3.302 \(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(g+h x)^5} \, dx\)

Optimal. Leaf size=389 \[ -\frac {B n (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right )}{4 (g+h x) (b g-a h)^3 (d g-c h)^3}-\frac {B n (b c-a d) \log (g+h x) (-a d h-b c h+2 b d g) \left (-a^2 d^2 h^2+2 a b d^2 g h-\left (b^2 \left (c^2 h^2-2 c d g h+2 d^2 g^2\right )\right )\right )}{4 (b g-a h)^4 (d g-c h)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{4 h (g+h x)^4}+\frac {b^4 B n \log (a+b x)}{4 h (b g-a h)^4}-\frac {B n (b c-a d) (-a d h-b c h+2 b d g)}{8 (g+h x)^2 (b g-a h)^2 (d g-c h)^2}-\frac {B n (b c-a d)}{12 (g+h x)^3 (b g-a h) (d g-c h)}-\frac {B d^4 n \log (c+d x)}{4 h (d g-c h)^4} \]

[Out]

-1/12*B*(-a*d+b*c)*n/(-a*h+b*g)/(-c*h+d*g)/(h*x+g)^3-1/8*B*(-a*d+b*c)*(-a*d*h-b*c*h+2*b*d*g)*n/(-a*h+b*g)^2/(-
c*h+d*g)^2/(h*x+g)^2-1/4*B*(-a*d+b*c)*(a^2*d^2*h^2-a*b*d*h*(-c*h+3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d^2*g^2))*n/(
-a*h+b*g)^3/(-c*h+d*g)^3/(h*x+g)+1/4*b^4*B*n*ln(b*x+a)/h/(-a*h+b*g)^4-1/4*B*d^4*n*ln(d*x+c)/h/(-c*h+d*g)^4+1/4
*(-A-B*ln(e*(b*x+a)^n/((d*x+c)^n)))/h/(h*x+g)^4-1/4*B*(-a*d+b*c)*(-a*d*h-b*c*h+2*b*d*g)*(2*a*b*d^2*g*h-a^2*d^2
*h^2-b^2*(c^2*h^2-2*c*d*g*h+2*d^2*g^2))*n*ln(h*x+g)/(-a*h+b*g)^4/(-c*h+d*g)^4

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Rubi [A]  time = 0.82, antiderivative size = 401, normalized size of antiderivative = 1.03, 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} \[ -\frac {B n (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right )}{4 (g+h x) (b g-a h)^3 (d g-c h)^3}-\frac {B n (b c-a d) \log (g+h x) (-a d h-b c h+2 b d g) \left (-a^2 d^2 h^2+2 a b d^2 g h+b^2 \left (-\left (c^2 h^2-2 c d g h+2 d^2 g^2\right )\right )\right )}{4 (b g-a h)^4 (d g-c h)^4}+\frac {b^4 B n \log (a+b x)}{4 h (b g-a h)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 h (g+h x)^4}-\frac {B n (b c-a d) (-a d h-b c h+2 b d g)}{8 (g+h x)^2 (b g-a h)^2 (d g-c h)^2}-\frac {B n (b c-a d)}{12 (g+h x)^3 (b g-a h) (d g-c h)}-\frac {A}{4 h (g+h x)^4}-\frac {B d^4 n \log (c+d x)}{4 h (d g-c h)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(g + h*x)^5,x]

[Out]

-A/(4*h*(g + h*x)^4) - (B*(b*c - a*d)*n)/(12*(b*g - a*h)*(d*g - c*h)*(g + h*x)^3) - (B*(b*c - a*d)*(2*b*d*g -
b*c*h - a*d*h)*n)/(8*(b*g - a*h)^2*(d*g - c*h)^2*(g + h*x)^2) - (B*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g -
c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n)/(4*(b*g - a*h)^3*(d*g - c*h)^3*(g + h*x)) + (b^4*B*n*Log[a +
b*x])/(4*h*(b*g - a*h)^4) - (B*d^4*n*Log[c + d*x])/(4*h*(d*g - c*h)^4) - (B*Log[(e*(a + b*x)^n)/(c + d*x)^n])
/(4*h*(g + h*x)^4) - (B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*(2*a*b*d^2*g*h - a^2*d^2*h^2 - b^2*(2*d^2*g^2 -
2*c*d*g*h + c^2*h^2))*n*Log[g + h*x])/(4*(b*g - a*h)^4*(d*g - c*h)^4)

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]

