3.3.100 \(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(g+h x)^3} \, dx\) [300]

Optimal. Leaf size=191 \[ -\frac {B (b c-a d) n}{2 (b g-a h) (d g-c h) (g+h x)}+\frac {b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac {B d^2 n \log (c+d x)}{2 h (d g-c h)^2}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{2 (b g-a h)^2 (d g-c h)^2} \]

[Out]

-1/2*B*(-a*d+b*c)*n/(-a*h+b*g)/(-c*h+d*g)/(h*x+g)+1/2*b^2*B*n*ln(b*x+a)/h/(-a*h+b*g)^2-1/2*B*d^2*n*ln(d*x+c)/h
/(-c*h+d*g)^2+1/2*(-A-B*ln(e*(b*x+a)^n/((d*x+c)^n)))/h/(h*x+g)^2+1/2*B*(-a*d+b*c)*(-a*d*h-b*c*h+2*b*d*g)*n*ln(
h*x+g)/(-a*h+b*g)^2/(-c*h+d*g)^2

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Rubi [A]
time = 0.15, antiderivative size = 191, 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, 84} \begin {gather*} -\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{2 h (g+h x)^2}+\frac {b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac {B n (b c-a d)}{2 (g+h x) (b g-a h) (d g-c h)}+\frac {B n (b c-a d) \log (g+h x) (-a d h-b c h+2 b d g)}{2 (b g-a h)^2 (d g-c h)^2}-\frac {B d^2 n \log (c+d x)}{2 h (d g-c h)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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

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Mathematica [A]
time = 0.25, size = 238, normalized size = 1.25 \begin {gather*} \frac {1}{2} \left (\frac {B (-b c+a d) n}{(b g-a h) (d g-c h) (g+h x)}+\frac {b^2 B n \log (a+b x)}{h (b g-a h)^2}-\frac {B n \log (a+b x)}{h (g+h x)^2}-\frac {B d^2 n \log (c+d x)}{h (d g-c h)^2}+\frac {B n \log (c+d x)}{h (g+h x)^2}-\frac {A+B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{h (g+h x)^2}-\frac {B (b c-a d) (-2 b d g+b c h+a d h) n \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.70, size = 4925, normalized size = 25.79

method	result	size
risch	\(\text {Expression too large to display}\)	\(4925\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^3,x,method=_RETURNVERBOSE)

[Out]

