b^3*c^3*d^3*g^2*i^3*n^2-90*B*ln(b*x+a)*a^2*b^4*c^4*d^2*g^2*i^3*n^2+36*B*ln(b*x+a)*a*b^5*c^5
*d*g^2*i^3*n^2+60*A*x^6*b^6*d^6*g^2*i^3*n+6*B*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^6*g^2*i^3*n-6*B*ln(b*x+a)*b^6*c^
6*g^2*i^3*n^2-6*B*ln(b*x+a)*a^6*d^6*g^2*i^3*n^2-6*B*a^6*d^6*g^2*i^3*n^2-6*B*b^6*c^6*g^2*i^3*n^2-12*B*x^5*b^6*c
*d^5*g^2*i^3*n^2+144*A*x^5*a*b^5*d^6*g^2*i^3*n+216*A*x^5*b^6*c*d^5*g^2*i^3*n+21*B*x^4*a^2*b^4*d^6*g^2*i^3*n^2-
39*B*x^4*b^6*c^2*d^4*g^2*i^3*n^2+90*A*x^4*a^2*b^4*d^6*g^2*i^3*n+270*A*x^4*b^6*c^2*d^4*g^2*i^3*n+2*B*x^3*a^3*b^
3*d^6*g^2*i^3*n^2-38*B*x^3*b^6*c^3*d^3*g^2*i^3*n^2+120*A*x^3*b^6*c^3*d^3*g^2*i^3*n-3*B*x^2*a^4*b^2*d^6*g^2*i^3
*n^2-3*B*x^2*b^6*c^4*d^2*g^2*i^3*n^2+6*B*x*a^5*b*d^6*g^2*i^3*n^2+6*B*x*b^6*c^5*d*g^2*i^3*n^2+60*B*x^6*ln(e*((b
*x+a)/(d*x+c))^n)*b^6*d^6*g^2*i^3*n)/d^3/b^4/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (367) = 734\).

Time = 0.66 (sec) , antiderivative size = 1075, normalized size of antiderivative = 2.78 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {60 \, A b^{6} d^{6} g^{2} i^{3} x^{6} + 6 \, {\left (20 \, B a^{3} b^{3} c^{3} d^{3} - 15 \, B a^{4} b^{2} c^{2} d^{4} + 6 \, B a^{5} b c d^{5} - B a^{6} d^{6}\right )} g^{2} i^{3} n \log \left (b x + a\right ) - 6 \, {\left (B b^{6} c^{6} - 6 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2}\right )} g^{2} i^{3} n \log \left (d x + c\right ) - 12 \, {\left ({\left (B b^{6} c d^{5} - B a b^{5} d^{6}\right )} g^{2} i^{3} n - 6 \, {\left (3 \, A b^{6} c d^{5} + 2 \, A a b^{5} d^{6}\right )} g^{2} i^{3}\right )} x^{5} - 3 \, {\left ({\left (13 \, B b^{6} c^{2} d^{4} - 6 \, B a b^{5} c d^{5} - 7 \, B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} n - 30 \, {\left (3 \, A b^{6} c^{2} d^{4} + 6 \, A a b^{5} c d^{5} + A a^{2} b^{4} d^{6}\right )} g^{2} i^{3}\right )} x^{4} - 2 \, {\left ({\left (19 \, B b^{6} c^{3} d^{3} + 21 \, B a b^{5} c^{2} d^{4} - 39 \, B a^{2} b^{4} c d^{5} - B a^{3} b^{3} d^{6}\right )} g^{2} i^{3} n - 60 \, {\left (A b^{6} c^{3} d^{3} + 6 \, A a b^{5} c^{2} d^{4} + 3 \, A a^{2} b^{4} c d^{5}\right )} g^{2} i^{3}\right )} x^{3} - 3 \, {\left ({\left (B b^{6} c^{4} d^{2} + 34 \, B a b^{5} c^{3} d^{3} - 30 \, B a^{2} b^{4} c^{2} d^{4} - 6 \, B a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{2} i^{3} n - 60 \, {\left (2 \, A a b^{5} c^{3} d^{3} + 3 \, A a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3}\right )} x^{2} + 6 \, {\left (60 \, A a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} + {\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} - 5 \, B a^{2} b^{4} c^{3} d^{3} + 15 \, B a^{3} b^{3} c^{2} d^{4} - 6 \, B a^{4} b^{2} c d^{5} + B a^{5} b d^{6}\right )} g^{2} i^{3} n\right )} x + 6 \, {\left (10 \, B b^{6} d^{6} g^{2} i^{3} x^{6} + 60 \, B