\(\int (a g+b g x)^2 (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [109]

Optimal result

Integrand size = 41, antiderivative size = 190 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {B (b c-a d)^3 g^2 i n x}{12 b d^2}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2}{24 b^2 d}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2}-\frac {B (b c-a d)^4 g^2 i n \log (c+d x)}{12 b^2 d^3} \]

[Out]

1/12*B*(-a*d+b*c)^3*g^2*i*n*x/b/d^2-1/24*B*(-a*d+b*c)^2*g^2*i*n*(b*x+a)^2/b^2/d+1/4*g^2*i*(b*x+a)^3*(d*x+c)*(A
+B*ln(e*((b*x+a)/(d*x+c))^n))/b+1/12*(-a*d+b*c)*g^2*i*(b*x+a)^3*(A-B*n+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2-1/12*B
*(-a*d+b*c)^4*g^2*i*n*ln(d*x+c)/b^2/d^3

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 190, 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} \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A-B n\right )}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b}-\frac {B g^2 i n (b c-a d)^4 \log (c+d x)}{12 b^2 d^3}-\frac {B g^2 i n (a+b x)^2 (b c-a d)^2}{24 b^2 d}+\frac {B g^2 i n x (b c-a d)^3}{12 b d^2} \]

[In]

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

[Out]

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

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^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b}+\frac {((b c-a d) i) \int (a g+b g x)^2 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{4 b} \\ & = \frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2}-\frac {\left (B (b c-a d)^2 i n\right ) \int \frac {(a g+b g x)^3}{(a+b x) (c+d x)} \, dx}{12 b^2 g} \\ & = \frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2}-\frac {\left (B (b c-a d)^2 g^2 i n\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{12 b^2} \\ & = \frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2}-\frac {\left (B (b c-a d)^2 g^2 i n\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}{12 b^2} \\ & = \frac {B (b c-a d)^3 g^2 i n x}{12 b d^2}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2}{24 b^2 d}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2}-\frac {B (b c-a d)^4 g^2 i n \log (c+d x)}{12 b^2 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.18 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^2 i \left (8 (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 d (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {4 B (b c-a d)^2 n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )}{d^3}-\frac {B (b c-a d) n \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )}{d^3}\right )}{24 b^2} \]

[In]

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

[Out]

(g^2*i*(8*(b*c - a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*d*(a + b*x)^4*(A + B*Log[e*((a +
b*x)/(c + d*x))^n]) + (4*B*(b*c - a*d)^2*n*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*
x]))/d^3 - (B*(b*c - a*d)*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^3))/(24*b^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(875\) vs. \(2(180)=360\).

Time = 5.00 (sec) , antiderivative size = 876, normalized size of antiderivative = 4.61