1/2*B/h/(h*x+g)^2*ln((d*x+c)^n)-1/4*(-2*I*B*Pi*a*b*c^2*g*h^3*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-2*I*B
*Pi*a*b*c^2*g*h^3*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-2*I*B*Pi*a*b*c^2*g*h^3*csgn(I/((d*x+c)^n))
*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-4*B*a^2*c*d*g*h^3*ln((b*x+a)^n)-4*B*a*b*c^2*g*h^3*ln((b*x+a)^n)-4*B*a*b*d^2*g
^3*h*ln((b*x+a)^n)-4*B*b^2*c*d*g^3*h*ln((b*x+a)^n)+2*B*ln(e)*a^2*d^2*g^2*h^2+2*B*ln(e)*b^2*c^2*g^2*h^2+I*B*Pi*
a^2*d^2*g^2*h^2*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+I*B*Pi*b^2*c^2*g^2*h^2*csgn(I*
e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-4*A*a^2*c*d*g*h^3-4*A*a*b*c^2*g*h^3-4*A*a*b*d^2*g^3*h-4*A*b^2*c*d*g^3*h-4
*B*ln(e)*a*b*c^2*g*h^3-4*B*ln(e)*a*b*d^2*g^3*h-2*B*a^2*c*d*h^4*n*x+2*B*a^2*d^2*g*h^3*n*x+2*B*a*b*c^2*h^4*n*x-2
*B*b^2*c^2*g*h^3*n*x+8*A*a*b*c*d*g^2*h^2-I*B*Pi*a^2*d^2*g^2*h^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-I*B*Pi*b^2*c
^2*g^2*h^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-2*I*B*Pi*a*b*d^2*g^3*h*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-
2*I*B*Pi*a*b*d^2*g^3*h*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+2*I*B*Pi*a*b*d^2*g^3*h*csgn(I*(b*x+a)
^n/((d*x+c)^n))^3-2*I*B*Pi*a*b*d^2*g^3*h*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-2*I*B*Pi*a*b*d^2*
g^3*h*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-4*I*B*Pi*a*b*c*d*g^2*h^2*csgn(I*(b*x+a)^
n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+2*A*a^2*d^2*g^2*h^2+2*A*b^2*c^2*g^2*h^2+I*B*Pi*b^2*c^2*g^
2*h^2*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*B*Pi*b^2*c^2*g^2*h^2*csgn(I*(b*x+a)^n/((d*x+c)^n))
*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-I*B*Pi*a^2*c^2*h^4*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/(
(d*x+c)^n))+2*I*B*Pi*a*b*d^2*g^3*h*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+I*B*Pi*a^2*d^2*g^2*h^2*csgn(I*(b*x+a)^n)*
csgn(I*(b*x+a)^n/((d*x+c)^n))^2-2*B*a^2*c*d*g*h^3*n+2*B*a*b*c^2*g*h^3*n-2*B*a*b*d^2*g^3*h*n+2*B*b^2*c*d*g^3*h*
n+2*I*B*Pi*a*b*c^2*g*h^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-2*I*B*Pi*b^2*c*d*g^3*h*csgn(I*(b*x+a)^n/((d*x+c)^n)
)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+2*I*B*Pi*a*b*c^2*g*h^3*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a
)^n/((d*x+c)^n))+2*I*B*Pi*a*b*d^2*g^3*h*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n
)-4*I*B*Pi*a*b*c*d*g^2*h^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)+2*I*B*Pi*a^
2*c*d*g*h^3*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+8*B*ln(e)*a*b*c*d*g^2*h^2+2*I*
B*Pi*b^2*c*d*g^3*h*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+2*I*B*Pi*b^2*c*d*g^3*h*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-I*