a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} x + 12 \, {\left (3 \, B b^{6} c d^{5} + 2 \, B a b^{5} d^{6}\right )} g^{2} i^{3} x^{5} + 15 \, {\left (3 \, B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} x^{4} + 20 \, {\left (B b^{6} c^{3} d^{3} + 6 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} g^{2} i^{3} x^{3} + 30 \, {\left (2 \, B a b^{5} c^{3} d^{3} + 3 \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3} x^{2}\right )} \log \left (e\right ) + 6 \, {\left (10 \, B b^{6} d^{6} g^{2} i^{3} n x^{6} + 60 \, B a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} n x + 12 \, {\left (3 \, B b^{6} c d^{5} + 2 \, B a b^{5} d^{6}\right )} g^{2} i^{3} n x^{5} + 15 \, {\left (3 \, B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} n x^{4} + 20 \, {\left (B b^{6} c^{3} d^{3} + 6 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} g^{2} i^{3} n x^{3} + 30 \, {\left (2 \, B a b^{5} c^{3} d^{3} + 3 \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{360 \, b^{4} d^{3}} \]

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

1/360*(60*A*b^6*d^6*g^2*i^3*x^6 + 6*(20*B*a^3*b^3*c^3*d^3 - 15*B*a^4*b^2*c^2*d^4 + 6*B*a^5*b*c*d^5 - B*a^6*d^6
)*g^2*i^3*n*log(b*x + a) - 6*(B*b^6*c^6 - 6*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2)*g^2*i^3*n*log(d*x + c) - 12*
((B*b^6*c*d^5 - B*a*b^5*d^6)*g^2*i^3*n - 6*(3*A*b^6*c*d^5 + 2*A*a*b^5*d^6)*g^2*i^3)*x^5 - 3*((13*B*b^6*c^2*d^4
- 6*B*a*b^5*c*d^5 - 7*B*a^2*b^4*d^6)*g^2*i^3*n - 30*(3*A*b^6*c^2*d^4 + 6*A*a*b^5*c*d^5 + A*a^2*b^4*d^6)*g^2*i
^3)*x^4 - 2*((19*B*b^6*c^3*d^3 + 21*B*a*b^5*c^2*d^4 - 39*B*a^2*b^4*c*d^5 - B*a^3*b^3*d^6)*g^2*i^3*n - 60*(A*b^
6*c^3*d^3 + 6*A*a*b^5*c^2*d^4 + 3*A*a^2*b^4*c*d^5)*g^2*i^3)*x^3 - 3*((B*b^6*c^4*d^2 + 34*B*a*b^5*c^3*d^3 - 30*
B*a^2*b^4*c^2*d^4 - 6*B*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^2*i^3*n - 60*(2*A*a*b^5*c^3*d^3 + 3*A*a^2*b^4*c^2*d^4
)*g^2*i^3)*x^2 + 6*(60*A*a^2*b^4*c^3*d^3*g^2*i^3 + (B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 - 5*B*a^2*b^4*c^3*d^3 + 15
*B*a^3*b^3*c^2*d^4 - 6*B*a^4*b^2*c*d^5 + B*a^5*b*d^6)*g^2*i^3*n)*x + 6*(10*B*b^6*d^6*g^2*i^3*x^6 + 60*B*a^2*b^
4*c^3*d^3*g^2*i^3*x + 12*(3*B*b^6*c*d^5 + 2*B*a*b^5*d^6)*g^2*i^3*x^5 + 15*(3*B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 +
B*a^2*b^4*d^6)*g^2*i^3*x^4 + 20*(B*b^6*c^3*d^3 + 6*B*a*b^5*c^2*d^4 + 3*B*a^2*b^4*c*d^5)*g^2*i^3*x^3 + 30*(2*B
*a*b^5*c^3*d^3 + 3*B*a^2*b^4*c^2*d^4)*g^2*i^3*x^2)*log(e) + 6*(10*B*b^6*d^6*g^2*i^3*n*x^6 + 60*B*a^2*b^4*c^3*d
^3*g^2*i^3*n*x + 12*(3*B*b^6*c*d^5 + 2*B*a*b^5*d^6)*g^2*i^3*n*x^5 + 15*(3*B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + B*
a^2*b^4*d^6)*g^2*i^3*n*x^4 + 20*(B*b^6*c^3*d^3 + 6*B*a*b^5*c^2*d^4 + 3*B*a^2*b^4*c*d^5)*g^2*i^3*n*x^3 + 30*(2*
