\(\int (a g+b g x)^3 (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [108]

Optimal result

Integrand size = 41, antiderivative size = 223 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^4 g^3 i n x}{20 b d^3}+\frac {B (b c-a d)^3 g^3 i n (a+b x)^2}{40 b^2 d^2}-\frac {B (b c-a d)^2 g^3 i n (a+b x)^3}{60 b^2 d}+\frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}+\frac {(b c-a d) g^3 i (a+b x)^4 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^2}+\frac {B (b c-a d)^5 g^3 i n \log (c+d x)}{20 b^2 d^4} \]

[Out]

-1/20*B*(-a*d+b*c)^4*g^3*i*n*x/b/d^3+1/40*B*(-a*d+b*c)^3*g^3*i*n*(b*x+a)^2/b^2/d^2-1/60*B*(-a*d+b*c)^2*g^3*i*n
*(b*x+a)^3/b^2/d+1/5*g^3*i*(b*x+a)^4*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b+1/20*(-a*d+b*c)*g^3*i*(b*x+a)^4
*(A-B*n+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2+1/20*B*(-a*d+b*c)^5*g^3*i*n*ln(d*x+c)/b^2/d^4

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2559, 2547, 21, 45} \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A-B n\right )}{20 b^2}+\frac {g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}+\frac {B g^3 i n (b c-a d)^5 \log (c+d x)}{20 b^2 d^4}+\frac {B g^3 i n (a+b x)^2 (b c-a d)^3}{40 b^2 d^2}-\frac {B g^3 i n (a+b x)^3 (b c-a d)^2}{60 b^2 d}-\frac {B g^3 i n x (b c-a d)^4}{20 b d^3} \]

[In]

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

[Out]

-1/20*(B*(b*c - a*d)^4*g^3*i*n*x)/(b*d^3) + (B*(b*c - a*d)^3*g^3*i*n*(a + b*x)^2)/(40*b^2*d^2) - (B*(b*c - a*d
)^2*g^3*i*n*(a + b*x)^3)/(60*b^2*d) + (g^3*i*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*
b) + ((b*c - a*d)*g^3*i*(a + b*x)^4*(A - B*n + B*Log[e*((a + b*x)/(c + d*x))^n]))/(20*b^2) + (B*(b*c - a*d)^5*
g^3*i*n*Log[c + d*x])/(20*b^2*d^4)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 2547

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x
_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 1))), x] - Dist[B*n*((b*c -
a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, m
, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, -2]

Rule 2559

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

Rubi steps \begin{align*} \text {integral}& = \frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}+\frac {((b c-a d) i) \int (a g+b g x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b} \\ & = \frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}+\frac {(b c-a d) g^3 i (a+b x)^4 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^2}-\frac {\left (B (b c-a d)^2 i n\right ) \int \frac {(a g+b g x)^4}{(a+b x) (c+d x)} \, dx}{20 b^2 g} \\ & = \frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}+\frac {(b c-a d) g^3 i (a+b x)^4 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^2}-\frac {\left (B (b c-a d)^2 g^3 i n\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{20 b^2} \\ & = \frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}+\frac {(b c-a d) g^3 i (a+b x)^4 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^2}-\frac {\left (B (b c-a d)^2 g^3 i n\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{20 b^2} \\ & = -\frac {B (b c-a d)^4 g^3 i n x}{20 b d^3}+\frac {B (b c-a d)^3 g^3 i n (a+b x)^2}{40 b^2 d^2}-\frac {B (b c-a d)^2 g^3 i n (a+b x)^3}{60 b^2 d}+\frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}+\frac {(b c-a d) g^3 i (a+b x)^4 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^2}+\frac {B (b c-a d)^5 g^3 i n \log (c+d x)}{20 b^2 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.21 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^3 i \left (30 (b c-a d) (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+24 d (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {5 B (b c-a d)^2 n \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )}{d^4}+\frac {2 B (b c-a d) n \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )}{d^4}\right )}{120 b^2} \]

[In]

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

[Out]

(g^3*i*(30*(b*c - a*d)*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*d*(a + b*x)^5*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]) - (5*B*(b*c - a*d)^2*n*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3
*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]))/d^4 + (2*B*(b*c - a*d)*n*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c -
a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a + b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]))/d^4))
/(120*b^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1194\) vs. \(2(211)=422\).