B*Pi*a^2*c^2*h^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)-8*B*ln(-d*x-c)*a*b*d^
2*g^2*h^2*n*x+8*B*ln(-h*x-g)*a*b*d^2*g^2*h^2*n*x-8*B*ln(-h*x-g)*b^2*c*d*g^2*h^2*n*x-4*B*ln(-d*x-c)*a*b*d^2*g*h
^3*n*x^2+4*B*ln(-h*x-g)*a*b*d^2*g*h^3*n*x^2-4*B*ln(-h*x-g)*b^2*c*d*g*h^3*n*x^2+4*B*ln(b*x+a)*b^2*c*d*g*h^3*n*x
^2+8*B*ln(b*x+a)*b^2*c*d*g^2*h^2*n*x+2*I*B*Pi*b^2*c*d*g^3*h*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/(
(d*x+c)^n)*(b*x+a)^n)+2*I*B*Pi*b^2*c*d*g^3*h*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n
))+I*B*Pi*a^2*c^2*h^4*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+I*B*Pi*a^2*c^2*h^4*csgn(I*(b*x+a)^n)*csgn(I*
(b*x+a)^n/((d*x+c)^n))^2+I*B*Pi*a^2*c^2*h^4*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*B*Pi*a^2*c^2
*h^4*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+2*B*a^2*c^2*h^4*ln((b*x+a)^n)+2*B*b^2*d^2
*g^4*ln((b*x+a)^n)+4*I*B*Pi*a*b*c*d*g^2*h^2*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-I*B*Pi*b^2*d^2*g^4*csg
n(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+I*B*Pi*a^2*d^2*g^2*h^2*csgn(I/((d*x+c)^n))*cs
gn(I*(b*x+a)^n/((d*x+c)^n))^2+I*B*Pi*b^2*c^2*g^2*h^2*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-2*I*B*P
i*b^2*c*d*g^3*h*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-2*I*B*Pi*b^2*c*d*g^3*h*csgn(I*(b*x+a)^n)*csgn(I*(b
*x+a)^n/((d*x+c)^n))^2-2*I*B*Pi*b^2*c*d*g^3*h*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-4*I*B*Pi*a*b
*c*d*g^2*h^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-2*I*B*Pi*a^2*c*d*g*h^3*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^
2-2*I*B*Pi*a*b*c^2*g*h^3*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-I*B*Pi*b^2*c^2*g^2*h^
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)+2*I*B*Pi*a*b*c^2*g*h^3*csgn(I*e)*csg
n(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-4*B*ln(e)*b^2*c*d*g^3*h-4*B*ln(e)*a^2*c*d*g*h^3-I*B
*Pi*a^2*c^2*h^4*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-I*B*Pi*a^2*c^2*h^4*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-I*B*Pi*b^
2*d^2*g^4*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+2*B*ln(-d*x-c)*b^2*d^2*g^2*h^2*n*x^2-2*B*ln(b*x+a)*b^2*d^2*g^2*h^2*n
*x^2+4*B*ln(-d*x-c)*a^2*d^2*g*h^3*n*x+4*B*ln(-d*x-c)*b^2*d^2*g^3*h*n*x-4*B*ln(-h*x-g)*a^2*d^2*g*h^3*n*x+4*B*ln
(-h*x-g)*b^2*c^2*g*h^3*n*x-4*B*ln(b*x+a)*b^2*c^2*g*h^3*n*x-4*B*ln(b*x+a)*b^2*d^2*g^3*h*n*x-4*B*ln(-d*x-c)*a*b*
d^2*g^3*h*n-4*I*B*Pi*a*b*c*d*g^2*h^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-I*B*Pi*a^2*d^2*g^2*h^2*csgn(I*(b*x+a)^n
/((d*x+c)^n))^3+2*B*a^2*d^2*g^2*h^2*ln((b*x+a)^...

