\(\int (a+b x)^3 \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [17]

Optimal result

Integrand size = 31, antiderivative size = 755 \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}-\frac {13 (b c-a d)^3 q^2 r^2 x}{24 d^3}-\frac {(b c-a d)^3 q (p+q) r^2 x}{2 d^3}+\frac {3 (b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}+\frac {13 (b c-a d)^2 q^2 r^2 (a+b x)^2}{48 b d^2}-\frac {7 (b c-a d) p q r^2 (a+b x)^3}{72 b d}-\frac {7 (b c-a d) q^2 r^2 (a+b x)^3}{72 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {25 (b c-a d)^4 q^2 r^2 \log (c+d x)}{24 b d^4}+\frac {(b c-a d)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^4 q^2 r^2 \log ^2(c+d x)}{4 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac {(b c-a d)^4 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 b d^4} \] Output:

-1/8*(-a*d+b*c)^3*p*q*r^2*x/d^3-13/24*(-a*d+b*c)^3*q^2*r^2*x/d^3-1/2*(-a*d 
+b*c)^3*q*(p+q)*r^2*x/d^3+3/16*(-a*d+b*c)^2*p*q*r^2*(b*x+a)^2/b/d^2+13/48* 
(-a*d+b*c)^2*q^2*r^2*(b*x+a)^2/b/d^2-7/72*(-a*d+b*c)*p*q*r^2*(b*x+a)^3/b/d 
-7/72*(-a*d+b*c)*q^2*r^2*(b*x+a)^3/b/d+1/32*p^2*r^2*(b*x+a)^4/b+1/16*p*q*r 
^2*(b*x+a)^4/b+1/32*q^2*r^2*(b*x+a)^4/b+1/8*(-a*d+b*c)^4*p*q*r^2*ln(d*x+c) 
/b/d^4+25/24*(-a*d+b*c)^4*q^2*r^2*ln(d*x+c)/b/d^4+1/2*(-a*d+b*c)^4*p*q*r^2 
*ln(-d*(b*x+a)/(-a*d+b*c))*ln(d*x+c)/b/d^4+1/4*(-a*d+b*c)^4*q^2*r^2*ln(d*x 
+c)^2/b/d^4+1/2*(-a*d+b*c)^3*q*r*(b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b 
/d^3-1/4*(-a*d+b*c)^2*q*r*(b*x+a)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^2+ 
1/6*(-a*d+b*c)*q*r*(b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d-1/8*p*r*( 
b*x+a)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b-1/8*q*r*(b*x+a)^4*ln(e*(f*(b*x+ 
a)^p*(d*x+c)^q)^r)/b-1/2*(-a*d+b*c)^4*q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x 
+c)^q)^r)/b/d^4+1/4*(b*x+a)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/b+1/2*(-a* 
d+b*c)^4*p*q*r^2*polylog(2,b*(d*x+c)/(-a*d+b*c))/b/d^4
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1853\) vs. \(2(755)=1510\).

Time = 1.63 (sec) , antiderivative size = 1853, normalized size of antiderivative = 2.45 \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]
 

Output:

(2*a^4*p*q*r^2)/b - (a*b^2*c^3*p*q*r^2)/(2*d^3) + (2*a^2*b*c^2*p*q*r^2)/d^ 
2 - (3*a^3*c*p*q*r^2)/d + (a^3*p^2*r^2*x)/8 + (37*a^3*p*q*r^2*x)/24 - (5*b 
^3*c^3*p*q*r^2*x)/(8*d^3) + (9*a*b^2*c^2*p*q*r^2*x)/(4*d^2) - (35*a^2*b*c* 
p*q*r^2*x)/(12*d) + 2*a^3*q^2*r^2*x - (25*b^3*c^3*q^2*r^2*x)/(24*d^3) + (1 
1*a*b^2*c^2*q^2*r^2*x)/(3*d^2) - (9*a^2*b*c*q^2*r^2*x)/(2*d) + (3*a^2*b*p^ 
2*r^2*x^2)/16 + (41*a^2*b*p*q*r^2*x^2)/48 + (3*b^3*c^2*p*q*r^2*x^2)/(16*d^ 
2) - (2*a*b^2*c*p*q*r^2*x^2)/(3*d) + (3*a^2*b*q^2*r^2*x^2)/4 + (13*b^3*c^2 
*q^2*r^2*x^2)/(48*d^2) - (5*a*b^2*c*q^2*r^2*x^2)/(6*d) + (a*b^2*p^2*r^2*x^ 
3)/8 + (25*a*b^2*p*q*r^2*x^3)/72 - (7*b^3*c*p*q*r^2*x^3)/(72*d) + (2*a*b^2 
*q^2*r^2*x^3)/9 - (7*b^3*c*q^2*r^2*x^3)/(72*d) + (b^3*p^2*r^2*x^4)/32 + (b 
^3*p*q*r^2*x^4)/16 + (b^3*q^2*r^2*x^4)/32 - (a^4*p^2*r^2*Log[a + b*x]^2)/( 
4*b) + (2*a^4*p*q*r^2*Log[c + d*x])/b + (b^3*c^4*p*q*r^2*Log[c + d*x])/(8* 
d^4) - (a*b^2*c^3*p*q*r^2*Log[c + d*x])/(2*d^3) + (3*a^2*b*c^2*p*q*r^2*Log 
[c + d*x])/(4*d^2) - (a^3*c*p*q*r^2*Log[c + d*x])/(2*d) + (25*b^3*c^4*q^2* 
r^2*Log[c + d*x])/(24*d^4) - (11*a*b^2*c^3*q^2*r^2*Log[c + d*x])/(3*d^3) + 
 (9*a^2*b*c^2*q^2*r^2*Log[c + d*x])/(2*d^2) - (2*a^3*c*q^2*r^2*Log[c + d*x 
])/d + (b^3*c^4*q^2*r^2*Log[c + d*x]^2)/(4*d^4) - (a*b^2*c^3*q^2*r^2*Log[c 
 + d*x]^2)/d^3 + (3*a^2*b*c^2*q^2*r^2*Log[c + d*x]^2)/(2*d^2) - (a^3*c*q^2 
*r^2*Log[c + d*x]^2)/d - (2*a^4*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/ 
b - (a^3*p*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/2 - 2*a^3*q*r*x*Lo...
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 723, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2984, 2981, 17, 49, 2009, 2994, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx\)

\(\Big \downarrow \) 2984

\(\displaystyle -\frac {1}{2} p r \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )dx-\frac {d q r \int \frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{2 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\)

\(\Big \downarrow \) 2981

\(\displaystyle -\frac {1}{2} p r \left (-\frac {d q r \int \frac {(a+b x)^4}{c+d x}dx}{4 b}-\frac {1}{4} p r \int (a+b x)^3dx+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\right )-\frac {d q r \int \frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{2 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {1}{2} p r \left (-\frac {d q r \int \frac {(a+b x)^4}{c+d x}dx}{4 b}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {p r (a+b x)^4}{16 b}\right )-\frac {d q r \int \frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{2 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {1}{2} p r \left (-\frac {d q r \int \left (\frac {(a d-b c)^4}{d^4 (c+d x)}-\frac {b (b c-a d)^3}{d^4}+\frac {b (a+b x)^3}{d}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (b c-a d)^2 (a+b x)}{d^3}\right )dx}{4 b}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {p r (a+b x)^4}{16 b}\right )-\frac {d q r \int \frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{2 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d q r \int \frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{2 b}-\frac {1}{2} p r \left (-\frac {d q r \left (\frac {(b c-a d)^4 \log (c+d x)}{d^5}-\frac {b x (b c-a d)^3}{d^4}+\frac {(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac {(a+b x)^3 (b c-a d)}{3 d^2}+\frac {(a+b x)^4}{4 d}\right )}{4 b}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {p r (a+b x)^4}{16 b}\right )+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\)

