3.2.54 \(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(a+b x)^4} \, dx\) [154]

Optimal. Leaf size=166 \[ -\frac {B n}{9 b (a+b x)^3}+\frac {B d n}{6 b (b c-a d) (a+b x)^2}-\frac {B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3} \]

[Out]

-1/9*B*n/b/(b*x+a)^3+1/6*B*d*n/b/(-a*d+b*c)/(b*x+a)^2-1/3*B*d^2*n/b/(-a*d+b*c)^2/(b*x+a)-1/3*B*d^3*n*ln(b*x+a)
/b/(-a*d+b*c)^3+1/3*B*d^3*n*ln(d*x+c)/b/(-a*d+b*c)^3+1/3*(-A-B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/(b*x+a)^3

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Rubi [A]
time = 0.07, antiderivative size = 166, 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}{3 b (a+b x)^3}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {B d^2 n}{3 b (a+b x) (b c-a d)^2}+\frac {B d n}{6 b (a+b x)^2 (b c-a d)}-\frac {B n}{9 b (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(a + b*x)^4,x]

[Out]

-1/9*(B*n)/(b*(a + b*x)^3) + (B*d*n)/(6*b*(b*c - a*d)*(a + b*x)^2) - (B*d^2*n)/(3*b*(b*c - a*d)^2*(a + b*x)) -
(B*d^3*n*Log[a + b*x])/(3*b*(b*c - a*d)^3) + (B*d^3*n*Log[c + d*x])/(3*b*(b*c - a*d)^3) - (A + B*Log[(e*(a +
b*x)^n)/(c + d*x)^n])/(3*b*(a + b*x)^3)

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)^4} \, dx &=\int \left (\frac {A}{(a+b x)^4}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}\right ) \, dx\\ &=-\frac {A}{3 b (a+b x)^3}+B \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx\\ &=-\frac {A}{3 b (a+b x)^3}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b}\\ &=-\frac {A}{3 b (a+b x)^3}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {(B (b c-a d) n) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b}\\ &=-\frac {A}{3 b (a+b x)^3}-\frac {B n}{9 b (a+b x)^3}+\frac {B d n}{6 b (b c-a d) (a+b x)^2}-\frac {B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 167, normalized size = 1.01 \begin {gather*} -\frac {6 B d^3 n (a+b x)^3 \log (a+b x)-6 B d^3 n (a+b x)^3 \log (c+d x)+(b c-a d) \left (6 A (b c-a d)^2+B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+6 B (b c-a d)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{18 b (b c-a d)^3 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(a + b*x)^4,x]

[Out]

-1/18*(6*B*d^3*n*(a + b*x)^3*Log[a + b*x] - 6*B*d^3*n*(a + b*x)^3*Log[c + d*x] + (b*c - a*d)*(6*A*(b*c - a*d)^
2 + B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + 6*B*(b*c - a*d)^2*Log[(e*(a
+ b*x)^n)/(c + d*x)^n]))/(b*(b*c - a*d)^3*(a + b*x)^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.33, size = 1976, normalized size = 11.90

method	result	size
risch	\(\text {Expression too large to display}\)	\(1976\)

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)^4,x,method=_RETURNVERBOSE)

[Out]

