3.2.55 \(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(a+b x)^5} \, dx\) [155]

Optimal. Leaf size=195 \[ -\frac {B n}{16 b (a+b x)^4}+\frac {B d n}{12 b (b c-a d) (a+b x)^3}-\frac {B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}+\frac {B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4} \]

[Out]

-1/16*B*n/b/(b*x+a)^4+1/12*B*d*n/b/(-a*d+b*c)/(b*x+a)^3-1/8*B*d^2*n/b/(-a*d+b*c)^2/(b*x+a)^2+1/4*B*d^3*n/b/(-a
*d+b*c)^3/(b*x+a)+1/4*B*d^4*n*ln(b*x+a)/b/(-a*d+b*c)^4-1/4*B*d^4*n*ln(d*x+c)/b/(-a*d+b*c)^4+1/4*(-A-B*ln(e*(b*
x+a)^n/((d*x+c)^n)))/b/(b*x+a)^4

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 46} \begin {gather*} -\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{4 b (a+b x)^4}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4}+\frac {B d^3 n}{4 b (a+b x) (b c-a d)^3}-\frac {B d^2 n}{8 b (a+b x)^2 (b c-a d)^2}+\frac {B d n}{12 b (a+b x)^3 (b c-a d)}-\frac {B n}{16 b (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(a + b*x)^5,x]

[Out]

-1/16*(B*n)/(b*(a + b*x)^4) + (B*d*n)/(12*b*(b*c - a*d)*(a + b*x)^3) - (B*d^2*n)/(8*b*(b*c - a*d)^2*(a + b*x)^
2) + (B*d^3*n)/(4*b*(b*c - a*d)^3*(a + b*x)) + (B*d^4*n*Log[a + b*x])/(4*b*(b*c - a*d)^4) - (B*d^4*n*Log[c + d
*x])/(4*b*(b*c - a*d)^4) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(4*b*(a + b*x)^4)

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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*} \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx &=\int \left (\frac {A}{(a+b x)^5}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}\right ) \, dx\\ &=-\frac {A}{4 b (a+b x)^4}+B \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx\\ &=-\frac {A}{4 b (a+b x)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b}\\ &=-\frac {A}{4 b (a+b x)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {(B (b c-a d) n) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b}\\ &=-\frac {A}{4 b (a+b x)^4}-\frac {B n}{16 b (a+b x)^4}+\frac {B d n}{12 b (b c-a d) (a+b x)^3}-\frac {B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}+\frac {B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 160, normalized size = 0.82 \begin {gather*} \frac {\frac {12 B d^4 n \log (a+b x)}{(b c-a d)^4}-\frac {12 B d^4 n \log (c+d x)}{(b c-a d)^4}+\frac {-12 A-3 B n+\frac {4 B d n (a+b x)}{b c-a d}-\frac {6 B d^2 n (a+b x)^2}{(b c-a d)^2}+\frac {12 B d^3 n (a+b x)^3}{(b c-a d)^3}-12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}}{48 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(a + b*x)^5,x]

[Out]

((12*B*d^4*n*Log[a + b*x])/(b*c - a*d)^4 - (12*B*d^4*n*Log[c + d*x])/(b*c - a*d)^4 + (-12*A - 3*B*n + (4*B*d*n
*(a + b*x))/(b*c - a*d) - (6*B*d^2*n*(a + b*x)^2)/(b*c - a*d)^2 + (12*B*d^3*n*(a + b*x)^3)/(b*c - a*d)^3 - 12*
B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(a + b*x)^4)/(48*b)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.39, size = 2583, normalized size = 13.25