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 6742

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

Rubi steps

\begin {align*} \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^5} \, dx &=\int \left (\frac {A}{(g+h x)^5}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^5}\right ) \, dx\\ &=-\frac {A}{4 h (g+h x)^4}+B \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^5} \, dx\\ &=-\frac {A}{4 h (g+h x)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 h (g+h x)^4}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x) (c+d x) (g+h x)^4} \, dx}{4 h}\\ &=-\frac {A}{4 h (g+h x)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 h (g+h x)^4}+\frac {(B (b c-a d) n) \int \left (\frac {b^5}{(b c-a d) (b g-a h)^4 (a+b x)}-\frac {d^5}{(b c-a d) (-d g+c h)^4 (c+d x)}+\frac {h^2}{(b g-a h) (d g-c h) (g+h x)^4}-\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)^3}+\frac {h^2 \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right )}{(b g-a h)^3 (d g-c h)^3 (g+h x)^2}+\frac {h^2 (2 b d g-b c h-a d h) \left (2 b^2 d^2 g^2-2 b^2 c d g h-2 a b d^2 g h+b^2 c^2 h^2+a^2 d^2 h^2\right )}{(b g-a h)^4 (d g-c h)^4 (g+h x)}\right ) \, dx}{4 h}\\ &=-\frac {A}{4 h (g+h x)^4}-\frac {B (b c-a d) n}{12 (b g-a h) (d g-c h) (g+h x)^3}-\frac {B (b c-a d) (2 b d g-b c h-a d h) n}{8 (b g-a h)^2 (d g-c h)^2 (g+h x)^2}-\frac {B (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n}{4 (b g-a h)^3 (d g-c h)^3 (g+h x)}+\frac {b^4 B n \log (a+b x)}{4 h (b g-a h)^4}-\frac {B d^4 n \log (c+d x)}{4 h (d g-c h)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 h (g+h x)^4}-\frac {B (b c-a d) (2 b d g-b c h-a d h) \left (2 a b d^2 g h-a^2 d^2 h^2-b^2 \left (2 d^2 g^2-2 c d g h+c^2 h^2\right )\right ) n \log (g+h x)}{4 (b g-a h)^4 (d g-c h)^4}\\ \end {align*}

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Mathematica [A]  time = 1.21, size = 366, normalized size = 0.94 \[ -\frac {-B n (b c-a d) \left (-\frac {h \left (a^2 d^2 h^2+a b d h (c h-3 d g)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right )}{(g+h x) (b g-a h)^3 (d g-c h)^3}-\frac {h \log (g+h x) (a d h+b c h-2 b d g) \left (a^2 d^2 h^2-2 a b d^2 g h+b^2 \left (c^2 h^2-2 c d g h+2 d^2 g^2\right )\right )}{(b g-a h)^4 (d g-c h)^4}+\frac {b^4 \log (a+b x)}{(b c-a d) (b g-a h)^4}-\frac {d^4 \log (c+d x)}{(b c-a d) (d g-c h)^4}+\frac {h (a d h+b c h-2 b d g)}{2 (g+h x)^2 (b g-a h)^2 (d g-c h)^2}-\frac {h}{3 (g+h x)^3 (b g-a h) (d g-c h)}\right )+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^4}+\frac {A}{(g+h x)^4}}{4 h} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(g + h*x)^5,x]

[Out]

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

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fricas [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*log(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^5,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 15.66, size = 3293, normalized size = 8.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^5,x, algorithm="giac")

[Out]

