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*(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))

________________________________________________________________________________________

Rubi [A]  time = 0.314674, 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 6742

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

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 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]

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*}

Mathematica [B]  time = 0.756876, size = 524, normalized size = 2.85 \[ \frac{-3 B d n (a+b x) \log (a+b x) \left (2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 B n \log (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+A^2+2 A B n+B^2 n^2 \log ^2(c+d x)+2 B^2 n^2\right )+3 B d n (a+b x) \log (c+d x) \left (2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+A^2+2 A B n+2 B^2 n^2\right )-(b c-a d) \left (3 B \left (A^2+2 A B n+2 B^2 n^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 B^2 (A+B n) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )+3 A^2 B n+A^3+6 A B^2 n^2+6 B^3 n^3\right )+3 B^2 d n^2 (a+b x) \log ^2(a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n \log (c+d x)+B n\right )+3 B^2 d n^2 (a+b x) \log ^2(c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n\right )+B^3 d n^3 (a+b x) \log ^3(c+d x)-B^3 d n^3 (a+b x) \log ^3(a+b x)}{b (a+b x) (b c-a d)} \]

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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Maple [C]  time = 11.812, size = 69354, normalized size = 376.9 \begin{align*} \text{output too large to display} \end{align*}

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 = 1.50655, size = 1524, normalized size = 8.28 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 1.23035, size = 1808, normalized size = 9.83 \begin{align*} \text{result too large to display} \end{align*}

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)