\(\int (c g+d g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [30]

Optimal result

Integrand size = 33, antiderivative size = 156 \[ \int (c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^3 g^3 n x}{4 b^3}-\frac {B (b c-a d)^2 g^3 n (c+d x)^2}{8 b^2 d}-\frac {B (b c-a d) g^3 n (c+d x)^3}{12 b d}-\frac {B (b c-a d)^4 g^3 n \log (a+b x)}{4 b^4 d}+\frac {g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d} \]

[Out]

-1/4*B*(-a*d+b*c)^3*g^3*n*x/b^3-1/8*B*(-a*d+b*c)^2*g^3*n*(d*x+c)^2/b^2/d-1/12*B*(-a*d+b*c)*g^3*n*(d*x+c)^3/b/d
-1/4*B*(-a*d+b*c)^4*g^3*n*ln(b*x+a)/b^4/d+1/4*g^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2547, 21, 45} \[ \int (c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^3 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d}-\frac {B g^3 n (b c-a d)^4 \log (a+b x)}{4 b^4 d}-\frac {B g^3 n x (b c-a d)^3}{4 b^3}-\frac {B g^3 n (c+d x)^2 (b c-a d)^2}{8 b^2 d}-\frac {B g^3 n (c+d x)^3 (b c-a d)}{12 b d} \]

[In]

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

[Out]

-1/4*(B*(b*c - a*d)^3*g^3*n*x)/b^3 - (B*(b*c - a*d)^2*g^3*n*(c + d*x)^2)/(8*b^2*d) - (B*(b*c - a*d)*g^3*n*(c +
d*x)^3)/(12*b*d) - (B*(b*c - a*d)^4*g^3*n*Log[a + b*x])/(4*b^4*d) + (g^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/
(c + d*x))^n]))/(4*d)

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]

Rubi steps \begin{align*} \text {integral}& = \frac {g^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}-\frac {(B (b c-a d) n) \int \frac {(c g+d g x)^4}{(a+b x) (c+d x)} \, dx}{4 d g} \\ & = \frac {g^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}-\frac {\left (B (b c-a d) g^3 n\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{4 d} \\ & = \frac {g^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}-\frac {\left (B (b c-a d) g^3 n\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{4 d} \\ & = -\frac {B (b c-a d)^3 g^3 n x}{4 b^3}-\frac {B (b c-a d)^2 g^3 n (c+d x)^2}{8 b^2 d}-\frac {B (b c-a d) g^3 n (c+d x)^3}{12 b d}-\frac {B (b c-a d)^4 g^3 n \log (a+b x)}{4 b^4 d}+\frac {g^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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.79 \[ \int (c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^3 \left (-\frac {B (b c-a d) n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )}{6 b^4}+(c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{4 d} \]

[In]

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

[Out]

(g^3*(-1/6*(B*(b*c - a*d)*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]))/b^4 + (c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(4*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(651\) vs. \(2(146)=292\).

Time = 7.14 (sec) , antiderivative size = 652, normalized size of antiderivative = 4.18

method	result	size
parallelrisch	\(\frac {24 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{3} g^{3} n +6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{4} g^{3} n -6 B \ln \left (b x +a \right ) a^{4} d^{4} g^{3} n^{2}-6 B \ln \left (b x +a \right ) b^{4} c^{4} g^{3} n^{2}+24 A x \,b^{4} c^{3} d \,g^{3} n -24 A \,b^{4} c^{4} g^{3} n +21 B \,a^{3} b c \,d^{3} g^{3} n^{2}-24 B \,a^{2} b^{2} c^{2} d^{2} g^{3} n^{2}-9 B a \,b^{3} c^{3} d \,g^{3} n^{2}+6 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{4} g^{3} n +2 B \,x^{3} a \,b^{3} d^{4} g^{3} n^{2}-2 B \,x^{3} b^{4} c \,d^{3} g^{3} n^{2}-3 B \,x^{2} a^{2} b^{2} d^{4} g^{3} n^{2}-9 B \,x^{2} b^{4} c^{2} d^{2} g^{3} n^{2}+6 B x \,a^{3} b \,d^{4} g^{3} n^{2}-18 B x \,b^{4} c^{3} d \,g^{3} n^{2}+36 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{2} d^{2} g^{3} n +24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{3} d \,g^{3} n -60 A a \,b^{3} c^{3} d \,g^{3} n +24 A \,x^{3} b^{4} c \,d^{3} g^{3} n +36 A \,x^{2} b^{4} c^{2} d^{2} g^{3} n +24 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} g^{3} n^{2}-36 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} g^{3} n^{2}+24 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d \,g^{3} n^{2}+12 B \,x^{2} a \,b^{3} c \,d^{3} g^{3} n^{2}-24 B x \,a^{2} b^{2} c \,d^{3} g^{3} n^{2}+36 B x a \,b^{3} c^{2} d^{2} g^{3} n^{2}-6 B \,a^{4} d^{4} g^{3} n^{2}+18 B \,b^{4} c^{4} g^{3} n^{2}+6 A \,x^{4} b^{4} d^{4} g^{3} n}{24 b^{4} d n}\)	\(652\)