1/4*B*b^5*n*log(abs(b*x + a))/(b^5*g^4*h - 4*a*b^4*g^3*h^2 + 6*a^2*b^3*g^2*h^3 - 4*a^3*b^2*g*h^4 + a^4*b*h^5)
- 1/4*B*d^5*n*log(abs(d*x + c))/(d^5*g^4*h - 4*c*d^4*g^3*h^2 + 6*c^2*d^3*g^2*h^3 - 4*c^3*d^2*g*h^4 + c^4*d*h^5
) - 1/4*B*n*log(b*x + a)/(h^5*x^4 + 4*g*h^4*x^3 + 6*g^2*h^3*x^2 + 4*g^3*h^2*x + g^4*h) + 1/4*B*n*log(d*x + c)/
(h^5*x^4 + 4*g*h^4*x^3 + 6*g^2*h^3*x^2 + 4*g^3*h^2*x + g^4*h) + 1/4*(4*B*b^4*c*d^3*g^3*n - 4*B*a*b^3*d^4*g^3*n
- 6*B*b^4*c^2*d^2*g^2*h*n + 6*B*a^2*b^2*d^4*g^2*h*n + 4*B*b^4*c^3*d*g*h^2*n - 4*B*a^3*b*d^4*g*h^2*n - B*b^4*c
^4*h^3*n + B*a^4*d^4*h^3*n)*log(h*x + g)/(b^4*d^4*g^8 - 4*b^4*c*d^3*g^7*h - 4*a*b^3*d^4*g^7*h + 6*b^4*c^2*d^2*
g^6*h^2 + 16*a*b^3*c*d^3*g^6*h^2 + 6*a^2*b^2*d^4*g^6*h^2 - 4*b^4*c^3*d*g^5*h^3 - 24*a*b^3*c^2*d^2*g^5*h^3 - 24
*a^2*b^2*c*d^3*g^5*h^3 - 4*a^3*b*d^4*g^5*h^3 + b^4*c^4*g^4*h^4 + 16*a*b^3*c^3*d*g^4*h^4 + 36*a^2*b^2*c^2*d^2*g
^4*h^4 + 16*a^3*b*c*d^3*g^4*h^4 + a^4*d^4*g^4*h^4 - 4*a*b^3*c^4*g^3*h^5 - 24*a^2*b^2*c^3*d*g^3*h^5 - 24*a^3*b*
c^2*d^2*g^3*h^5 - 4*a^4*c*d^3*g^3*h^5 + 6*a^2*b^2*c^4*g^2*h^6 + 16*a^3*b*c^3*d*g^2*h^6 + 6*a^4*c^2*d^2*g^2*h^6
- 4*a^3*b*c^4*g*h^7 - 4*a^4*c^3*d*g*h^7 + a^4*c^4*h^8) - 1/24*(18*B*b^3*c*d^2*g^2*h^4*n*x^3 - 18*B*a*b^2*d^3*
g^2*h^4*n*x^3 - 18*B*b^3*c^2*d*g*h^5*n*x^3 + 18*B*a^2*b*d^3*g*h^5*n*x^3 + 6*B*b^3*c^3*h^6*n*x^3 - 6*B*a^3*d^3*
h^6*n*x^3 + 60*B*b^3*c*d^2*g^3*h^3*n*x^2 - 60*B*a*b^2*d^3*g^3*h^3*n*x^2 - 63*B*b^3*c^2*d*g^2*h^4*n*x^2 + 63*B*
a^2*b*d^3*g^2*h^4*n*x^2 + 21*B*b^3*c^3*g*h^5*n*x^2 + 9*B*a*b^2*c^2*d*g*h^5*n*x^2 - 9*B*a^2*b*c*d^2*g*h^5*n*x^2
- 21*B*a^3*d^3*g*h^5*n*x^2 - 3*B*a*b^2*c^3*h^6*n*x^2 + 3*B*a^3*c*d^2*h^6*n*x^2 + 68*B*b^3*c*d^2*g^4*h^2*n*x -
