\(\int (g+h x)^4 \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [25]

Optimal result

Integrand size = 29, antiderivative size = 334 \[ \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {(b g-a h)^4 p r x}{5 b^4}-\frac {(d g-c h)^4 q r x}{5 d^4}-\frac {(b g-a h)^3 p r (g+h x)^2}{10 b^3 h}-\frac {(d g-c h)^3 q r (g+h x)^2}{10 d^3 h}-\frac {(b g-a h)^2 p r (g+h x)^3}{15 b^2 h}-\frac {(d g-c h)^2 q r (g+h x)^3}{15 d^2 h}-\frac {(b g-a h) p r (g+h x)^4}{20 b h}-\frac {(d g-c h) q r (g+h x)^4}{20 d h}-\frac {p r (g+h x)^5}{25 h}-\frac {q r (g+h x)^5}{25 h}-\frac {(b g-a h)^5 p r \log (a+b x)}{5 b^5 h}-\frac {(d g-c h)^5 q r \log (c+d x)}{5 d^5 h}+\frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h} \]

[Out]

-1/5*(-a*h+b*g)^4*p*r*x/b^4-1/5*(-c*h+d*g)^4*q*r*x/d^4-1/10*(-a*h+b*g)^3*p*r*(h*x+g)^2/b^3/h-1/10*(-c*h+d*g)^3
*q*r*(h*x+g)^2/d^3/h-1/15*(-a*h+b*g)^2*p*r*(h*x+g)^3/b^2/h-1/15*(-c*h+d*g)^2*q*r*(h*x+g)^3/d^2/h-1/20*(-a*h+b*
g)*p*r*(h*x+g)^4/b/h-1/20*(-c*h+d*g)*q*r*(h*x+g)^4/d/h-1/25*p*r*(h*x+g)^5/h-1/25*q*r*(h*x+g)^5/h-1/5*(-a*h+b*g
)^5*p*r*ln(b*x+a)/b^5/h-1/5*(-c*h+d*g)^5*q*r*ln(d*x+c)/d^5/h+1/5*(h*x+g)^5*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/h

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 334, 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, 45} \[ \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {p r (b g-a h)^5 \log (a+b x)}{5 b^5 h}-\frac {p r x (b g-a h)^4}{5 b^4}-\frac {p r (g+h x)^2 (b g-a h)^3}{10 b^3 h}-\frac {p r (g+h x)^3 (b g-a h)^2}{15 b^2 h}+\frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac {p r (g+h x)^4 (b g-a h)}{20 b h}-\frac {q r (d g-c h)^5 \log (c+d x)}{5 d^5 h}-\frac {q r x (d g-c h)^4}{5 d^4}-\frac {q r (g+h x)^2 (d g-c h)^3}{10 d^3 h}-\frac {q r (g+h x)^3 (d g-c h)^2}{15 d^2 h}-\frac {q r (g+h x)^4 (d g-c h)}{20 d h}-\frac {p r (g+h x)^5}{25 h}-\frac {q r (g+h x)^5}{25 h} \]

[In]

Int[(g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

-1/5*((b*g - a*h)^4*p*r*x)/b^4 - ((d*g - c*h)^4*q*r*x)/(5*d^4) - ((b*g - a*h)^3*p*r*(g + h*x)^2)/(10*b^3*h) -
((d*g - c*h)^3*q*r*(g + h*x)^2)/(10*d^3*h) - ((b*g - a*h)^2*p*r*(g + h*x)^3)/(15*b^2*h) - ((d*g - c*h)^2*q*r*(
g + h*x)^3)/(15*d^2*h) - ((b*g - a*h)*p*r*(g + h*x)^4)/(20*b*h) - ((d*g - c*h)*q*r*(g + h*x)^4)/(20*d*h) - (p*
r*(g + h*x)^5)/(25*h) - (q*r*(g + h*x)^5)/(25*h) - ((b*g - a*h)^5*p*r*Log[a + b*x])/(5*b^5*h) - ((d*g - c*h)^5
*q*r*Log[c + d*x])/(5*d^5*h) + ((g + h*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*h)