method	result	size
risch	\(\text {Expression too large to display}\)	\(2583\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/4*B/b/(b*x+a)^4*ln((d*x+c)^n)+1/48*(-12*A*b^4*c^4+16*B*a*c^3*d*n*b^3-36*B*a^2*c^2*d^2*n*b^2+48*B*a^3*c*d^3*n
*b-6*I*B*Pi*b^4*c^4*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-6*I*B*Pi*b^4*c^4*csgn(I*(b*x+a)^n/((d*
x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-24*I*B*Pi*a*b^3*c^3*d*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*
(b*x+a)^n/((d*x+c)^n))-24*I*B*Pi*a^3*b*c*d^3*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x
+a)^n)-12*B*b^4*c^4*ln((b*x+a)^n)+6*I*B*Pi*b^4*c^4*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+36*I*B*Pi*a^2*b^2*c^2*d^2
*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-3*B*c^4*n*b^4-25*B*a^4*d^4*n+6*I*B*Pi*b^4*c^4*csgn(I*e)*csgn(I*(b*x+a)^n/((d*
x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-24*I*B*Pi*a^3*b*c*d^3*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b
*x+a)^n/((d*x+c)^n))-72*A*a^2*b^2*c^2*d^2-12*B*ln(d*x+c)*b^4*d^4*n*x^4+12*B*ln(-b*x-a)*b^4*d^4*n*x^4-12*B*a*b^
3*d^4*n*x^3+12*B*b^4*c*d^3*n*x^3-42*B*a^2*b^2*d^4*n*x^2-6*B*b^4*c^2*d^2*n*x^2-52*B*a^3*b*d^4*n*x+4*B*b^4*c^3*d
*n*x+48*A*a^3*b*c*d^3+48*A*a*b^3*c^3*d-36*I*B*Pi*a^2*b^2*c^2*d^2*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)
^n))^2-12*A*a^4*d^4+6*I*B*Pi*a^4*d^4*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-24*I*
B*Pi*a^3*b*c*d^3*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-24*I*B*Pi*a^3*b*c*d^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+48*B*
a*b^3*c*d^3*n*x^2+72*B*a^2*b^2*c*d^3*n*x-24*B*a*b^3*c^2*d^2*n*x+48*B*ln(e)*a^3*b*c*d^3-72*B*ln(e)*a^2*b^2*c^2*
d^2+12*B*ln(-b*x-a)*a^4*d^4*n-12*B*a^4*d^4*ln((b*x+a)^n)-24*I*B*Pi*a*b^3*c^3*d*csgn(I*(b*x+a)^n/((d*x+c)^n))^3
+24*I*B*Pi*a^3*b*c*d^3*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+24*I*B*Pi*a^3*b*c*d^3*csgn(I*(b*x+a
)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-36*I*B*Pi*a^2*b^2*c^2*d^2*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b
*x+a)^n)^2-36*I*B*Pi*a^2*b^2*c^2*d^2*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+36*I*B*Pi*a^2*b^2*c^2*d
^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+24*I*B*Pi*a*b^3*c^3*d*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*
(b*x+a)^n)^2+48*B*ln(e)*a*b^3*c^3*d+6*I*B*Pi*a^4*d^4*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)
^n)*(b*x+a)^n)-6*I*B*Pi*a^4*d^4*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-12*B*ln(e)*a^4*d^4-12*B*ln
(e)*b^4*c^4+6*I*B*Pi*b^4*c^4*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-6*I*B*Pi*a^4*d^4*csgn(I*(b*x+a)^n/((d*x+c)^n))*cs
gn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+24*I*B*Pi*a*b^3*c^3*d*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-24*I
*B*Pi*a*b^3*c^3*d*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+6*I*B*Pi*b^4*c^4*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csg
n(I*(b*x+a)^n/((d*x+c)^n))+48*B*a^3*b*c*d^3*ln((b*x+a)^n)-72*B*a^2*b^2*c^2*d^2*ln((b*x+a)^n)+48*B*a*b^3*c^3*d*
ln((b*x+a)^n)+6*I*B*Pi*a^4*d^4*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+6*I*B*Pi*a^4*d^4*csgn(I*e/((d*x+c)^n)*(b*x+a)^n
)^3-12*B*ln(d*x+c)*a^4*d^4*n-36*I*B*Pi*a^2*b^2*c^2*d^2*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x
+a)^n)^2+24*I*B*Pi*a*b^3*c^3*d*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+24*I*B*Pi*a*b^3*c^3*d*csgn(I*(b*x+a
)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-6*I*B*Pi*b^4*c^4*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-6*I*B*Pi*b^4
*c^4*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-6*I*B*Pi*a^4*d^4*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d
*x+c)^n))^2+36*I*B*Pi*a^2*b^2*c^2*d^2*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)+
36*I*B*Pi*a^2*b^2*c^2*d^2*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-24*I*B*Pi*a*b^3*
c^3*d*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-6*I*B*Pi*a^4*d^4*csgn(I*e)*csgn(
I*e/((d*x+c)^n)*(b*x+a)^n)^2+24*I*B*Pi*a^3*b*c*d^3*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+24*I*B*Pi*a^3*b
*c*d^3*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-48*B*ln(d*x+c)*a*b^3*d^4*n*x^3+48*B*ln(-b*x-a)*a*b^3*
d^4*n*x^3-72*B*ln(d*x+c)*a^2*b^2*d^4*n*x^2+72*B*ln(-b*x-a)*a^2*b^2*d^4*n*x^2-48*B*ln(d*x+c)*a^3*b*d^4*n*x+48*B
*ln(-b*x-a)*a^3*b*d^4*n*x)/(b*x+a)^4/(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*c^3)/(-a*d+b*c)/b