method	result	size
parallelrisch	\(\frac {2 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{4} g^{2} i n +6 A \,x^{4} b^{4} d^{4} g^{2} i n -2 B \ln \left (b x +a \right ) b^{4} c^{4} g^{2} i \,n^{2}-2 B \ln \left (b x +a \right ) a^{4} d^{4} g^{2} i \,n^{2}+24 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} c \,d^{3} g^{2} i n +24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{2} c \,d^{3} g^{2} i n +16 A \,x^{3} a \,b^{3} d^{4} g^{2} i n +8 A \,x^{3} b^{4} c \,d^{3} g^{2} i n +5 B \,x^{2} a^{2} b^{2} d^{4} g^{2} i \,n^{2}-B \,x^{2} b^{4} c^{2} d^{2} g^{2} i \,n^{2}+12 A \,x^{2} a^{2} b^{2} d^{4} g^{2} i n +2 B x \,a^{3} b \,d^{4} g^{2} i \,n^{2}+2 B x \,b^{4} c^{3} d \,g^{2} i \,n^{2}+6 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{4} g^{2} i n -12 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} g^{2} i \,n^{2}+8 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d \,g^{2} i \,n^{2}-8 B x a \,b^{3} c^{2} d^{2} g^{2} i \,n^{2}+24 A x \,a^{2} b^{2} c \,d^{3} g^{2} i n +8 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} g^{2} i \,n^{2}+12 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{2} c^{2} d^{2} g^{2} i n -8 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} c^{3} d \,g^{2} i n +16 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} d^{4} g^{2} i n +8 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{3} g^{2} i n +12 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{2} d^{4} g^{2} i n -4 B \,x^{2} a \,b^{3} c \,d^{3} g^{2} i \,n^{2}+24 A \,x^{2} a \,b^{3} c \,d^{3} g^{2} i n +4 B x \,a^{2} b^{2} c \,d^{3} g^{2} i \,n^{2}-11 B \,a^{3} b c \,d^{3} g^{2} i \,n^{2}+8 B \,a^{2} b^{2} c^{2} d^{2} g^{2} i \,n^{2}+7 B a \,b^{3} c^{3} d \,g^{2} i \,n^{2}-36 A \,a^{3} b c \,d^{3} g^{2} i n -48 A \,a^{2} b^{2} c^{2} d^{2} g^{2} i n +2 B \,x^{3} a \,b^{3} d^{4} g^{2} i \,n^{2}-2 B \,x^{3} b^{4} c \,d^{3} g^{2} i \,n^{2}-2 B \,b^{4} c^{4} g^{2} i \,n^{2}-2 B \,a^{4} d^{4} g^{2} i \,n^{2}}{24 b^{2} d^{3} n}\)	\(876\)

[In]

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

[Out]

1/24*(2*B*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^4*g^2*i*n+6*A*x^4*b^4*d^4*g^2*i*n-2*B*ln(b*x+a)*b^4*c^4*g^2*i*n^2-2*
B*ln(b*x+a)*a^4*d^4*g^2*i*n^2+24*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^3*c*d^3*g^2*i*n+24*B*x*ln(e*((b*x+a)/(d*x
+c))^n)*a^2*b^2*c*d^3*g^2*i*n+16*A*x^3*a*b^3*d^4*g^2*i*n+8*A*x^3*b^4*c*d^3*g^2*i*n+5*B*x^2*a^2*b^2*d^4*g^2*i*n
^2-B*x^2*b^4*c^2*d^2*g^2*i*n^2+12*A*x^2*a^2*b^2*d^4*g^2*i*n+2*B*x*a^3*b*d^4*g^2*i*n^2+2*B*x*b^4*c^3*d*g^2*i*n^
2+6*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^4*d^4*g^2*i*n-12*B*ln(b*x+a)*a^2*b^2*c^2*d^2*g^2*i*n^2+8*B*ln(b*x+a)*a*b
^3*c^3*d*g^2*i*n^2-8*B*x*a*b^3*c^2*d^2*g^2*i*n^2+24*A*x*a^2*b^2*c*d^3*g^2*i*n+8*B*ln(b*x+a)*a^3*b*c*d^3*g^2*i*
n^2+12*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c^2*d^2*g^2*i*n-8*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^3*c^3*d*g^2*i*n+1
6*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a*b^3*d^4*g^2*i*n+8*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c*d^3*g^2*i*n+12*B*x
^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*d^4*g^2*i*n-4*B*x^2*a*b^3*c*d^3*g^2*i*n^2+24*A*x^2*a*b^3*c*d^3*g^2*i*n+4*
B*x*a^2*b^2*c*d^3*g^2*i*n^2-11*B*a^3*b*c*d^3*g^2*i*n^2+8*B*a^2*b^2*c^2*d^2*g^2*i*n^2+7*B*a*b^3*c^3*d*g^2*i*n^2
-36*A*a^3*b*c*d^3*g^2*i*n-48*A*a^2*b^2*c^2*d^2*g^2*i*n+2*B*x^3*a*b^3*d^4*g^2*i*n^2-2*B*x^3*b^4*c*d^3*g^2*i*n^2
-2*B*b^4*c^4*g^2*i*n^2-2*B*a^4*d^4*g^2*i*n^2)/b^2/d^3/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (182) = 364\).

