\(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r)}{(g+h x)^5} \, dx\) [34]

Optimal result

Integrand size = 29, antiderivative size = 318 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx=\frac {b p r}{12 h (b g-a h) (g+h x)^3}+\frac {d q r}{12 h (d g-c h) (g+h x)^3}+\frac {b^2 p r}{8 h (b g-a h)^2 (g+h x)^2}+\frac {d^2 q r}{8 h (d g-c h)^2 (g+h x)^2}+\frac {b^3 p r}{4 h (b g-a h)^3 (g+h x)}+\frac {d^3 q r}{4 h (d g-c h)^3 (g+h x)}+\frac {b^4 p r \log (a+b x)}{4 h (b g-a h)^4}+\frac {d^4 q r \log (c+d x)}{4 h (d g-c h)^4}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}-\frac {b^4 p r \log (g+h x)}{4 h (b g-a h)^4}-\frac {d^4 q r \log (g+h x)}{4 h (d g-c h)^4} \]

[Out]

1/12*b*p*r/h/(-a*h+b*g)/(h*x+g)^3+1/12*d*q*r/h/(-c*h+d*g)/(h*x+g)^3+1/8*b^2*p*r/h/(-a*h+b*g)^2/(h*x+g)^2+1/8*d
^2*q*r/h/(-c*h+d*g)^2/(h*x+g)^2+1/4*b^3*p*r/h/(-a*h+b*g)^3/(h*x+g)+1/4*d^3*q*r/h/(-c*h+d*g)^3/(h*x+g)+1/4*b^4*
p*r*ln(b*x+a)/h/(-a*h+b*g)^4+1/4*d^4*q*r*ln(d*x+c)/h/(-c*h+d*g)^4-1/4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/h/(h*x+g
)^4-1/4*b^4*p*r*ln(h*x+g)/h/(-a*h+b*g)^4-1/4*d^4*q*r*ln(h*x+g)/h/(-c*h+d*g)^4

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2581, 46} \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx=\frac {b^4 p r \log (a+b x)}{4 h (b g-a h)^4}-\frac {b^4 p r \log (g+h x)}{4 h (b g-a h)^4}+\frac {b^3 p r}{4 h (g+h x) (b g-a h)^3}+\frac {b^2 p r}{8 h (g+h x)^2 (b g-a h)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}+\frac {b p r}{12 h (g+h x)^3 (b g-a h)}+\frac {d^4 q r \log (c+d x)}{4 h (d g-c h)^4}-\frac {d^4 q r \log (g+h x)}{4 h (d g-c h)^4}+\frac {d^3 q r}{4 h (g+h x) (d g-c h)^3}+\frac {d^2 q r}{8 h (g+h x)^2 (d g-c h)^2}+\frac {d q r}{12 h (g+h x)^3 (d g-c h)} \]

[In]

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^5,x]

[Out]

(b*p*r)/(12*h*(b*g - a*h)*(g + h*x)^3) + (d*q*r)/(12*h*(d*g - c*h)*(g + h*x)^3) + (b^2*p*r)/(8*h*(b*g - a*h)^2
*(g + h*x)^2) + (d^2*q*r)/(8*h*(d*g - c*h)^2*(g + h*x)^2) + (b^3*p*r)/(4*h*(b*g - a*h)^3*(g + h*x)) + (d^3*q*r
)/(4*h*(d*g - c*h)^3*(g + h*x)) + (b^4*p*r*Log[a + b*x])/(4*h*(b*g - a*h)^4) + (d^4*q*r*Log[c + d*x])/(4*h*(d*
g - c*h)^4) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(4*h*(g + h*x)^4) - (b^4*p*r*Log[g + h*x])/(4*h*(b*g - a*h)
^4) - (d^4*q*r*Log[g + h*x])/(4*h*(d*g - c*h)^4)