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (182) = 364\).
time = 0.36, size = 621, normalized size = 3.18 \begin {gather*} \frac {1}{48} \, {\left (\frac {12 \, d^{4} n e \log \left (b x + a\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {12 \, d^{4} n e \log \left (d x + c\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {12 \, b^{3} d^{3} n x^{3} e - 6 \, {\left (b^{3} c d^{2} n - 7 \, a b^{2} d^{3} n\right )} x^{2} e + 4 \, {\left (b^{3} c^{2} d n - 5 \, a b^{2} c d^{2} n + 13 \, a^{2} b d^{3} n\right )} x e - {\left (3 \, b^{3} c^{3} n - 13 \, a b^{2} c^{2} d n + 23 \, a^{2} b c d^{2} n - 25 \, a^{3} d^{3} n\right )} e}{a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x}\right )} B e^{\left (-1\right )} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {A}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(b*x+a)^5,x, algorithm="maxima")

[Out]

1/48*(12*d^4*n*e*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - 12
*d^4*n*e*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) + (12*b^3*d^
3*n*x^3*e - 6*(b^3*c*d^2*n - 7*a*b^2*d^3*n)*x^2*e + 4*(b^3*c^2*d*n - 5*a*b^2*c*d^2*n + 13*a^2*b*d^3*n)*x*e - (
3*b^3*c^3*n - 13*a*b^2*c^2*d*n + 23*a^2*b*c*d^2*n - 25*a^3*d^3*n)*e)/(a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^
2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6
*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3
)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*x))*B*e^(-1) - 1/4*B*log((b*x + a)^n
*e/(d*x + c)^n)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) - 1/4*A/(b^5*x^4 + 4*a*b^4*x^3 +
6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (182) = 364\).
time = 0.37, size = 772, normalized size = 3.96 \begin {gather*} -\frac {12 \, {\left (A + B\right )} b^{4} c^{4} - 48 \, {\left (A + B\right )} a b^{3} c^{3} d + 72 \, {\left (A + B\right )} a^{2} b^{2} c^{2} d^{2} - 48 \, {\left (A + B\right )} a^{3} b c d^{3} + 12 \, {\left (A + B\right )} a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} n x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} n x^{2} - 4 \, {\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} n x + {\left (3 \, B b^{4} c^{4} - 16 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 48 \, B a^{3} b c d^{3} + 25 \, B a^{4} d^{4}\right )} n - 12 \, {\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 (b x + a\right ) + 12 \, {\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 )}{48 \, {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4} + {\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/48*(12*(A + B)*b^4*c^4 - 48*(A + B)*a*b^3*c^3*d + 72*(A + B)*a^2*b^2*c^2*d^2 - 48*(A + B)*a^3*b*c*d^3 + 12*