Time = 0.43 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.78 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} g^{2} i x^{4} + 2 \, {\left (4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} g^{2} i n \log \left (b x + a\right ) - 2 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} g^{2} i n \log \left (d x + c\right ) - 2 \, {\left ({\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{2} i n - 4 \, {\left (A b^{4} c d^{3} + 2 \, A a b^{3} d^{4}\right )} g^{2} i\right )} x^{3} - {\left ({\left (B b^{4} c^{2} d^{2} + 4 \, B a b^{3} c d^{3} - 5 \, B a^{2} b^{2} d^{4}\right )} g^{2} i n - 12 \, {\left (2 \, A a b^{3} c d^{3} + A a^{2} b^{2} d^{4}\right )} g^{2} i\right )} x^{2} + 2 \, {\left (12 \, A a^{2} b^{2} c d^{3} g^{2} i + {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 2 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g^{2} i n\right )} x + 2 \, {\left (3 \, B b^{4} d^{4} g^{2} i x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i x + 4 \, {\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i x^{3} + 6 \, {\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i x^{2}\right )} \log \left (e\right ) + 2 \, {\left (3 \, B b^{4} d^{4} g^{2} i n x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i n x + 4 \, {\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i n x^{3} + 6 \, {\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{24 \, b^{2} d^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1013 vs. \(2 (173) = 346\).