1/3*B/b/(b*x+a)^3*ln((d*x+c)^n)-1/18*(-6*B*a^3*d^3*ln((b*x+a)^n)+6*B*b^3*c^3*ln((b*x+a)^n)-6*B*ln(d*x+c)*a^3*d
^3*n-6*A*a^3*d^3+6*A*b^3*c^3+18*B*a^2*b*c*d^2*ln((b*x+a)^n)-3*I*B*Pi*a^3*d^3*csgn(I*(b*x+a)^n/((d*x+c)^n))*csg
n(I*e/((d*x+c)^n)*(b*x+a)^n)^2+9*I*B*Pi*a^2*b*c*d^2*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+9*I*B*Pi
*a^2*b*c*d^2*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+9*I*B*Pi*a^2*b*c*d^2*csgn(I*(b*x+a)^n/((d*x+c
)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-18*A*a*b^2*c^2*d+3*I*B*Pi*a^3*d^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+18
*A*a^2*b*c*d^2+2*B*c^3*n*b^3-11*B*a^3*d^3*n+9*I*B*Pi*a*b^2*c^2*d*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-3*I*B*Pi*b^3*
c^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)-3*I*B*Pi*a^3*d^3*csgn(I*(b*x+a)^n)
*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-3*I*B*Pi*a^3*d^3*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-6*B*ln(e
)*a^3*d^3+6*B*ln(e)*b^3*c^3+3*I*B*Pi*a^3*d^3*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+3*I*B*Pi*b^3*c^3*csgn(I*e)*csgn(I
*e/((d*x+c)^n)*(b*x+a)^n)^2+6*B*ln(-b*x-a)*a^3*d^3*n-9*I*B*Pi*a*b^2*c^2*d*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+
a)^n)^2-9*I*B*Pi*a*b^2*c^2*d*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-9*I*B*Pi*a^2*b*c*d^2*csgn(I*e/(
(d*x+c)^n)*(b*x+a)^n)^3-9*I*B*Pi*a*b^2*c^2*d*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+9*I*B*Pi*a*b^
2*c^2*d*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+3*I*B*Pi*a^3*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)+18*B*a*b^2*c*d^2*n*x-3*I*B*Pi*b^3*c^3*csgn(I*e/((d*x+c)
^n)*(b*x+a)^n)^3-3*I*B*Pi*a^3*d^3*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+3*I*B*Pi*b^3*c^3*csgn(I*(b*x+a)^
n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+3*I*B*Pi*b^3*c^3*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+3*I*B*
Pi*b^3*c^3*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-9*I*B*Pi*a^2*b*c*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)-9*I*B*Pi*a^2*b*c*d^2*csgn(I*(b*x+a)^n)*csgn(I/((d*x+
c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+9*I*B*Pi*a*b^2*c^2*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)+3*I*B*Pi*a^3*d^3*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-9*B*a
*c^2*d*n*b^2+18*B*a^2*c*d^2*n*b+18*B*ln(e)*a^2*b*c*d^2-18*B*ln(e)*a*b^2*c^2*d-6*B*a*b^2*d^3*n*x^2+6*B*b^3*c*d^
2*n*x^2-15*B*a^2*b*d^3*n*x-3*B*b^3*c^2*d*n*x+9*I*B*Pi*a*b^2*c^2*d*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-9*I*B*Pi*a
^2*b*c*d^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-9*I*B*Pi*a*b^2*c^2*d*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c
)^n)*(b*x+a)^n)^2+9*I*B*Pi*a^2*b*c*d^2*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-3*I*B*Pi*b^3*c^3*csgn(I*(b*
x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-18*B*a*b^2*c^2*d*ln((b*x+a)^n)-3*I*B*Pi*b^3*c^3*csgn
(I*(b*x+a)^n/((d*x+c)^n))^3+18*B*ln(-b*x-a)*a*b^2*d^3*n*x^2-18*B*ln(d*x+c)*a^2*b*d^3*n*x+18*B*ln(-b*x-a)*a^2*b
*d^3*n*x-18*B*ln(d*x+c)*a*b^2*d^3*n*x^2-6*B*ln(d*x+c)*b^3*d^3*n*x^3+6*B*ln(-b*x-a)*b^3*d^3*n*x^3)/(b*x+a)^3/(a
^2*d^2-2*a*b*c*d+b^2*c^2)/(-a*d+b*c)/b

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (155) = 310\).
time = 0.31, size = 404, normalized size = 2.43 \begin {gather*} -\frac {1}{18} \, {\left (\frac {6 \, d^{3} n e \log \left (b x + a\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {6 \, d^{3} n e \log \left (d x + c\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {6 \, b^{2} d^{2} n x^{2} e - 3 \, {\left (b^{2} c d n - 5 \, a b d^{2} n\right )} x e + {\left (2 \, b^{2} c^{2} n - 7 \, a b c d n + 11 \, a^{2} d^{2} n\right )} e}{a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\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 )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {A}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} 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)^4,x, algorithm="maxima")

[Out]

-1/18*(6*d^3*n*e*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 6*d^3*n*e*log(d*x + c)
/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) + (6*b^2*d^2*n*x^2*e - 3*(b^2*c*d*n - 5*a*b*d^2*n)*x*
e + (2*b^2*c^2*n - 7*a*b*c*d*n + 11*a^2*d^2*n)*e)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^
5*c*d + a^2*b^4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d +
a^4*b^2*d^2)*x))*B*e^(-1) - 1/3*B*log((b*x + a)^n*e/(d*x + c)^n)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)
- 1/3*A/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (155) = 310\).
time = 0.37, size = 504, normalized size = 3.04 \begin {gather*} -\frac {6 \, {\left (A + B\right )} b^{3} c^{3} - 18 \, {\left (A + B\right )} a b^{2} c^{2} d + 18 \, {\left (A + B\right )} a^{2} b c d^{2} - 6 \, {\left (A + B\right )} a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} n x + {\left (2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} - 11 \, B a^{3} d^{3}\right )} n + 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (b x + a\right ) - 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (d x + c\right )}{18 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3} + {\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\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)^4,x, algorithm="fricas")

[Out]

