3.168 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(a+b x)^2} \, dx\)

Optimal. Leaf size=184 \[ -\frac {6 B^2 n^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)}-\frac {3 B n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)}-\frac {(c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x) (b c-a d)}-\frac {6 B^3 n^3 (c+d x)}{(a+b x) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.31, antiderivative size = 360, normalized size of antiderivative = 1.96, number of steps used = 11, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6742, 2490, 32} \[ -\frac {3 A^2 B (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {3 A^2 B n}{b (a+b x)}-\frac {A^3}{b (a+b x)}-\frac {3 A B^2 (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {6 A B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {6 A B^2 n^2}{b (a+b x)}-\frac {6 B^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {B^3 (c+d x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {3 B^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)}-\frac {6 B^3 n^3}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^2,x]

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2490

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_))^
2, x_Symbol] :> Simp[((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/((b*g - a*h)*(g + h*x)), x] - Dist[(p*
r*s*(b*c - a*d))/(b*g - a*h), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((c + d*x)*(g + h*x)), x], x] /
; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[b*g - a*h, 0] &&
IGtQ[s, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [B]  time = 0.79, size = 524, normalized size = 2.85 \[ \frac {-3 B d n (a+b x) \log (a+b x) \left (2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 B n \log (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+A^2+2 A B n+B^2 n^2 \log ^2(c+d x)+2 B^2 n^2\right )+3 B d n (a+b x) \log (c+d x) \left (2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+A^2+2 A B n+2 B^2 n^2\right )-(b c-a d) \left (3 B \left (A^2+2 A B n+2 B^2 n^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 B^2 (A+B n) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )+A^3+3 A^2 B n+6 A B^2 n^2+6 B^3 n^3\right )+3 B^2 d n^2 (a+b x) \log ^2(a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n \log (c+d x)+B n\right )+3 B^2 d n^2 (a+b x) \log ^2(c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n\right )+B^3 d n^3 (a+b x) \log ^3(c+d x)-B^3 d n^3 (a+b x) \log ^3(a+b x)}{b (a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^2,x]

[Out]

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

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fricas [B]  time = 0.92, size = 825, normalized size = 4.48 \[ -\frac {A^{3} b c - A^{3} a d + 6 \, {\left (B^{3} b c - B^{3} a d\right )} n^{3} + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right )^{3} - {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (d x + c\right )^{3} + {\left (B^{3} b c - B^{3} a d\right )} \log \relax (e)^{3} + 6 \, {\left (A B^{2} b c - A B^{2} a d\right )} n^{2} + 3 \, {\left (B^{3} b c n^{3} + A B^{2} b c n^{2} + {\left (B^{3} b d n^{3} + A B^{2} b d n^{2}\right )} x + {\left (B^{3} b d n^{2} x + B^{3} b c n^{2}\right )} \log \relax (e)\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (B^{3} b c n^{3} + A B^{2} b c n^{2} + {\left (B^{3} b d n^{3} + A B^{2} b d n^{2}\right )} x + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right ) + {\left (B^{3} b d n^{2} x + B^{3} b c n^{2}\right )} \log \relax (e)\right )} \log \left (d x + c\right )^{2} + 3 \, {\left (A B^{2} b c - A B^{2} a d + {\left (B^{3} b c - B^{3} a d\right )} n\right )} \log \relax (e)^{2} + 3 \, {\left (A^{2} B b c - A^{2} B a d\right )} n + 3 \, {\left (2 \, B^{3} b c n^{3} + 2 \, A B^{2} b c n^{2} + A^{2} B b c n + {\left (B^{3} b d n x + B^{3} b c n\right )} \log \relax (e)^{2} + {\left (2 \, B^{3} b d n^{3} + 2 \, A B^{2} b d n^{2} + A^{2} B b d n\right )} x + 2 \, {\left (B^{3} b c n^{2} + A B^{2} b c n + {\left (B^{3} b d n^{2} + A B^{2} b d n\right )} x\right )} \log \relax (e)\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, B^{3} b c n^{3} + 2 \, A B^{2} b c n^{2} + A^{2} B b c n + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right )^{2} + {\left (B^{3} b d n x + B^{3} b c n\right )} \log \relax (e)^{2} + {\left (2 \, B^{3} b d n^{3} + 2 \, A B^{2} b d n^{2} + A^{2} B b d n\right )} x + 2 \, {\left (B^{3} b c n^{3} + A B^{2} b c n^{2} + {\left (B^{3} b d n^{3} + A B^{2} b d n^{2}\right )} x + {\left (B^{3} b d n^{2} x + B^{3} b c n^{2}\right )} \log \relax (e)\right )} \log \left (b x + a\right ) + 2 \, {\left (B^{3} b c n^{2} + A B^{2} b c n + {\left (B^{3} b d n^{2} + A B^{2} b d n\right )} x\right )} \log \relax (e)\right )} \log \left (d x + c\right ) + 3 \, {\left (A^{2} B b c - A^{2} B a d + 2 \, {\left (B^{3} b c - B^{3} a d\right )} n^{2} + 2 \, {\left (A B^{2} b c - A B^{2} a d\right )} n\right )} \log \relax (e)}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^3/(b*x + a)^2, x)

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maple [C]  time = 20.96, size = 69354, normalized size = 376.92 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^2,x)

[Out]

result too large to display

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maxima [B]  time = 2.16, size = 1129, normalized size = 6.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.06, size = 474, normalized size = 2.58 \[ -\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (\frac {3\,B\,b\,d\,A^2\,x^2+3\,B\,\left (a\,d+b\,c\right )\,A^2\,x+3\,B\,a\,c\,A^2}{b\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}+\frac {6\,d\,\left (n\,B^3+A\,B^2\right )\,\left (b^2\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,b\,c\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{d}\right )}{b^2\,\left (a\,d-b\,c\right )\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}\right )-\frac {A^3+3\,A^2\,B\,n+6\,A\,B^2\,n^2+6\,B^3\,n^3}{x\,b^2+a\,b}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\,\left (\frac {3\,A\,B^2}{x\,b^2+a\,b}+\frac {3\,B^3\,n}{x\,b^2+a\,b}-\frac {3\,d\,\left (n\,B^3+A\,B^2\right )}{b\,\left (a\,d-b\,c\right )}\right )-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^3\,\left (\frac {B^3}{b\,\left (a+b\,x\right )}-\frac {B^3\,d}{b\,\left (a\,d-b\,c\right )}\right )-\frac {B\,d\,n\,\mathrm {atan}\left (\frac {B\,d\,n\,\left (\frac {c\,b^2+a\,d\,b}{b}+2\,b\,d\,x\right )\,\left (A^2+2\,A\,B\,n+2\,B^2\,n^2\right )\,3{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (3\,d\,A^2\,B\,n+6\,d\,A\,B^2\,n^2+6\,d\,B^3\,n^3\right )}\right )\,\left (A^2+2\,A\,B\,n+2\,B^2\,n^2\right )\,6{}\mathrm {i}}{b\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3/(a + b*x)^2,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3/(b*x+a)**2,x)

[Out]

Timed out

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