Integrand size = 29, antiderivative size = 510 \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {(b c-a d) p q r^2 x}{2 d}-\frac {(b c-a d) q^2 r^2 x}{2 d}-\frac {(b c-a d) q (p+q) r^2 x}{d}+\frac {p^2 r^2 (a+b x)^2}{4 b}+\frac {p q r^2 (a+b x)^2}{2 b}+\frac {q^2 r^2 (a+b x)^2}{4 b}+\frac {(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac {3 (b c-a d)^2 q^2 r^2 \log (c+d x)}{2 b d^2}+\frac {(b c-a d)^2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {(b c-a d)^2 q^2 r^2 \log ^2(c+d x)}{2 b d^2}+\frac {(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(b c-a d)^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {(b c-a d)^2 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b d^2} \] Output:
-1/2*(-a*d+b*c)*p*q*r^2*x/d-1/2*(-a*d+b*c)*q^2*r^2*x/d-(-a*d+b*c)*q*(p+q)* r^2*x/d+1/4*p^2*r^2*(b*x+a)^2/b+1/2*p*q*r^2*(b*x+a)^2/b+1/4*q^2*r^2*(b*x+a )^2/b+1/2*(-a*d+b*c)^2*p*q*r^2*ln(d*x+c)/b/d^2+3/2*(-a*d+b*c)^2*q^2*r^2*ln (d*x+c)/b/d^2+(-a*d+b*c)^2*p*q*r^2*ln(-d*(b*x+a)/(-a*d+b*c))*ln(d*x+c)/b/d ^2+1/2*(-a*d+b*c)^2*q^2*r^2*ln(d*x+c)^2/b/d^2+(-a*d+b*c)*q*r*(b*x+a)*ln(e* (f*(b*x+a)^p*(d*x+c)^q)^r)/b/d-1/2*p*r*(b*x+a)^2*ln(e*(f*(b*x+a)^p*(d*x+c) ^q)^r)/b-1/2*q*r*(b*x+a)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b-(-a*d+b*c)^2* q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^2+1/2*(b*x+a)^2*ln(e*(f* (b*x+a)^p*(d*x+c)^q)^r)^2/b+(-a*d+b*c)^2*p*q*r^2*polylog(2,b*(d*x+c)/(-a*d +b*c))/b/d^2
Time = 0.73 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.53 \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {-4 a b c d p q r^2+8 a^2 d^2 p q r^2+2 a b d^2 p^2 r^2 x-6 b^2 c d p q r^2 x+10 a b d^2 p q r^2 x-6 b^2 c d q^2 r^2 x+8 a b d^2 q^2 r^2 x+b^2 d^2 p^2 r^2 x^2+2 b^2 d^2 p q r^2 x^2+b^2 d^2 q^2 r^2 x^2-2 a^2 d^2 p^2 r^2 \log ^2(a+b x)+2 b^2 c^2 p q r^2 \log (c+d x)-4 a b c d p q r^2 \log (c+d x)+8 a^2 d^2 p q r^2 \log (c+d x)+6 b^2 c^2 q^2 r^2 \log (c+d x)-8 a b c d q^2 r^2 \log (c+d x)+2 b^2 c^2 q^2 r^2 \log ^2(c+d x)-4 a b c d q^2 r^2 \log ^2(c+d x)-8 a^2 d^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a b d^2 p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 b^2 c d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-8 a b d^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b^2 d^2 p r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b^2 d^2 q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 b^2 c^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+8 a b c d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 a b d^2 x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b^2 d^2 x^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 p r \log (a+b x) \left (2 b c (b c-2 a d) q r \log (c+d x)-2 (b c-a d)^2 q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (3 a d (p-q) r+2 b c q r+2 a d \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )-4 (b c-a d)^2 p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{4 b d^2} \] Input:
Integrate[(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]
Output:
(-4*a*b*c*d*p*q*r^2 + 8*a^2*d^2*p*q*r^2 + 2*a*b*d^2*p^2*r^2*x - 6*b^2*c*d* p*q*r^2*x + 10*a*b*d^2*p*q*r^2*x - 6*b^2*c*d*q^2*r^2*x + 8*a*b*d^2*q^2*r^2 *x + b^2*d^2*p^2*r^2*x^2 + 2*b^2*d^2*p*q*r^2*x^2 + b^2*d^2*q^2*r^2*x^2 - 2 *a^2*d^2*p^2*r^2*Log[a + b*x]^2 + 2*b^2*c^2*p*q*r^2*Log[c + d*x] - 4*a*b*c *d*p*q*r^2*Log[c + d*x] + 8*a^2*d^2*p*q*r^2*Log[c + d*x] + 6*b^2*c^2*q^2*r ^2*Log[c + d*x] - 8*a*b*c*d*q^2*r^2*Log[c + d*x] + 2*b^2*c^2*q^2*r^2*Log[c + d*x]^2 - 4*a*b*c*d*q^2*r^2*Log[c + d*x]^2 - 8*a^2*d^2*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 4*a*b*d^2*p*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q )^r] + 4*b^2*c*d*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 8*a*b*d^2*q* r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 2*b^2*d^2*p*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 2*b^2*d^2*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x) ^q)^r] - 4*b^2*c^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 8*a*b*c*d*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 4*a*b*d ^2*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + 2*b^2*d^2*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + 2*p*r*Log[a + b*x]*(2*b*c*(b*c - 2*a*d)*q*r*Log [c + d*x] - 2*(b*c - a*d)^2*q*r*Log[(b*(c + d*x))/(b*c - a*d)] + a*d*(3*a* d*(p - q)*r + 2*b*c*q*r + 2*a*d*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])) - 4 *(b*c - a*d)^2*p*q*r^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(4*b*d^2)
Time = 0.92 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, 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.