Time = 84.02 (sec) , antiderivative size = 1013, normalized size of antiderivative = 5.33 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\begin {cases} a^{2} c g^{2} i x \left (A + B \log {\left (e \left (\frac {a}{c}\right )^{n} \right )}\right ) & \text {for}\: b = 0 \wedge d = 0 \\a^{2} g^{2} \left (A c i x + \frac {A d i x^{2}}{2} + \frac {B c^{2} i \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )}}{2 d} + \frac {B c i n x}{2} + B c i x \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )} + \frac {B d i n x^{2}}{4} + \frac {B d i x^{2} \log {\left (e \left (\frac {a}{c + d x}\right )^{n} \right )}}{2}\right ) & \text {for}\: b = 0 \\c i \left (A a^{2} g^{2} x + A a b g^{2} x^{2} + \frac {A b^{2} g^{2} x^{3}}{3} + \frac {B a^{3} g^{2} \log {\left (e \left (\frac {a}{c} + \frac {b x}{c}\right )^{n} \right )}}{3 b} - \frac {B a^{2} g^{2} n x}{3} + B a^{2} g^{2} x \log {\left (e \left (\frac {a}{c} + \frac {b x}{c}\right )^{n} \right )} - \frac {B a b g^{2} n x^{2}}{3} + B a b g^{2} x^{2} \log {\left (e \left (\frac {a}{c} + \frac {b x}{c}\right )^{n} \right )} - \frac {B b^{2} g^{2} n x^{3}}{9} + \frac {B b^{2} g^{2} x^{3} \log {\left (e \left (\frac {a}{c} + \frac {b x}{c}\right )^{n} \right )}}{3}\right ) & \text {for}\: d = 0 \\A a^{2} c g^{2} i x + \frac {A a^{2} d g^{2} i x^{2}}{2} + A a b c g^{2} i x^{2} + \frac {2 A a b d g^{2} i x^{3}}{3} + \frac {A b^{2} c g^{2} i x^{3}}{3} + \frac {A b^{2} d g^{2} i x^{4}}{4} - \frac {B a^{4} d g^{2} i n \log {\left (\frac {c}{d} + x \right )}}{12 b^{2}} - \frac {B a^{4} d g^{2} i \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{12 b^{2}} + \frac {B a^{3} c g^{2} i n \log {\left (\frac {c}{d} + x \right )}}{3 b} + \frac {B a^{3} c g^{2} i \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{3 b} + \frac {B a^{3} d g^{2} i n x}{12 b} - \frac {B a^{2} c^{2} g^{2} i n \log {\left (\frac {c}{d} + x \right )}}{2 d} + \frac {B a^{2} c g^{2} i n x}{6} + B a^{2} c g^{2} i x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )} + \frac {5 B a^{2} d g^{2} i n x^{2}}{24} + \frac {B a^{2} d g^{2} i x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{2} + \frac {B a b c^{3} g^{2} i n \log {\left (\frac {c}{d} + x \right )}}{3 d^{2}} - \frac {B a b c^{2} g^{2} i n x}{3 d} - \frac {B a b c g^{2} i n x^{2}}{6} + B a b c g^{2} i x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )} + \frac {B a b d g^{2} i n x^{3}}{12} + \frac {2 B a b d g^{2} i x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{3} - \frac {B b^{2} c^{4} g^{2} i n \log {\left (\frac {c}{d} + x \right )}}{12 d^{3}} + \frac {B b^{2} c^{3} g^{2} i n x}{12 d^{2}} - \frac {B b^{2} c^{2} g^{2} i n x^{2}}{24 d} - \frac {B b^{2} c g^{2} i n x^{3}}{12} + \frac {B b^{2} c g^{2} i x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{3} + \frac {B b^{2} d g^{2} i x^{4} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{4} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*c*g**2*i*x*(A + B*log(e*(a/c)**n)), Eq(b, 0) & Eq(d, 0)), (a**2*g**2*(A*c*i*x + A*d*i*x**2/2 +
B*c**2*i*log(e*(a/(c + d*x))**n)/(2*d) + B*c*i*n*x/2 + B*c*i*x*log(e*(a/(c + d*x))**n) + B*d*i*n*x**2/4 + B*d
*i*x**2*log(e*(a/(c + d*x))**n)/2), Eq(b, 0)), (c*i*(A*a**2*g**2*x + A*a*b*g**2*x**2 + A*b**2*g**2*x**3/3 + B*
a**3*g**2*log(e*(a/c + b*x/c)**n)/(3*b) - B*a**2*g**2*n*x/3 + B*a**2*g**2*x*log(e*(a/c + b*x/c)**n) - B*a*b*g*
*2*n*x**2/3 + B*a*b*g**2*x**2*log(e*(a/c + b*x/c)**n) - B*b**2*g**2*n*x**3/9 + B*b**2*g**2*x**3*log(e*(a/c + b
*x/c)**n)/3), Eq(d, 0)), (A*a**2*c*g**2*i*x + A*a**2*d*g**2*i*x**2/2 + A*a*b*c*g**2*i*x**2 + 2*A*a*b*d*g**2*i*
x**3/3 + A*b**2*c*g**2*i*x**3/3 + A*b**2*d*g**2*i*x**4/4 - B*a**4*d*g**2*i*n*log(c/d + x)/(12*b**2) - B*a**4*d
*g**2*i*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(12*b**2) + B*a**3*c*g**2*i*n*log(c/d + x)/(3*b) + B*a**3*c*g*
*2*i*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(3*b) + B*a**3*d*g**2*i*n*x/(12*b) - B*a**2*c**2*g**2*i*n*log(c/d
+ x)/(2*d) + B*a**2*c*g**2*i*n*x/6 + B*a**2*c*g**2*i*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n) + 5*B*a**2*d*g
**2*i*n*x**2/24 + B*a**2*d*g**2*i*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/2 + B*a*b*c**3*g**2*i*n*log(c/d
+ x)/(3*d**2) - B*a*b*c**2*g**2*i*n*x/(3*d) - B*a*b*c*g**2*i*n*x**2/6 + B*a*b*c*g**2*i*x**2*log(e*(a/(c + d*x
) + b*x/(c + d*x))**n) + B*a*b*d*g**2*i*n*x**3/12 + 2*B*a*b*d*g**2*i*x**3*log(e*(a/(c + d*x) + b*x/(c + d*x))*
*n)/3 - B*b**2*c**4*g**2*i*n*log(c/d + x)/(12*d**3) + B*b**2*c**3*g**2*i*n*x/(12*d**2) - B*b**2*c**2*g**2*i*n*
x**2/(24*d) - B*b**2*c*g**2*i*n*x**3/12 + B*b**2*c*g**2*i*x**3*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/3 + B*b
**2*d*g**2*i*x**4*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (182) = 364\).