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2581

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}+\frac {(b p r) \int \frac {1}{(a+b x) (g+h x)^4} \, dx}{4 h}+\frac {(d q r) \int \frac {1}{(c+d x) (g+h x)^4} \, dx}{4 h} \\ & = -\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}+\frac {(b p r) \int \left (\frac {b^4}{(b g-a h)^4 (a+b x)}-\frac {h}{(b g-a h) (g+h x)^4}-\frac {b h}{(b g-a h)^2 (g+h x)^3}-\frac {b^2 h}{(b g-a h)^3 (g+h x)^2}-\frac {b^3 h}{(b g-a h)^4 (g+h x)}\right ) \, dx}{4 h}+\frac {(d q r) \int \left (\frac {d^4}{(d g-c h)^4 (c+d x)}-\frac {h}{(d g-c h) (g+h x)^4}-\frac {d h}{(d g-c h)^2 (g+h x)^3}-\frac {d^2 h}{(d g-c h)^3 (g+h x)^2}-\frac {d^3 h}{(d g-c h)^4 (g+h x)}\right ) \, dx}{4 h} \\ & = \frac {b p r}{12 h (b g-a h) (g+h x)^3}+\frac {d q r}{12 h (d g-c h) (g+h x)^3}+\frac {b^2 p r}{8 h (b g-a h)^2 (g+h x)^2}+\frac {d^2 q r}{8 h (d g-c h)^2 (g+h x)^2}+\frac {b^3 p r}{4 h (b g-a h)^3 (g+h x)}+\frac {d^3 q r}{4 h (d g-c h)^3 (g+h x)}+\frac {b^4 p r \log (a+b x)}{4 h (b g-a h)^4}+\frac {d^4 q r \log (c+d x)}{4 h (d g-c h)^4}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}-\frac {b^4 p r \log (g+h x)}{4 h (b g-a h)^4}-\frac {d^4 q r \log (g+h x)}{4 h (d g-c h)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.51 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx=\frac {-6 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {r (g+h x) \left (2 (b g-a h)^3 (d g-c h)^3 (b d g (p+q)-h (b c p+a d q))-(g+h x) \left ((b g-a h)^2 (d g-c h)^2 \left (6 a b d^2 g h q-3 a^2 d^2 h^2 q-3 b^2 \left (-2 c d g h p+c^2 h^2 p+d^2 g^2 (p+q)\right )\right )+6 (g+h x) \left (-\left ((b g-a h) (-d g+c h) \left (3 a b^2 d^3 g^2 h q-3 a^2 b d^3 g h^2 q+a^3 d^3 h^3 q-b^3 \left (-3 c d^2 g^2 h p+3 c^2 d g h^2 p-c^3 h^3 p+d^3 g^3 (p+q)\right )\right )\right )-(g+h x) \left (b^4 (d g-c h)^4 p \log (a+b x)+d^4 (b g-a h)^4 q \log (c+d x)-\left (-4 a b^3 d^4 g^3 h q+6 a^2 b^2 d^4 g^2 h^2 q-4 a^3 b d^4 g h^3 q+a^4 d^4 h^4 q+b^4 \left (-4 c d^3 g^3 h p+6 c^2 d^2 g^2 h^2 p-4 c^3 d g h^3 p+c^4 h^4 p+d^4 g^4 (p+q)\right )\right ) \log (g+h x)\right )\right )\right )\right )}{(b g-a h)^4 (d g-c h)^4}}{24 h (g+h x)^4} \]

[In]

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^5,x]

[Out]

(-6*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (r*(g + h*x)*(2*(b*g - a*h)^3*(d*g - c*h)^3*(b*d*g*(p + q) - h*(b*c
*p + a*d*q)) - (g + h*x)*((b*g - a*h)^2*(d*g - c*h)^2*(6*a*b*d^2*g*h*q - 3*a^2*d^2*h^2*q - 3*b^2*(-2*c*d*g*h*p
+ c^2*h^2*p + d^2*g^2*(p + q))) + 6*(g + h*x)*(-((b*g - a*h)*(-(d*g) + c*h)*(3*a*b^2*d^3*g^2*h*q - 3*a^2*b*d^
3*g*h^2*q + a^3*d^3*h^3*q - b^3*(-3*c*d^2*g^2*h*p + 3*c^2*d*g*h^2*p - c^3*h^3*p + d^3*g^3*(p + q)))) - (g + h*
x)*(b^4*(d*g - c*h)^4*p*Log[a + b*x] + d^4*(b*g - a*h)^4*q*Log[c + d*x] - (-4*a*b^3*d^4*g^3*h*q + 6*a^2*b^2*d^
4*g^2*h^2*q - 4*a^3*b*d^4*g*h^3*q + a^4*d^4*h^4*q + b^4*(-4*c*d^3*g^3*h*p + 6*c^2*d^2*g^2*h^2*p - 4*c^3*d*g*h^
3*p + c^4*h^4*p + d^4*g^4*(p + q)))*Log[g + h*x])))))/((b*g - a*h)^4*(d*g - c*h)^4))/(24*h*(g + h*x)^4)

Maple [F]

\[\int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (h x +g \right )^{5}}d x\]

[In]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx=\text {Timed out} \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx=\text {Timed out} \]

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/(h*x+g)**5,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (296) = 592\).