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac {(b p r) \int \frac {(g+h x)^5}{a+b x} \, dx}{5 h}-\frac {(d q r) \int \frac {(g+h x)^5}{c+d x} \, dx}{5 h} \\ & = \frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac {(b p r) \int \left (\frac {h (b g-a h)^4}{b^5}+\frac {(b g-a h)^5}{b^5 (a+b x)}+\frac {h (b g-a h)^3 (g+h x)}{b^4}+\frac {h (b g-a h)^2 (g+h x)^2}{b^3}+\frac {h (b g-a h) (g+h x)^3}{b^2}+\frac {h (g+h x)^4}{b}\right ) \, dx}{5 h}-\frac {(d q r) \int \left (\frac {h (d g-c h)^4}{d^5}+\frac {(d g-c h)^5}{d^5 (c+d x)}+\frac {h (d g-c h)^3 (g+h x)}{d^4}+\frac {h (d g-c h)^2 (g+h x)^2}{d^3}+\frac {h (d g-c h) (g+h x)^3}{d^2}+\frac {h (g+h x)^4}{d}\right ) \, dx}{5 h} \\ & = -\frac {(b g-a h)^4 p r x}{5 b^4}-\frac {(d g-c h)^4 q r x}{5 d^4}-\frac {(b g-a h)^3 p r (g+h x)^2}{10 b^3 h}-\frac {(d g-c h)^3 q r (g+h x)^2}{10 d^3 h}-\frac {(b g-a h)^2 p r (g+h x)^3}{15 b^2 h}-\frac {(d g-c h)^2 q r (g+h x)^3}{15 d^2 h}-\frac {(b g-a h) p r (g+h x)^4}{20 b h}-\frac {(d g-c h) q r (g+h x)^4}{20 d h}-\frac {p r (g+h x)^5}{25 h}-\frac {q r (g+h x)^5}{25 h}-\frac {(b g-a h)^5 p r \log (a+b x)}{5 b^5 h}-\frac {(d g-c h)^5 q r \log (c+d x)}{5 d^5 h}+\frac {(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.82 \[ \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {-\frac {p r \left (60 b h (b g-a h)^4 x+30 b^2 (b g-a h)^3 (g+h x)^2+20 b^3 (b g-a h)^2 (g+h x)^3+15 b^4 (b g-a h) (g+h x)^4+12 b^5 (g+h x)^5+60 (b g-a h)^5 \log (a+b x)\right )}{60 b^5}-\frac {q r \left (60 d h (d g-c h)^4 x+30 d^2 (d g-c h)^3 (g+h x)^2+20 d^3 (d g-c h)^2 (g+h x)^3+15 d^4 (d g-c h) (g+h x)^4+12 d^5 (g+h x)^5+60 (d g-c h)^5 \log (c+d x)\right )}{60 d^5}+(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h} \]

[In]

Integrate[(g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

(-1/60*(p*r*(60*b*h*(b*g - a*h)^4*x + 30*b^2*(b*g - a*h)^3*(g + h*x)^2 + 20*b^3*(b*g - a*h)^2*(g + h*x)^3 + 15
*b^4*(b*g - a*h)*(g + h*x)^4 + 12*b^5*(g + h*x)^5 + 60*(b*g - a*h)^5*Log[a + b*x]))/b^5 - (q*r*(60*d*h*(d*g -
c*h)^4*x + 30*d^2*(d*g - c*h)^3*(g + h*x)^2 + 20*d^3*(d*g - c*h)^2*(g + h*x)^3 + 15*d^4*(d*g - c*h)*(g + h*x)^
4 + 12*d^5*(g + h*x)^5 + 60*(d*g - c*h)^5*Log[c + d*x]))/(60*d^5) + (g + h*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)
^q)^r])/(5*h)

Maple [F]

\[\int \left (h x +g \right )^{4} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )d x\]

[In]

int((h*x+g)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

[Out]

int((h*x+g)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 945 vs. \(2 (308) = 616\).