Time = 11.44 (sec) , antiderivative size = 1195, normalized size of antiderivative = 5.36

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1195\)

[In]

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

[Out]

1/120*(22*B*x^3*a^2*b^3*d^5*g^3*i*n^2-2*B*x^3*b^5*c^2*d^3*g^3*i*n^2+120*A*x^3*a^2*b^3*d^5*g^3*i*n+27*B*x^2*a^3
*b^2*d^5*g^3*i*n^2+3*B*x^2*b^5*c^3*d^2*g^3*i*n^2+60*A*x^2*a^3*b^2*d^5*g^3*i*n+6*B*x*a^4*b*d^5*g^3*i*n^2-6*B*x*
b^5*c^4*d*g^3*i*n^2+24*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^5*d^5*g^3*i*n+6*B*x^4*a*b^4*d^5*g^3*i*n^2-6*B*x^4*b^5
*c*d^4*g^3*i*n^2+90*A*x^4*a*b^4*d^5*g^3*i*n+30*A*x^4*b^5*c*d^4*g^3*i*n+120*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a*b
^4*c*d^4*g^3*i*n+180*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*c*d^4*g^3*i*n+120*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a
^3*b^2*c*d^4*g^3*i*n-6*B*ln(b*x+a)*a^5*d^5*g^3*i*n^2+24*A*x^5*b^5*d^5*g^3*i*n-6*B*ln(e*((b*x+a)/(d*x+c))^n)*b^
5*c^5*g^3*i*n+6*B*ln(b*x+a)*b^5*c^5*g^3*i*n^2+30*B*x*a^3*b^2*c*d^4*g^3*i*n^2-60*B*x*a^2*b^3*c^2*d^3*g^3*i*n^2+
30*B*x*a*b^4*c^3*d^2*g^3*i*n^2+120*A*x*a^3*b^2*c*d^4*g^3*i*n+60*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^2*d^3*g^
3*i*n-60*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*c^3*d^2*g^3*i*n+30*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^4*c^4*d*g^3*i*
n+30*B*ln(b*x+a)*a^4*b*c*d^4*g^3*i*n^2-60*B*ln(b*x+a)*a^3*b^2*c^2*d^3*g^3*i*n^2+60*B*ln(b*x+a)*a^2*b^3*c^3*d^2
*g^3*i*n^2+90*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^4*d^5*g^3*i*n+30*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c*d^4*g
^3*i*n+120*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*d^5*g^3*i*n-20*B*x^3*a*b^4*c*d^4*g^3*i*n^2-30*B*ln(b*x+a)*a
*b^4*c^4*d*g^3*i*n^2+120*A*x^3*a*b^4*c*d^4*g^3*i*n+60*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*d^5*g^3*i*n-15*B
*x^2*a^2*b^3*c*d^4*g^3*i*n^2-15*B*x^2*a*b^4*c^2*d^3*g^3*i*n^2+180*A*x^2*a^2*b^3*c*d^4*g^3*i*n-63*B*a^4*b*c*d^4
*g^3*i*n^2+45*B*a^3*b^2*c^2*d^3*g^3*i*n^2+45*B*a^2*b^3*c^3*d^2*g^3*i*n^2-27*B*a*b^4*c^4*d*g^3*i*n^2-180*A*a^4*
b*c*d^4*g^3*i*n-300*A*a^3*b^2*c^2*d^3*g^3*i*n-6*B*a^5*d^5*g^3*i*n^2+6*B*b^5*c^5*g^3*i*n^2)/d^4/n/b^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (213) = 426\).