Time = 0.23 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.44 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx=\frac {{\left ({\left (\frac {6 \, b^{3} \log \left (b x + a\right )}{b^{4} g^{4} - 4 \, a b^{3} g^{3} h + 6 \, a^{2} b^{2} g^{2} h^{2} - 4 \, a^{3} b g h^{3} + a^{4} h^{4}} - \frac {6 \, b^{3} \log \left (h x + g\right )}{b^{4} g^{4} - 4 \, a b^{3} g^{3} h + 6 \, a^{2} b^{2} g^{2} h^{2} - 4 \, a^{3} b g h^{3} + a^{4} h^{4}} + \frac {6 \, b^{2} h^{2} x^{2} + 11 \, b^{2} g^{2} - 7 \, a b g h + 2 \, a^{2} h^{2} + 3 \, {\left (5 \, b^{2} g h - a b h^{2}\right )} x}{b^{3} g^{6} - 3 \, a b^{2} g^{5} h + 3 \, a^{2} b g^{4} h^{2} - a^{3} g^{3} h^{3} + {\left (b^{3} g^{3} h^{3} - 3 \, a b^{2} g^{2} h^{4} + 3 \, a^{2} b g h^{5} - a^{3} h^{6}\right )} x^{3} + 3 \, {\left (b^{3} g^{4} h^{2} - 3 \, a b^{2} g^{3} h^{3} + 3 \, a^{2} b g^{2} h^{4} - a^{3} g h^{5}\right )} x^{2} + 3 \, {\left (b^{3} g^{5} h - 3 \, a b^{2} g^{4} h^{2} + 3 \, a^{2} b g^{3} h^{3} - a^{3} g^{2} h^{4}\right )} x}\right )} b f p + {\left (\frac {6 \, d^{3} \log \left (d x + c\right )}{d^{4} g^{4} - 4 \, c d^{3} g^{3} h + 6 \, c^{2} d^{2} g^{2} h^{2} - 4 \, c^{3} d g h^{3} + c^{4} h^{4}} - \frac {6 \, d^{3} \log \left (h x + g\right )}{d^{4} g^{4} - 4 \, c d^{3} g^{3} h + 6 \, c^{2} d^{2} g^{2} h^{2} - 4 \, c^{3} d g h^{3} + c^{4} h^{4}} + \frac {6 \, d^{2} h^{2} x^{2} + 11 \, d^{2} g^{2} - 7 \, c d g h + 2 \, c^{2} h^{2} + 3 \, {\left (5 \, d^{2} g h - c d h^{2}\right )} x}{d^{3} g^{6} - 3 \, c d^{2} g^{5} h + 3 \, c^{2} d g^{4} h^{2} - c^{3} g^{3} h^{3} + {\left (d^{3} g^{3} h^{3} - 3 \, c d^{2} g^{2} h^{4} + 3 \, c^{2} d g h^{5} - c^{3} h^{6}\right )} x^{3} + 3 \, {\left (d^{3} g^{4} h^{2} - 3 \, c d^{2} g^{3} h^{3} + 3 \, c^{2} d g^{2} h^{4} - c^{3} g h^{5}\right )} x^{2} + 3 \, {\left (d^{3} g^{5} h - 3 \, c d^{2} g^{4} h^{2} + 3 \, c^{2} d g^{3} h^{3} - c^{3} g^{2} h^{4}\right )} x}\right )} d f q\right )} r}{24 \, f h} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{4 \, {\left (h x + g\right )}^{4} h} \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x, algorithm="maxima")

[Out]