-1/18*(6*(A + B)*b^3*c^3 - 18*(A + B)*a*b^2*c^2*d + 18*(A + B)*a^2*b*c*d^2 - 6*(A + B)*a^3*d^3 + 6*(B*b^3*c*d^
2 - B*a*b^2*d^3)*n*x^2 - 3*(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b*d^3)*n*x + (2*B*b^3*c^3 - 9*B*a*b^2*c^2*
d + 18*B*a^2*b*c*d^2 - 11*B*a^3*d^3)*n + 6*(B*b^3*d^3*n*x^3 + 3*B*a*b^2*d^3*n*x^2 + 3*B*a^2*b*d^3*n*x + (B*b^3
*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*n)*log(b*x + a) - 6*(B*b^3*d^3*n*x^3 + 3*B*a*b^2*d^3*n*x^2 + 3*B*a^2
*b*d^3*n*x + (B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*n)*log(d*x + c))/(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d +
3*a^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 -
3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^
5*b^2*d^3)*x)

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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)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (155) = 310\).
time = 4.21, size = 448, normalized size = 2.70 \begin {gather*} -\frac {B d^{3} n \log \left (b x + a\right )}{3 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} + \frac {B d^{3} n \log \left (d x + c\right )}{3 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {B n \log \left (b x + a\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {B n \log \left (d x + c\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {6 \, B b^{2} d^{2} n x^{2} - 3 \, B b^{2} c d n x + 15 \, B a b d^{2} n x + 2 \, B b^{2} c^{2} n - 7 \, B a b c d n + 11 \, B a^{2} d^{2} n + 6 \, A b^{2} c^{2} + 6 \, B b^{2} c^{2} - 12 \, A a b c d - 12 \, B a b c d + 6 \, A a^{2} d^{2} + 6 \, B a^{2} d^{2}}{18 \, {\left (b^{6} c^{2} x^{3} - 2 \, a b^{5} c d x^{3} + a^{2} b^{4} d^{2} x^{3} + 3 \, a b^{5} c^{2} x^{2} - 6 \, a^{2} b^{4} c d x^{2} + 3 \, a^{3} b^{3} d^{2} x^{2} + 3 \, a^{2} b^{4} c^{2} x - 6 \, a^{3} b^{3} c d x + 3 \, a^{4} b^{2} d^{2} x + a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\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)^4,x, algorithm="giac")

[Out]

-1/3*B*d^3*n*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) + 1/3*B*d^3*n*log(d*x + c)/(
b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 1/3*B*n*log(b*x + a)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b
^2*x + a^3*b) + 1/3*B*n*log(d*x + c)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/18*(6*B*b^2*d^2*n*x^2 -
3*B*b^2*c*d*n*x + 15*B*a*b*d^2*n*x + 2*B*b^2*c^2*n - 7*B*a*b*c*d*n + 11*B*a^2*d^2*n + 6*A*b^2*c^2 + 6*B*b^2*c
^2 - 12*A*a*b*c*d - 12*B*a*b*c*d + 6*A*a^2*d^2 + 6*B*a^2*d^2)/(b^6*c^2*x^3 - 2*a*b^5*c*d*x^3 + a^2*b^4*d^2*x^3
+ 3*a*b^5*c^2*x^2 - 6*a^2*b^4*c*d*x^2 + 3*a^3*b^3*d^2*x^2 + 3*a^2*b^4*c^2*x - 6*a^3*b^3*c*d*x + 3*a^4*b^2*d^2
*x + a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)

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Mupad [B]
time = 4.91, size = 317, normalized size = 1.91 \begin {gather*} \frac {2\,A\,a\,c\,d}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,b\,c^2}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{3\,b\,{\left (a+b\,x\right )}^3}-\frac {A\,a^2\,d^2}{3\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,c^2\,n}{9\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {5\,B\,a\,d^2\,n\,x}{6\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,d^2\,n\,x^2}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {7\,B\,a\,c\,d\,n}{18\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {11\,B\,a^2\,d^2\,n}{18\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,b\,c\,d\,n\,x}{6\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,d^3\,n\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{3\,b\,{\left (a\,d-b\,c\right )}^3} \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)^4,x)

[Out]

(2*A*a*c*d)/(3*(a*d - b*c)^2*(a + b*x)^3) - (A*b*c^2)/(3*(a*d - b*c)^2*(a + b*x)^3) - (B*log((e*(a + b*x)^n)/(
c + d*x)^n))/(3*b*(a + b*x)^3) - (A*a^2*d^2)/(3*b*(a*d - b*c)^2*(a + b*x)^3) - (B*b*c^2*n)/(9*(a*d - b*c)^2*(a
+ b*x)^3) - (B*d^3*n*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(3*b*(a*d - b*c)^3) - (5*B*a*d^2*n*x)
/(6*(a*d - b*c)^2*(a + b*x)^3) - (B*b*d^2*n*x^2)/(3*(a*d - b*c)^2*(a + b*x)^3) + (7*B*a*c*d*n)/(18*(a*d - b*c)
^2*(a + b*x)^3) - (11*B*a^2*d^2*n)/(18*b*(a*d - b*c)^2*(a + b*x)^3) + (B*b*c*d*n*x)/(6*(a*d - b*c)^2*(a + b*x)
^3)

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