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (182) = 364\).
time = 0.30, size = 382, normalized size = 2.00 \begin {gather*} \frac {1}{2} \, {\left (\frac {b^{2} n e \log \left (b x + a\right )}{b^{2} g^{2} h - 2 \, a b g h^{2} + a^{2} h^{3}} - \frac {d^{2} n e \log \left (d x + c\right )}{d^{2} g^{2} h - 2 \, c d g h^{2} + c^{2} h^{3}} - \frac {{\left (2 \, a b d^{2} g n - a^{2} d^{2} h n - {\left (2 \, c d g n - c^{2} h n\right )} b^{2}\right )} e \log \left (h x + g\right )}{{\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} a^{2} - 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} a b + {\left (d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2}\right )} b^{2}} + \frac {{\left (b c n - a d n\right )} e}{{\left (d g^{2} h - c g h^{2}\right )} a - {\left (d g^{3} - c g^{2} h\right )} b + {\left ({\left (d g h^{2} - c h^{3}\right )} a - {\left (d g^{2} h - c g h^{2}\right )} b\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 )}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} - \frac {A}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\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)))/(h*x+g)^3,x, algorithm="maxima")

[Out]

1/2*(b^2*n*e*log(b*x + a)/(b^2*g^2*h - 2*a*b*g*h^2 + a^2*h^3) - d^2*n*e*log(d*x + c)/(d^2*g^2*h - 2*c*d*g*h^2
+ c^2*h^3) - (2*a*b*d^2*g*n - a^2*d^2*h*n - (2*c*d*g*n - c^2*h*n)*b^2)*e*log(h*x + g)/((d^2*g^2*h^2 - 2*c*d*g*
h^3 + c^2*h^4)*a^2 - 2*(d^2*g^3*h - 2*c*d*g^2*h^2 + c^2*g*h^3)*a*b + (d^2*g^4 - 2*c*d*g^3*h + c^2*g^2*h^2)*b^2
) + (b*c*n - a*d*n)*e/((d*g^2*h - c*g*h^2)*a - (d*g^3 - c*g^2*h)*b + ((d*g*h^2 - c*h^3)*a - (d*g^2*h - c*g*h^2
)*b)*x))*B*e^(-1) - 1/2*B*log((b*x + a)^n*e/(d*x + c)^n)/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 1/2*A/(h^3*x^2 + 2*g*
h^2*x + g^2*h)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (182) = 364\).
time = 45.46, size = 1046, normalized size = 5.48 \begin {gather*} -\frac {{\left (A + B\right )} b^{2} d^{2} g^{4} + {\left (A + B\right )} a^{2} c^{2} h^{4} - 2 \, {\left ({\left (A + B\right )} b^{2} c d + {\left (A + B\right )} a b d^{2}\right )} g^{3} h + {\left ({\left (A + B\right )} b^{2} c^{2} + 4 \, {\left (A + B\right )} a b c d + {\left (A + B\right )} a^{2} d^{2}\right )} g^{2} h^{2} - 2 \, {\left ({\left (A + B\right )} a b c^{2} + {\left (A + B\right )} a^{2} c d\right )} g h^{3} + {\left ({\left (B b^{2} c d - B a b d^{2}\right )} g^{2} h^{2} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} g h^{3} + {\left (B a b c^{2} - B a^{2} c d\right )} h^{4}\right )} n x + {\left ({\left (B b^{2} c d - B a b d^{2}\right )} g^{3} h - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} g^{2} h^{2} + {\left (B a b c^{2} - B a^{2} c d\right )} g h^{3}\right )} n - {\left ({\left (B b^{2} d^{2} g^{2} h^{2} - 2 \, B b^{2} c d g h^{3} + B b^{2} c^{2} h^{4}\right )} n x^{2} + 2 \, {\left (B b^{2} d^{2} g^{3} h - 2 \, B b^{2} c d g^{2} h^{2} + B b^{2} c^{2} g h^{3}\right )} n x + {\left (2 \, B a b d^{2} g^{3} h - B a^{2} c^{2} h^{4} - {\left (4 \, B a b c d + B a^{2} d^{2}\right )} g^{2} h^{2} + 2 \, {\left (B a b c^{2} + B a^{2} c d\right )} g h^{3}\right )} n\right )} \log \left (b x + a\right ) + {\left ({\left (B b^{2} d^{2} g^{2} h^{2} - 2 \, B a b d^{2} g h^{3} + B a^{2} d^{2} h^{4}\right )} n x^{2} + 2 \, {\left (B b^{2} d^{2} g^{3} h - 2 \, B a b d^{2} g^{2} h^{2} + B a^{2} d^{2} g h^{3}\right )} n x + {\left (2 \, B b^{2} c d g^{3} h - B a^{2} c^{2} h^{4} - {\left (B b^{2} c^{2} + 4 \, B a b c d\right )} g^{2} h^{2} + 2 \, {\left (B a b c^{2} + B a^{2} c d\right )} g h^{3}\right )} n\right )} \log \left (d x + c\right ) - {\left ({\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} g h^{3} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} h^{4}\right )} n x^{2} + 2 \, {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} g^{2} h^{2} - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} g h^{3}\right )} n x + {\left (2 \, {\left (B b^{2} c d - B a b d^{2}\right )} g^{3} h - {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} g^{2} h^{2}\right )} n\right )} \log \left (h x + g\right )}{2 \, {\left (b^{2} d^{2} g^{6} h + a^{2} c^{2} g^{2} h^{5} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} g^{5} h^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} g^{4} h^{3} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} g^{3} h^{4} + {\left (b^{2} d^{2} g^{4} h^{3} + a^{2} c^{2} h^{7} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} g^{3} h^{4} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} g^{2} h^{5} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} g h^{6}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} g^{5} h^{2} + a^{2} c^{2} g h^{6} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} g^{4} h^{3} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} g^{3} h^{4} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} g^{2} h^{5}\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)))/(h*x+g)^3,x, algorithm="fricas")