(A + B)*a^4*d^4 - 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*n*x^3 + 6*(B*b^4*c^2*d^2 - 8*B*a*b^3*c*d^3 + 7*B*a^2*b^2*d^4)
*n*x^2 - 4*(B*b^4*c^3*d - 6*B*a*b^3*c^2*d^2 + 18*B*a^2*b^2*c*d^3 - 13*B*a^3*b*d^4)*n*x + (3*B*b^4*c^4 - 16*B*a
*b^3*c^3*d + 36*B*a^2*b^2*c^2*d^2 - 48*B*a^3*b*c*d^3 + 25*B*a^4*d^4)*n - 12*(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(b*x + a) + 12*(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))/(a^4*b^5*c^4 - 4
*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4 + (b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^
2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3
+ a^5*b^4*d^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*x^2
+ 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*x)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))/(b*x+a)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (182) = 364\).
time = 4.65, size = 710, normalized size = 3.64 \begin {gather*} \frac {B d^{4} n \log \left (b x + a\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {B d^{4} n \log \left (d x + c\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {B n \log \left (b x + a\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {B n \log \left (d x + c\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {12 \, B b^{3} d^{3} n x^{3} - 6 \, B b^{3} c d^{2} n x^{2} + 42 \, B a b^{2} d^{3} n x^{2} + 4 \, B b^{3} c^{2} d n x - 20 \, B a b^{2} c d^{2} n x + 52 \, B a^{2} b d^{3} n x - 3 \, B b^{3} c^{3} n + 13 \, B a b^{2} c^{2} d n - 23 \, B a^{2} b c d^{2} n + 25 \, B a^{3} d^{3} n - 12 \, A b^{3} c^{3} - 12 \, B b^{3} c^{3} + 36 \, A a b^{2} c^{2} d + 36 \, B a b^{2} c^{2} d - 36 \, A a^{2} b c d^{2} - 36 \, B a^{2} b c d^{2} + 12 \, A a^{3} d^{3} + 12 \, B a^{3} d^{3}}{48 \, {\left (b^{8} c^{3} x^{4} - 3 \, a b^{7} c^{2} d x^{4} + 3 \, a^{2} b^{6} c d^{2} x^{4} - a^{3} b^{5} d^{3} x^{4} + 4 \, a b^{7} c^{3} x^{3} - 12 \, a^{2} b^{6} c^{2} d x^{3} + 12 \, a^{3} b^{5} c d^{2} x^{3} - 4 \, a^{4} b^{4} d^{3} x^{3} + 6 \, a^{2} b^{6} c^{3} x^{2} - 18 \, a^{3} b^{5} c^{2} d x^{2} + 18 \, a^{4} b^{4} c d^{2} x^{2} - 6 \, a^{5} b^{3} d^{3} x^{2} + 4 \, a^{3} b^{5} c^{3} x - 12 \, a^{4} b^{4} c^{2} d x + 12 \, a^{5} b^{3} c d^{2} x - 4 \, a^{6} b^{2} d^{3} x + a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(b*x+a)^5,x, algorithm="giac")

[Out]