\(\Big \downarrow \) 2994

\(\displaystyle -\frac {d q r \int \left (\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (a d-b c)^4}{d^4 (c+d x)}-\frac {b (b c-a d)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4}+\frac {b (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}-\frac {b (b c-a d) (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {b (b c-a d)^2 (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}\right )dx}{2 b}-\frac {1}{2} p r \left (-\frac {d q r \left (\frac {(b c-a d)^4 \log (c+d x)}{d^5}-\frac {b x (b c-a d)^3}{d^4}+\frac {(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac {(a+b x)^3 (b c-a d)}{3 d^2}+\frac {(a+b x)^4}{4 d}\right )}{4 b}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {p r (a+b x)^4}{16 b}\right )+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d q r \left (\frac {(b c-a d)^4 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^5}-\frac {p r (b c-a d)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^5}-\frac {p r (b c-a d)^4 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^5}-\frac {q r (b c-a d)^4 \log ^2(c+d x)}{2 d^5}-\frac {25 q r (b c-a d)^4 \log (c+d x)}{12 d^5}-\frac {(a+b x) (b c-a d)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4}+\frac {b r x (p+q) (b c-a d)^3}{d^4}+\frac {13 b q r x (b c-a d)^3}{12 d^4}+\frac {(a+b x)^2 (b c-a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 d^3}-\frac {p r (a+b x)^2 (b c-a d)^2}{4 d^3}-\frac {13 q r (a+b x)^2 (b c-a d)^2}{24 d^3}-\frac {(a+b x)^3 (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 d^2}+\frac {p r (a+b x)^3 (b c-a d)}{9 d^2}+\frac {7 q r (a+b x)^3 (b c-a d)}{36 d^2}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 d}-\frac {p r (a+b x)^4}{16 d}-\frac {q r (a+b x)^4}{16 d}\right )}{2 b}-\frac {1}{2} p r \left (-\frac {d q r \left (\frac {(b c-a d)^4 \log (c+d x)}{d^5}-\frac {b x (b c-a d)^3}{d^4}+\frac {(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac {(a+b x)^3 (b c-a d)}{3 d^2}+\frac {(a+b x)^4}{4 d}\right )}{4 b}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {p r (a+b x)^4}{16 b}\right )+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\)

Input:

Int[(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]
 

Output:

((a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(4*b) - (p*r*(-1/16*( 
p*r*(a + b*x)^4)/b - (d*q*r*(-((b*(b*c - a*d)^3*x)/d^4) + ((b*c - a*d)^2*( 
a + b*x)^2)/(2*d^3) - ((b*c - a*d)*(a + b*x)^3)/(3*d^2) + (a + b*x)^4/(4*d 
) + ((b*c - a*d)^4*Log[c + d*x])/d^5))/(4*b) + ((a + b*x)^4*Log[e*(f*(a + 
b*x)^p*(c + d*x)^q)^r])/(4*b)))/2 - (d*q*r*((13*b*(b*c - a*d)^3*q*r*x)/(12 
*d^4) + (b*(b*c - a*d)^3*(p + q)*r*x)/d^4 - ((b*c - a*d)^2*p*r*(a + b*x)^2 
)/(4*d^3) - (13*(b*c - a*d)^2*q*r*(a + b*x)^2)/(24*d^3) + ((b*c - a*d)*p*r 
*(a + b*x)^3)/(9*d^2) + (7*(b*c - a*d)*q*r*(a + b*x)^3)/(36*d^2) - (p*r*(a 
 + b*x)^4)/(16*d) - (q*r*(a + b*x)^4)/(16*d) - (25*(b*c - a*d)^4*q*r*Log[c 
 + d*x])/(12*d^5) - ((b*c - a*d)^4*p*r*Log[-((d*(a + b*x))/(b*c - a*d))]*L 
og[c + d*x])/d^5 - ((b*c - a*d)^4*q*r*Log[c + d*x]^2)/(2*d^5) - ((b*c - a* 
d)^3*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^4 + ((b*c - a*d)^2* 
(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*d^3) - ((b*c - a*d)*( 
a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*d^2) + ((a + b*x)^4*Lo 
g[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*d) + ((b*c - a*d)^4*Log[c + d*x]*Lo 
g[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^5 - ((b*c - a*d)^4*p*r*PolyLog[2, (b 
*(c + d*x))/(b*c - a*d)])/d^5))/(2*b)
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 49

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] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2981

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)*(Lo 
g[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(h*(m + 1))), x] + (-Simp[b*p*(r/(h*(m + 
 1)))   Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Simp[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]
 

rule 2984

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] + (-Simp[b*p*r*( 
s/(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)), x], x] - Simp[d*q*r*(s/(h*(m + 1)))   Int[(g + h*x)^( 
m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(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] && 
IGtQ[s, 0] && NeQ[m, -1]
 

rule 2994

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]^(s_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[Log[e*(f*(a + 
b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c 
, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]
 

Maple [F]

\[\int \left (b x +a \right )^{3} {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]

Input:

int((b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)
 

Output:

int((b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)
 

Fricas [F]

\[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{3} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \] Input:

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="frica 
s")
 

Output:

integral((b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*log(((b*x + a)^p*(d*x + 
 c)^q*f)^r*e)^2, x)
 

Sympy [F]

\[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int \left (a + b x\right )^{3} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}\, dx \] Input:

integrate((b*x+a)**3*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2,x)
 

Output:

Integral((a + b*x)**3*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 1071, normalized size of antiderivative = 1.42 \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="maxim 
a")
 

Output:

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)*log(((b*x + a)^p*(d*x 
+ c)^q*f)^r*e)^2 + 1/24*(12*a^4*f*p*log(b*x + a)/b - (3*b^3*d^3*f*(p + q)* 
x^4 + 4*(a*b^2*d^3*f*(3*p + 4*q) - b^3*c*d^2*f*q)*x^3 + 6*(3*a^2*b*d^3*f*( 
p + 2*q) + b^3*c^2*d*f*q - 4*a*b^2*c*d^2*f*q)*x^2 + 12*(a^3*d^3*f*(p + 4*q 
) - b^3*c^3*f*q + 4*a*b^2*c^2*d*f*q - 6*a^2*b*c*d^2*f*q)*x)/d^3 - 12*(b^3* 
c^4*f*q - 4*a*b^2*c^3*d*f*q + 6*a^2*b*c^2*d^2*f*q - 4*a^3*c*d^3*f*q)*log(d 
*x + c)/d^4)*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f + 1/288*r^2*(12*((3* 
p*q + 25*q^2)*b^3*c^4*f^2 - 4*(3*p*q + 22*q^2)*a*b^2*c^3*d*f^2 + 18*(p*q + 
 6*q^2)*a^2*b*c^2*d^2*f^2 - 12*(p*q + 4*q^2)*a^3*c*d^3*f^2)*log(d*x + c)/d 
^4 - 144*(b^4*c^4*f^2*p*q - 4*a*b^3*c^3*d*f^2*p*q + 6*a^2*b^2*c^2*d^2*f^2* 
p*q - 4*a^3*b*c*d^3*f^2*p*q + a^4*d^4*f^2*p*q)*(log(b*x + a)*log((b*d*x + 
a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))/(b*d^4) + (9*(p 
^2 + 2*p*q + q^2)*b^4*d^4*f^2*x^4 - 72*a^4*d^4*f^2*p^2*log(b*x + a)^2 - 4* 
(7*(p*q + q^2)*b^4*c*d^3*f^2 - (9*p^2 + 25*p*q + 16*q^2)*a*b^3*d^4*f^2)*x^ 
3 + 6*((9*p*q + 13*q^2)*b^4*c^2*d^2*f^2 - 8*(4*p*q + 5*q^2)*a*b^3*c*d^3*f^ 
2 + (9*p^2 + 41*p*q + 36*q^2)*a^2*b^2*d^4*f^2)*x^2 + 144*(b^4*c^4*f^2*p*q 
- 4*a*b^3*c^3*d*f^2*p*q + 6*a^2*b^2*c^2*d^2*f^2*p*q - 4*a^3*b*c*d^3*f^2*p* 
q)*log(b*x + a)*log(d*x + c) + 72*(b^4*c^4*f^2*q^2 - 4*a*b^3*c^3*d*f^2*q^2 
 + 6*a^2*b^2*c^2*d^2*f^2*q^2 - 4*a^3*b*c*d^3*f^2*q^2)*log(d*x + c)^2 - 12* 
(5*(3*p*q + 5*q^2)*b^4*c^3*d*f^2 - 2*(27*p*q + 44*q^2)*a*b^3*c^2*d^2*f^...
 

Giac [F]

\[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{3} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \] Input:

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="giac" 
)
 

Output:

integrate((b*x + a)^3*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,{\left (a+b\,x\right )}^3 \,d x \] Input:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2*(a + b*x)^3,x)
 

Output:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2*(a + b*x)^3, x)
 

Reduce [F]

\[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {too large to display} \] Input:

int((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)
                                                                                    
                                                                                    
 

Output:

( - 144*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + 
 a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a**5* 
d**5*p**2*q*r - 144*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c 
*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x* 
*2),x)*a**5*d**5*p*q**2*r + 720*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**( 
p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x** 
2 + b*d*q*x**2),x)*a**4*b*c*d**4*p**2*q*r + 720*int(log(f**r*(c + d*x)**(q 
*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c 
*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a**4*b*c*d**4*p*q**2*r - 1440*int(log(f 
**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q* 
x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a**3*b**2*c**2*d**3*p* 
*2*q*r - 1440*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a 
*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x) 
*a**3*b**2*c**2*d**3*p*q**2*r + 1440*int(log(f**r*(c + d*x)**(q*r)*(a + b* 
x)**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d* 
p*x**2 + b*d*q*x**2),x)*a**2*b**3*c**3*d**2*p**2*q*r + 1440*int(log(f**r*( 
c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b 
*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a**2*b**3*c**3*d**2*p*q**2* 
r - 720*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + 
 a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a*...