Time = 0.48 (sec) , antiderivative size = 720, normalized size of antiderivative = 3.23 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {24 \, A b^{5} d^{5} g^{3} i x^{5} + 6 \, {\left (5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g^{3} i n \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3}\right )} g^{3} i n \log \left (d x + c\right ) - 6 \, {\left ({\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{3} i n - 5 \, {\left (A b^{5} c d^{4} + 3 \, A a b^{4} d^{5}\right )} g^{3} i\right )} x^{4} - 2 \, {\left ({\left (B b^{5} c^{2} d^{3} + 10 \, B a b^{4} c d^{4} - 11 \, B a^{2} b^{3} d^{5}\right )} g^{3} i n - 60 \, {\left (A a b^{4} c d^{4} + A a^{2} b^{3} d^{5}\right )} g^{3} i\right )} x^{3} + 3 \, {\left ({\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} - 5 \, B a^{2} b^{3} c d^{4} + 9 \, B a^{3} b^{2} d^{5}\right )} g^{3} i n + 20 \, {\left (3 \, A a^{2} b^{3} c d^{4} + A a^{3} b^{2} d^{5}\right )} g^{3} i\right )} x^{2} + 6 \, {\left (20 \, A a^{3} b^{2} c d^{4} g^{3} i - {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{3} i n\right )} x + 6 \, {\left (4 \, B b^{5} d^{5} g^{3} i x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i x + 5 \, {\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i x^{4} + 20 \, {\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 10 \, {\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i x^{2}\right )} \log \left (e\right ) + 6 \, {\left (4 \, B b^{5} d^{5} g^{3} i n x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i n x + 5 \, {\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i n x^{4} + 20 \, {\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i n x^{3} + 10 \, {\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{120 \, b^{2} d^{4}} \]

[In]

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

[Out]

1/120*(24*A*b^5*d^5*g^3*i*x^5 + 6*(5*B*a^4*b*c*d^4 - B*a^5*d^5)*g^3*i*n*log(b*x + a) + 6*(B*b^5*c^5 - 5*B*a*b^
4*c^4*d + 10*B*a^2*b^3*c^3*d^2 - 10*B*a^3*b^2*c^2*d^3)*g^3*i*n*log(d*x + c) - 6*((B*b^5*c*d^4 - B*a*b^4*d^5)*g
^3*i*n - 5*(A*b^5*c*d^4 + 3*A*a*b^4*d^5)*g^3*i)*x^4 - 2*((B*b^5*c^2*d^3 + 10*B*a*b^4*c*d^4 - 11*B*a^2*b^3*d^5)
*g^3*i*n - 60*(A*a*b^4*c*d^4 + A*a^2*b^3*d^5)*g^3*i)*x^3 + 3*((B*b^5*c^3*d^2 - 5*B*a*b^4*c^2*d^3 - 5*B*a^2*b^3
*c*d^4 + 9*B*a^3*b^2*d^5)*g^3*i*n + 20*(3*A*a^2*b^3*c*d^4 + A*a^3*b^2*d^5)*g^3*i)*x^2 + 6*(20*A*a^3*b^2*c*d^4*
g^3*i - (B*b^5*c^4*d - 5*B*a*b^4*c^3*d^2 + 10*B*a^2*b^3*c^2*d^3 - 5*B*a^3*b^2*c*d^4 - B*a^4*b*d^5)*g^3*i*n)*x
+ 6*(4*B*b^5*d^5*g^3*i*x^5 + 20*B*a^3*b^2*c*d^4*g^3*i*x + 5*(B*b^5*c*d^4 + 3*B*a*b^4*d^5)*g^3*i*x^4 + 20*(B*a*
b^4*c*d^4 + B*a^2*b^3*d^5)*g^3*i*x^3 + 10*(3*B*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*g^3*i*x^2)*log(e) + 6*(4*B*b^5*d
^5*g^3*i*n*x^5 + 20*B*a^3*b^2*c*d^4*g^3*i*n*x + 5*(B*b^5*c*d^4 + 3*B*a*b^4*d^5)*g^3*i*n*x^4 + 20*(B*a*b^4*c*d^
4 + B*a^2*b^3*d^5)*g^3*i*n*x^3 + 10*(3*B*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*g^3*i*n*x^2)*log((b*x + a)/(d*x + c)))
/(b^2*d^4)

