3.2.68 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(a+b x)^2} \, dx\) [168]

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

[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 = 0.10, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2573, 2549, 2342, 2341} \begin {gather*} -\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)} \end {gather*}

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]

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2549

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/b)^m, Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x]
, x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[m,
p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

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] Leaf count is larger than twice the leaf count of optimal. \(524\) vs. \(2(184)=368\).
time = 0.47, size = 524, normalized size = 2.85 \begin {gather*} \frac {-B^3 d n^3 (a+b x) \log ^3(a+b x)+B^3 d n^3 (a+b x) \log ^3(c+d x)+3 B^2 d n^2 (a+b x) \log ^2(c+d x) \left (A+B n+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+3 B^2 d n^2 (a+b x) \log ^2(a+b x) \left (A+B n+B n \log (c+d x)+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+3 B d n (a+b x) \log (c+d x) \left (A^2+2 A B n+2 B^2 n^2+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 )\right )-(b c-a d) \left (A^3+3 A^2 B n+6 A B^2 n^2+6 B^3 n^3+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 )\right )-3 B d n (a+b x) \log (a+b x) \left (A^2+2 A B n+2 B^2 n^2+B^2 n^2 \log ^2(c+d x)+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 )+2 B n \log (c+d x) \left (A+B n+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{b (b c-a d) (a+b x)} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 14.08, size = 69354, normalized size = 376.92

method	result	size
risch	\(\text {Expression too large to display}\)	\(69354\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1092 vs. \(2 (187) = 374\).
time = 0.36, size = 1092, normalized size = 5.93 \begin {gather*} -3 \, {\left (\frac {d n e \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d n e \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {n e}{b^{2} x + a b}\right )} A^{2} B e^{\left (-1\right )} - \frac {B^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3}}{b^{2} x + a b} - 3 \, {\left (2 \, {\left (\frac {d n e \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d n e \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {n e}{b^{2} x + a b}\right )} e^{\left (-1\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) - \frac {{\left ({\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right )^{2} + {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (d x + c\right )^{2} - 2 \, {\left (b c n^{2} - a d n^{2}\right )} e^{2} - 2 \, {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b d n^{2} x e^{2} + a d n^{2} e^{2} - {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} e^{\left (-2\right )}}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x}\right )} A B^{2} - {\left (3 \, {\left (\frac {d n e \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d n e \log \left (d x + c\right )}{b^{2} c - a b d} + \frac {n e}{b^{2} x + a b}\right )} e^{\left (-1\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} - {\left (\frac {3 \, {\left ({\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right )^{2} + {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (d x + c\right )^{2} - 2 \, {\left (b c n^{2} - a d n^{2}\right )} e^{2} - 2 \, {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b d n^{2} x e^{2} + a d n^{2} e^{2} - {\left (b d n^{2} x e^{2} + a d n^{2} e^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} e^{\left (-1\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} - \frac {{\left ({\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )^{3} - {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (d x + c\right )^{3} - 3 \, {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )^{2} - 3 \, {\left (b d n^{3} x e^{3} + a d n^{3} e^{3} - {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 6 \, {\left (b c n^{3} - a d n^{3}\right )} e^{3} + 6 \, {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b d n^{3} x e^{3} + 2 \, a d n^{3} e^{3} + {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )^{2} - 2 \, {\left (b d n^{3} x e^{3} + a d n^{3} e^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} e^{\left (-2\right )}}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x}\right )} e^{\left (-1\right )}\right )} B^{3} - \frac {3 \, A B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2}}{b^{2} x + a b} - \frac {3 \, A^{2} B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{b^{2} x + a b} - \frac {A^{3}}{b^{2} x + a b} \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)))^3/(b*x+a)^2,x, algorithm="maxima")

[Out]