68*B*a*b^2*d^3*g^4*h^2*n*x - 76*B*b^3*c^2*d*g^3*h^3*n*x + 76*B*a^2*b*d^3*g^3*h^3*n*x + 26*B*b^3*c^3*g^2*h^4*n
*x + 24*B*a*b^2*c^2*d*g^2*h^4*n*x - 24*B*a^2*b*c*d^2*g^2*h^4*n*x - 26*B*a^3*d^3*g^2*h^4*n*x - 10*B*a*b^2*c^3*g
*h^5*n*x + 10*B*a^3*c*d^2*g*h^5*n*x + 2*B*a^2*b*c^3*h^6*n*x - 2*B*a^3*c^2*d*h^6*n*x + 26*B*b^3*c*d^2*g^5*h*n -
26*B*a*b^2*d^3*g^5*h*n - 31*B*b^3*c^2*d*g^4*h^2*n + 31*B*a^2*b*d^3*g^4*h^2*n + 11*B*b^3*c^3*g^3*h^3*n + 15*B*
a*b^2*c^2*d*g^3*h^3*n - 15*B*a^2*b*c*d^2*g^3*h^3*n - 11*B*a^3*d^3*g^3*h^3*n - 7*B*a*b^2*c^3*g^2*h^4*n + 7*B*a^
3*c*d^2*g^2*h^4*n + 2*B*a^2*b*c^3*g*h^5*n - 2*B*a^3*c^2*d*g*h^5*n + 6*A*b^3*d^3*g^6 + 6*B*b^3*d^3*g^6 - 18*A*b
^3*c*d^2*g^5*h - 18*B*b^3*c*d^2*g^5*h - 18*A*a*b^2*d^3*g^5*h - 18*B*a*b^2*d^3*g^5*h + 18*A*b^3*c^2*d*g^4*h^2 +
18*B*b^3*c^2*d*g^4*h^2 + 54*A*a*b^2*c*d^2*g^4*h^2 + 54*B*a*b^2*c*d^2*g^4*h^2 + 18*A*a^2*b*d^3*g^4*h^2 + 18*B*
a^2*b*d^3*g^4*h^2 - 6*A*b^3*c^3*g^3*h^3 - 6*B*b^3*c^3*g^3*h^3 - 54*A*a*b^2*c^2*d*g^3*h^3 - 54*B*a*b^2*c^2*d*g^
3*h^3 - 54*A*a^2*b*c*d^2*g^3*h^3 - 54*B*a^2*b*c*d^2*g^3*h^3 - 6*A*a^3*d^3*g^3*h^3 - 6*B*a^3*d^3*g^3*h^3 + 18*A
*a*b^2*c^3*g^2*h^4 + 18*B*a*b^2*c^3*g^2*h^4 + 54*A*a^2*b*c^2*d*g^2*h^4 + 54*B*a^2*b*c^2*d*g^2*h^4 + 18*A*a^3*c
*d^2*g^2*h^4 + 18*B*a^3*c*d^2*g^2*h^4 - 18*A*a^2*b*c^3*g*h^5 - 18*B*a^2*b*c^3*g*h^5 - 18*A*a^3*c^2*d*g*h^5 - 1
8*B*a^3*c^2*d*g*h^5 + 6*A*a^3*c^3*h^6 + 6*B*a^3*c^3*h^6)/(b^3*d^3*g^6*h^5*x^4 - 3*b^3*c*d^2*g^5*h^6*x^4 - 3*a*
b^2*d^3*g^5*h^6*x^4 + 3*b^3*c^2*d*g^4*h^7*x^4 + 9*a*b^2*c*d^2*g^4*h^7*x^4 + 3*a^2*b*d^3*g^4*h^7*x^4 - b^3*c^3*
g^3*h^8*x^4 - 9*a*b^2*c^2*d*g^3*h^8*x^4 - 9*a^2*b*c*d^2*g^3*h^8*x^4 - a^3*d^3*g^3*h^8*x^4 + 3*a*b^2*c^3*g^2*h^