Sympy [F(-1)]

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

[In]

integrate((b*g*x+a*g)**3*(d*i*x+c*i)*(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. 1118 vs. \(2 (213) = 426\).

Time = 0.22 (sec) , antiderivative size = 1118, normalized size of antiderivative = 5.01 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/5*B*b^3*d*g^3*i*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A*b^3*d*g^3*i*x^5 + 1/4*B*b^3*c*g^3*i*x^4*l
og(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*B*a*b^2*d*g^3*i*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A
*b^3*c*g^3*i*x^4 + 3/4*A*a*b^2*d*g^3*i*x^4 + B*a*b^2*c*g^3*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + B*a^
2*b*d*g^3*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a*b^2*c*g^3*i*x^3 + A*a^2*b*d*g^3*i*x^3 + 3/2*B*a^2
*b*c*g^3*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*B*a^3*d*g^3*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x +
c))^n) + 3/2*A*a^2*b*c*g^3*i*x^2 + 1/2*A*a^3*d*g^3*i*x^2 + 1/60*B*b^3*d*g^3*i*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)) - 1/24*B*b^3*c*g^3*i*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)) - 1/8*B*a*b^2*d*g^3*i*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)) + 1/2*B*a*b^2*c*g^3*i*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*b*d*g^3*i*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)) - 3/2*B*a^2*b*c*g^3*i*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d))
- 1/2*B*a^3*d*g^3*i*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^3*c*g^3*i*n*(a
*log(b*x + a)/b - c*log(d*x + c)/d) + B*a^3*c*g^3*i*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^3*c*g^3*i*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3945 vs. \(2 (213) = 426\).