Time = 0.33 (sec) , antiderivative size = 945, normalized size of antiderivative = 2.83 \[ \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {12 \, {\left (b^{5} d^{5} h^{4} p + b^{5} d^{5} h^{4} q\right )} r x^{5} + 15 \, {\left ({\left (5 \, b^{5} d^{5} g h^{3} - a b^{4} d^{5} h^{4}\right )} p + {\left (5 \, b^{5} d^{5} g h^{3} - b^{5} c d^{4} h^{4}\right )} q\right )} r x^{4} + 20 \, {\left ({\left (10 \, b^{5} d^{5} g^{2} h^{2} - 5 \, a b^{4} d^{5} g h^{3} + a^{2} b^{3} d^{5} h^{4}\right )} p + {\left (10 \, b^{5} d^{5} g^{2} h^{2} - 5 \, b^{5} c d^{4} g h^{3} + b^{5} c^{2} d^{3} h^{4}\right )} q\right )} r x^{3} + 30 \, {\left ({\left (10 \, b^{5} d^{5} g^{3} h - 10 \, a b^{4} d^{5} g^{2} h^{2} + 5 \, a^{2} b^{3} d^{5} g h^{3} - a^{3} b^{2} d^{5} h^{4}\right )} p + {\left (10 \, b^{5} d^{5} g^{3} h - 10 \, b^{5} c d^{4} g^{2} h^{2} + 5 \, b^{5} c^{2} d^{3} g h^{3} - b^{5} c^{3} d^{2} h^{4}\right )} q\right )} r x^{2} + 60 \, {\left ({\left (5 \, b^{5} d^{5} g^{4} - 10 \, a b^{4} d^{5} g^{3} h + 10 \, a^{2} b^{3} d^{5} g^{2} h^{2} - 5 \, a^{3} b^{2} d^{5} g h^{3} + a^{4} b d^{5} h^{4}\right )} p + {\left (5 \, b^{5} d^{5} g^{4} - 10 \, b^{5} c d^{4} g^{3} h + 10 \, b^{5} c^{2} d^{3} g^{2} h^{2} - 5 \, b^{5} c^{3} d^{2} g h^{3} + b^{5} c^{4} d h^{4}\right )} q\right )} r x - 60 \, {\left (b^{5} d^{5} h^{4} p r x^{5} + 5 \, b^{5} d^{5} g h^{3} p r x^{4} + 10 \, b^{5} d^{5} g^{2} h^{2} p r x^{3} + 10 \, b^{5} d^{5} g^{3} h p r x^{2} + 5 \, b^{5} d^{5} g^{4} p r x + {\left (5 \, a b^{4} d^{5} g^{4} - 10 \, a^{2} b^{3} d^{5} g^{3} h + 10 \, a^{3} b^{2} d^{5} g^{2} h^{2} - 5 \, a^{4} b d^{5} g h^{3} + a^{5} d^{5} h^{4}\right )} p r\right )} \log \left (b x + a\right ) - 60 \, {\left (b^{5} d^{5} h^{4} q r x^{5} + 5 \, b^{5} d^{5} g h^{3} q r x^{4} + 10 \, b^{5} d^{5} g^{2} h^{2} q r x^{3} + 10 \, b^{5} d^{5} g^{3} h q r x^{2} + 5 \, b^{5} d^{5} g^{4} q r x + {\left (5 \, b^{5} c d^{4} g^{4} - 10 \, b^{5} c^{2} d^{3} g^{3} h + 10 \, b^{5} c^{3} d^{2} g^{2} h^{2} - 5 \, b^{5} c^{4} d g h^{3} + b^{5} c^{5} h^{4}\right )} q r\right )} \log \left (d x + c\right ) - 60 \, {\left (b^{5} d^{5} h^{4} x^{5} + 5 \, b^{5} d^{5} g h^{3} x^{4} + 10 \, b^{5} d^{5} g^{2} h^{2} x^{3} + 10 \, b^{5} d^{5} g^{3} h x^{2} + 5 \, b^{5} d^{5} g^{4} x\right )} \log \left (e\right ) - 60 \, {\left (b^{5} d^{5} h^{4} r x^{5} + 5 \, b^{5} d^{5} g h^{3} r x^{4} + 10 \, b^{5} d^{5} g^{2} h^{2} r x^{3} + 10 \, b^{5} d^{5} g^{3} h r x^{2} + 5 \, b^{5} d^{5} g^{4} r x\right )} \log \left (f\right )}{300 \, b^{5} d^{5}} \]

[In]

integrate((h*x+g)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="fricas")

[Out]

