\(\int (a+b x)^3 (A+B \log (e (a+b x)^n (c+d x)^{-n})) \, dx\) [148]

Optimal result

Integrand size = 31, antiderivative size = 142 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=-\frac {B (b c-a d)^3 n x}{4 d^3}+\frac {B (b c-a d)^2 n (a+b x)^2}{8 b d^2}-\frac {B (b c-a d) n (a+b x)^3}{12 b d}+\frac {B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac {(a+b x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 45} \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {(a+b x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{4 b}+\frac {B n (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac {B n x (b c-a d)^3}{4 d^3}+\frac {B n (a+b x)^2 (b c-a d)^2}{8 b d^2}-\frac {B n (a+b x)^3 (b c-a d)}{12 b d} \]

[In]

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

[Out]

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

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 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(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] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.92 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {b d x \left (6 a^3 d^3 (4 A+3 B n)+9 a^2 b d^2 (-4 B c n+4 A d x+B d n x)+b^3 \left (6 A d^3 x^3+B c n \left (-6 c^2+3 c d x-2 d^2 x^2\right )\right )+2 a b^2 d \left (12 A d^2 x^2+B n \left (12 c^2-6 c d x+d^2 x^2\right )\right )\right )-18 a^4 B d^4 n \log (a+b x)+6 B \left (b^4 c^4-4 a b^3 c^3 d+6 a^2 b^2 c^2 d^2-4 a^3 b c d^3+4 a^4 d^4\right ) n \log (c+d x)+6 B d^4 \left (4 a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{24 b d^4} \]

[In]

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

[Out]

(b*d*x*(6*a^3*d^3*(4*A + 3*B*n) + 9*a^2*b*d^2*(-4*B*c*n + 4*A*d*x + B*d*n*x) + b^3*(6*A*d^3*x^3 + B*c*n*(-6*c^
2 + 3*c*d*x - 2*d^2*x^2)) + 2*a*b^2*d*(12*A*d^2*x^2 + B*n*(12*c^2 - 6*c*d*x + d^2*x^2))) - 18*a^4*B*d^4*n*Log[
a + b*x] + 6*B*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 4*a^4*d^4)*n*Log[c + d*x] + 6*B*
d^4*(4*a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a*b^3*x^3 + b^4*x^4)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(24*b*d^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(663\) vs. \(2(132)=264\).

Time = 52.80 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.68

method	result	size
parallelrisch	\(\frac {-6 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{4} c^{4} n +3 B \,x^{2} b^{4} c^{2} d^{2} n^{2}+18 B x \,a^{3} b \,d^{4} n^{2}-6 B x \,b^{4} c^{3} d \,n^{2}+24 A x \,a^{3} b \,d^{4} n +36 A \,x^{2} a^{2} b^{2} d^{4} n +9 B \,a^{3} b c \,d^{3} n^{2}+24 B \,a^{2} b^{2} c^{2} d^{2} n^{2}-21 B a \,b^{3} c^{3} d \,n^{2}-60 A \,a^{3} b c \,d^{3} n +2 B \,x^{3} a \,b^{3} d^{4} n^{2}-2 B \,x^{3} b^{4} c \,d^{3} n^{2}+24 A \,x^{3} a \,b^{3} d^{4} n +9 B \,x^{2} a^{2} b^{2} d^{4} n^{2}+6 B \,x^{4} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{4} d^{4} n -18 B \,a^{4} d^{4} n^{2}+6 B \,b^{4} c^{4} n^{2}-24 A \,a^{4} d^{4} n -12 B \,x^{2} a \,b^{3} c \,d^{3} n^{2}-36 B x \,a^{2} b^{2} c \,d^{3} n^{2}+24 B x a \,b^{3} c^{2} d^{2} n^{2}-24 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} n^{2}+36 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} n^{2}-24 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d \,n^{2}+24 B \,x^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a \,b^{3} d^{4} n +36 B \,x^{2} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a^{2} b^{2} d^{4} n +24 B x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a^{3} b \,d^{4} n +24 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a^{3} b c \,d^{3} n -36 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a^{2} b^{2} c^{2} d^{2} n +24 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a \,b^{3} c^{3} d n +6 A \,x^{4} b^{4} d^{4} n +6 B \ln \left (b x +a \right ) a^{4} d^{4} n^{2}+6 B \ln \left (b x +a \right ) b^{4} c^{4} n^{2}}{24 d^{4} n b}\)	\(664\)
risch	\(\text {Expression too large to display}\)	\(1838\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (132) = 264\).

Time = 0.28 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.94 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} x^{4} + 2 \, {\left (12 \, A a b^{3} d^{4} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} n\right )} x^{3} + 3 \, {\left (12 \, A a^{2} b^{2} d^{4} + {\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + 3 \, B a^{2} b^{2} d^{4}\right )} n\right )} x^{2} + 6 \, {\left (4 \, A a^{3} b d^{4} - {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} - 3 \, B a^{3} b d^{4}\right )} n\right )} x + 6 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x + B a^{4} d^{4} n\right )} \log \left (b x + a\right ) - 6 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x - {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} n\right )} \log \left (d x + c\right ) + 6 \, {\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x\right )} \log \left (e\right )}{24 \, b d^{4}} \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (132) = 264\).