B*a*b^5*c^3*d^3 + 3*B*a^2*b^4*c^2*d^4)*g^2*i^3*n*x^2)*log((b*x + a)/(d*x + c)))/(b^4*d^3)

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1978 vs. \(2 (367) = 734\).

Time = 0.25 (sec) , antiderivative size = 1978, normalized size of antiderivative = 5.11 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

1/6*B*b^2*d^3*g^2*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A*b^2*d^3*g^2*i^3*x^6 + 3/5*B*b^2*c*d^2
*g^2*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2/5*B*a*b*d^3*g^2*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + 3/5*A*b^2*c*d^2*g^2*i^3*x^5 + 2/5*A*a*b*d^3*g^2*i^3*x^5 + 3/4*B*b^2*c^2*d*g^2*i^3*x^4*log(e*(b*x/(d
*x + c) + a/(d*x + c))^n) + 3/2*B*a*b*c*d^2*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*B*a^2*d^3
*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*A*b^2*c^2*d*g^2*i^3*x^4 + 3/2*A*a*b*c*d^2*g^2*i^3*x^
4 + 1/4*A*a^2*d^3*g^2*i^3*x^4 + 1/3*B*b^2*c^3*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*B*a*b*c^2
*d*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + B*a^2*c*d^2*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + 1/3*A*b^2*c^3*g^2*i^3*x^3 + 2*A*a*b*c^2*d*g^2*i^3*x^3 + A*a^2*c*d^2*g^2*i^3*x^3 + B*a*b*c^3*g^2*i^3
*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*B*a^2*c^2*d*g^2*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^
n) + A*a*b*c^3*g^2*i^3*x^2 + 3/2*A*a^2*c^2*d*g^2*i^3*x^2 - 1/360*B*b^2*d^3*g^2*i^3*n*(60*a^6*log(b*x + a)/b^6
- 60*c^6*log(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3
*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) + 1/20*B*b^2*c
*d^2*g^2*i^3*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^
2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4)) + 1/30*B*a*b*d
^3*g^2*i^3*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*
d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4)) - 1/8*B*b^2*c^2*
d*g^2*i^3*n*(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)) - 1/4*B*a*b*c*d^2*g^2*i^3*n*(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)) - 1/24*B*a^2*d^3*g^2*i^3*n*(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)) + 1/6*B*b^2*c^3*g^2*i^3*n*(
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
)) + B*a*b*c^2*d*g^2*i^3*n*(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)) + 1/2*B*a^2*c*d^2*g^2*i^3*n*(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)) - B*a*b*c^3*g^2*i^3*n*(a^2*log(b*x + a)/b^2 - c
^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) - 3/2*B*a^2*c^2*d*g^2*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c
)/d^2 + (b*c - a*d)*x/(b*d)) + B*a^2*c^3*g^2*i^3*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + B*a^2*c^3*g^2*i^3*x
*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^2*c^3*g^2*i^3*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4300 vs. \(2 (367) = 734\).