1/24*((6*b^3*log(b*x + a)/(b^4*g^4 - 4*a*b^3*g^3*h + 6*a^2*b^2*g^2*h^2 - 4*a^3*b*g*h^3 + a^4*h^4) - 6*b^3*log(
h*x + g)/(b^4*g^4 - 4*a*b^3*g^3*h + 6*a^2*b^2*g^2*h^2 - 4*a^3*b*g*h^3 + a^4*h^4) + (6*b^2*h^2*x^2 + 11*b^2*g^2
- 7*a*b*g*h + 2*a^2*h^2 + 3*(5*b^2*g*h - a*b*h^2)*x)/(b^3*g^6 - 3*a*b^2*g^5*h + 3*a^2*b*g^4*h^2 - a^3*g^3*h^3
+ (b^3*g^3*h^3 - 3*a*b^2*g^2*h^4 + 3*a^2*b*g*h^5 - a^3*h^6)*x^3 + 3*(b^3*g^4*h^2 - 3*a*b^2*g^3*h^3 + 3*a^2*b*
g^2*h^4 - a^3*g*h^5)*x^2 + 3*(b^3*g^5*h - 3*a*b^2*g^4*h^2 + 3*a^2*b*g^3*h^3 - a^3*g^2*h^4)*x))*b*f*p + (6*d^3*
log(d*x + c)/(d^4*g^4 - 4*c*d^3*g^3*h + 6*c^2*d^2*g^2*h^2 - 4*c^3*d*g*h^3 + c^4*h^4) - 6*d^3*log(h*x + g)/(d^4
*g^4 - 4*c*d^3*g^3*h + 6*c^2*d^2*g^2*h^2 - 4*c^3*d*g*h^3 + c^4*h^4) + (6*d^2*h^2*x^2 + 11*d^2*g^2 - 7*c*d*g*h
+ 2*c^2*h^2 + 3*(5*d^2*g*h - c*d*h^2)*x)/(d^3*g^6 - 3*c*d^2*g^5*h + 3*c^2*d*g^4*h^2 - c^3*g^3*h^3 + (d^3*g^3*h
^3 - 3*c*d^2*g^2*h^4 + 3*c^2*d*g*h^5 - c^3*h^6)*x^3 + 3*(d^3*g^4*h^2 - 3*c*d^2*g^3*h^3 + 3*c^2*d*g^2*h^4 - c^3
*g*h^5)*x^2 + 3*(d^3*g^5*h - 3*c*d^2*g^4*h^2 + 3*c^2*d*g^3*h^3 - c^3*g^2*h^4)*x))*d*f*q)*r/(f*h) - 1/4*log(((b
*x + a)^p*(d*x + c)^q*f)^r*e)/((h*x + g)^4*h)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3943 vs. \(2 (296) = 592\).

Time = 0.60 (sec) , antiderivative size = 3943, normalized size of antiderivative = 12.40 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx=\text {Too large to display} \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x, algorithm="giac")

[Out]

