\(\int (g+h x)^3 \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [26]

Optimal result

Integrand size = 29, antiderivative size = 276 \[ \int (g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {(b g-a h)^3 p r x}{4 b^3}-\frac {(d g-c h)^3 q r x}{4 d^3}-\frac {(b g-a h)^2 p r (g+h x)^2}{8 b^2 h}-\frac {(d g-c h)^2 q r (g+h x)^2}{8 d^2 h}-\frac {(b g-a h) p r (g+h x)^3}{12 b h}-\frac {(d g-c h) q r (g+h x)^3}{12 d h}-\frac {p r (g+h x)^4}{16 h}-\frac {q r (g+h x)^4}{16 h}-\frac {(b g-a h)^4 p r \log (a+b x)}{4 b^4 h}-\frac {(d g-c h)^4 q r \log (c+d x)}{4 d^4 h}+\frac {(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h} \]

[Out]

-1/4*(-a*h+b*g)^3*p*r*x/b^3-1/4*(-c*h+d*g)^3*q*r*x/d^3-1/8*(-a*h+b*g)^2*p*r*(h*x+g)^2/b^2/h-1/8*(-c*h+d*g)^2*q
*r*(h*x+g)^2/d^2/h-1/12*(-a*h+b*g)*p*r*(h*x+g)^3/b/h-1/12*(-c*h+d*g)*q*r*(h*x+g)^3/d/h-1/16*p*r*(h*x+g)^4/h-1/
16*q*r*(h*x+g)^4/h-1/4*(-a*h+b*g)^4*p*r*ln(b*x+a)/b^4/h-1/4*(-c*h+d*g)^4*q*r*ln(d*x+c)/d^4/h+1/4*(h*x+g)^4*ln(
e*(f*(b*x+a)^p*(d*x+c)^q)^r)/h

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 276, 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} \[ \int (g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {p r (b g-a h)^4 \log (a+b x)}{4 b^4 h}-\frac {p r x (b g-a h)^3}{4 b^3}-\frac {p r (g+h x)^2 (b g-a h)^2}{8 b^2 h}+\frac {(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac {p r (g+h x)^3 (b g-a h)}{12 b h}-\frac {q r (d g-c h)^4 \log (c+d x)}{4 d^4 h}-\frac {q r x (d g-c h)^3}{4 d^3}-\frac {q r (g+h x)^2 (d g-c h)^2}{8 d^2 h}-\frac {q r (g+h x)^3 (d g-c h)}{12 d h}-\frac {p r (g+h x)^4}{16 h}-\frac {q r (g+h x)^4}{16 h} \]

[In]

Int[(g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

-1/4*((b*g - a*h)^3*p*r*x)/b^3 - ((d*g - c*h)^3*q*r*x)/(4*d^3) - ((b*g - a*h)^2*p*r*(g + h*x)^2)/(8*b^2*h) - (
(d*g - c*h)^2*q*r*(g + h*x)^2)/(8*d^2*h) - ((b*g - a*h)*p*r*(g + h*x)^3)/(12*b*h) - ((d*g - c*h)*q*r*(g + h*x)
^3)/(12*d*h) - (p*r*(g + h*x)^4)/(16*h) - (q*r*(g + h*x)^4)/(16*h) - ((b*g - a*h)^4*p*r*Log[a + b*x])/(4*b^4*h
) - ((d*g - c*h)^4*q*r*Log[c + d*x])/(4*d^4*h) + ((g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*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)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac {(b p r) \int \frac {(g+h x)^4}{a+b x} \, dx}{4 h}-\frac {(d q r) \int \frac {(g+h x)^4}{c+d x} \, dx}{4 h} \\ & = \frac {(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac {(b p r) \int \left (\frac {h (b g-a h)^3}{b^4}+\frac {(b g-a h)^4}{b^4 (a+b x)}+\frac {h (b g-a h)^2 (g+h x)}{b^3}+\frac {h (b g-a h) (g+h x)^2}{b^2}+\frac {h (g+h x)^3}{b}\right ) \, dx}{4 h}-\frac {(d q r) \int \left (\frac {h (d g-c h)^3}{d^4}+\frac {(d g-c h)^4}{d^4 (c+d x)}+\frac {h (d g-c h)^2 (g+h x)}{d^3}+\frac {h (d g-c h) (g+h x)^2}{d^2}+\frac {h (g+h x)^3}{d}\right ) \, dx}{4 h} \\ & = -\frac {(b g-a h)^3 p r x}{4 b^3}-\frac {(d g-c h)^3 q r x}{4 d^3}-\frac {(b g-a h)^2 p r (g+h x)^2}{8 b^2 h}-\frac {(d g-c h)^2 q r (g+h x)^2}{8 d^2 h}-\frac {(b g-a h) p r (g+h x)^3}{12 b h}-\frac {(d g-c h) q r (g+h x)^3}{12 d h}-\frac {p r (g+h x)^4}{16 h}-\frac {q r (g+h x)^4}{16 h}-\frac {(b g-a h)^4 p r \log (a+b x)}{4 b^4 h}-\frac {(d g-c h)^4 q r \log (c+d x)}{4 d^4 h}+\frac {(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.84 \[ \int (g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {\frac {1}{12} r \left (-\frac {p \left (12 b h (b g-a h)^3 x+6 b^2 (b g-a h)^2 (g+h x)^2+4 b^3 (b g-a h) (g+h x)^3+3 b^4 (g+h x)^4+12 (b g-a h)^4 \log (a+b x)\right )}{b^4}-\frac {q \left (12 d h (d g-c h)^3 x+6 d^2 (d g-c h)^2 (g+h x)^2+4 d^3 (d g-c h) (g+h x)^3+3 d^4 (g+h x)^4+12 (d g-c h)^4 \log (c+d x)\right )}{d^4}\right )+(g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h} \]