Time = 1.78 (sec) , antiderivative size = 4300, normalized size of antiderivative = 11.11 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

1/360*(6*(B*b^9*c^7*g^2*i^3*n - 7*B*a*b^8*c^6*d*g^2*i^3*n - 6*(b*x + a)*B*b^8*c^7*d*g^2*i^3*n/(d*x + c) + 21*B
*a^2*b^7*c^5*d^2*g^2*i^3*n + 42*(b*x + a)*B*a*b^7*c^6*d^2*g^2*i^3*n/(d*x + c) + 15*(b*x + a)^2*B*b^7*c^7*d^2*g
^2*i^3*n/(d*x + c)^2 - 35*B*a^3*b^6*c^4*d^3*g^2*i^3*n - 126*(b*x + a)*B*a^2*b^6*c^5*d^3*g^2*i^3*n/(d*x + c) -
105*(b*x + a)^2*B*a*b^6*c^6*d^3*g^2*i^3*n/(d*x + c)^2 + 35*B*a^4*b^5*c^3*d^4*g^2*i^3*n + 210*(b*x + a)*B*a^3*b
^5*c^4*d^4*g^2*i^3*n/(d*x + c) + 315*(b*x + a)^2*B*a^2*b^5*c^5*d^4*g^2*i^3*n/(d*x + c)^2 - 21*B*a^5*b^4*c^2*d^
5*g^2*i^3*n - 210*(b*x + a)*B*a^4*b^4*c^3*d^5*g^2*i^3*n/(d*x + c) - 525*(b*x + a)^2*B*a^3*b^4*c^4*d^5*g^2*i^3*
n/(d*x + c)^2 + 7*B*a^6*b^3*c*d^6*g^2*i^3*n + 126*(b*x + a)*B*a^5*b^3*c^2*d^6*g^2*i^3*n/(d*x + c) + 525*(b*x +
a)^2*B*a^4*b^3*c^3*d^6*g^2*i^3*n/(d*x + c)^2 - B*a^7*b^2*d^7*g^2*i^3*n - 42*(b*x + a)*B*a^6*b^2*c*d^7*g^2*i^3
*n/(d*x + c) - 315*(b*x + a)^2*B*a^5*b^2*c^2*d^7*g^2*i^3*n/(d*x + c)^2 + 6*(b*x + a)*B*a^7*b*d^8*g^2*i^3*n/(d*
x + c) + 105*(b*x + a)^2*B*a^6*b*c*d^8*g^2*i^3*n/(d*x + c)^2 - 15*(b*x + a)^2*B*a^7*d^9*g^2*i^3*n/(d*x + c)^2)
*log((b*x + a)/(d*x + c))/(b^6*d^3 - 6*(b*x + a)*b^5*d^4/(d*x + c) + 15*(b*x + a)^2*b^4*d^5/(d*x + c)^2 - 20*(
b*x + a)^3*b^3*d^6/(d*x + c)^3 + 15*(b*x + a)^4*b^2*d^7/(d*x + c)^4 - 6*(b*x + a)^5*b*d^8/(d*x + c)^5 + (b*x +
a)^6*d^9/(d*x + c)^6) - (2*B*b^12*c^7*g^2*i^3*n - 14*B*a*b^11*c^6*d*g^2*i^3*n - 18*(b*x + a)*B*b^11*c^7*d*g^2
*i^3*n/(d*x + c) + 42*B*a^2*b^10*c^5*d^2*g^2*i^3*n + 126*(b*x + a)*B*a*b^10*c^6*d^2*g^2*i^3*n/(d*x + c) + 63*(
b*x + a)^2*B*b^10*c^7*d^2*g^2*i^3*n/(d*x + c)^2 - 70*B*a^3*b^9*c^4*d^3*g^2*i^3*n - 378*(b*x + a)*B*a^2*b^9*c^5
*d^3*g^2*i^3*n/(d*x + c) - 441*(b*x + a)^2*B*a*b^9*c^6*d^3*g^2*i^3*n/(d*x + c)^2 - 74*(b*x + a)^3*B*b^9*c^7*d^
3*g^2*i^3*n/(d*x + c)^3 + 70*B*a^4*b^8*c^3*d^4*g^2*i^3*n + 630*(b*x + a)*B*a^3*b^8*c^4*d^4*g^2*i^3*n/(d*x + c)
+ 1323*(b*x + a)^2*B*a^2*b^8*c^5*d^4*g^2*i^3*n/(d*x + c)^2 + 518*(b*x + a)^3*B*a*b^8*c^6*d^4*g^2*i^3*n/(d*x +