1/4*b^5*p*r*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*d^5*q*r*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*p*r*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*q*r*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*(b^4*d^4*g^4*p*r - 4*b^4*c*d^3*g^3*h*p*r +
6*b^4*c^2*d^2*g^2*h^2*p*r - 4*b^4*c^3*d*g*h^3*p*r + b^4*c^4*h^4*p*r + b^4*d^4*g^4*q*r - 4*a*b^3*d^4*g^3*h*q*r
+ 6*a^2*b^2*d^4*g^2*h^2*q*r - 4*a^3*b*d^4*g*h^3*q*r + a^4*d^4*h^4*q*r)*log(h*x + g)/(b^4*d^4*g^8*h - 4*b^4*c*
d^3*g^7*h^2 - 4*a*b^3*d^4*g^7*h^2 + 6*b^4*c^2*d^2*g^6*h^3 + 16*a*b^3*c*d^3*g^6*h^3 + 6*a^2*b^2*d^4*g^6*h^3 - 4
*b^4*c^3*d*g^5*h^4 - 24*a*b^3*c^2*d^2*g^5*h^4 - 24*a^2*b^2*c*d^3*g^5*h^4 - 4*a^3*b*d^4*g^5*h^4 + b^4*c^4*g^4*h
^5 + 16*a*b^3*c^3*d*g^4*h^5 + 36*a^2*b^2*c^2*d^2*g^4*h^5 + 16*a^3*b*c*d^3*g^4*h^5 + a^4*d^4*g^4*h^5 - 4*a*b^3*
c^4*g^3*h^6 - 24*a^2*b^2*c^3*d*g^3*h^6 - 24*a^3*b*c^2*d^2*g^3*h^6 - 4*a^4*c*d^3*g^3*h^6 + 6*a^2*b^2*c^4*g^2*h^
7 + 16*a^3*b*c^3*d*g^2*h^7 + 6*a^4*c^2*d^2*g^2*h^7 - 4*a^3*b*c^4*g*h^8 - 4*a^4*c^3*d*g*h^8 + a^4*c^4*h^9) + 1/
24*(6*b^3*d^3*g^3*h^3*p*r*x^3 - 18*b^3*c*d^2*g^2*h^4*p*r*x^3 + 18*b^3*c^2*d*g*h^5*p*r*x^3 - 6*b^3*c^3*h^6*p*r*
x^3 + 6*b^3*d^3*g^3*h^3*q*r*x^3 - 18*a*b^2*d^3*g^2*h^4*q*r*x^3 + 18*a^2*b*d^3*g*h^5*q*r*x^3 - 6*a^3*d^3*h^6*q*
r*x^3 + 21*b^3*d^3*g^4*h^2*p*r*x^2 - 63*b^3*c*d^2*g^3*h^3*p*r*x^2 - 3*a*b^2*d^3*g^3*h^3*p*r*x^2 + 63*b^3*c^2*d
*g^2*h^4*p*r*x^2 + 9*a*b^2*c*d^2*g^2*h^4*p*r*x^2 - 21*b^3*c^3*g*h^5*p*r*x^2 - 9*a*b^2*c^2*d*g*h^5*p*r*x^2 + 3*
a*b^2*c^3*h^6*p*r*x^2 + 21*b^3*d^3*g^4*h^2*q*r*x^2 - 3*b^3*c*d^2*g^3*h^3*q*r*x^2 - 63*a*b^2*d^3*g^3*h^3*q*r*x^
2 + 9*a*b^2*c*d^2*g^2*h^4*q*r*x^2 + 63*a^2*b*d^3*g^2*h^4*q*r*x^2 - 9*a^2*b*c*d^2*g*h^5*q*r*x^2 - 21*a^3*d^3*g*
h^5*q*r*x^2 + 3*a^3*c*d^2*h^6*q*r*x^2 + 26*b^3*d^3*g^5*h*p*r*x - 78*b^3*c*d^2*g^4*h^2*p*r*x - 10*a*b^2*d^3*g^4
*h^2*p*r*x + 78*b^3*c^2*d*g^3*h^3*p*r*x + 30*a*b^2*c*d^2*g^3*h^3*p*r*x + 2*a^2*b*d^3*g^3*h^3*p*r*x - 26*b^3*c^
3*g^2*h^4*p*r*x - 30*a*b^2*c^2*d*g^2*h^4*p*r*x - 6*a^2*b*c*d^2*g^2*h^4*p*r*x + 10*a*b^2*c^3*g*h^5*p*r*x + 6*a^
