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

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

[Out]

-(B*(b*c - a*d)*n)/(2*(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)

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Rubi [A]  time = 0.300838, antiderivative size = 203, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {6742, 2492, 72} \[ \frac{b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac{B n (b c-a d)}{2 (g+h x) (b g-a h) (d g-c h)}+\frac{B n (b c-a d) \log (g+h x) (-a d h-b c h+2 b d g)}{2 (b g-a h)^2 (d g-c h)^2}-\frac{A}{2 h (g+h x)^2}-\frac{B d^2 n \log (c+d x)}{2 h (d g-c h)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(2*h*(g + h*x)^2) - (B*(b*c - a*d)*n)/(2*(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) - (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 6742

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

Rule 2492

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

Rule 72

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]

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*}

Mathematica [A]  time = 0.733684, size = 178, normalized size = 0.93 \[ -\frac{B n (b c-a d) \left (\frac{\frac{d^2 \log (c+d x)}{b c-a d}+\frac{h \left (\frac{(b g-a h) (d g-c h)}{g+h x}+\log (g+h x) (a d h+b c h-2 b d g)\right )}{(b g-a h)^2}}{(d g-c h)^2}-\frac{b^2 \log (a+b x)}{(b c-a d) (b g-a h)^2}\right )+\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}+\frac{A}{(g+h x)^2}}{2 h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.69, size = 4925, normalized size = 25.8 \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)))/(h*x+g)^3,x)

[Out]

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

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Maxima [B]  time = 2.42682, size = 516, normalized size = 2.7 \begin{align*} \frac{{\left (\frac{b^{2} e n \log \left (b x + a\right )}{b^{2} g^{2} h - 2 \, a b g h^{2} + a^{2} h^{3}} - \frac{d^{2} e n \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} e g n - a^{2} d^{2} e h n -{\left (2 \, c d e g n - c^{2} e h n\right )} b^{2}\right )} \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{b c e n - a d e n}{{\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}{2 \, e} - \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{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)))/(h*x+g)^3,x, algorithm="maxima")

[Out]

1/2*(b^2*e*n*log(b*x + a)/(b^2*g^2*h - 2*a*b*g*h^2 + a^2*h^3) - d^2*e*n*log(d*x + c)/(d^2*g^2*h - 2*c*d*g*h^2
+ c^2*h^3) - (2*a*b*d^2*e*g*n - a^2*d^2*e*h*n - (2*c*d*e*g*n - c^2*e*h*n)*b^2)*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*e*n - a*d*e*n)/((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 - 1/2*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 [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*log(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^3,x, algorithm="fricas")

[Out]

Timed out

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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)))/(h*x+g)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.87964, size = 1197, normalized size = 6.27 \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)))/(h*x+g)^3,x, algorithm="giac")

[Out]

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