Time = 0.21 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.89 \[ \int (a g+b g x)^2 (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 {1}{4} \, B b^{2} d g^{2} i x^{4} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{4} \, A b^{2} d g^{2} i x^{4} + \frac {1}{3} \, B b^{2} c g^{2} i x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {2}{3} \, B a b d g^{2} i x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{3} \, A b^{2} c g^{2} i x^{3} + \frac {2}{3} \, A a b d g^{2} i x^{3} + B a b c g^{2} i x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{2} \, B a^{2} d g^{2} i x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a b c g^{2} i x^{2} + \frac {1}{2} \, A a^{2} d g^{2} i x^{2} - \frac {1}{24} \, B b^{2} d g^{2} i n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + \frac {1}{6} \, B b^{2} c g^{2} i n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} + \frac {1}{3} \, B a b d g^{2} i n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - B a b c g^{2} i n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} - \frac {1}{2} \, B a^{2} d g^{2} i n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B a^{2} c g^{2} i n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B a^{2} c g^{2} i x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a^{2} c g^{2} i x \]

[In]

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

[Out]

1/4*B*b^2*d*g^2*i*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b^2*d*g^2*i*x^4 + 1/3*B*b^2*c*g^2*i*x^3*l
og(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2/3*B*a*b*d*g^2*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*A*b
^2*c*g^2*i*x^3 + 2/3*A*a*b*d*g^2*i*x^3 + B*a*b*c*g^2*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*B*a^2*
d*g^2*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a*b*c*g^2*i*x^2 + 1/2*A*a^2*d*g^2*i*x^2 - 1/24*B*b^2*d*
g^2*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)) + 1/6*B*b^2*c*g^2*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)) + 1/3*B*a*b*d*g^2*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)) - B*a*b*
c*g^2*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^2*d*g^2*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^2*c*g^2*i*n*(a*log(b*x + a)/b - c*log(d*x + c)
/d) + B*a^2*c*g^2*i*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^2*c*g^2*i*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2571 vs. \(2 (182) = 364\).