2*b*c^2*d*g*h^5*p*r*x - 2*a^2*b*c^3*h^6*p*r*x + 26*b^3*d^3*g^5*h*q*r*x - 10*b^3*c*d^2*g^4*h^2*q*r*x - 78*a*b^2
*d^3*g^4*h^2*q*r*x + 2*b^3*c^2*d*g^3*h^3*q*r*x + 30*a*b^2*c*d^2*g^3*h^3*q*r*x + 78*a^2*b*d^3*g^3*h^3*q*r*x - 6
*a*b^2*c^2*d*g^2*h^4*q*r*x - 30*a^2*b*c*d^2*g^2*h^4*q*r*x - 26*a^3*d^3*g^2*h^4*q*r*x + 6*a^2*b*c^2*d*g*h^5*q*r
*x + 10*a^3*c*d^2*g*h^5*q*r*x - 2*a^3*c^2*d*h^6*q*r*x + 11*b^3*d^3*g^6*p*r - 33*b^3*c*d^2*g^5*h*p*r - 7*a*b^2*
d^3*g^5*h*p*r + 33*b^3*c^2*d*g^4*h^2*p*r + 21*a*b^2*c*d^2*g^4*h^2*p*r + 2*a^2*b*d^3*g^4*h^2*p*r - 11*b^3*c^3*g
^3*h^3*p*r - 21*a*b^2*c^2*d*g^3*h^3*p*r - 6*a^2*b*c*d^2*g^3*h^3*p*r + 7*a*b^2*c^3*g^2*h^4*p*r + 6*a^2*b*c^2*d*
g^2*h^4*p*r - 2*a^2*b*c^3*g*h^5*p*r + 11*b^3*d^3*g^6*q*r - 7*b^3*c*d^2*g^5*h*q*r - 33*a*b^2*d^3*g^5*h*q*r + 2*
b^3*c^2*d*g^4*h^2*q*r + 21*a*b^2*c*d^2*g^4*h^2*q*r + 33*a^2*b*d^3*g^4*h^2*q*r - 6*a*b^2*c^2*d*g^3*h^3*q*r - 21
*a^2*b*c*d^2*g^3*h^3*q*r - 11*a^3*d^3*g^3*h^3*q*r + 6*a^2*b*c^2*d*g^2*h^4*q*r + 7*a^3*c*d^2*g^2*h^4*q*r - 2*a^
3*c^2*d*g*h^5*q*r - 6*b^3*d^3*g^6*r*log(f) + 18*b^3*c*d^2*g^5*h*r*log(f) + 18*a*b^2*d^3*g^5*h*r*log(f) - 18*b^
3*c^2*d*g^4*h^2*r*log(f) - 54*a*b^2*c*d^2*g^4*h^2*r*log(f) - 18*a^2*b*d^3*g^4*h^2*r*log(f) + 6*b^3*c^3*g^3*h^3
*r*log(f) + 54*a*b^2*c^2*d*g^3*h^3*r*log(f) + 54*a^2*b*c*d^2*g^3*h^3*r*log(f) + 6*a^3*d^3*g^3*h^3*r*log(f) - 1
8*a*b^2*c^3*g^2*h^4*r*log(f) - 54*a^2*b*c^2*d*g^2*h^4*r*log(f) - 18*a^3*c*d^2*g^2*h^4*r*log(f) + 18*a^2*b*c^3*
g*h^5*r*log(f) + 18*a^3*c^2*d*g*h^5*r*log(f) - 6*a^3*c^3*h^6*r*log(f) - 6*b^3*d^3*g^6*log(e) + 18*b^3*c*d^2*g^
5*h*log(e) + 18*a*b^2*d^3*g^5*h*log(e) - 18*b^3*c^2*d*g^4*h^2*log(e) - 54*a*b^2*c*d^2*g^4*h^2*log(e) - 18*a^2*
b*d^3*g^4*h^2*log(e) + 6*b^3*c^3*g^3*h^3*log(e) + 54*a*b^2*c^2*d*g^3*h^3*log(e) + 54*a^2*b*c*d^2*g^3*h^3*log(e
) + 6*a^3*d^3*g^3*h^3*log(e) - 18*a*b^2*c^3*g^2*h^4*log(e) - 54*a^2*b*c^2*d*g^2*h^4*log(e) - 18*a^3*c*d^2*g^2*
h^4*log(e) + 18*a^2*b*c^3*g*h^5*log(e) + 18*a^3*c^2*d*g*h^5*log(e) - 6*a^3*c^3*h^6*log(e))/(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^10*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 + 1
2*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)