-1/300*(12*(b^5*d^5*h^4*p + b^5*d^5*h^4*q)*r*x^5 + 15*((5*b^5*d^5*g*h^3 - a*b^4*d^5*h^4)*p + (5*b^5*d^5*g*h^3
- b^5*c*d^4*h^4)*q)*r*x^4 + 20*((10*b^5*d^5*g^2*h^2 - 5*a*b^4*d^5*g*h^3 + a^2*b^3*d^5*h^4)*p + (10*b^5*d^5*g^2
*h^2 - 5*b^5*c*d^4*g*h^3 + b^5*c^2*d^3*h^4)*q)*r*x^3 + 30*((10*b^5*d^5*g^3*h - 10*a*b^4*d^5*g^2*h^2 + 5*a^2*b^
3*d^5*g*h^3 - a^3*b^2*d^5*h^4)*p + (10*b^5*d^5*g^3*h - 10*b^5*c*d^4*g^2*h^2 + 5*b^5*c^2*d^3*g*h^3 - b^5*c^3*d^
2*h^4)*q)*r*x^2 + 60*((5*b^5*d^5*g^4 - 10*a*b^4*d^5*g^3*h + 10*a^2*b^3*d^5*g^2*h^2 - 5*a^3*b^2*d^5*g*h^3 + a^4
*b*d^5*h^4)*p + (5*b^5*d^5*g^4 - 10*b^5*c*d^4*g^3*h + 10*b^5*c^2*d^3*g^2*h^2 - 5*b^5*c^3*d^2*g*h^3 + b^5*c^4*d
*h^4)*q)*r*x - 60*(b^5*d^5*h^4*p*r*x^5 + 5*b^5*d^5*g*h^3*p*r*x^4 + 10*b^5*d^5*g^2*h^2*p*r*x^3 + 10*b^5*d^5*g^3
*h*p*r*x^2 + 5*b^5*d^5*g^4*p*r*x + (5*a*b^4*d^5*g^4 - 10*a^2*b^3*d^5*g^3*h + 10*a^3*b^2*d^5*g^2*h^2 - 5*a^4*b*
d^5*g*h^3 + a^5*d^5*h^4)*p*r)*log(b*x + a) - 60*(b^5*d^5*h^4*q*r*x^5 + 5*b^5*d^5*g*h^3*q*r*x^4 + 10*b^5*d^5*g^
2*h^2*q*r*x^3 + 10*b^5*d^5*g^3*h*q*r*x^2 + 5*b^5*d^5*g^4*q*r*x + (5*b^5*c*d^4*g^4 - 10*b^5*c^2*d^3*g^3*h + 10*
b^5*c^3*d^2*g^2*h^2 - 5*b^5*c^4*d*g*h^3 + b^5*c^5*h^4)*q*r)*log(d*x + c) - 60*(b^5*d^5*h^4*x^5 + 5*b^5*d^5*g*h
^3*x^4 + 10*b^5*d^5*g^2*h^2*x^3 + 10*b^5*d^5*g^3*h*x^2 + 5*b^5*d^5*g^4*x)*log(e) - 60*(b^5*d^5*h^4*r*x^5 + 5*b
^5*d^5*g*h^3*r*x^4 + 10*b^5*d^5*g^2*h^2*r*x^3 + 10*b^5*d^5*g^3*h*r*x^2 + 5*b^5*d^5*g^4*r*x)*log(f))/(b^5*d^5)

Sympy [F(-1)]

Timed out. \[ \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Timed out} \]

[In]

integrate((h*x+g)**4*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (308) = 616\).