Time = 1.23 (sec) , antiderivative size = 3945, normalized size of antiderivative = 17.69 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/120*(6*(B*b^9*c^6*g^3*i*n - 6*B*a*b^8*c^5*d*g^3*i*n - 5*(b*x + a)*B*b^8*c^6*d*g^3*i*n/(d*x + c) + 15*B*a^2*
b^7*c^4*d^2*g^3*i*n + 30*(b*x + a)*B*a*b^7*c^5*d^2*g^3*i*n/(d*x + c) + 10*(b*x + a)^2*B*b^7*c^6*d^2*g^3*i*n/(d
*x + c)^2 - 20*B*a^3*b^6*c^3*d^3*g^3*i*n - 75*(b*x + a)*B*a^2*b^6*c^4*d^3*g^3*i*n/(d*x + c) - 60*(b*x + a)^2*B
*a*b^6*c^5*d^3*g^3*i*n/(d*x + c)^2 - 10*(b*x + a)^3*B*b^6*c^6*d^3*g^3*i*n/(d*x + c)^3 + 15*B*a^4*b^5*c^2*d^4*g
^3*i*n + 100*(b*x + a)*B*a^3*b^5*c^3*d^4*g^3*i*n/(d*x + c) + 150*(b*x + a)^2*B*a^2*b^5*c^4*d^4*g^3*i*n/(d*x +
c)^2 + 60*(b*x + a)^3*B*a*b^5*c^5*d^4*g^3*i*n/(d*x + c)^3 - 6*B*a^5*b^4*c*d^5*g^3*i*n - 75*(b*x + a)*B*a^4*b^4
*c^2*d^5*g^3*i*n/(d*x + c) - 200*(b*x + a)^2*B*a^3*b^4*c^3*d^5*g^3*i*n/(d*x + c)^2 - 150*(b*x + a)^3*B*a^2*b^4
*c^4*d^5*g^3*i*n/(d*x + c)^3 + B*a^6*b^3*d^6*g^3*i*n + 30*(b*x + a)*B*a^5*b^3*c*d^6*g^3*i*n/(d*x + c) + 150*(b
*x + a)^2*B*a^4*b^3*c^2*d^6*g^3*i*n/(d*x + c)^2 + 200*(b*x + a)^3*B*a^3*b^3*c^3*d^6*g^3*i*n/(d*x + c)^3 - 5*(b
*x + a)*B*a^6*b^2*d^7*g^3*i*n/(d*x + c) - 60*(b*x + a)^2*B*a^5*b^2*c*d^7*g^3*i*n/(d*x + c)^2 - 150*(b*x + a)^3
*B*a^4*b^2*c^2*d^7*g^3*i*n/(d*x + c)^3 + 10*(b*x + a)^2*B*a^6*b*d^8*g^3*i*n/(d*x + c)^2 + 60*(b*x + a)^3*B*a^5
*b*c*d^8*g^3*i*n/(d*x + c)^3 - 10*(b*x + a)^3*B*a^6*d^9*g^3*i*n/(d*x + c)^3)*log((b*x + a)/(d*x + c))/(b^5*d^4
- 5*(b*x + a)*b^4*d^5/(d*x + c) + 10*(b*x + a)^2*b^3*d^6/(d*x + c)^2 - 10*(b*x + a)^3*b^2*d^7/(d*x + c)^3 + 5
*(b*x + a)^4*b*d^8/(d*x + c)^4 - (b*x + a)^5*d^9/(d*x + c)^5) + (5*B*b^10*c^6*g^3*i*n - 30*B*a*b^9*c^5*d*g^3*i
*n - 19*(b*x + a)*B*b^9*c^6*d*g^3*i*n/(d*x + c) + 75*B*a^2*b^8*c^4*d^2*g^3*i*n + 114*(b*x + a)*B*a*b^8*c^5*d^2
*g^3*i*n/(d*x + c) + 23*(b*x + a)^2*B*b^8*c^6*d^2*g^3*i*n/(d*x + c)^2 - 100*B*a^3*b^7*c^3*d^3*g^3*i*n - 285*(b
*x + a)*B*a^2*b^7*c^4*d^3*g^3*i*n/(d*x + c) - 138*(b*x + a)^2*B*a*b^7*c^5*d^3*g^3*i*n/(d*x + c)^2 - 3*(b*x + a
)^3*B*b^7*c^6*d^3*g^3*i*n/(d*x + c)^3 + 75*B*a^4*b^6*c^2*d^4*g^3*i*n + 380*(b*x + a)*B*a^3*b^6*c^3*d^4*g^3*i*n
/(d*x + c) + 345*(b*x + a)^2*B*a^2*b^6*c^4*d^4*g^3*i*n/(d*x + c)^2 + 18*(b*x + a)^3*B*a*b^6*c^5*d^4*g^3*i*n/(d