[In]

Integrate[(g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

((r*(-((p*(12*b*h*(b*g - a*h)^3*x + 6*b^2*(b*g - a*h)^2*(g + h*x)^2 + 4*b^3*(b*g - a*h)*(g + h*x)^3 + 3*b^4*(g
+ h*x)^4 + 12*(b*g - a*h)^4*Log[a + b*x]))/b^4) - (q*(12*d*h*(d*g - c*h)^3*x + 6*d^2*(d*g - c*h)^2*(g + h*x)^
2 + 4*d^3*(d*g - c*h)*(g + h*x)^3 + 3*d^4*(g + h*x)^4 + 12*(d*g - c*h)^4*Log[c + d*x]))/d^4))/12 + (g + h*x)^4
*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*h)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1078\) vs. \(2(254)=508\).

Time = 291.42 (sec) , antiderivative size = 1079, normalized size of antiderivative = 3.91

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1079\)

[In]

int((h*x+g)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x,method=_RETURNVERBOSE)

[Out]

-1/48*(-24*x^2*b^4*c*d^3*g*h^2*q*r-48*a^3*g*h^2*p*r*d^4*b-48*b^4*c^3*d*g*h^2*q*r+6*a^3*b*c*d^3*h^3*p*r+6*a*b^3
*c^3*d*h^3*q*r+6*x^2*a^2*b^2*d^4*h^3*p*r+6*x^2*b^4*c^2*d^2*h^3*q*r+36*x^2*b^4*d^4*g^2*h*p*r+36*x^2*b^4*d^4*g^2
*h*q*r-96*ln(b*x+a)*a*b^3*d^4*g^3*p*r-12*x*a^3*b*d^4*h^3*p*r-12*x*b^4*c^3*d*h^3*q*r+72*ln(e*(f*(b*x+a)^p*(d*x+
c)^q)^r)*a*b^3*c*d^3*g^2*h-48*ln(b*x+a)*b^4*c*d^3*g^3*p*r-48*ln(d*x+c)*a*b^3*d^4*g^3*q*r-96*ln(d*x+c)*b^4*c*d^
3*g^3*q*r-72*ln(b*x+a)*a*b^3*c*d^3*g^2*h*p*r-72*ln(d*x+c)*a*b^3*c*d^3*g^2*h*q*r-4*x^3*a*b^3*d^4*h^3*p*r-4*x^3*
b^4*c*d^3*h^3*q*r+16*x^3*b^4*d^4*g*h^2*p*r+16*x^3*b^4*d^4*g*h^2*q*r+72*a^2*b^2*d^4*p*r*g^2*h+72*b^4*c^2*d^2*q*
r*g^2*h-24*a^2*b^2*c*d^3*g*h^2*p*r-24*a*b^3*c^2*d^2*g*h^2*q*r+36*a*b^3*c*d^3*g^2*h*p*r+36*a*b^3*c*d^3*g^2*h*q*
r+12*a^4*h^3*p*r*d^4+12*b^4*c^4*h^3*q*r-12*x^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*b^4*d^4*h^3-48*x*ln(e*(f*(b*x+a
)^p*(d*x+c)^q)^r)*b^4*d^4*g^3+48*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*a*b^3*d^4*g^3+48*ln(e*(f*(b*x+a)^p*(d*x+c)^q)
^r)*b^4*c*d^3*g^3-48*a*b^3*d^4*p*r*g^3-48*a*b^3*d^4*q*r*g^3-48*b^4*c*d^3*p*r*g^3-48*b^4*c*d^3*q*r*g^3-72*x^2*l
n(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*b^4*d^4*g^2*h+48*x*b^4*d^4*g^3*p*r+48*x*b^4*d^4*g^3*q*r+12*ln(b*x+a)*a^4*d^4*h^
3*p*r+12*ln(d*x+c)*b^4*c^4*h^3*q*r+3*x^4*b^4*d^4*h^3*p*r+3*x^4*b^4*d^4*h^3*q*r-48*x^3*ln(e*(f*(b*x+a)^p*(d*x+c