Time = 1.32 (sec) , antiderivative size = 2571, normalized size of antiderivative = 13.53 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/24*(2*(B*b^7*c^5*g^2*i*n - 5*B*a*b^6*c^4*d*g^2*i*n - 4*(b*x + a)*B*b^6*c^5*d*g^2*i*n/(d*x + c) + 10*B*a^2*b^
5*c^3*d^2*g^2*i*n + 20*(b*x + a)*B*a*b^5*c^4*d^2*g^2*i*n/(d*x + c) + 6*(b*x + a)^2*B*b^5*c^5*d^2*g^2*i*n/(d*x
+ c)^2 - 10*B*a^3*b^4*c^2*d^3*g^2*i*n - 40*(b*x + a)*B*a^2*b^4*c^3*d^3*g^2*i*n/(d*x + c) - 30*(b*x + a)^2*B*a*
b^4*c^4*d^3*g^2*i*n/(d*x + c)^2 + 5*B*a^4*b^3*c*d^4*g^2*i*n + 40*(b*x + a)*B*a^3*b^3*c^2*d^4*g^2*i*n/(d*x + c)
+ 60*(b*x + a)^2*B*a^2*b^3*c^3*d^4*g^2*i*n/(d*x + c)^2 - B*a^5*b^2*d^5*g^2*i*n - 20*(b*x + a)*B*a^4*b^2*c*d^5
*g^2*i*n/(d*x + c) - 60*(b*x + a)^2*B*a^3*b^2*c^2*d^5*g^2*i*n/(d*x + c)^2 + 4*(b*x + a)*B*a^5*b*d^6*g^2*i*n/(d
*x + c) + 30*(b*x + a)^2*B*a^4*b*c*d^6*g^2*i*n/(d*x + c)^2 - 6*(b*x + a)^2*B*a^5*d^7*g^2*i*n/(d*x + c)^2)*log(
(b*x + a)/(d*x + c))/(b^4*d^3 - 4*(b*x + a)*b^3*d^4/(d*x + c) + 6*(b*x + a)^2*b^2*d^5/(d*x + c)^2 - 4*(b*x + a
)^3*b*d^6/(d*x + c)^3 + (b*x + a)^4*d^7/(d*x + c)^4) + (B*b^8*c^5*g^2*i*n - 5*B*a*b^7*c^4*d*g^2*i*n - 2*(b*x +
a)*B*b^7*c^5*d*g^2*i*n/(d*x + c) + 10*B*a^2*b^6*c^3*d^2*g^2*i*n + 10*(b*x + a)*B*a*b^6*c^4*d^2*g^2*i*n/(d*x +
c) - (b*x + a)^2*B*b^6*c^5*d^2*g^2*i*n/(d*x + c)^2 - 10*B*a^3*b^5*c^2*d^3*g^2*i*n - 20*(b*x + a)*B*a^2*b^5*c^
3*d^3*g^2*i*n/(d*x + c) + 5*(b*x + a)^2*B*a*b^5*c^4*d^3*g^2*i*n/(d*x + c)^2 + 2*(b*x + a)^3*B*b^5*c^5*d^3*g^2*
i*n/(d*x + c)^3 + 5*B*a^4*b^4*c*d^4*g^2*i*n + 20*(b*x + a)*B*a^3*b^4*c^2*d^4*g^2*i*n/(d*x + c) - 10*(b*x + a)^
2*B*a^2*b^4*c^3*d^4*g^2*i*n/(d*x + c)^2 - 10*(b*x + a)^3*B*a*b^4*c^4*d^4*g^2*i*n/(d*x + c)^3 - B*a^5*b^3*d^5*g
^2*i*n - 10*(b*x + a)*B*a^4*b^3*c*d^5*g^2*i*n/(d*x + c) + 10*(b*x + a)^2*B*a^3*b^3*c^2*d^5*g^2*i*n/(d*x + c)^2
+ 20*(b*x + a)^3*B*a^2*b^3*c^3*d^5*g^2*i*n/(d*x + c)^3 + 2*(b*x + a)*B*a^5*b^2*d^6*g^2*i*n/(d*x + c) - 5*(b*x
+ a)^2*B*a^4*b^2*c*d^6*g^2*i*n/(d*x + c)^2 - 20*(b*x + a)^3*B*a^3*b^2*c^2*d^6*g^2*i*n/(d*x + c)^3 + (b*x + a)
^2*B*a^5*b*d^7*g^2*i*n/(d*x + c)^2 + 10*(b*x + a)^3*B*a^4*b*c*d^7*g^2*i*n/(d*x + c)^3 - 2*(b*x + a)^3*B*a^5*d^