\(\displaystyle \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx\) |
\(\Big \downarrow \) 2984 |
\(\displaystyle -p r \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )dx-\frac {d q r \int \frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{b}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}\) |
\(\Big \downarrow \) 2981 |
\(\displaystyle -p r \left (-\frac {d q r \int \frac {(a+b x)^2}{c+d x}dx}{2 b}-\frac {1}{2} p r \int (a+b x)dx+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}\right )-\frac {d q r \int \frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{b}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -p r \left (-\frac {d q r \int \frac {(a+b x)^2}{c+d x}dx}{2 b}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {p r (a+b x)^2}{4 b}\right )-\frac {d q r \int \frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{b}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -p r \left (-\frac {d q r \int \left (\frac {(a d-b c)^2}{d^2 (c+d x)}-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}\right )dx}{2 b}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {p r (a+b x)^2}{4 b}\right )-\frac {d q r \int \frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{b}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d q r \int \frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{b}-p r \left (-\frac {d q r \left (\frac {(b c-a d)^2 \log (c+d x)}{d^3}-\frac {b x (b c-a d)}{d^2}+\frac {(a+b x)^2}{2 d}\right )}{2 b}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {p r (a+b x)^2}{4 b}\right )+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}\) |
\(\Big \downarrow \) 2994 |
\(\displaystyle -\frac {d q r \int \left (\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (a d-b c)^2}{d^2 (c+d x)}-\frac {b (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {b (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}\right )dx}{b}-p r \left (-\frac {d q r \left (\frac {(b c-a d)^2 \log (c+d x)}{d^3}-\frac {b x (b c-a d)}{d^2}+\frac {(a+b x)^2}{2 d}\right )}{2 b}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {p r (a+b x)^2}{4 b}\right )+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d q r \left (\frac {(b c-a d)^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}-\frac {p r (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^3}-\frac {p r (b c-a d)^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^3}-\frac {q r (b c-a d)^2 \log ^2(c+d x)}{2 d^3}-\frac {3 q r (b c-a d)^2 \log (c+d x)}{2 d^3}-\frac {(a+b x) (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {b r x (p+q) (b c-a d)}{d^2}+\frac {b q r x (b c-a d)}{2 d^2}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 d}-\frac {p r (a+b x)^2}{4 d}-\frac {q r (a+b x)^2}{4 d}\right )}{b}-p r \left (-\frac {d q r \left (\frac {(b c-a d)^2 \log (c+d x)}{d^3}-\frac {b x (b c-a d)}{d^2}+\frac {(a+b x)^2}{2 d}\right )}{2 b}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {p r (a+b x)^2}{4 b}\right )+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}\) |
Input:
Int[(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]
Output:
((a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(2*b) - p*r*(-1/4*(p* r*(a + b*x)^2)/b - (d*q*r*(-((b*(b*c - a*d)*x)/d^2) + (a + b*x)^2/(2*d) + ((b*c - a*d)^2*Log[c + d*x])/d^3))/(2*b) + ((a + b*x)^2*Log[e*(f*(a + b*x) ^p*(c + d*x)^q)^r])/(2*b)) - (d*q*r*((b*(b*c - a*d)*q*r*x)/(2*d^2) + (b*(b *c - a*d)*(p + q)*r*x)/d^2 - (p*r*(a + b*x)^2)/(4*d) - (q*r*(a + b*x)^2)/( 4*d) - (3*(b*c - a*d)^2*q*r*Log[c + d*x])/(2*d^3) - ((b*c - a*d)^2*p*r*Log [-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/d^3 - ((b*c - a*d)^2*q*r*Log[ c + d*x]^2)/(2*d^3) - ((b*c - a*d)*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x )^q)^r])/d^2 + ((a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*d) + ((b*c - a*d)^2*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^3 - (( b*c - a*d)^2*p*r*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/d^3))/b
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]
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]
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]
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]
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]
\[\int \left (b x +a \right ) {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]
Input:
int((b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)
Output:
int((b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)