c)^3 + 33*(b*x + a)^4*B*b^8*c^7*d^4*g^2*i^3*n/(d*x + c)^4 - 42*B*a^5*b^7*c^2*d^5*g^2*i^3*n - 630*(b*x + a)*B*
a^4*b^7*c^3*d^5*g^2*i^3*n/(d*x + c) - 2205*(b*x + a)^2*B*a^3*b^7*c^4*d^5*g^2*i^3*n/(d*x + c)^2 - 1554*(b*x + a
)^3*B*a^2*b^7*c^5*d^5*g^2*i^3*n/(d*x + c)^3 - 231*(b*x + a)^4*B*a*b^7*c^6*d^5*g^2*i^3*n/(d*x + c)^4 - 6*(b*x +
a)^5*B*b^7*c^7*d^5*g^2*i^3*n/(d*x + c)^5 + 14*B*a^6*b^6*c*d^6*g^2*i^3*n + 378*(b*x + a)*B*a^5*b^6*c^2*d^6*g^2
*i^3*n/(d*x + c) + 2205*(b*x + a)^2*B*a^4*b^6*c^3*d^6*g^2*i^3*n/(d*x + c)^2 + 2590*(b*x + a)^3*B*a^3*b^6*c^4*d
^6*g^2*i^3*n/(d*x + c)^3 + 693*(b*x + a)^4*B*a^2*b^6*c^5*d^6*g^2*i^3*n/(d*x + c)^4 + 42*(b*x + a)^5*B*a*b^6*c^
6*d^6*g^2*i^3*n/(d*x + c)^5 - 2*B*a^7*b^5*d^7*g^2*i^3*n - 126*(b*x + a)*B*a^6*b^5*c*d^7*g^2*i^3*n/(d*x + c) -
1323*(b*x + a)^2*B*a^5*b^5*c^2*d^7*g^2*i^3*n/(d*x + c)^2 - 2590*(b*x + a)^3*B*a^4*b^5*c^3*d^7*g^2*i^3*n/(d*x +
c)^3 - 1155*(b*x + a)^4*B*a^3*b^5*c^4*d^7*g^2*i^3*n/(d*x + c)^4 - 126*(b*x + a)^5*B*a^2*b^5*c^5*d^7*g^2*i^3*n
/(d*x + c)^5 + 18*(b*x + a)*B*a^7*b^4*d^8*g^2*i^3*n/(d*x + c) + 441*(b*x + a)^2*B*a^6*b^4*c*d^8*g^2*i^3*n/(d*x
+ c)^2 + 1554*(b*x + a)^3*B*a^5*b^4*c^2*d^8*g^2*i^3*n/(d*x + c)^3 + 1155*(b*x + a)^4*B*a^4*b^4*c^3*d^8*g^2*i^
3*n/(d*x + c)^4 + 210*(b*x + a)^5*B*a^3*b^4*c^4*d^8*g^2*i^3*n/(d*x + c)^5 - 63*(b*x + a)^2*B*a^7*b^3*d^9*g^2*i
^3*n/(d*x + c)^2 - 518*(b*x + a)^3*B*a^6*b^3*c*d^9*g^2*i^3*n/(d*x + c)^3 - 693*(b*x + a)^4*B*a^5*b^3*c^2*d^9*g
^2*i^3*n/(d*x + c)^4 - 210*(b*x + a)^5*B*a^4*b^3*c^3*d^9*g^2*i^3*n/(d*x + c)^5 + 74*(b*x + a)^3*B*a^7*b^2*d^10
*g^2*i^3*n/(d*x + c)^3 + 231*(b*x + a)^4*B*a^6*b^2*c*d^10*g^2*i^3*n/(d*x + c)^4 + 126*(b*x + a)^5*B*a^5*b^2*c^
2*d^10*g^2*i^3*n/(d*x + c)^5 - 33*(b*x + a)^4*B*a^7*b*d^11*g^2*i^3*n/(d*x + c)^4 - 42*(b*x + a)^5*B*a^6*b*c*d^
11*g^2*i^3*n/(d*x + c)^5 + 6*(b*x + a)^5*B*a^7*d^12*g^2*i^3*n/(d*x + c)^5 - 6*B*b^12*c^7*g^2*i^3*log(e) + 42*B
*a*b^11*c^6*d*g^2*i^3*log(e) + 36*(b*x + a)*B*b^11*c^7*d*g^2*i^3*log(e)/(d*x + c) - 126*B*a^2*b^10*c^5*d^2*g^2
*i^3*log(e) - 252*(b*x + a)*B*a*b^10*c^6*d^2*g^2*i^3*log(e)/(d*x + c) - 90*(b*x + a)^2*B*b^10*c^7*d^2*g^2*i^3*