[Out]

-1/2*((A + B)*b^2*d^2*g^4 + (A + B)*a^2*c^2*h^4 - 2*((A + B)*b^2*c*d + (A + B)*a*b*d^2)*g^3*h + ((A + B)*b^2*c
^2 + 4*(A + B)*a*b*c*d + (A + B)*a^2*d^2)*g^2*h^2 - 2*((A + B)*a*b*c^2 + (A + B)*a^2*c*d)*g*h^3 + ((B*b^2*c*d
- B*a*b*d^2)*g^2*h^2 - (B*b^2*c^2 - B*a^2*d^2)*g*h^3 + (B*a*b*c^2 - B*a^2*c*d)*h^4)*n*x + ((B*b^2*c*d - B*a*b*
d^2)*g^3*h - (B*b^2*c^2 - B*a^2*d^2)*g^2*h^2 + (B*a*b*c^2 - B*a^2*c*d)*g*h^3)*n - ((B*b^2*d^2*g^2*h^2 - 2*B*b^
2*c*d*g*h^3 + B*b^2*c^2*h^4)*n*x^2 + 2*(B*b^2*d^2*g^3*h - 2*B*b^2*c*d*g^2*h^2 + B*b^2*c^2*g*h^3)*n*x + (2*B*a*
b*d^2*g^3*h - B*a^2*c^2*h^4 - (4*B*a*b*c*d + B*a^2*d^2)*g^2*h^2 + 2*(B*a*b*c^2 + B*a^2*c*d)*g*h^3)*n)*log(b*x
+ a) + ((B*b^2*d^2*g^2*h^2 - 2*B*a*b*d^2*g*h^3 + B*a^2*d^2*h^4)*n*x^2 + 2*(B*b^2*d^2*g^3*h - 2*B*a*b*d^2*g^2*h
^2 + B*a^2*d^2*g*h^3)*n*x + (2*B*b^2*c*d*g^3*h - B*a^2*c^2*h^4 - (B*b^2*c^2 + 4*B*a*b*c*d)*g^2*h^2 + 2*(B*a*b*
c^2 + B*a^2*c*d)*g*h^3)*n)*log(d*x + c) - ((2*(B*b^2*c*d - B*a*b*d^2)*g*h^3 - (B*b^2*c^2 - B*a^2*d^2)*h^4)*n*x
^2 + 2*(2*(B*b^2*c*d - B*a*b*d^2)*g^2*h^2 - (B*b^2*c^2 - B*a^2*d^2)*g*h^3)*n*x + (2*(B*b^2*c*d - B*a*b*d^2)*g^
3*h - (B*b^2*c^2 - B*a^2*d^2)*g^2*h^2)*n)*log(h*x + g))/(b^2*d^2*g^6*h + a^2*c^2*g^2*h^5 - 2*(b^2*c*d + a*b*d^
2)*g^5*h^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*g^4*h^3 - 2*(a*b*c^2 + a^2*c*d)*g^3*h^4 + (b^2*d^2*g^4*h^3 + a^2*
c^2*h^7 - 2*(b^2*c*d + a*b*d^2)*g^3*h^4 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*g^2*h^5 - 2*(a*b*c^2 + a^2*c*d)*g*h^
6)*x^2 + 2*(b^2*d^2*g^5*h^2 + a^2*c^2*g*h^6 - 2*(b^2*c*d + a*b*d^2)*g^4*h^3 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*
g^3*h^4 - 2*(a*b*c^2 + a^2*c*d)*g^2*h^5)*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)))/(h*x+g)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (182) = 364\).
time = 2.72, size = 523, normalized size = 2.74 \begin {gather*} \frac {B b^{3} n \log \left ({\left | b x + a \right |}\right )}{2 \, {\left (b^{3} g^{2} h - 2 \, a b^{2} g h^{2} + a^{2} b h^{3}\right )}} - \frac {B d^{3} n \log \left ({\left | d x + c \right |}\right )}{2 \, {\left (d^{3} g^{2} h - 2 \, c d^{2} g h^{2} + c^{2} d h^{3}\right )}} - \frac {B n \log \left (b x + a\right )}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} + \frac {B n \log \left (d x + c\right )}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} + \frac {{\left (2 \, B b^{2} c d g n - 2 \, B a b d^{2} g n - B b^{2} c^{2} h n + B a^{2} d^{2} h n\right )} \log \left (h x + g\right )}{2 \, {\left (b^{2} d^{2} g^{4} - 2 \, b^{2} c d g^{3} h - 2 \, a b d^{2} g^{3} h + b^{2} c^{2} g^{2} h^{2} + 4 \, a b c d g^{2} h^{2} + a^{2} d^{2} g^{2} h^{2} - 2 \, a b c^{2} g h^{3} - 2 \, a^{2} c d g h^{3} + a^{2} c^{2} h^{4}\right )}} - \frac {B b c h^{2} n x - B a d h^{2} n x + B b c g h n - B a d g h n + A b d g^{2} + B b d g^{2} - A b c g h - B b c g h - A a d g h - B a d g h + A a c h^{2} + B a c h^{2}}{2 \, {\left (b d g^{2} h^{3} x^{2} - b c g h^{4} x^{2} - a d g h^{4} x^{2} + a c h^{5} x^{2} + 2 \, b d g^{3} h^{2} x - 2 \, b c g^{2} h^{3} x - 2 \, a d g^{2} h^{3} x + 2 \, a c g h^{4} x + b d g^{4} h - b c g^{3} h^{2} - a d g^{3} h^{2} + a c g^{2} h^{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)))/(h*x+g)^3,x, algorithm="giac")