)^q)^r)*b^4*d^4*g*h^2+48*x*a^2*b^2*d^4*g*h^2*p*r-72*x*a*b^3*d^4*g^2*h*p*r+48*x*b^4*c^2*d^2*g*h^2*q*r-72*x*b^4*
c*d^3*g^2*h*q*r-48*ln(b*x+a)*a^3*b*d^4*g*h^2*p*r+72*ln(b*x+a)*a^2*b^2*d^4*g^2*h*p*r-48*ln(d*x+c)*b^4*c^3*d*g*h
^2*q*r+72*ln(d*x+c)*b^4*c^2*d^2*g^2*h*q*r-24*x^2*a*b^3*d^4*g*h^2*p*r)/b^4/d^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (254) = 508\).

Time = 0.30 (sec) , antiderivative size = 679, normalized size of antiderivative = 2.46 \[ \int (g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {3 \, {\left (b^{4} d^{4} h^{3} p + b^{4} d^{4} h^{3} q\right )} r x^{4} + 4 \, {\left ({\left (4 \, b^{4} d^{4} g h^{2} - a b^{3} d^{4} h^{3}\right )} p + {\left (4 \, b^{4} d^{4} g h^{2} - b^{4} c d^{3} h^{3}\right )} q\right )} r x^{3} + 6 \, {\left ({\left (6 \, b^{4} d^{4} g^{2} h - 4 \, a b^{3} d^{4} g h^{2} + a^{2} b^{2} d^{4} h^{3}\right )} p + {\left (6 \, b^{4} d^{4} g^{2} h - 4 \, b^{4} c d^{3} g h^{2} + b^{4} c^{2} d^{2} h^{3}\right )} q\right )} r x^{2} + 12 \, {\left ({\left (4 \, b^{4} d^{4} g^{3} - 6 \, a b^{3} d^{4} g^{2} h + 4 \, a^{2} b^{2} d^{4} g h^{2} - a^{3} b d^{4} h^{3}\right )} p + {\left (4 \, b^{4} d^{4} g^{3} - 6 \, b^{4} c d^{3} g^{2} h + 4 \, b^{4} c^{2} d^{2} g h^{2} - b^{4} c^{3} d h^{3}\right )} q\right )} r x - 12 \, {\left (b^{4} d^{4} h^{3} p r x^{4} + 4 \, b^{4} d^{4} g h^{2} p r x^{3} + 6 \, b^{4} d^{4} g^{2} h p r x^{2} + 4 \, b^{4} d^{4} g^{3} p r x + {\left (4 \, a b^{3} d^{4} g^{3} - 6 \, a^{2} b^{2} d^{4} g^{2} h + 4 \, a^{3} b d^{4} g h^{2} - a^{4} d^{4} h^{3}\right )} p r\right )} \log \left (b x + a\right ) - 12 \, {\left (b^{4} d^{4} h^{3} q r x^{4} + 4 \, b^{4} d^{4} g h^{2} q r x^{3} + 6 \, b^{4} d^{4} g^{2} h q r x^{2} + 4 \, b^{4} d^{4} g^{3} q r x + {\left (4 \, b^{4} c d^{3} g^{3} - 6 \, b^{4} c^{2} d^{2} g^{2} h + 4 \, b^{4} c^{3} d g h^{2} - b^{4} c^{4} h^{3}\right )} q r\right )} \log \left (d x + c\right ) - 12 \, {\left (b^{4} d^{4} h^{3} x^{4} + 4 \, b^{4} d^{4} g h^{2} x^{3} + 6 \, b^{4} d^{4} g^{2} h x^{2} + 4 \, b^{4} d^{4} g^{3} x\right )} \log \left (e\right ) - 12 \, {\left (b^{4} d^{4} h^{3} r x^{4} + 4 \, b^{4} d^{4} g h^{2} r x^{3} + 6 \, b^{4} d^{4} g^{2} h r x^{2} + 4 \, b^{4} d^{4} g^{3} r x\right )} \log \left (f\right )}{48 \, b^{4} d^{4}} \]