*x + c)^3 - 6*(b*x + a)^4*B*b^6*c^6*d^4*g^3*i*n/(d*x + c)^4 - 30*B*a^5*b^5*c*d^5*g^3*i*n - 285*(b*x + a)*B*a^4
*b^5*c^2*d^5*g^3*i*n/(d*x + c) - 460*(b*x + a)^2*B*a^3*b^5*c^3*d^5*g^3*i*n/(d*x + c)^2 - 45*(b*x + a)^3*B*a^2*
b^5*c^4*d^5*g^3*i*n/(d*x + c)^3 + 36*(b*x + a)^4*B*a*b^5*c^5*d^5*g^3*i*n/(d*x + c)^4 + 5*B*a^6*b^4*d^6*g^3*i*n
+ 114*(b*x + a)*B*a^5*b^4*c*d^6*g^3*i*n/(d*x + c) + 345*(b*x + a)^2*B*a^4*b^4*c^2*d^6*g^3*i*n/(d*x + c)^2 + 6
0*(b*x + a)^3*B*a^3*b^4*c^3*d^6*g^3*i*n/(d*x + c)^3 - 90*(b*x + a)^4*B*a^2*b^4*c^4*d^6*g^3*i*n/(d*x + c)^4 - 1
9*(b*x + a)*B*a^6*b^3*d^7*g^3*i*n/(d*x + c) - 138*(b*x + a)^2*B*a^5*b^3*c*d^7*g^3*i*n/(d*x + c)^2 - 45*(b*x +
a)^3*B*a^4*b^3*c^2*d^7*g^3*i*n/(d*x + c)^3 + 120*(b*x + a)^4*B*a^3*b^3*c^3*d^7*g^3*i*n/(d*x + c)^4 + 23*(b*x +
a)^2*B*a^6*b^2*d^8*g^3*i*n/(d*x + c)^2 + 18*(b*x + a)^3*B*a^5*b^2*c*d^8*g^3*i*n/(d*x + c)^3 - 90*(b*x + a)^4*
B*a^4*b^2*c^2*d^8*g^3*i*n/(d*x + c)^4 - 3*(b*x + a)^3*B*a^6*b*d^9*g^3*i*n/(d*x + c)^3 + 36*(b*x + a)^4*B*a^5*b
*c*d^9*g^3*i*n/(d*x + c)^4 - 6*(b*x + a)^4*B*a^6*d^10*g^3*i*n/(d*x + c)^4 + 6*B*b^10*c^6*g^3*i*log(e) - 36*B*a
*b^9*c^5*d*g^3*i*log(e) - 30*(b*x + a)*B*b^9*c^6*d*g^3*i*log(e)/(d*x + c) + 90*B*a^2*b^8*c^4*d^2*g^3*i*log(e)
+ 180*(b*x + a)*B*a*b^8*c^5*d^2*g^3*i*log(e)/(d*x + c) + 60*(b*x + a)^2*B*b^8*c^6*d^2*g^3*i*log(e)/(d*x + c)^2
- 120*B*a^3*b^7*c^3*d^3*g^3*i*log(e) - 450*(b*x + a)*B*a^2*b^7*c^4*d^3*g^3*i*log(e)/(d*x + c) - 360*(b*x + a)
^2*B*a*b^7*c^5*d^3*g^3*i*log(e)/(d*x + c)^2 - 60*(b*x + a)^3*B*b^7*c^6*d^3*g^3*i*log(e)/(d*x + c)^3 + 90*B*a^4
*b^6*c^2*d^4*g^3*i*log(e) + 600*(b*x + a)*B*a^3*b^6*c^3*d^4*g^3*i*log(e)/(d*x + c) + 900*(b*x + a)^2*B*a^2*b^6
*c^4*d^4*g^3*i*log(e)/(d*x + c)^2 + 360*(b*x + a)^3*B*a*b^6*c^5*d^4*g^3*i*log(e)/(d*x + c)^3 - 36*B*a^5*b^5*c*
d^5*g^3*i*log(e) - 450*(b*x + a)*B*a^4*b^5*c^2*d^5*g^3*i*log(e)/(d*x + c) - 1200*(b*x + a)^2*B*a^3*b^5*c^3*d^5
*g^3*i*log(e)/(d*x + c)^2 - 900*(b*x + a)^3*B*a^2*b^5*c^4*d^5*g^3*i*log(e)/(d*x + c)^3 + 6*B*a^6*b^4*d^6*g^3*i
*log(e) + 180*(b*x + a)*B*a^5*b^4*c*d^6*g^3*i*log(e)/(d*x + c) + 900*(b*x + a)^2*B*a^4*b^4*c^2*d^6*g^3*i*log(e
)/(d*x + c)^2 + 1200*(b*x + a)^3*B*a^3*b^4*c^3*d^6*g^3*i*log(e)/(d*x + c)^3 - 30*(b*x + a)*B*a^6*b^3*d^7*g^3*i