[Out]

1/2*B*b^3*n*log(abs(b*x + a))/(b^3*g^2*h - 2*a*b^2*g*h^2 + a^2*b*h^3) - 1/2*B*d^3*n*log(abs(d*x + c))/(d^3*g^2
*h - 2*c*d^2*g*h^2 + c^2*d*h^3) - 1/2*B*n*log(b*x + a)/(h^3*x^2 + 2*g*h^2*x + g^2*h) + 1/2*B*n*log(d*x + c)/(h
^3*x^2 + 2*g*h^2*x + g^2*h) + 1/2*(2*B*b^2*c*d*g*n - 2*B*a*b*d^2*g*n - B*b^2*c^2*h*n + B*a^2*d^2*h*n)*log(h*x
+ g)/(b^2*d^2*g^4 - 2*b^2*c*d*g^3*h - 2*a*b*d^2*g^3*h + b^2*c^2*g^2*h^2 + 4*a*b*c*d*g^2*h^2 + a^2*d^2*g^2*h^2
- 2*a*b*c^2*g*h^3 - 2*a^2*c*d*g*h^3 + a^2*c^2*h^4) - 1/2*(B*b*c*h^2*n*x - B*a*d*h^2*n*x + B*b*c*g*h*n - B*a*d*
g*h*n + A*b*d*g^2 + B*b*d*g^2 - A*b*c*g*h - B*b*c*g*h - A*a*d*g*h - B*a*d*g*h + A*a*c*h^2 + B*a*c*h^2)/(b*d*g^
2*h^3*x^2 - b*c*g*h^4*x^2 - a*d*g*h^4*x^2 + a*c*h^5*x^2 + 2*b*d*g^3*h^2*x - 2*b*c*g^2*h^3*x - 2*a*d*g^2*h^3*x
+ 2*a*c*g*h^4*x + b*d*g^4*h - b*c*g^3*h^2 - a*d*g^3*h^2 + a*c*g^2*h^3)