9*x^4 + 9*a^2*b*c^2*d*g^2*h^9*x^4 + 3*a^3*c*d^2*g^2*h^9*x^4 - 3*a^2*b*c^3*g*h^10*x^4 - 3*a^3*c^2*d*g*h^10*x^4
+ a^3*c^3*h^11*x^4 + 4*b^3*d^3*g^7*h^4*x^3 - 12*b^3*c*d^2*g^6*h^5*x^3 - 12*a*b^2*d^3*g^6*h^5*x^3 + 12*b^3*c^2*
d*g^5*h^6*x^3 + 36*a*b^2*c*d^2*g^5*h^6*x^3 + 12*a^2*b*d^3*g^5*h^6*x^3 - 4*b^3*c^3*g^4*h^7*x^3 - 36*a*b^2*c^2*d
*g^4*h^7*x^3 - 36*a^2*b*c*d^2*g^4*h^7*x^3 - 4*a^3*d^3*g^4*h^7*x^3 + 12*a*b^2*c^3*g^3*h^8*x^3 + 36*a^2*b*c^2*d*
g^3*h^8*x^3 + 12*a^3*c*d^2*g^3*h^8*x^3 - 12*a^2*b*c^3*g^2*h^9*x^3 - 12*a^3*c^2*d*g^2*h^9*x^3 + 4*a^3*c^3*g*h^1
0*x^3 + 6*b^3*d^3*g^8*h^3*x^2 - 18*b^3*c*d^2*g^7*h^4*x^2 - 18*a*b^2*d^3*g^7*h^4*x^2 + 18*b^3*c^2*d*g^6*h^5*x^2
+ 54*a*b^2*c*d^2*g^6*h^5*x^2 + 18*a^2*b*d^3*g^6*h^5*x^2 - 6*b^3*c^3*g^5*h^6*x^2 - 54*a*b^2*c^2*d*g^5*h^6*x^2
- 54*a^2*b*c*d^2*g^5*h^6*x^2 - 6*a^3*d^3*g^5*h^6*x^2 + 18*a*b^2*c^3*g^4*h^7*x^2 + 54*a^2*b*c^2*d*g^4*h^7*x^2 +
18*a^3*c*d^2*g^4*h^7*x^2 - 18*a^2*b*c^3*g^3*h^8*x^2 - 18*a^3*c^2*d*g^3*h^8*x^2 + 6*a^3*c^3*g^2*h^9*x^2 + 4*b^
3*d^3*g^9*h^2*x - 12*b^3*c*d^2*g^8*h^3*x - 12*a*b^2*d^3*g^8*h^3*x + 12*b^3*c^2*d*g^7*h^4*x + 36*a*b^2*c*d^2*g^
7*h^4*x + 12*a^2*b*d^3*g^7*h^4*x - 4*b^3*c^3*g^6*h^5*x - 36*a*b^2*c^2*d*g^6*h^5*x - 36*a^2*b*c*d^2*g^6*h^5*x -
4*a^3*d^3*g^6*h^5*x + 12*a*b^2*c^3*g^5*h^6*x + 36*a^2*b*c^2*d*g^5*h^6*x + 12*a^3*c*d^2*g^5*h^6*x - 12*a^2*b*c
^3*g^4*h^7*x - 12*a^3*c^2*d*g^4*h^7*x + 4*a^3*c^3*g^3*h^8*x + b^3*d^3*g^10*h - 3*b^3*c*d^2*g^9*h^2 - 3*a*b^2*d
^3*g^9*h^2 + 3*b^3*c^2*d*g^8*h^3 + 9*a*b^2*c*d^2*g^8*h^3 + 3*a^2*b*d^3*g^8*h^3 - b^3*c^3*g^7*h^4 - 9*a*b^2*c^2
*d*g^7*h^4 - 9*a^2*b*c*d^2*g^7*h^4 - a^3*d^3*g^7*h^4 + 3*a*b^2*c^3*g^6*h^5 + 9*a^2*b*c^2*d*g^6*h^5 + 3*a^3*c*d
^2*g^6*h^5 - 3*a^2*b*c^3*g^5*h^6 - 3*a^3*c^2*d*g^5*h^6 + a^3*c^3*g^4*h^7)