[In]

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

[Out]

1/24*(24*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c*d^3*g^3*n+6*B*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^4*g^3*n-6*B*ln(b*
x+a)*a^4*d^4*g^3*n^2-6*B*ln(b*x+a)*b^4*c^4*g^3*n^2+24*A*x*b^4*c^3*d*g^3*n-24*A*b^4*c^4*g^3*n+21*B*a^3*b*c*d^3*
g^3*n^2-24*B*a^2*b^2*c^2*d^2*g^3*n^2-9*B*a*b^3*c^3*d*g^3*n^2+6*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^4*d^4*g^3*n+2
*B*x^3*a*b^3*d^4*g^3*n^2-2*B*x^3*b^4*c*d^3*g^3*n^2-3*B*x^2*a^2*b^2*d^4*g^3*n^2-9*B*x^2*b^4*c^2*d^2*g^3*n^2+6*B
*x*a^3*b*d^4*g^3*n^2-18*B*x*b^4*c^3*d*g^3*n^2+36*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^2*d^2*g^3*n+24*B*x*ln(e
*((b*x+a)/(d*x+c))^n)*b^4*c^3*d*g^3*n-60*A*a*b^3*c^3*d*g^3*n+24*A*x^3*b^4*c*d^3*g^3*n+36*A*x^2*b^4*c^2*d^2*g^3
*n+24*B*ln(b*x+a)*a^3*b*c*d^3*g^3*n^2-36*B*ln(b*x+a)*a^2*b^2*c^2*d^2*g^3*n^2+24*B*ln(b*x+a)*a*b^3*c^3*d*g^3*n^
2+12*B*x^2*a*b^3*c*d^3*g^3*n^2-24*B*x*a^2*b^2*c*d^3*g^3*n^2+36*B*x*a*b^3*c^2*d^2*g^3*n^2-6*B*a^4*d^4*g^3*n^2+1
8*B*b^4*c^4*g^3*n^2+6*A*x^4*b^4*d^4*g^3*n)/b^4/d/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (146) = 292\).

Time = 0.30 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.75 \[ \int (c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} g^{3} x^{4} - 6 \, B b^{4} c^{4} g^{3} n \log \left (d x + c\right ) + 6 \, {\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} g^{3} n \log \left (b x + a\right ) + 2 \, {\left (12 \, A b^{4} c d^{3} g^{3} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{3} n\right )} x^{3} + 3 \, {\left (12 \, A b^{4} c^{2} d^{2} g^{3} - {\left (3 \, B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{3} n\right )} x^{2} + 6 \, {\left (4 \, A b^{4} c^{3} d g^{3} - {\left (3 \, B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 4 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} g^{3} n\right )} x + 6 \, {\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B b^{4} c d^{3} g^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} g^{3} x^{2} + 4 \, B b^{4} c^{3} d g^{3} x\right )} \log \left (e\right ) + 6 \, {\left (B b^{4} d^{4} g^{3} n x^{4} + 4 \, B b^{4} c d^{3} g^{3} n x^{3} + 6 \, B b^{4} c^{2} d^{2} g^{3} n x^{2} + 4 \, B b^{4} c^{3} d g^{3} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{24 \, b^{4} d} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

integrate((d*g*x+c*g)**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. 479 vs. \(2 (146) = 292\).

Time = 0.20 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.07 \[ \int (c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {1}{4} \, B d^{3} g^{3} x^{4} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{4} \, A d^{3} g^{3} x^{4} + B c d^{2} g^{3} x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c d^{2} g^{3} x^{3} + \frac {3}{2} \, B c^{2} d g^{3} x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {3}{2} \, A c^{2} d g^{3} x^{2} - \frac {1}{24} \, B d^{3} g^{3} n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + \frac {1}{2} \, B c d^{2} g^{3} n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - \frac {3}{2} \, B c^{2} d g^{3} n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B c^{3} g^{3} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B c^{3} g^{3} x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c^{3} g^{3} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1402 vs. \(2 (146) = 292\).