Time = 0.20 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.87 \[ \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{5} \, {\left (h^{4} x^{5} + 5 \, g h^{3} x^{4} + 10 \, g^{2} h^{2} x^{3} + 10 \, g^{3} h x^{2} + 5 \, g^{4} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {r {\left (\frac {60 \, {\left (5 \, a b^{4} f g^{4} p - 10 \, a^{2} b^{3} f g^{3} h p + 10 \, a^{3} b^{2} f g^{2} h^{2} p - 5 \, a^{4} b f g h^{3} p + a^{5} f h^{4} p\right )} \log \left (b x + a\right )}{b^{5}} + \frac {60 \, {\left (5 \, c d^{4} f g^{4} q - 10 \, c^{2} d^{3} f g^{3} h q + 10 \, c^{3} d^{2} f g^{2} h^{2} q - 5 \, c^{4} d f g h^{3} q + c^{5} f h^{4} q\right )} \log \left (d x + c\right )}{d^{5}} - \frac {12 \, b^{4} d^{4} f h^{4} {\left (p + q\right )} x^{5} - 15 \, {\left (a b^{3} d^{4} f h^{4} p - {\left (5 \, d^{4} f g h^{3} {\left (p + q\right )} - c d^{3} f h^{4} q\right )} b^{4}\right )} x^{4} - 20 \, {\left (5 \, a b^{3} d^{4} f g h^{3} p - a^{2} b^{2} d^{4} f h^{4} p - {\left (10 \, d^{4} f g^{2} h^{2} {\left (p + q\right )} - 5 \, c d^{3} f g h^{3} q + c^{2} d^{2} f h^{4} q\right )} b^{4}\right )} x^{3} - 30 \, {\left (10 \, a b^{3} d^{4} f g^{2} h^{2} p - 5 \, a^{2} b^{2} d^{4} f g h^{3} p + a^{3} b d^{4} f h^{4} p - {\left (10 \, d^{4} f g^{3} h {\left (p + q\right )} - 10 \, c d^{3} f g^{2} h^{2} q + 5 \, c^{2} d^{2} f g h^{3} q - c^{3} d f h^{4} q\right )} b^{4}\right )} x^{2} - 60 \, {\left (10 \, a b^{3} d^{4} f g^{3} h p - 10 \, a^{2} b^{2} d^{4} f g^{2} h^{2} p + 5 \, a^{3} b d^{4} f g h^{3} p - a^{4} d^{4} f h^{4} p - {\left (5 \, d^{4} f g^{4} {\left (p + q\right )} - 10 \, c d^{3} f g^{3} h q + 10 \, c^{2} d^{2} f g^{2} h^{2} q - 5 \, c^{3} d f g h^{3} q + c^{4} f h^{4} q\right )} b^{4}\right )} x}{b^{4} d^{4}}\right )}}{300 \, f} \]

[In]

integrate((h*x+g)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="maxima")

[Out]

1/5*(h^4*x^5 + 5*g*h^3*x^4 + 10*g^2*h^2*x^3 + 10*g^3*h*x^2 + 5*g^4*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e) + 1
/300*r*(60*(5*a*b^4*f*g^4*p - 10*a^2*b^3*f*g^3*h*p + 10*a^3*b^2*f*g^2*h^2*p - 5*a^4*b*f*g*h^3*p + a^5*f*h^4*p)
*log(b*x + a)/b^5 + 60*(5*c*d^4*f*g^4*q - 10*c^2*d^3*f*g^3*h*q + 10*c^3*d^2*f*g^2*h^2*q - 5*c^4*d*f*g*h^3*q +
c^5*f*h^4*q)*log(d*x + c)/d^5 - (12*b^4*d^4*f*h^4*(p + q)*x^5 - 15*(a*b^3*d^4*f*h^4*p - (5*d^4*f*g*h^3*(p + q)
- c*d^3*f*h^4*q)*b^4)*x^4 - 20*(5*a*b^3*d^4*f*g*h^3*p - a^2*b^2*d^4*f*h^4*p - (10*d^4*f*g^2*h^2*(p + q) - 5*c
*d^3*f*g*h^3*q + c^2*d^2*f*h^4*q)*b^4)*x^3 - 30*(10*a*b^3*d^4*f*g^2*h^2*p - 5*a^2*b^2*d^4*f*g*h^3*p + a^3*b*d^
4*f*h^4*p - (10*d^4*f*g^3*h*(p + q) - 10*c*d^3*f*g^2*h^2*q + 5*c^2*d^2*f*g*h^3*q - c^3*d*f*h^4*q)*b^4)*x^2 - 6
0*(10*a*b^3*d^4*f*g^3*h*p - 10*a^2*b^2*d^4*f*g^2*h^2*p + 5*a^3*b*d^4*f*g*h^3*p - a^4*d^4*f*h^4*p - (5*d^4*f*g^
4*(p + q) - 10*c*d^3*f*g^3*h*q + 10*c^2*d^2*f*g^2*h^2*q - 5*c^3*d*f*g*h^3*q + c^4*f*h^4*q)*b^4)*x)/(b^4*d^4))/
f