________________________________________________________________________________________

maple [C]  time = 1.73, size = 16077, normalized size = 41.33 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^5,x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [B]  time = 2.97, size = 1912, normalized size = 4.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(h*x+g)^5,x, algorithm="maxima")

[Out]

1/24*(6*b^4*e*n*log(b*x + a)/(b^4*g^4*h - 4*a*b^3*g^3*h^2 + 6*a^2*b^2*g^2*h^3 - 4*a^3*b*g*h^4 + a^4*h^5) - 6*d
^4*e*n*log(d*x + c)/(d^4*g^4*h - 4*c*d^3*g^3*h^2 + 6*c^2*d^2*g^2*h^3 - 4*c^3*d*g*h^4 + c^4*h^5) - 6*(4*a*b^3*d
^4*e*g^3*n - 6*a^2*b^2*d^4*e*g^2*h*n + 4*a^3*b*d^4*e*g*h^2*n - a^4*d^4*e*h^3*n - (4*c*d^3*e*g^3*n - 6*c^2*d^2*
e*g^2*h*n + 4*c^3*d*e*g*h^2*n - c^4*e*h^3*n)*b^4)*log(h*x + g)/((d^4*g^4*h^4 - 4*c*d^3*g^3*h^5 + 6*c^2*d^2*g^2
*h^6 - 4*c^3*d*g*h^7 + c^4*h^8)*a^4 - 4*(d^4*g^5*h^3 - 4*c*d^3*g^4*h^4 + 6*c^2*d^2*g^3*h^5 - 4*c^3*d*g^2*h^6 +
c^4*g*h^7)*a^3*b + 6*(d^4*g^6*h^2 - 4*c*d^3*g^5*h^3 + 6*c^2*d^2*g^4*h^4 - 4*c^3*d*g^3*h^5 + c^4*g^2*h^6)*a^2*
b^2 - 4*(d^4*g^7*h - 4*c*d^3*g^6*h^2 + 6*c^2*d^2*g^5*h^3 - 4*c^3*d*g^4*h^4 + c^4*g^3*h^5)*a*b^3 + (d^4*g^8 - 4
*c*d^3*g^7*h + 6*c^2*d^2*g^6*h^2 - 4*c^3*d*g^5*h^3 + c^4*g^4*h^4)*b^4) - ((11*d^3*e*g^2*h^2*n - 7*c*d^2*e*g*h^
3*n + 2*c^2*d*e*h^4*n)*a^3 - (31*d^3*e*g^3*h*n - 15*c*d^2*e*g^2*h^2*n + 2*c^3*e*h^4*n)*a^2*b + (26*d^3*e*g^4*n
- 15*c^2*d*e*g^2*h^2*n + 7*c^3*e*g*h^3*n)*a*b^2 - (26*c*d^2*e*g^4*n - 31*c^2*d*e*g^3*h*n + 11*c^3*e*g^2*h^2*n
)*b^3 + 6*(3*a*b^2*d^3*e*g^2*h^2*n - 3*a^2*b*d^3*e*g*h^3*n + a^3*d^3*e*h^4*n - (3*c*d^2*e*g^2*h^2*n - 3*c^2*d*
e*g*h^3*n + c^3*e*h^4*n)*b^3)*x^2 + 3*((5*d^3*e*g*h^3*n - c*d^2*e*h^4*n)*a^3 - 3*(5*d^3*e*g^2*h^2*n - c*d^2*e*
g*h^3*n)*a^2*b + (14*d^3*e*g^3*h*n - 3*c^2*d*e*g*h^3*n + c^3*e*h^4*n)*a*b^2 - (14*c*d^2*e*g^3*h*n - 15*c^2*d*e
*g^2*h^2*n + 5*c^3*e*g*h^3*n)*b^3)*x)/((d^3*g^6*h^3 - 3*c*d^2*g^5*h^4 + 3*c^2*d*g^4*h^5 - c^3*g^3*h^6)*a^3 - 3
*(d^3*g^7*h^2 - 3*c*d^2*g^6*h^3 + 3*c^2*d*g^5*h^4 - c^3*g^4*h^5)*a^2*b + 3*(d^3*g^8*h - 3*c*d^2*g^7*h^2 + 3*c^
2*d*g^6*h^3 - c^3*g^5*h^4)*a*b^2 - (d^3*g^9 - 3*c*d^2*g^8*h + 3*c^2*d*g^7*h^2 - c^3*g^6*h^3)*b^3 + ((d^3*g^3*h
^6 - 3*c*d^2*g^2*h^7 + 3*c^2*d*g*h^8 - c^3*h^9)*a^3 - 3*(d^3*g^4*h^5 - 3*c*d^2*g^3*h^6 + 3*c^2*d*g^2*h^7 - c^3
*g*h^8)*a^2*b + 3*(d^3*g^5*h^4 - 3*c*d^2*g^4*h^5 + 3*c^2*d*g^3*h^6 - c^3*g^2*h^7)*a*b^2 - (d^3*g^6*h^3 - 3*c*d
^2*g^5*h^4 + 3*c^2*d*g^4*h^5 - c^3*g^3*h^6)*b^3)*x^3 + 3*((d^3*g^4*h^5 - 3*c*d^2*g^3*h^6 + 3*c^2*d*g^2*h^7 - c
^3*g*h^8)*a^3 - 3*(d^3*g^5*h^4 - 3*c*d^2*g^4*h^5 + 3*c^2*d*g^3*h^6 - c^3*g^2*h^7)*a^2*b + 3*(d^3*g^6*h^3 - 3*c
*d^2*g^5*h^4 + 3*c^2*d*g^4*h^5 - c^3*g^3*h^6)*a*b^2 - (d^3*g^7*h^2 - 3*c*d^2*g^6*h^3 + 3*c^2*d*g^5*h^4 - c^3*g
^4*h^5)*b^3)*x^2 + 3*((d^3*g^5*h^4 - 3*c*d^2*g^4*h^5 + 3*c^2*d*g^3*h^6 - c^3*g^2*h^7)*a^3 - 3*(d^3*g^6*h^3 - 3
*c*d^2*g^5*h^4 + 3*c^2*d*g^4*h^5 - c^3*g^3*h^6)*a^2*b + 3*(d^3*g^7*h^2 - 3*c*d^2*g^6*h^3 + 3*c^2*d*g^5*h^4 - c
^3*g^4*h^5)*a*b^2 - (d^3*g^8*h - 3*c*d^2*g^7*h^2 + 3*c^2*d*g^6*h^3 - c^3*g^5*h^4)*b^3)*x))*B/e - 1/4*B*log((b*
x + a)^n*e/(d*x + c)^n)/(h^5*x^4 + 4*g*h^4*x^3 + 6*g^2*h^3*x^2 + 4*g^3*h^2*x + g^4*h) - 1/4*A/(h^5*x^4 + 4*g*h
^4*x^3 + 6*g^2*h^3*x^2 + 4*g^3*h^2*x + g^4*h)

________________________________________________________________________________________

mupad [B]  time = 14.28, size = 2570, normalized size = 6.61 \[ \text {result too large to display} \]

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)^5,x)

[Out]