Time = 0.84 (sec) , antiderivative size = 1402, normalized size of antiderivative = 8.99 \[ \int (c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/24*(6*(B*b^5*c^5*g^3*n - 5*B*a*b^4*c^4*d*g^3*n + 10*B*a^2*b^3*c^3*d^2*g^3*n - 10*B*a^3*b^2*c^2*d^3*g^3*n + 5
*B*a^4*b*c*d^4*g^3*n - B*a^5*d^5*g^3*n)*log((b*x + a)/(d*x + c))/(b^4*d - 4*(b*x + a)*b^3*d^2/(d*x + c) + 6*(b
*x + a)^2*b^2*d^3/(d*x + c)^2 - 4*(b*x + a)^3*b*d^4/(d*x + c)^3 + (b*x + a)^4*d^5/(d*x + c)^4) - (11*B*b^8*c^5
*g^3*n - 55*B*a*b^7*c^4*d*g^3*n - 26*(b*x + a)*B*b^7*c^5*d*g^3*n/(d*x + c) + 110*B*a^2*b^6*c^3*d^2*g^3*n + 130
*(b*x + a)*B*a*b^6*c^4*d^2*g^3*n/(d*x + c) + 21*(b*x + a)^2*B*b^6*c^5*d^2*g^3*n/(d*x + c)^2 - 110*B*a^3*b^5*c^
2*d^3*g^3*n - 260*(b*x + a)*B*a^2*b^5*c^3*d^3*g^3*n/(d*x + c) - 105*(b*x + a)^2*B*a*b^5*c^4*d^3*g^3*n/(d*x + c
)^2 - 6*(b*x + a)^3*B*b^5*c^5*d^3*g^3*n/(d*x + c)^3 + 55*B*a^4*b^4*c*d^4*g^3*n + 260*(b*x + a)*B*a^3*b^4*c^2*d
^4*g^3*n/(d*x + c) + 210*(b*x + a)^2*B*a^2*b^4*c^3*d^4*g^3*n/(d*x + c)^2 + 30*(b*x + a)^3*B*a*b^4*c^4*d^4*g^3*
n/(d*x + c)^3 - 11*B*a^5*b^3*d^5*g^3*n - 130*(b*x + a)*B*a^4*b^3*c*d^5*g^3*n/(d*x + c) - 210*(b*x + a)^2*B*a^3
*b^3*c^2*d^5*g^3*n/(d*x + c)^2 - 60*(b*x + a)^3*B*a^2*b^3*c^3*d^5*g^3*n/(d*x + c)^3 + 26*(b*x + a)*B*a^5*b^2*d
^6*g^3*n/(d*x + c) + 105*(b*x + a)^2*B*a^4*b^2*c*d^6*g^3*n/(d*x + c)^2 + 60*(b*x + a)^3*B*a^3*b^2*c^2*d^6*g^3*
n/(d*x + c)^3 - 21*(b*x + a)^2*B*a^5*b*d^7*g^3*n/(d*x + c)^2 - 30*(b*x + a)^3*B*a^4*b*c*d^7*g^3*n/(d*x + c)^3
+ 6*(b*x + a)^3*B*a^5*d^8*g^3*n/(d*x + c)^3 - 6*B*b^8*c^5*g^3*log(e) + 30*B*a*b^7*c^4*d*g^3*log(e) - 60*B*a^2*
b^6*c^3*d^2*g^3*log(e) + 60*B*a^3*b^5*c^2*d^3*g^3*log(e) - 30*B*a^4*b^4*c*d^4*g^3*log(e) + 6*B*a^5*b^3*d^5*g^3
*log(e) - 6*A*b^8*c^5*g^3 + 30*A*a*b^7*c^4*d*g^3 - 60*A*a^2*b^6*c^3*d^2*g^3 + 60*A*a^3*b^5*c^2*d^3*g^3 - 30*A*
a^4*b^4*c*d^4*g^3 + 6*A*a^5*b^3*d^5*g^3)/(b^7*d - 4*(b*x + a)*b^6*d^2/(d*x + c) + 6*(b*x + a)^2*b^5*d^3/(d*x +
c)^2 - 4*(b*x + a)^3*b^4*d^4/(d*x + c)^3 + (b*x + a)^4*b^3*d^5/(d*x + c)^4) + 6*(B*b^5*c^5*g^3*n - 5*B*a*b^4*
c^4*d*g^3*n + 10*B*a^2*b^3*c^3*d^2*g^3*n - 10*B*a^3*b^2*c^2*d^3*g^3*n + 5*B*a^4*b*c*d^4*g^3*n - B*a^5*d^5*g^3*
n)*log(-b + (b*x + a)*d/(d*x + c))/(b^4*d) - 6*(B*b^5*c^5*g^3*n - 5*B*a*b^4*c^4*d*g^3*n + 10*B*a^2*b^3*c^3*d^2
*g^3*n - 10*B*a^3*b^2*c^2*d^3*g^3*n + 5*B*a^4*b*c*d^4*g^3*n - B*a^5*d^5*g^3*n)*log((b*x + a)/(d*x + c))/(b^4*d
))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