[In]

integrate((h*x+g)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="fricas")

[Out]

-1/48*(3*(b^4*d^4*h^3*p + b^4*d^4*h^3*q)*r*x^4 + 4*((4*b^4*d^4*g*h^2 - a*b^3*d^4*h^3)*p + (4*b^4*d^4*g*h^2 - b
^4*c*d^3*h^3)*q)*r*x^3 + 6*((6*b^4*d^4*g^2*h - 4*a*b^3*d^4*g*h^2 + a^2*b^2*d^4*h^3)*p + (6*b^4*d^4*g^2*h - 4*b
^4*c*d^3*g*h^2 + b^4*c^2*d^2*h^3)*q)*r*x^2 + 12*((4*b^4*d^4*g^3 - 6*a*b^3*d^4*g^2*h + 4*a^2*b^2*d^4*g*h^2 - a^
3*b*d^4*h^3)*p + (4*b^4*d^4*g^3 - 6*b^4*c*d^3*g^2*h + 4*b^4*c^2*d^2*g*h^2 - b^4*c^3*d*h^3)*q)*r*x - 12*(b^4*d^
4*h^3*p*r*x^4 + 4*b^4*d^4*g*h^2*p*r*x^3 + 6*b^4*d^4*g^2*h*p*r*x^2 + 4*b^4*d^4*g^3*p*r*x + (4*a*b^3*d^4*g^3 - 6
*a^2*b^2*d^4*g^2*h + 4*a^3*b*d^4*g*h^2 - a^4*d^4*h^3)*p*r)*log(b*x + a) - 12*(b^4*d^4*h^3*q*r*x^4 + 4*b^4*d^4*
g*h^2*q*r*x^3 + 6*b^4*d^4*g^2*h*q*r*x^2 + 4*b^4*d^4*g^3*q*r*x + (4*b^4*c*d^3*g^3 - 6*b^4*c^2*d^2*g^2*h + 4*b^4
*c^3*d*g*h^2 - b^4*c^4*h^3)*q*r)*log(d*x + c) - 12*(b^4*d^4*h^3*x^4 + 4*b^4*d^4*g*h^2*x^3 + 6*b^4*d^4*g^2*h*x^
2 + 4*b^4*d^4*g^3*x)*log(e) - 12*(b^4*d^4*h^3*r*x^4 + 4*b^4*d^4*g*h^2*r*x^3 + 6*b^4*d^4*g^2*h*r*x^2 + 4*b^4*d^
4*g^3*r*x)*log(f))/(b^4*d^4)

Sympy [F(-1)]

Timed out. \[ \int (g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Timed out} \]

[In]