*log(e)/(d*x + c) - 360*(b*x + a)^2*B*a^5*b^3*c*d^7*g^3*i*log(e)/(d*x + c)^2 - 900*(b*x + a)^3*B*a^4*b^3*c^2*d
^7*g^3*i*log(e)/(d*x + c)^3 + 60*(b*x + a)^2*B*a^6*b^2*d^8*g^3*i*log(e)/(d*x + c)^2 + 360*(b*x + a)^3*B*a^5*b^
2*c*d^8*g^3*i*log(e)/(d*x + c)^3 - 60*(b*x + a)^3*B*a^6*b*d^9*g^3*i*log(e)/(d*x + c)^3 + 6*A*b^10*c^6*g^3*i -
36*A*a*b^9*c^5*d*g^3*i - 30*(b*x + a)*A*b^9*c^6*d*g^3*i/(d*x + c) + 90*A*a^2*b^8*c^4*d^2*g^3*i + 180*(b*x + a)
*A*a*b^8*c^5*d^2*g^3*i/(d*x + c) + 60*(b*x + a)^2*A*b^8*c^6*d^2*g^3*i/(d*x + c)^2 - 120*A*a^3*b^7*c^3*d^3*g^3*
i - 450*(b*x + a)*A*a^2*b^7*c^4*d^3*g^3*i/(d*x + c) - 360*(b*x + a)^2*A*a*b^7*c^5*d^3*g^3*i/(d*x + c)^2 - 60*(
b*x + a)^3*A*b^7*c^6*d^3*g^3*i/(d*x + c)^3 + 90*A*a^4*b^6*c^2*d^4*g^3*i + 600*(b*x + a)*A*a^3*b^6*c^3*d^4*g^3*
i/(d*x + c) + 900*(b*x + a)^2*A*a^2*b^6*c^4*d^4*g^3*i/(d*x + c)^2 + 360*(b*x + a)^3*A*a*b^6*c^5*d^4*g^3*i/(d*x
+ c)^3 - 36*A*a^5*b^5*c*d^5*g^3*i - 450*(b*x + a)*A*a^4*b^5*c^2*d^5*g^3*i/(d*x + c) - 1200*(b*x + a)^2*A*a^3*
b^5*c^3*d^5*g^3*i/(d*x + c)^2 - 900*(b*x + a)^3*A*a^2*b^5*c^4*d^5*g^3*i/(d*x + c)^3 + 6*A*a^6*b^4*d^6*g^3*i +
180*(b*x + a)*A*a^5*b^4*c*d^6*g^3*i/(d*x + c) + 900*(b*x + a)^2*A*a^4*b^4*c^2*d^6*g^3*i/(d*x + c)^2 + 1200*(b*
x + a)^3*A*a^3*b^4*c^3*d^6*g^3*i/(d*x + c)^3 - 30*(b*x + a)*A*a^6*b^3*d^7*g^3*i/(d*x + c) - 360*(b*x + a)^2*A*
a^5*b^3*c*d^7*g^3*i/(d*x + c)^2 - 900*(b*x + a)^3*A*a^4*b^3*c^2*d^7*g^3*i/(d*x + c)^3 + 60*(b*x + a)^2*A*a^6*b
^2*d^8*g^3*i/(d*x + c)^2 + 360*(b*x + a)^3*A*a^5*b^2*c*d^8*g^3*i/(d*x + c)^3 - 60*(b*x + a)^3*A*a^6*b*d^9*g^3*
i/(d*x + c)^3)/(b^6*d^4 - 5*(b*x + a)*b^5*d^5/(d*x + c) + 10*(b*x + a)^2*b^4*d^6/(d*x + c)^2 - 10*(b*x + a)^3*
b^3*d^7/(d*x + c)^3 + 5*(b*x + a)^4*b^2*d^8/(d*x + c)^4 - (b*x + a)^5*b*d^9/(d*x + c)^5) + 6*(B*b^6*c^6*g^3*i*
n - 6*B*a*b^5*c^5*d*g^3*i*n + 15*B*a^2*b^4*c^4*d^2*g^3*i*n - 20*B*a^3*b^3*c^3*d^3*g^3*i*n + 15*B*a^4*b^2*c^2*d
^4*g^3*i*n - 6*B*a^5*b*c*d^5*g^3*i*n + B*a^6*d^6*g^3*i*n)*log(b - (b*x + a)*d/(d*x + c))/(b^2*d^4) - 6*(B*b^6*
c^6*g^3*i*n - 6*B*a*b^5*c^5*d*g^3*i*n + 15*B*a^2*b^4*c^4*d^2*g^3*i*n - 20*B*a^3*b^3*c^3*d^3*g^3*i*n + 15*B*a^4
*b^2*c^2*d^4*g^3*i*n - 6*B*a^5*b*c*d^5*g^3*i*n + B*a^6*d^6*g^3*i*n)*log((b*x + a)/(d*x + c))/(b^2*d^4))*(b*c/(
b*c - a*d)^2 - a*d/(b*c - a*d)^2)