log(e)/(d*x + c)^2 + 210*B*a^3*b^9*c^4*d^3*g^2*i^3*log(e) + 756*(b*x + a)*B*a^2*b^9*c^5*d^3*g^2*i^3*log(e)/(d*
x + c) + 630*(b*x + a)^2*B*a*b^9*c^6*d^3*g^2*i^3*log(e)/(d*x + c)^2 - 210*B*a^4*b^8*c^3*d^4*g^2*i^3*log(e) - 1
260*(b*x + a)*B*a^3*b^8*c^4*d^4*g^2*i^3*log(e)/(d*x + c) - 1890*(b*x + a)^2*B*a^2*b^8*c^5*d^4*g^2*i^3*log(e)/(
d*x + c)^2 + 126*B*a^5*b^7*c^2*d^5*g^2*i^3*log(e) + 1260*(b*x + a)*B*a^4*b^7*c^3*d^5*g^2*i^3*log(e)/(d*x + c)
+ 3150*(b*x + a)^2*B*a^3*b^7*c^4*d^5*g^2*i^3*log(e)/(d*x + c)^2 - 42*B*a^6*b^6*c*d^6*g^2*i^3*log(e) - 756*(b*x
+ a)*B*a^5*b^6*c^2*d^6*g^2*i^3*log(e)/(d*x + c) - 3150*(b*x + a)^2*B*a^4*b^6*c^3*d^6*g^2*i^3*log(e)/(d*x + c)
^2 + 6*B*a^7*b^5*d^7*g^2*i^3*log(e) + 252*(b*x + a)*B*a^6*b^5*c*d^7*g^2*i^3*log(e)/(d*x + c) + 1890*(b*x + a)^
2*B*a^5*b^5*c^2*d^7*g^2*i^3*log(e)/(d*x + c)^2 - 36*(b*x + a)*B*a^7*b^4*d^8*g^2*i^3*log(e)/(d*x + c) - 630*(b*
x + a)^2*B*a^6*b^4*c*d^8*g^2*i^3*log(e)/(d*x + c)^2 + 90*(b*x + a)^2*B*a^7*b^3*d^9*g^2*i^3*log(e)/(d*x + c)^2
- 6*A*b^12*c^7*g^2*i^3 + 42*A*a*b^11*c^6*d*g^2*i^3 + 36*(b*x + a)*A*b^11*c^7*d*g^2*i^3/(d*x + c) - 126*A*a^2*b
^10*c^5*d^2*g^2*i^3 - 252*(b*x + a)*A*a*b^10*c^6*d^2*g^2*i^3/(d*x + c) - 90*(b*x + a)^2*A*b^10*c^7*d^2*g^2*i^3
/(d*x + c)^2 + 210*A*a^3*b^9*c^4*d^3*g^2*i^3 + 756*(b*x + a)*A*a^2*b^9*c^5*d^3*g^2*i^3/(d*x + c) + 630*(b*x +
a)^2*A*a*b^9*c^6*d^3*g^2*i^3/(d*x + c)^2 - 210*A*a^4*b^8*c^3*d^4*g^2*i^3 - 1260*(b*x + a)*A*a^3*b^8*c^4*d^4*g^
2*i^3/(d*x + c) - 1890*(b*x + a)^2*A*a^2*b^8*c^5*d^4*g^2*i^3/(d*x + c)^2 + 126*A*a^5*b^7*c^2*d^5*g^2*i^3 + 126
0*(b*x + a)*A*a^4*b^7*c^3*d^5*g^2*i^3/(d*x + c) + 3150*(b*x + a)^2*A*a^3*b^7*c^4*d^5*g^2*i^3/(d*x + c)^2 - 42*
A*a^6*b^6*c*d^6*g^2*i^3 - 756*(b*x + a)*A*a^5*b^6*c^2*d^6*g^2*i^3/(d*x + c) - 3150*(b*x + a)^2*A*a^4*b^6*c^3*d
^6*g^2*i^3/(d*x + c)^2 + 6*A*a^7*b^5*d^7*g^2*i^3 + 252*(b*x + a)*A*a^6*b^5*c*d^7*g^2*i^3/(d*x + c) + 1890*(b*x
+ a)^2*A*a^5*b^5*c^2*d^7*g^2*i^3/(d*x + c)^2 - 36*(b*x + a)*A*a^7*b^4*d^8*g^2*i^3/(d*x + c) - 630*(b*x + a)^2
*A*a^6*b^4*c*d^8*g^2*i^3/(d*x + c)^2 + 90*(b*x + a)^2*A*a^7*b^3*d^9*g^2*i^3/(d*x + c)^2)/(b^9*d^3 - 6*(b*x + a