1/4*B*d^4*n*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - 1/4*B*d
^4*n*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - 1/4*B*n*log(b*
x + a)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) + 1/4*B*n*log(d*x + c)/(b^5*x^4 + 4*a*b^4
*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) + 1/48*(12*B*b^3*d^3*n*x^3 - 6*B*b^3*c*d^2*n*x^2 + 42*B*a*b^2*d^3*
n*x^2 + 4*B*b^3*c^2*d*n*x - 20*B*a*b^2*c*d^2*n*x + 52*B*a^2*b*d^3*n*x - 3*B*b^3*c^3*n + 13*B*a*b^2*c^2*d*n - 2
3*B*a^2*b*c*d^2*n + 25*B*a^3*d^3*n - 12*A*b^3*c^3 - 12*B*b^3*c^3 + 36*A*a*b^2*c^2*d + 36*B*a*b^2*c^2*d - 36*A*
a^2*b*c*d^2 - 36*B*a^2*b*c*d^2 + 12*A*a^3*d^3 + 12*B*a^3*d^3)/(b^8*c^3*x^4 - 3*a*b^7*c^2*d*x^4 + 3*a^2*b^6*c*d
^2*x^4 - a^3*b^5*d^3*x^4 + 4*a*b^7*c^3*x^3 - 12*a^2*b^6*c^2*d*x^3 + 12*a^3*b^5*c*d^2*x^3 - 4*a^4*b^4*d^3*x^3 +
6*a^2*b^6*c^3*x^2 - 18*a^3*b^5*c^2*d*x^2 + 18*a^4*b^4*c*d^2*x^2 - 6*a^5*b^3*d^3*x^2 + 4*a^3*b^5*c^3*x - 12*a^
4*b^4*c^2*d*x + 12*a^5*b^3*c*d^2*x - 4*a^6*b^2*d^3*x + a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b
*d^3)

________________________________________________________________________________________

Mupad [B]
time = 5.33, size = 555, normalized size = 2.85 \begin {gather*} -\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3+25\,B\,a^3\,d^3\,n-3\,B\,b^3\,c^3\,n+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2+13\,B\,a\,b^2\,c^2\,d\,n-23\,B\,a^2\,b\,c\,d^2\,n}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x\,\left (13\,B\,n\,a^2\,b\,d^2-5\,B\,n\,a\,b^2\,c\,d+B\,n\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d^2\,x^2\,\left (B\,b^3\,c\,n-7\,B\,a\,b^2\,d\,n\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,n\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b+16\,a^3\,b^2\,x+24\,a^2\,b^3\,x^2+16\,a\,b^4\,x^3+4\,b^5\,x^4}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{4\,b\,\left (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 )}-\frac {B\,d^4\,n\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4+8\,a^3\,b^2\,c\,d^3-8\,a\,b^4\,c^3\,d+4\,b^5\,c^4}{4\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,{\left (a\,d-b\,c\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))/(a + b*x)^5,x)

[Out]

- ((12*A*a^3*d^3 - 12*A*b^3*c^3 + 25*B*a^3*d^3*n - 3*B*b^3*c^3*n + 36*A*a*b^2*c^2*d - 36*A*a^2*b*c*d^2 + 13*B*
a*b^2*c^2*d*n - 23*B*a^2*b*c*d^2*n)/(12*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (d*x*(B*b^3*c^2
*n + 13*B*a^2*b*d^2*n - 5*B*a*b^2*c*d*n))/(3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (d^2*x^2*(
B*b^3*c*n - 7*B*a*b^2*d*n))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B*b^3*d^3*n*x^3)/(a^3*d
^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(4*a^4*b + 4*b^5*x^4 + 16*a^3*b^2*x + 16*a*b^4*x^3 + 24*a^2*b^3
*x^2) - (B*log((e*(a + b*x)^n)/(c + d*x)^n))/(4*b*(a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x)) -
(B*d^4*n*atanh((4*b^5*c^4 - 4*a^4*b*d^4 + 8*a^3*b^2*c*d^3 - 8*a*b^4*c^3*d)/(4*b*(a*d - b*c)^4) - (2*b*d*x*(a^
3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*d - b*c)^4))/(2*b*(a*d - b*c)^4)

________________________________________________________________________________________