integrate((h*x+g)**3*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.56 \[ \int (g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{4} \, {\left (h^{3} x^{4} + 4 \, g h^{2} x^{3} + 6 \, g^{2} h x^{2} + 4 \, g^{3} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {r {\left (\frac {12 \, {\left (4 \, a b^{3} f g^{3} p - 6 \, a^{2} b^{2} f g^{2} h p + 4 \, a^{3} b f g h^{2} p - a^{4} f h^{3} p\right )} \log \left (b x + a\right )}{b^{4}} + \frac {12 \, {\left (4 \, c d^{3} f g^{3} q - 6 \, c^{2} d^{2} f g^{2} h q + 4 \, c^{3} d f g h^{2} q - c^{4} f h^{3} q\right )} \log \left (d x + c\right )}{d^{4}} - \frac {3 \, b^{3} d^{3} f h^{3} {\left (p + q\right )} x^{4} - 4 \, {\left (a b^{2} d^{3} f h^{3} p - {\left (4 \, d^{3} f g h^{2} {\left (p + q\right )} - c d^{2} f h^{3} q\right )} b^{3}\right )} x^{3} - 6 \, {\left (4 \, a b^{2} d^{3} f g h^{2} p - a^{2} b d^{3} f h^{3} p - {\left (6 \, d^{3} f g^{2} h {\left (p + q\right )} - 4 \, c d^{2} f g h^{2} q + c^{2} d f h^{3} q\right )} b^{3}\right )} x^{2} - 12 \, {\left (6 \, a b^{2} d^{3} f g^{2} h p - 4 \, a^{2} b d^{3} f g h^{2} p + a^{3} d^{3} f h^{3} p - {\left (4 \, d^{3} f g^{3} {\left (p + q\right )} - 6 \, c d^{2} f g^{2} h q + 4 \, c^{2} d f g h^{2} q - c^{3} f h^{3} q\right )} b^{3}\right )} x}{b^{3} d^{3}}\right )}}{48 \, f} \]

[In]

integrate((h*x+g)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="maxima")

[Out]

1/4*(h^3*x^4 + 4*g*h^2*x^3 + 6*g^2*h*x^2 + 4*g^3*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e) + 1/48*r*(12*(4*a*b^3
*f*g^3*p - 6*a^2*b^2*f*g^2*h*p + 4*a^3*b*f*g*h^2*p - a^4*f*h^3*p)*log(b*x + a)/b^4 + 12*(4*c*d^3*f*g^3*q - 6*c
^2*d^2*f*g^2*h*q + 4*c^3*d*f*g*h^2*q - c^4*f*h^3*q)*log(d*x + c)/d^4 - (3*b^3*d^3*f*h^3*(p + q)*x^4 - 4*(a*b^2
*d^3*f*h^3*p - (4*d^3*f*g*h^2*(p + q) - c*d^2*f*h^3*q)*b^3)*x^3 - 6*(4*a*b^2*d^3*f*g*h^2*p - a^2*b*d^3*f*h^3*p
- (6*d^3*f*g^2*h*(p + q) - 4*c*d^2*f*g*h^2*q + c^2*d*f*h^3*q)*b^3)*x^2 - 12*(6*a*b^2*d^3*f*g^2*h*p - 4*a^2*b*
d^3*f*g*h^2*p + a^3*d^3*f*h^3*p - (4*d^3*f*g^3*(p + q) - 6*c*d^2*f*g^2*h*q + 4*c^2*d*f*g*h^2*q - c^3*f*h^3*q)*
b^3)*x)/(b^3*d^3))/f

Giac [F(-1)]

Timed out. \[ \int (g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Timed out} \]

[In]