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Mupad [B]
time = 6.35, size = 431, normalized size = 2.26 \begin {gather*} \frac {\ln \left (g+h\,x\right )\,\left (h\,\left (B\,a^2\,d^2\,n-B\,b^2\,c^2\,n\right )-2\,B\,a\,b\,d^2\,g\,n+2\,B\,b^2\,c\,d\,g\,n\right )}{2\,a^2\,c^2\,h^4-4\,a^2\,c\,d\,g\,h^3+2\,a^2\,d^2\,g^2\,h^2-4\,a\,b\,c^2\,g\,h^3+8\,a\,b\,c\,d\,g^2\,h^2-4\,a\,b\,d^2\,g^3\,h+2\,b^2\,c^2\,g^2\,h^2-4\,b^2\,c\,d\,g^3\,h+2\,b^2\,d^2\,g^4}-\frac {\frac {A\,a\,c\,h^2+A\,b\,d\,g^2-A\,a\,d\,g\,h-A\,b\,c\,g\,h-B\,a\,d\,g\,h\,n+B\,b\,c\,g\,h\,n}{a\,c\,h^2+b\,d\,g^2-a\,d\,g\,h-b\,c\,g\,h}-\frac {x\,\left (B\,a\,d\,h^2\,n-B\,b\,c\,h^2\,n\right )}{a\,c\,h^2+b\,d\,g^2-a\,d\,g\,h-b\,c\,g\,h}}{2\,g^2\,h+4\,g\,h^2\,x+2\,h^3\,x^2}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{2\,h\,\left (g^2+2\,g\,h\,x+h^2\,x^2\right )}+\frac {B\,b^2\,n\,\ln \left (a+b\,x\right )}{2\,a^2\,h^3-4\,a\,b\,g\,h^2+2\,b^2\,g^2\,h}-\frac {B\,d^2\,n\,\ln \left (c+d\,x\right )}{2\,c^2\,h^3-4\,c\,d\,g\,h^2+2\,d^2\,g^2\,h} \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))/(g + h*x)^3,x)

[Out]

(log(g + h*x)*(h*(B*a^2*d^2*n - B*b^2*c^2*n) - 2*B*a*b*d^2*g*n + 2*B*b^2*c*d*g*n))/(2*a^2*c^2*h^4 + 2*b^2*d^2*
g^4 + 2*a^2*d^2*g^2*h^2 + 2*b^2*c^2*g^2*h^2 - 4*a*b*c^2*g*h^3 - 4*a*b*d^2*g^3*h - 4*a^2*c*d*g*h^3 - 4*b^2*c*d*
g^3*h + 8*a*b*c*d*g^2*h^2) - ((A*a*c*h^2 + A*b*d*g^2 - A*a*d*g*h - A*b*c*g*h - B*a*d*g*h*n + B*b*c*g*h*n)/(a*c
*h^2 + b*d*g^2 - a*d*g*h - b*c*g*h) - (x*(B*a*d*h^2*n - B*b*c*h^2*n))/(a*c*h^2 + b*d*g^2 - a*d*g*h - b*c*g*h))
/(2*g^2*h + 2*h^3*x^2 + 4*g*h^2*x) - (B*log((e*(a + b*x)^n)/(c + d*x)^n))/(2*h*(g^2 + h^2*x^2 + 2*g*h*x)) + (B
*b^2*n*log(a + b*x))/(2*a^2*h^3 + 2*b^2*g^2*h - 4*a*b*g*h^2) - (B*d^2*n*log(c + d*x))/(2*c^2*h^3 + 2*d^2*g^2*h
- 4*c*d*g*h^2)

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