Giac [F]

\[ \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (h x + g\right )}^{4} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \,d x } \]

[In]

integrate((h*x+g)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 2.08 (sec) , antiderivative size = 1128, normalized size of antiderivative = 3.38 \[ \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \]

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(g + h*x)^4,x)

[Out]

log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(g^4*x + (h^4*x^5)/5 + 2*g^3*h*x^2 + g*h^3*x^4 + 2*g^2*h^2*x^3) - x^2*(((
5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*((h^3*r*(b*c*h*p + 5*b*d*g*p + a*d*h*q + 5*b*d*g*q))/(5*b*d) - (h^4*r*(p + q)
*(5*a*d + 5*b*c))/(25*b*d)))/(5*b*d) - (g*h^2*r*(b*c*h*p + 2*b*d*g*p + a*d*h*q + 2*b*d*g*q))/(b*d) + (a*c*h^4*
r*(p + q))/(5*b*d)))/(10*b*d) - (a*c*((h^3*r*(b*c*h*p + 5*b*d*g*p + a*d*h*q + 5*b*d*g*q))/(5*b*d) - (h^4*r*(p
+ q)*(5*a*d + 5*b*c))/(25*b*d)))/(2*b*d) + (g^2*h*r*(b*c*h*p + b*d*g*p + a*d*h*q + b*d*g*q))/(b*d)) - x^4*((h^
3*r*(b*c*h*p + 5*b*d*g*p + a*d*h*q + 5*b*d*g*q))/(20*b*d) - (h^4*r*(p + q)*(5*a*d + 5*b*c))/(100*b*d)) - x*((a
*c*(((5*a*d + 5*b*c)*((h^3*r*(b*c*h*p + 5*b*d*g*p + a*d*h*q + 5*b*d*g*q))/(5*b*d) - (h^4*r*(p + q)*(5*a*d + 5*
b*c))/(25*b*d)))/(5*b*d) - (g*h^2*r*(b*c*h*p + 2*b*d*g*p + a*d*h*q + 2*b*d*g*q))/(b*d) + (a*c*h^4*r*(p + q))/(
5*b*d)))/(b*d) - ((5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*((h^3*r*(b*c*h*p + 5*b*d*g*p + a*d*h*q +
5*b*d*g*q))/(5*b*d) - (h^4*r*(p + q)*(5*a*d + 5*b*c))/(25*b*d)))/(5*b*d) - (g*h^2*r*(b*c*h*p + 2*b*d*g*p + a*
d*h*q + 2*b*d*g*q))/(b*d) + (a*c*h^4*r*(p + q))/(5*b*d)))/(5*b*d) - (a*c*((h^3*r*(b*c*h*p + 5*b*d*g*p + a*d*h*
q + 5*b*d*g*q))/(5*b*d) - (h^4*r*(p + q)*(5*a*d + 5*b*c))/(25*b*d)))/(b*d) + (2*g^2*h*r*(b*c*h*p + b*d*g*p + a
*d*h*q + b*d*g*q))/(b*d)))/(5*b*d) + (g^3*r*(2*b*c*h*p + b*d*g*p + 2*a*d*h*q + b*d*g*q))/(b*d)) + x^3*(((5*a*d
+ 5*b*c)*((h^3*r*(b*c*h*p + 5*b*d*g*p + a*d*h*q + 5*b*d*g*q))/(5*b*d) - (h^4*r*(p + q)*(5*a*d + 5*b*c))/(25*b
*d)))/(15*b*d) - (g*h^2*r*(b*c*h*p + 2*b*d*g*p + a*d*h*q + 2*b*d*g*q))/(3*b*d) + (a*c*h^4*r*(p + q))/(15*b*d))
+ (log(a + b*x)*((a^5*h^4*p*r)/5 + a*b^4*g^4*p*r + 2*a^3*b^2*g^2*h^2*p*r - a^4*b*g*h^3*p*r - 2*a^2*b^3*g^3*h*
p*r))/b^5 + (log(c + d*x)*((c^5*h^4*q*r)/5 + c*d^4*g^4*q*r + 2*c^3*d^2*g^2*h^2*q*r - c^4*d*g*h^3*q*r - 2*c^2*d
^3*g^3*h*q*r))/d^5 - (h^4*r*x^5*(p + q))/25