Time = 0.21 (sec) , antiderivative size = 467, normalized size of antiderivative = 3.29 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {1}{4} \, B b^{3} x^{4} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {1}{4} \, A b^{3} x^{4} + B a b^{2} x^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a b^{2} x^{3} + \frac {3}{2} \, B a^{2} b x^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {3}{2} \, A a^{2} b x^{2} + B a^{3} x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a^{3} x + \frac {{\left (\frac {a e n \log \left (b x + a\right )}{b} - \frac {c e n \log \left (d x + c\right )}{d}\right )} B a^{3}}{e} - \frac {3 \, {\left (\frac {a^{2} e n \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} e n \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c e n - a d e n\right )} x}{b d}\right )} B a^{2} b}{2 \, e} + \frac {{\left (\frac {2 \, a^{3} e n \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} e n \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d e n - a b d^{2} e n\right )} x^{2} - 2 \, {\left (b^{2} c^{2} e n - a^{2} d^{2} e n\right )} x}{b^{2} d^{2}}\right )} B a b^{2}}{2 \, e} - \frac {{\left (\frac {6 \, a^{4} e n \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} e n \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} e n - a b^{2} d^{3} e n\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d e n - a^{2} b d^{3} e n\right )} x^{2} + 6 \, {\left (b^{3} c^{3} e n - a^{3} d^{3} e n\right )} x}{b^{3} d^{3}}\right )} B b^{3}}{24 \, e} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (132) = 264\).

Time = 2.07 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.56 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=\frac {B a^{4} n \log \left (b x + a\right )}{4 \, b} + \frac {1}{4} \, {\left (B b^{3} \log \left (e\right ) + A b^{3}\right )} x^{4} - \frac {{\left (B b^{3} c n - B a b^{2} d n - 12 \, B a b^{2} d \log \left (e\right ) - 12 \, A a b^{2} d\right )} x^{3}}{12 \, d} + \frac {1}{4} \, {\left (B b^{3} n x^{4} + 4 \, B a b^{2} n x^{3} + 6 \, B a^{2} b n x^{2} + 4 \, B a^{3} n x\right )} \log \left (b x + a\right ) - \frac {1}{4} \, {\left (B b^{3} n x^{4} + 4 \, B a b^{2} n x^{3} + 6 \, B a^{2} b n x^{2} + 4 \, B a^{3} n x\right )} \log \left (d x + c\right ) + \frac {{\left (B b^{3} c^{2} n - 4 \, B a b^{2} c d n + 3 \, B a^{2} b d^{2} n + 12 \, B a^{2} b d^{2} \log \left (e\right ) + 12 \, A a^{2} b d^{2}\right )} x^{2}}{8 \, d^{2}} - \frac {{\left (B b^{3} c^{3} n - 4 \, B a b^{2} c^{2} d n + 6 \, B a^{2} b c d^{2} n - 3 \, B a^{3} d^{3} n - 4 \, B a^{3} d^{3} \log \left (e\right ) - 4 \, A a^{3} d^{3}\right )} x}{4 \, d^{3}} + \frac {{\left (B b^{3} c^{4} n - 4 \, B a b^{2} c^{3} d n + 6 \, B a^{2} b c^{2} d^{2} n - 4 \, B a^{3} c d^{3} n\right )} \log \left (d x + c\right )}{4 \, d^{4}} \]

[In]

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

[Out]

1/4*B*a^4*n*log(b*x + a)/b + 1/4*(B*b^3*log(e) + A*b^3)*x^4 - 1/12*(B*b^3*c*n - B*a*b^2*d*n - 12*B*a*b^2*d*log
(e) - 12*A*a*b^2*d)*x^3/d + 1/4*(B*b^3*n*x^4 + 4*B*a*b^2*n*x^3 + 6*B*a^2*b*n*x^2 + 4*B*a^3*n*x)*log(b*x + a) -
1/4*(B*b^3*n*x^4 + 4*B*a*b^2*n*x^3 + 6*B*a^2*b*n*x^2 + 4*B*a^3*n*x)*log(d*x + c) + 1/8*(B*b^3*c^2*n - 4*B*a*b
^2*c*d*n + 3*B*a^2*b*d^2*n + 12*B*a^2*b*d^2*log(e) + 12*A*a^2*b*d^2)*x^2/d^2 - 1/4*(B*b^3*c^3*n - 4*B*a*b^2*c^
2*d*n + 6*B*a^2*b*c*d^2*n - 3*B*a^3*d^3*n - 4*B*a^3*d^3*log(e) - 4*A*a^3*d^3)*x/d^3 + 1/4*(B*b^3*c^4*n - 4*B*a
*b^2*c^3*d*n + 6*B*a^2*b*c^2*d^2*n - 4*B*a^3*c*d^3*n)*log(d*x + c)/d^4

Mupad [B] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 520, normalized size of antiderivative = 3.66 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx=x^3\,\left (\frac {b^2\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12\,d}-\frac {A\,b^2\,\left (4\,a\,d+4\,b\,c\right )}{12\,d}\right )+\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (B\,a^3\,x+\frac {3\,B\,a^2\,b\,x^2}{2}+B\,a\,b^2\,x^3+\frac {B\,b^3\,x^4}{4}\right )+x\,\left (\frac {a^2\,\left (8\,A\,a\,d+12\,A\,b\,c+3\,B\,a\,d\,n-3\,B\,b\,c\,n\right )}{2\,d}+\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {b^2\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{4\,b\,d}-\frac {a\,b\,\left (6\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^2\,c}{d}\right )}{4\,b\,d}-\frac {a\,c\,\left (\frac {b^2\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {b^2\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{8\,b\,d}-\frac {a\,b\,\left (6\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{2\,d}+\frac {A\,a\,b^2\,c}{2\,d}\right )+\frac {A\,b^3\,x^4}{4}+\frac {\ln \left (c+d\,x\right )\,\left (-4\,B\,n\,a^3\,c\,d^3+6\,B\,n\,a^2\,b\,c^2\,d^2-4\,B\,n\,a\,b^2\,c^3\,d+B\,n\,b^3\,c^4\right )}{4\,d^4}+\frac {B\,a^4\,n\,\ln \left (a+b\,x\right )}{4\,b} \]

[In]

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

[Out]

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