)*b^8*d^4/(d*x + c) + 15*(b*x + a)^2*b^7*d^5/(d*x + c)^2 - 20*(b*x + a)^3*b^6*d^6/(d*x + c)^3 + 15*(b*x + a)^4
*b^5*d^7/(d*x + c)^4 - 6*(b*x + a)^5*b^4*d^8/(d*x + c)^5 + (b*x + a)^6*b^3*d^9/(d*x + c)^6) + 6*(B*b^7*c^7*g^2
*i^3*n - 7*B*a*b^6*c^6*d*g^2*i^3*n + 21*B*a^2*b^5*c^5*d^2*g^2*i^3*n - 35*B*a^3*b^4*c^4*d^3*g^2*i^3*n + 35*B*a^
4*b^3*c^3*d^4*g^2*i^3*n - 21*B*a^5*b^2*c^2*d^5*g^2*i^3*n + 7*B*a^6*b*c*d^6*g^2*i^3*n - B*a^7*d^7*g^2*i^3*n)*lo
g(-b + (b*x + a)*d/(d*x + c))/(b^4*d^3) - 6*(B*b^7*c^7*g^2*i^3*n - 7*B*a*b^6*c^6*d*g^2*i^3*n + 21*B*a^2*b^5*c^
5*d^2*g^2*i^3*n - 35*B*a^3*b^4*c^4*d^3*g^2*i^3*n + 35*B*a^4*b^3*c^3*d^4*g^2*i^3*n - 21*B*a^5*b^2*c^2*d^5*g^2*i
^3*n + 7*B*a^6*b*c*d^6*g^2*i^3*n - B*a^7*d^7*g^2*i^3*n)*log((b*x + a)/(d*x + c))/(b^4*d^3))*(b*c/(b*c - a*d)^2
- a*d/(b*c - a*d)^2)

Mupad [B] (verification not implemented)

Time = 2.83 (sec) , antiderivative size = 2547, normalized size of antiderivative = 6.58 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

x^2*((a*c*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))
/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 6
0*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(2*b*d) - ((60*a*d + 60*b*c)*((g^2*i^3*(4*A*a^3*d^3 + 16
*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b
*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^
2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a
^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(60*b*d) - (a*c*((b*d^2*g^2*
i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60))/(b*d)))/(120*b*d)
+ (c*g^2*i^3*(12*A*a^3*d^3 + 3*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^3*n + 36*A*a*b^2*c^2*d + 54*A*a^2*b*c*d^2 -
5*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d^2*n))/(6*b*d)) + x^3*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n -
3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(12*b) + ((60*a*
d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c
))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n +
60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(180*b*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c