8*g^2*i*n/(d*x + c)^3 + 2*B*b^8*c^5*g^2*i*log(e) - 10*B*a*b^7*c^4*d*g^2*i*log(e) - 8*(b*x + a)*B*b^7*c^5*d*g^2
*i*log(e)/(d*x + c) + 20*B*a^2*b^6*c^3*d^2*g^2*i*log(e) + 40*(b*x + a)*B*a*b^6*c^4*d^2*g^2*i*log(e)/(d*x + c)
+ 12*(b*x + a)^2*B*b^6*c^5*d^2*g^2*i*log(e)/(d*x + c)^2 - 20*B*a^3*b^5*c^2*d^3*g^2*i*log(e) - 80*(b*x + a)*B*a
^2*b^5*c^3*d^3*g^2*i*log(e)/(d*x + c) - 60*(b*x + a)^2*B*a*b^5*c^4*d^3*g^2*i*log(e)/(d*x + c)^2 + 10*B*a^4*b^4
*c*d^4*g^2*i*log(e) + 80*(b*x + a)*B*a^3*b^4*c^2*d^4*g^2*i*log(e)/(d*x + c) + 120*(b*x + a)^2*B*a^2*b^4*c^3*d^
4*g^2*i*log(e)/(d*x + c)^2 - 2*B*a^5*b^3*d^5*g^2*i*log(e) - 40*(b*x + a)*B*a^4*b^3*c*d^5*g^2*i*log(e)/(d*x + c
) - 120*(b*x + a)^2*B*a^3*b^3*c^2*d^5*g^2*i*log(e)/(d*x + c)^2 + 8*(b*x + a)*B*a^5*b^2*d^6*g^2*i*log(e)/(d*x +
c) + 60*(b*x + a)^2*B*a^4*b^2*c*d^6*g^2*i*log(e)/(d*x + c)^2 - 12*(b*x + a)^2*B*a^5*b*d^7*g^2*i*log(e)/(d*x +
c)^2 + 2*A*b^8*c^5*g^2*i - 10*A*a*b^7*c^4*d*g^2*i - 8*(b*x + a)*A*b^7*c^5*d*g^2*i/(d*x + c) + 20*A*a^2*b^6*c^
3*d^2*g^2*i + 40*(b*x + a)*A*a*b^6*c^4*d^2*g^2*i/(d*x + c) + 12*(b*x + a)^2*A*b^6*c^5*d^2*g^2*i/(d*x + c)^2 -
20*A*a^3*b^5*c^2*d^3*g^2*i - 80*(b*x + a)*A*a^2*b^5*c^3*d^3*g^2*i/(d*x + c) - 60*(b*x + a)^2*A*a*b^5*c^4*d^3*g
^2*i/(d*x + c)^2 + 10*A*a^4*b^4*c*d^4*g^2*i + 80*(b*x + a)*A*a^3*b^4*c^2*d^4*g^2*i/(d*x + c) + 120*(b*x + a)^2
*A*a^2*b^4*c^3*d^4*g^2*i/(d*x + c)^2 - 2*A*a^5*b^3*d^5*g^2*i - 40*(b*x + a)*A*a^4*b^3*c*d^5*g^2*i/(d*x + c) -
120*(b*x + a)^2*A*a^3*b^3*c^2*d^5*g^2*i/(d*x + c)^2 + 8*(b*x + a)*A*a^5*b^2*d^6*g^2*i/(d*x + c) + 60*(b*x + a)
^2*A*a^4*b^2*c*d^6*g^2*i/(d*x + c)^2 - 12*(b*x + a)^2*A*a^5*b*d^7*g^2*i/(d*x + c)^2)/(b^5*d^3 - 4*(b*x + a)*b^
4*d^4/(d*x + c) + 6*(b*x + a)^2*b^3*d^5/(d*x + c)^2 - 4*(b*x + a)^3*b^2*d^6/(d*x + c)^3 + (b*x + a)^4*b*d^7/(d
*x + c)^4) + 2*(B*b^5*c^5*g^2*i*n - 5*B*a*b^4*c^4*d*g^2*i*n + 10*B*a^2*b^3*c^3*d^2*g^2*i*n - 10*B*a^3*b^2*c^2*
d^3*g^2*i*n + 5*B*a^4*b*c*d^4*g^2*i*n - B*a^5*d^5*g^2*i*n)*log(-b + (b*x + a)*d/(d*x + c))/(b^2*d^3) - 2*(B*b^
5*c^5*g^2*i*n - 5*B*a*b^4*c^4*d*g^2*i*n + 10*B*a^2*b^3*c^3*d^2*g^2*i*n - 10*B*a^3*b^2*c^2*d^3*g^2*i*n + 5*B*a^
4*b*c*d^4*g^2*i*n - B*a^5*d^5*g^2*i*n)*log((b*x + a)/(d*x + c))/(b^2*d^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d
)^2)