-3*(d*n*e*log(b*x + a)/(b^2*c - a*b*d) - d*n*e*log(d*x + c)/(b^2*c - a*b*d) + n*e/(b^2*x + a*b))*A^2*B*e^(-1)
- B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^2*x + a*b) - 3*(2*(d*n*e*log(b*x + a)/(b^2*c - a*b*d) - d*n*e*log(d*
x + c)/(b^2*c - a*b*d) + n*e/(b^2*x + a*b))*e^(-1)*log((b*x + a)^n*e/(d*x + c)^n) - ((b*d*n^2*x*e^2 + a*d*n^2*
e^2)*log(b*x + a)^2 + (b*d*n^2*x*e^2 + a*d*n^2*e^2)*log(d*x + c)^2 - 2*(b*c*n^2 - a*d*n^2)*e^2 - 2*(b*d*n^2*x*
e^2 + a*d*n^2*e^2)*log(b*x + a) + 2*(b*d*n^2*x*e^2 + a*d*n^2*e^2 - (b*d*n^2*x*e^2 + a*d*n^2*e^2)*log(b*x + a))
*log(d*x + c))*e^(-2)/(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x))*A*B^2 - (3*(d*n*e*log(b*x + a)/(b^2*c - a*b*d
) - d*n*e*log(d*x + c)/(b^2*c - a*b*d) + n*e/(b^2*x + a*b))*e^(-1)*log((b*x + a)^n*e/(d*x + c)^n)^2 - (3*((b*d
*n^2*x*e^2 + a*d*n^2*e^2)*log(b*x + a)^2 + (b*d*n^2*x*e^2 + a*d*n^2*e^2)*log(d*x + c)^2 - 2*(b*c*n^2 - a*d*n^2
)*e^2 - 2*(b*d*n^2*x*e^2 + a*d*n^2*e^2)*log(b*x + a) + 2*(b*d*n^2*x*e^2 + a*d*n^2*e^2 - (b*d*n^2*x*e^2 + a*d*n
^2*e^2)*log(b*x + a))*log(d*x + c))*e^(-1)*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) - ((b*d*n^3*x*e^3 + a*d*n^3*e^3)*log(b*x + a)^3 - (b*d*n^3*x*e^3 + a*d*n^3*e^3)*log(d*x + c)^3 - 3*(b*d*
n^3*x*e^3 + a*d*n^3*e^3)*log(b*x + a)^2 - 3*(b*d*n^3*x*e^3 + a*d*n^3*e^3 - (b*d*n^3*x*e^3 + a*d*n^3*e^3)*log(b
*x + a))*log(d*x + c)^2 + 6*(b*c*n^3 - a*d*n^3)*e^3 + 6*(b*d*n^3*x*e^3 + a*d*n^3*e^3)*log(b*x + a) - 3*(2*b*d*
n^3*x*e^3 + 2*a*d*n^3*e^3 + (b*d*n^3*x*e^3 + a*d*n^3*e^3)*log(b*x + a)^2 - 2*(b*d*n^3*x*e^3 + a*d*n^3*e^3)*log
(b*x + a))*log(d*x + c))*e^(-2)/(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x))*e^(-1))*B^3 - 3*A*B^2*log((b*x + a)
^n*e/(d*x + c)^n)^2/(b^2*x + a*b) - 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (187) = 374\).
time = 0.42, size = 673, normalized size = 3.66 \begin {gather*} -\frac {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 (A^{3} + 3 \, A^{2} B + 3 \, A B^{2} + B^{3}\right )} b c - {\left (A^{3} + 3 \, A^{2} B + 3 \, A B^{2} + B^{3}\right )} a d + 6 \, {\left ({\left (A B^{2} + B^{3}\right )} b c - {\left (A B^{2} + B^{3}\right )} a d\right )} n^{2} + 3 \, {\left (B^{3} b c n^{3} + {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (B^{3} b d n^{3} + {\left (A B^{2} + B^{3}\right )} b d n^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (B^{3} b c n^{3} + {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (B^{3} b d n^{3} + {\left (A B^{2} + B^{3}\right )} 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 )\right )} \log \left (d x + c\right )^{2} + 3 \, {\left ({\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b c - {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} a d\right )} n + 3 \, {\left (2 \, B^{3} b c n^{3} + 2 \, {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b c n + {\left (2 \, B^{3} b d n^{3} + 2 \, {\left (A B^{2} + B^{3}\right )} b d n^{2} + {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b d n\right )} x\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, B^{3} b c n^{3} + 2 \, {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} 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 (2 \, B^{3} b d n^{3} + 2 \, {\left (A B^{2} + B^{3}\right )} b d n^{2} + {\left (A^{2} B + 2 \, A B^{2} + B^{3}\right )} b d n\right )} x + 2 \, {\left (B^{3} b c n^{3} + {\left (A B^{2} + B^{3}\right )} b c n^{2} + {\left (B^{3} b d n^{3} + {\left (A B^{2} + B^{3}\right )} b d n^{2}\right )} x\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} \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)))^3/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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)))**3/(b*x+a)**2,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)))^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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Mupad [B]
time = 6.06, size = 474, normalized size = 2.58 \begin {gather*} -\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 )} \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))^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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