+ B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60))/(3*b*d)) - x^4*((((b*d^2*g^2*i^3*(18*A*a*d
+ 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(240*b*d) - (d
*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/20 + (A*a
*b*c*d^2*g^2*i^3)/4) + x^5*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/30 - (A*b*d^2*g^2*i^3*(6
0*a*d + 60*b*c))/300) - x*(((60*a*d + 60*b*c)*((a*c*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n
))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b
^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(b*d) - ((60*a
*d + 60*b*c)*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2
*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24
*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*
i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^
2*g^2*i^3))/(60*b*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(6
0*a*d + 60*b*c))/60))/(b*d)))/(60*b*d) + (c*g^2*i^3*(12*A*a^3*d^3 + 3*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^3*n
+ 36*A*a*b^2*c^2*d + 54*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d^2*n))/(3*b*d)))/(60*b*d) + (a*c*((g^
2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*
b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n
- B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2
+ 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(60*b
*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/
60))/(b*d)))/(b*d) - (a*c^2*g^2*i^3*(12*A*a^2*d^2 + 6*A*b^2*c^2 + 3*B*a^2*d^2*n - 2*B*b^2*c^2*n + 24*A*a*b*c*d
- B*a*b*c*d*n))/(2*b*d)) + log(e*((a + b*x)/(c + d*x))^n)*(B*a^2*c^3*g^2*i^3*x + (B*c*g^2*i^3*x^3*(3*a^2*d^2
+ b^2*c^2 + 6*a*b*c*d))/3 + (B*d*g^2*i^3*x^4*(a^2*d^2 + 3*b^2*c^2 + 6*a*b*c*d))/4 + (B*b^2*d^3*g^2*i^3*x^6)/6
+ (B*a*c^2*g^2*i^3*x^2*(3*a*d + 2*b*c))/2 + (B*b*d^2*g^2*i^3*x^5*(2*a*d + 3*b*c))/5) - (log(a + b*x)*(B*a^6*d^
3*g^2*i^3*n - 20*B*a^3*b^3*c^3*g^2*i^3*n + 15*B*a^4*b^2*c^2*d*g^2*i^3*n - 6*B*a^5*b*c*d^2*g^2*i^3*n))/(60*b^4)
- (log(c + d*x)*(B*b^2*c^6*g^2*i^3*n + 15*B*a^2*c^4*d^2*g^2*i^3*n - 6*B*a*b*c^5*d*g^2*i^3*n))/(60*d^3) + (A*b
^2*d^3*g^2*i^3*x^6)/6