Mupad [B] (verification not implemented)

Time = 2.41 (sec) , antiderivative size = 1237, normalized size of antiderivative = 5.55 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

x*((a*c*(((20*a*d + 20*b*c)*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d +
20*b*c))/20))/(20*b*d) - (b*g^3*i*(24*A*a^2*d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d - 2
*B*a*b*c*d*n))/(4*d) + A*a*b^2*c*g^3*i))/(b*d) - ((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((
b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(20*b*d) - (b*g^
3*i*(24*A*a^2*d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d - 2*B*a*b*c*d*n))/(4*d) + A*a*b^2
*c*g^3*i))/(20*b*d) - (a*c*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d + 2
0*b*c))/20))/(b*d) + (a*g^3*i*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 12*A*a*b*c*d))/d))/(20*
b*d) + (a^2*g^3*i*(2*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 16*A*a*b*c*d + 2*B*a*b*c*d*n))/(
2*b*d)) + x^2*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/5
- (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(20*b*d) - (b*g^3*i*(24*A*a^2*d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2*n - B*b^
2*c^2*n + 32*A*a*b*c*d - 2*B*a*b*c*d*n))/(4*d) + A*a*b^2*c*g^3*i))/(40*b*d) - (a*c*((b^2*g^3*i*(20*A*a*d + 10*
A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(2*b*d) + (a*g^3*i*(4*A*a^2*d^2 + 4*A*b^2
*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 12*A*a*b*c*d))/(2*d)) - x^3*(((20*a*d + 20*b*c)*((b^2*g^3*i*(20*A*a*d + 10*
A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(60*b*d) - (b*g^3*i*(24*A*a^2*d^2 + 4*A*b
^2*c^2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d - 2*B*a*b*c*d*n))/(12*d) + (A*a*b^2*c*g^3*i)/3) + log(e*((
a + b*x)/(c + d*x))^n)*((B*a^2*g^3*i*x^2*(a*d + 3*b*c))/2 + (B*b^2*g^3*i*x^4*(3*a*d + b*c))/4 + B*a^3*c*g^3*i*
x + (B*b^3*d*g^3*i*x^5)/5 + B*a*b*g^3*i*x^3*(a*d + b*c)) + x^4*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n - B*
b*c*n))/20 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/80) - (log(a + b*x)*(B*a^5*d*g^3*i*n - 5*B*a^4*b*c*g^3*i*n))/(20*
b^2) + (log(c + d*x)*(B*b^3*c^5*g^3*i*n - 10*B*a^3*c^2*d^3*g^3*i*n - 5*B*a*b^2*c^4*d*g^3*i*n + 10*B*a^2*b*c^3*
d^2*g^3*i*n))/(20*d^4) + (A*b^3*d*g^3*i*x^5)/5