Mupad [B] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 588, normalized size of antiderivative = 3.77 \[ \int (c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=x^3\,\left (\frac {d^2\,g^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12\,b}-\frac {A\,d^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,b}\right )-x^2\,\left (\frac {\left (\frac {d^2\,g^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {c\,d\,g^3\,\left (4\,A\,a\,d+6\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{2\,b}+\frac {A\,a\,c\,d^2\,g^3}{2\,b}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,c^3\,g^3\,x+\frac {3\,B\,c^2\,d\,g^3\,x^2}{2}+B\,c\,d^2\,g^3\,x^3+\frac {B\,d^3\,g^3\,x^4}{4}\right )+x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {d^2\,g^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {c\,d\,g^3\,\left (4\,A\,a\,d+6\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}+\frac {A\,a\,c\,d^2\,g^3}{b}\right )}{4\,b\,d}+\frac {c^2\,g^3\,\left (12\,A\,a\,d+8\,A\,b\,c+3\,B\,a\,d\,n-3\,B\,b\,c\,n\right )}{2\,b}-\frac {a\,c\,\left (\frac {d^2\,g^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )}{b\,d}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,n\,a^4\,d^3\,g^3-4\,B\,n\,a^3\,b\,c\,d^2\,g^3+6\,B\,n\,a^2\,b^2\,c^2\,d\,g^3-4\,B\,n\,a\,b^3\,c^3\,g^3\right )}{4\,b^4}+\frac {A\,d^3\,g^3\,x^4}{4}-\frac {B\,c^4\,g^3\,n\,\ln \left (c+d\,x\right )}{4\,d} \]

[In]

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

[Out]

x^3*((d^2*g^3*(4*A*a*d + 16*A*b*c + B*a*d*n - B*b*c*n))/(12*b) - (A*d^2*g^3*(4*a*d + 4*b*c))/(12*b)) - x^2*(((
(d^2*g^3*(4*A*a*d + 16*A*b*c + B*a*d*n - B*b*c*n))/(4*b) - (A*d^2*g^3*(4*a*d + 4*b*c))/(4*b))*(4*a*d + 4*b*c))
/(8*b*d) - (c*d*g^3*(4*A*a*d + 6*A*b*c + B*a*d*n - B*b*c*n))/(2*b) + (A*a*c*d^2*g^3)/(2*b)) + log(e*((a + b*x)
/(c + d*x))^n)*((B*d^3*g^3*x^4)/4 + B*c^3*g^3*x + (3*B*c^2*d*g^3*x^2)/2 + B*c*d^2*g^3*x^3) + x*(((4*a*d + 4*b*
c)*((((d^2*g^3*(4*A*a*d + 16*A*b*c + B*a*d*n - B*b*c*n))/(4*b) - (A*d^2*g^3*(4*a*d + 4*b*c))/(4*b))*(4*a*d + 4
*b*c))/(4*b*d) - (c*d*g^3*(4*A*a*d + 6*A*b*c + B*a*d*n - B*b*c*n))/b + (A*a*c*d^2*g^3)/b))/(4*b*d) + (c^2*g^3*
(12*A*a*d + 8*A*b*c + 3*B*a*d*n - 3*B*b*c*n))/(2*b) - (a*c*((d^2*g^3*(4*A*a*d + 16*A*b*c + B*a*d*n - B*b*c*n))
/(4*b) - (A*d^2*g^3*(4*a*d + 4*b*c))/(4*b)))/(b*d)) - (log(a + b*x)*(B*a^4*d^3*g^3*n - 4*B*a*b^3*c^3*g^3*n - 4
*B*a^3*b*c*d^2*g^3*n + 6*B*a^2*b^2*c^2*d*g^3*n))/(4*b^4) + (A*d^3*g^3*x^4)/4 - (B*c^4*g^3*n*log(c + d*x))/(4*d
)