integrate((h*x+g)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 1.88 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.32 \[ \int (g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (g^3\,x+\frac {3\,g^2\,h\,x^2}{2}+g\,h^2\,x^3+\frac {h^3\,x^4}{4}\right )-x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {h^2\,r\,\left (b\,c\,h\,p+4\,b\,d\,g\,p+a\,d\,h\,q+4\,b\,d\,g\,q\right )}{4\,b\,d}-\frac {h^3\,r\,\left (p+q\right )\,\left (4\,a\,d+4\,b\,c\right )}{16\,b\,d}\right )}{4\,b\,d}-\frac {g\,h\,r\,\left (2\,b\,c\,h\,p+3\,b\,d\,g\,p+2\,a\,d\,h\,q+3\,b\,d\,g\,q\right )}{2\,b\,d}+\frac {a\,c\,h^3\,r\,\left (p+q\right )}{4\,b\,d}\right )}{4\,b\,d}+\frac {g^2\,r\,\left (3\,b\,c\,h\,p+2\,b\,d\,g\,p+3\,a\,d\,h\,q+2\,b\,d\,g\,q\right )}{2\,b\,d}-\frac {a\,c\,\left (\frac {h^2\,r\,\left (b\,c\,h\,p+4\,b\,d\,g\,p+a\,d\,h\,q+4\,b\,d\,g\,q\right )}{4\,b\,d}-\frac {h^3\,r\,\left (p+q\right )\,\left (4\,a\,d+4\,b\,c\right )}{16\,b\,d}\right )}{b\,d}\right )-x^3\,\left (\frac {h^2\,r\,\left (b\,c\,h\,p+4\,b\,d\,g\,p+a\,d\,h\,q+4\,b\,d\,g\,q\right )}{12\,b\,d}-\frac {h^3\,r\,\left (p+q\right )\,\left (4\,a\,d+4\,b\,c\right )}{48\,b\,d}\right )+x^2\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {h^2\,r\,\left (b\,c\,h\,p+4\,b\,d\,g\,p+a\,d\,h\,q+4\,b\,d\,g\,q\right )}{4\,b\,d}-\frac {h^3\,r\,\left (p+q\right )\,\left (4\,a\,d+4\,b\,c\right )}{16\,b\,d}\right )}{8\,b\,d}-\frac {g\,h\,r\,\left (2\,b\,c\,h\,p+3\,b\,d\,g\,p+2\,a\,d\,h\,q+3\,b\,d\,g\,q\right )}{4\,b\,d}+\frac {a\,c\,h^3\,r\,\left (p+q\right )}{8\,b\,d}\right )-\frac {\ln \left (a+b\,x\right )\,\left (p\,r\,a^4\,h^3-4\,p\,r\,a^3\,b\,g\,h^2+6\,p\,r\,a^2\,b^2\,g^2\,h-4\,p\,r\,a\,b^3\,g^3\right )}{4\,b^4}-\frac {\ln \left (c+d\,x\right )\,\left (q\,r\,c^4\,h^3-4\,q\,r\,c^3\,d\,g\,h^2+6\,q\,r\,c^2\,d^2\,g^2\,h-4\,q\,r\,c\,d^3\,g^3\right )}{4\,d^4}-\frac {h^3\,r\,x^4\,\left (p+q\right )}{16} \]

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(g + h*x)^3,x)

[Out]

log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(g^3*x + (h^3*x^4)/4 + (3*g^2*h*x^2)/2 + g*h^2*x^3) - x*(((4*a*d + 4*b*c)
*(((4*a*d + 4*b*c)*((h^2*r*(b*c*h*p + 4*b*d*g*p + a*d*h*q + 4*b*d*g*q))/(4*b*d) - (h^3*r*(p + q)*(4*a*d + 4*b*
c))/(16*b*d)))/(4*b*d) - (g*h*r*(2*b*c*h*p + 3*b*d*g*p + 2*a*d*h*q + 3*b*d*g*q))/(2*b*d) + (a*c*h^3*r*(p + q))
/(4*b*d)))/(4*b*d) + (g^2*r*(3*b*c*h*p + 2*b*d*g*p + 3*a*d*h*q + 2*b*d*g*q))/(2*b*d) - (a*c*((h^2*r*(b*c*h*p +
4*b*d*g*p + a*d*h*q + 4*b*d*g*q))/(4*b*d) - (h^3*r*(p + q)*(4*a*d + 4*b*c))/(16*b*d)))/(b*d)) - x^3*((h^2*r*(
b*c*h*p + 4*b*d*g*p + a*d*h*q + 4*b*d*g*q))/(12*b*d) - (h^3*r*(p + q)*(4*a*d + 4*b*c))/(48*b*d)) + x^2*(((4*a*
d + 4*b*c)*((h^2*r*(b*c*h*p + 4*b*d*g*p + a*d*h*q + 4*b*d*g*q))/(4*b*d) - (h^3*r*(p + q)*(4*a*d + 4*b*c))/(16*
b*d)))/(8*b*d) - (g*h*r*(2*b*c*h*p + 3*b*d*g*p + 2*a*d*h*q + 3*b*d*g*q))/(4*b*d) + (a*c*h^3*r*(p + q))/(8*b*d)
) - (log(a + b*x)*(a^4*h^3*p*r - 4*a*b^3*g^3*p*r - 4*a^3*b*g*h^2*p*r + 6*a^2*b^2*g^2*h*p*r))/(4*b^4) - (log(c
+ d*x)*(c^4*h^3*q*r - 4*c*d^3*g^3*q*r - 4*c^3*d*g*h^2*q*r + 6*c^2*d^2*g^2*h*q*r))/(4*d^4) - (h^3*r*x^4*(p + q)
)/16