\[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \] Input:
integrate((b*x+a)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="fricas" )
Output:
integral((b*x + a)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2, x)
\[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int \left (a + b x\right ) \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}\, dx \] Input:
integrate((b*x+a)*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2,x)
Output:
Integral((a + b*x)*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)**2, x)
Time = 0.06 (sec) , antiderivative size = 504, normalized size of antiderivative = 0.99 \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} + \frac {{\left (\frac {2 \, a^{2} f p \log \left (b x + a\right )}{b} - \frac {b d f {\left (p + q\right )} x^{2} + 2 \, {\left (a d f {\left (p + 2 \, q\right )} - b c f q\right )} x}{d} - \frac {2 \, {\left (b c^{2} f q - 2 \, a c d f q\right )} \log \left (d x + c\right )}{d^{2}}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{2 \, f} + \frac {r^{2} {\left (\frac {2 \, {\left ({\left (p q + 3 \, q^{2}\right )} b c^{2} f^{2} - 2 \, {\left (p q + 2 \, q^{2}\right )} a c d f^{2}\right )} \log \left (d x + c\right )}{d^{2}} - \frac {4 \, {\left (b^{2} c^{2} f^{2} p q - 2 \, a b c d f^{2} p q + a^{2} d^{2} f^{2} p q\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b d^{2}} - \frac {2 \, a^{2} d^{2} f^{2} p^{2} \log \left (b x + a\right )^{2} - {\left (p^{2} + 2 \, p q + q^{2}\right )} b^{2} d^{2} f^{2} x^{2} - 4 \, {\left (b^{2} c^{2} f^{2} p q - 2 \, a b c d f^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - 2 \, {\left (b^{2} c^{2} f^{2} q^{2} - 2 \, a b c d f^{2} q^{2}\right )} \log \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (p q + q^{2}\right )} b^{2} c d f^{2} - {\left (p^{2} + 5 \, p q + 4 \, q^{2}\right )} a b d^{2} f^{2}\right )} x - 2 \, {\left (2 \, a b c d f^{2} p q - {\left (p^{2} + 3 \, p q\right )} a^{2} d^{2} f^{2}\right )} \log \left (b x + a\right )}{b d^{2}}\right )}}{4 \, f^{2}} \] Input:
integrate((b*x+a)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="maxima" )
Output:
1/2*(b*x^2 + 2*a*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/2*(2*a^2*f* p*log(b*x + a)/b - (b*d*f*(p + q)*x^2 + 2*(a*d*f*(p + 2*q) - b*c*f*q)*x)/d - 2*(b*c^2*f*q - 2*a*c*d*f*q)*log(d*x + c)/d^2)*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f + 1/4*r^2*(2*((p*q + 3*q^2)*b*c^2*f^2 - 2*(p*q + 2*q^2)*a* c*d*f^2)*log(d*x + c)/d^2 - 4*(b^2*c^2*f^2*p*q - 2*a*b*c*d*f^2*p*q + a^2*d ^2*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^2) - (2*a^2*d^2*f^2*p^2*log(b*x + a)^2 - (p^2 + 2*p*q + q^2)*b^2*d^2*f^2*x^2 - 4*(b^2*c^2*f^2*p*q - 2*a*b*c*d*f^2*p*q)* log(b*x + a)*log(d*x + c) - 2*(b^2*c^2*f^2*q^2 - 2*a*b*c*d*f^2*q^2)*log(d* x + c)^2 + 2*(3*(p*q + q^2)*b^2*c*d*f^2 - (p^2 + 5*p*q + 4*q^2)*a*b*d^2*f^ 2)*x - 2*(2*a*b*c*d*f^2*p*q - (p^2 + 3*p*q)*a^2*d^2*f^2)*log(b*x + a))/(b* d^2))/f^2
\[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \] Input:
integrate((b*x+a)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="giac")
Output:
integrate((b*x + a)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2, x)
Timed out. \[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,\left (a+b\,x\right ) \,d x \] Input:
int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2*(a + b*x),x)
Output:
int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2*(a + b*x), x)
\[ \int (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {too large to display} \] Input:
int((b*x+a)*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)
Output:
( - 4*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*d* *3*p**2*q*r - 4*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*d**3*p*q**2*r + 12*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*c*d**2*p**2*q*r + 12*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*c*d**2*p*q**2*r - 12*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*b**2*c**2*d*p**2*q*r - 12*in t(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*b**2*c**2*d*p *q**2*r + 4*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)*b **3*c**3*p**2*q*r + 4*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)*b**3*c**3*p*q**2*r + 2*log(c + d*x)*a**2*d**2*p**2*q*r**2 + 8*log (c + d*x)*a**2*d**2*p*q**2*r**2 + 6*log(c + d*x)*a**2*d**2*q**3*r**2 - 4*l og(c + d*x)*a*b*c*d*p**2*q*r**2 - 16*log(c + d*x)*a*b*c*d*p*q**2*r**2 -...