((x*(13*B*a^3*d^3*g^2*h^4*n - 13*B*b^3*c^3*g^2*h^4*n - B*a^2*b*c^3*h^6*n + B*a^3*c^2*d*h^6*n + 5*B*a*b^2*c^3*g
*h^5*n - 5*B*a^3*c*d^2*g*h^5*n + 34*B*a*b^2*d^3*g^4*h^2*n - 38*B*a^2*b*d^3*g^3*h^3*n - 34*B*b^3*c*d^2*g^4*h^2*
n + 38*B*b^3*c^2*d*g^3*h^3*n - 12*B*a*b^2*c^2*d*g^2*h^4*n + 12*B*a^2*b*c*d^2*g^2*h^4*n))/(3*(a^3*c^3*h^6 + b^3
*d^3*g^6 - a^3*d^3*g^3*h^3 - b^3*c^3*g^3*h^3 - 3*a^2*b*c^3*g*h^5 - 3*a*b^2*d^3*g^5*h - 3*a^3*c^2*d*g*h^5 - 3*b
^3*c*d^2*g^5*h + 3*a*b^2*c^3*g^2*h^4 + 3*a^2*b*d^3*g^4*h^2 + 3*a^3*c*d^2*g^2*h^4 + 3*b^3*c^2*d*g^4*h^2 + 9*a*b
^2*c*d^2*g^4*h^2 - 9*a*b^2*c^2*d*g^3*h^3 - 9*a^2*b*c*d^2*g^3*h^3 + 9*a^2*b*c^2*d*g^2*h^4)) - (6*A*a^3*c^3*h^6
+ 6*A*b^3*d^3*g^6 - 6*A*a^3*d^3*g^3*h^3 - 6*A*b^3*c^3*g^3*h^3 + 18*A*a*b^2*c^3*g^2*h^4 + 18*A*a^2*b*d^3*g^4*h^
2 + 18*A*a^3*c*d^2*g^2*h^4 + 18*A*b^3*c^2*d*g^4*h^2 - 11*B*a^3*d^3*g^3*h^3*n + 11*B*b^3*c^3*g^3*h^3*n - 18*A*a
^2*b*c^3*g*h^5 - 18*A*a*b^2*d^3*g^5*h - 18*A*a^3*c^2*d*g*h^5 - 18*A*b^3*c*d^2*g^5*h + 2*B*a^2*b*c^3*g*h^5*n -
26*B*a*b^2*d^3*g^5*h*n - 2*B*a^3*c^2*d*g*h^5*n + 26*B*b^3*c*d^2*g^5*h*n + 54*A*a*b^2*c*d^2*g^4*h^2 - 54*A*a*b^
2*c^2*d*g^3*h^3 - 54*A*a^2*b*c*d^2*g^3*h^3 + 54*A*a^2*b*c^2*d*g^2*h^4 - 7*B*a*b^2*c^3*g^2*h^4*n + 31*B*a^2*b*d
^3*g^4*h^2*n + 7*B*a^3*c*d^2*g^2*h^4*n - 31*B*b^3*c^2*d*g^4*h^2*n + 15*B*a*b^2*c^2*d*g^3*h^3*n - 15*B*a^2*b*c*
d^2*g^3*h^3*n)/(6*(a^3*c^3*h^6 + b^3*d^3*g^6 - a^3*d^3*g^3*h^3 - b^3*c^3*g^3*h^3 - 3*a^2*b*c^3*g*h^5 - 3*a*b^2
*d^3*g^5*h - 3*a^3*c^2*d*g*h^5 - 3*b^3*c*d^2*g^5*h + 3*a*b^2*c^3*g^2*h^4 + 3*a^2*b*d^3*g^4*h^2 + 3*a^3*c*d^2*g
^2*h^4 + 3*b^3*c^2*d*g^4*h^2 + 9*a*b^2*c*d^2*g^4*h^2 - 9*a*b^2*c^2*d*g^3*h^3 - 9*a^2*b*c*d^2*g^3*h^3 + 9*a^2*b
*c^2*d*g^2*h^4)) + (x^3*(B*a^3*d^3*h^6*n - B*b^3*c^3*h^6*n - 3*B*a^2*b*d^3*g*h^5*n + 3*B*b^3*c^2*d*g*h^5*n + 3
*B*a*b^2*d^3*g^2*h^4*n - 3*B*b^3*c*d^2*g^2*h^4*n))/(a^3*c^3*h^6 + b^3*d^3*g^6 - a^3*d^3*g^3*h^3 - b^3*c^3*g^3*
h^3 - 3*a^2*b*c^3*g*h^5 - 3*a*b^2*d^3*g^5*h - 3*a^3*c^2*d*g*h^5 - 3*b^3*c*d^2*g^5*h + 3*a*b^2*c^3*g^2*h^4 + 3*
a^2*b*d^3*g^4*h^2 + 3*a^3*c*d^2*g^2*h^4 + 3*b^3*c^2*d*g^4*h^2 + 9*a*b^2*c*d^2*g^4*h^2 - 9*a*b^2*c^2*d*g^3*h^3