Mupad [B] (verification not implemented)

Time = 11.66 (sec) , antiderivative size = 2215, normalized size of antiderivative = 6.97 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx=\text {Too large to display} \]

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)/(g + h*x)^5,x)

[Out]

((11*b^3*d^3*g^5*p*r + 11*b^3*d^3*g^5*q*r - 11*b^3*c^3*g^2*h^3*p*r - 11*a^3*d^3*g^2*h^3*q*r - 2*a^2*b*c^3*h^5*
p*r - 2*a^3*c^2*d*h^5*q*r + 7*a*b^2*c^3*g*h^4*p*r - 7*a*b^2*d^3*g^4*h*p*r - 33*a*b^2*d^3*g^4*h*q*r + 7*a^3*c*d
^2*g*h^4*q*r - 33*b^3*c*d^2*g^4*h*p*r - 7*b^3*c*d^2*g^4*h*q*r + 2*a^2*b*d^3*g^3*h^2*p*r + 33*a^2*b*d^3*g^3*h^2
*q*r + 33*b^3*c^2*d*g^3*h^2*p*r + 2*b^3*c^2*d*g^3*h^2*q*r + 21*a*b^2*c*d^2*g^3*h^2*p*r - 21*a*b^2*c^2*d*g^2*h^
3*p*r - 6*a^2*b*c*d^2*g^2*h^3*p*r + 21*a*b^2*c*d^2*g^3*h^2*q*r - 6*a*b^2*c^2*d*g^2*h^3*q*r - 21*a^2*b*c*d^2*g^
2*h^3*q*r + 6*a^2*b*c^2*d*g*h^4*p*r + 6*a^2*b*c^2*d*g*h^4*q*r)/(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^2*(b^3*c^3*h^5*p*r + a^3*d^3*h^5*q*r - b
^3*d^3*g^3*h^2*p*r - b^3*d^3*g^3*h^2*q*r - 3*a^2*b*d^3*g*h^4*q*r - 3*b^3*c^2*d*g*h^4*p*r + 3*a*b^2*d^3*g^2*h^3
*q*r + 3*b^3*c*d^2*g^2*h^3*p*r))/(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*(a*b^2*c^3*h^5*p*r + a^3*c*d^2*h^5*q*r - 5*b^3*c^3*g*h^4*p*r - 5*a^3*d^3*g
*h^4*q*r + 5*b^3*d^3*g^4*h*p*r + 5*b^3*d^3*g^4*h*q*r - a*b^2*d^3*g^3*h^2*p*r - 15*a*b^2*d^3*g^3*h^2*q*r + 15*a
^2*b*d^3*g^2*h^3*q*r - 15*b^3*c*d^2*g^3*h^2*p*r + 15*b^3*c^2*d*g^2*h^3*p*r - b^3*c*d^2*g^3*h^2*q*r + 3*a*b^2*c
*d^2*g^2*h^3*p*r + 3*a*b^2*c*d^2*g^2*h^3*q*r - 3*a*b^2*c^2*d*g*h^4*p*r - 3*a^2*b*c*d^2*g*h^4*q*r))/(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^3*h
+ 4*h^4*x^3 + 12*g^2*h^2*x + 12*g*h^3*x^2) - (log(g + h*x)*(h^4*(b^4*c^4*p*r + a^4*d^4*q*r) - h*(4*a*b^3*d^4*
g^3*q*r + 4*b^4*c*d^3*g^3*p*r) + h^2*(6*a^2*b^2*d^4*g^2*q*r + 6*b^4*c^2*d^2*g^2*p*r) - h^3*(4*a^3*b*d^4*g*q*r
+ 4*b^4*c^3*d*g*p*r) + b^4*d^4*g^4*p*r + b^4*d^4*g^4*q*r))/(4*a^4*c^4*h^9 + 4*b^4*d^4*g^8*h + 4*a^4*d^4*g^4*h^
5 + 4*b^4*c^4*g^4*h^5 + 24*a^2*b^2*c^4*g^2*h^7 + 24*a^2*b^2*d^4*g^6*h^3 + 24*a^4*c^2*d^2*g^2*h^7 + 24*b^4*c^2*
d^2*g^6*h^3 - 16*a^3*b*c^4*g*h^8 - 16*a^4*c^3*d*g*h^8 - 16*a*b^3*c^4*g^3*h^6 - 16*a*b^3*d^4*g^7*h^2 - 16*a^3*b
*d^4*g^5*h^4 - 16*a^4*c*d^3*g^3*h^6 - 16*b^4*c*d^3*g^7*h^2 - 16*b^4*c^3*d*g^5*h^4 + 64*a*b^3*c*d^3*g^6*h^3 + 6
4*a*b^3*c^3*d*g^4*h^5 + 64*a^3*b*c*d^3*g^4*h^5 + 64*a^3*b*c^3*d*g^2*h^7 - 96*a*b^3*c^2*d^2*g^5*h^4 - 96*a^2*b^
2*c*d^3*g^5*h^4 - 96*a^2*b^2*c^3*d*g^3*h^6 - 96*a^3*b*c^2*d^2*g^3*h^6 + 144*a^2*b^2*c^2*d^2*g^4*h^5) - (log(e*
(f*(a + b*x)^p*(c + d*x)^q)^r)*(x/4 + g/(4*h)))/(g + h*x)^5 + (b^4*p*r*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) + (d^4*q*r*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)