Mupad [B] (verification not implemented)

Time = 1.81 (sec) , antiderivative size = 663, normalized size of antiderivative = 3.49 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,a^2\,c\,g^2\,i\,x+\frac {B\,a\,g^2\,i\,x^2\,\left (a\,d+2\,b\,c\right )}{2}+\frac {B\,b\,g^2\,i\,x^3\,\left (2\,a\,d+b\,c\right )}{3}+\frac {B\,b^2\,d\,g^2\,i\,x^4}{4}\right )+x^3\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{36}\right )+x\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{12\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\,n\right )}{3\,d}+A\,a\,b\,c\,g^2\,i\right )}{12\,b\,d}-\frac {a\,c\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{b\,d}+\frac {a\,g^2\,i\,\left (2\,A\,a^2\,d^2+6\,A\,b^2\,c^2+B\,a^2\,d^2\,n-2\,B\,b^2\,c^2\,n+12\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\,n\right )}{2\,b\,d}\right )-x^2\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{24\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\,n\right )}{6\,d}+\frac {A\,a\,b\,c\,g^2\,i}{2}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4\,d\,g^2\,i\,n-4\,B\,a^3\,b\,c\,g^2\,i\,n\right )}{12\,b^2}-\frac {\ln \left (c+d\,x\right )\,\left (6\,B\,i\,n\,a^2\,c^2\,d^2\,g^2-4\,B\,i\,n\,a\,b\,c^3\,d\,g^2+B\,i\,n\,b^2\,c^4\,g^2\right )}{12\,d^3}+\frac {A\,b^2\,d\,g^2\,i\,x^4}{4} \]

[In]

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

[Out]

log(e*((a + b*x)/(c + d*x))^n)*(B*a^2*c*g^2*i*x + (B*a*g^2*i*x^2*(a*d + 2*b*c))/2 + (B*b*g^2*i*x^3*(2*a*d + b*
c))/3 + (B*b^2*d*g^2*i*x^4)/4) + x^3*((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d*n - B*b*c*n))/12 - (A*b*g^2*i*(12*a
*d + 12*b*c))/36) + x*(((12*a*d + 12*b*c)*(((12*a*d + 12*b*c)*((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d*n - B*b*c*
n))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12))/(12*b*d) - (g^2*i*(9*A*a^2*d^2 + 3*A*b^2*c^2 + 2*B*a^2*d^2*n - B*b^
2*c^2*n + 18*A*a*b*c*d - B*a*b*c*d*n))/(3*d) + A*a*b*c*g^2*i))/(12*b*d) - (a*c*((b*g^2*i*(12*A*a*d + 8*A*b*c +
B*a*d*n - B*b*c*n))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12))/(b*d) + (a*g^2*i*(2*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^
2*d^2*n - 2*B*b^2*c^2*n + 12*A*a*b*c*d + B*a*b*c*d*n))/(2*b*d)) - x^2*(((12*a*d + 12*b*c)*((b*g^2*i*(12*A*a*d
+ 8*A*b*c + B*a*d*n - B*b*c*n))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12))/(24*b*d) - (g^2*i*(9*A*a^2*d^2 + 3*A*b^
2*c^2 + 2*B*a^2*d^2*n - B*b^2*c^2*n + 18*A*a*b*c*d - B*a*b*c*d*n))/(6*d) + (A*a*b*c*g^2*i)/2) - (log(a + b*x)*
(B*a^4*d*g^2*i*n - 4*B*a^3*b*c*g^2*i*n))/(12*b^2) - (log(c + d*x)*(B*b^2*c^4*g^2*i*n + 6*B*a^2*c^2*d^2*g^2*i*n
- 4*B*a*b*c^3*d*g^2*i*n))/(12*d^3) + (A*b^2*d*g^2*i*x^4)/4