- 9*a^2*b*c*d^2*g^3*h^3 + 9*a^2*b*c^2*d*g^2*h^4) + (x^2*(B*a*b^2*c^3*h^6*n - B*a^3*c*d^2*h^6*n + 7*B*a^3*d^3*g
*h^5*n - 7*B*b^3*c^3*g*h^5*n + 20*B*a*b^2*d^3*g^3*h^3*n - 21*B*a^2*b*d^3*g^2*h^4*n - 20*B*b^3*c*d^2*g^3*h^3*n
+ 21*B*b^3*c^2*d*g^2*h^4*n - 3*B*a*b^2*c^2*d*g*h^5*n + 3*B*a^2*b*c*d^2*g*h^5*n))/(2*(a^3*c^3*h^6 + b^3*d^3*g^6
- a^3*d^3*g^3*h^3 - b^3*c^3*g^3*h^3 - 3*a^2*b*c^3*g*h^5 - 3*a*b^2*d^3*g^5*h - 3*a^3*c^2*d*g*h^5 - 3*b^3*c*d^2
*g^5*h + 3*a*b^2*c^3*g^2*h^4 + 3*a^2*b*d^3*g^4*h^2 + 3*a^3*c*d^2*g^2*h^4 + 3*b^3*c^2*d*g^4*h^2 + 9*a*b^2*c*d^2
*g^4*h^2 - 9*a*b^2*c^2*d*g^3*h^3 - 9*a^2*b*c*d^2*g^3*h^3 + 9*a^2*b*c^2*d*g^2*h^4)))/(4*g^4*h + 4*h^5*x^4 + 16*
g^3*h^2*x + 16*g*h^4*x^3 + 24*g^2*h^3*x^2) + (log(g + h*x)*(h*(6*B*a^2*b^2*d^4*g^2*n - 6*B*b^4*c^2*d^2*g^2*n)
- h^2*(4*B*a^3*b*d^4*g*n - 4*B*b^4*c^3*d*g*n) + h^3*(B*a^4*d^4*n - B*b^4*c^4*n) - 4*B*a*b^3*d^4*g^3*n + 4*B*b^
4*c*d^3*g^3*n))/(4*a^4*c^4*h^8 + 4*b^4*d^4*g^8 + 4*a^4*d^4*g^4*h^4 + 4*b^4*c^4*g^4*h^4 + 24*a^2*b^2*c^4*g^2*h^
6 + 24*a^2*b^2*d^4*g^6*h^2 + 24*a^4*c^2*d^2*g^2*h^6 + 24*b^4*c^2*d^2*g^6*h^2 - 16*a^3*b*c^4*g*h^7 - 16*a*b^3*d
^4*g^7*h - 16*a^4*c^3*d*g*h^7 - 16*b^4*c*d^3*g^7*h - 16*a*b^3*c^4*g^3*h^5 - 16*a^3*b*d^4*g^5*h^3 - 16*a^4*c*d^
3*g^3*h^5 - 16*b^4*c^3*d*g^5*h^3 + 64*a*b^3*c*d^3*g^6*h^2 + 64*a*b^3*c^3*d*g^4*h^4 + 64*a^3*b*c*d^3*g^4*h^4 +
64*a^3*b*c^3*d*g^2*h^6 - 96*a*b^3*c^2*d^2*g^5*h^3 - 96*a^2*b^2*c*d^3*g^5*h^3 - 96*a^2*b^2*c^3*d*g^3*h^5 - 96*a
^3*b*c^2*d^2*g^3*h^5 + 144*a^2*b^2*c^2*d^2*g^4*h^4) - (B*log((e*(a + b*x)^n)/(c + d*x)^n))/(4*h*(g^4 + h^4*x^4
+ 4*g^3*h*x + 4*g*h^3*x^3 + 6*g^2*h^2*x^2)) + (B*b^4*n*log(a + b*x))/(4*a^4*h^5 + 4*b^4*g^4*h - 16*a*b^3*g^3*
h^2 + 24*a^2*b^2*g^2*h^3 - 16*a^3*b*g*h^4) - (B*d^4*n*log(c + d*x))/(4*c^4*h^5 + 4*d^4*g^4*h - 16*c*d^3*g^3*h^
2 + 24*c^2*d^2*g^2*h^3 - 16*c^3*d*g*h^4)

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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)))/(h*x+g)**5,x)

[Out]

Timed out

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