Integrand size = 38, antiderivative size = 180 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^3 g^2 i x}{12 b d^2}-\frac {B (b c-a d)^2 g^2 i (a+b x)^2}{24 b^2 d}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b^2}-\frac {B (b c-a d)^4 g^2 i \log (c+d x)}{12 b^2 d^3} \]
1/12*B*(-a*d+b*c)^3*g^2*i*x/b/d^2-1/24*B*(-a*d+b*c)^2*g^2*i*(b*x+a)^2/b^2/ d+1/4*g^2*i*(b*x+a)^3*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b+1/12*(-a*d+b*c )*g^2*i*(b*x+a)^3*(A-B+B*ln(e*(b*x+a)/(d*x+c)))/b^2-1/12*B*(-a*d+b*c)^4*g^ 2*i*ln(d*x+c)/b^2/d^3
Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.21 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^2 i \left (8 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 d (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )}{d^3}-\frac {B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )}{d^3}\right )}{24 b^2} \]
(g^2*i*(8*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6 *d*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + (4*B*(b*c - a*d)^2*( 2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]))/d^3 - (B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^ 2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]))/d^3))/(24*b^2)
Time = 0.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2960, 27, 2948, 49, 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.
\(\displaystyle \int (a g+b g x)^2 (c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right ) \, dx\) |
\(\Big \downarrow \) 2960 |
\(\displaystyle \frac {i (b c-a d) \int g^2 (a+b x)^2 \left (A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )dx}{4 b}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {g^2 i (b c-a d) \int (a+b x)^2 \left (A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )dx}{4 b}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b}\) |
\(\Big \downarrow \) 2948 |
\(\displaystyle \frac {g^2 i (b c-a d) \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{3 b}-\frac {B (b c-a d) \int \frac {(a+b x)^2}{c+d x}dx}{3 b}\right )}{4 b}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {g^2 i (b c-a d) \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{3 b}-\frac {B (b c-a d) \int \left (\frac {(a d-b c)^2}{d^2 (c+d x)}-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}\right )dx}{3 b}\right )}{4 b}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {g^2 i (b c-a d) \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{3 b}-\frac {B (b c-a d) \left (\frac {(b c-a d)^2 \log (c+d x)}{d^3}-\frac {b x (b c-a d)}{d^2}+\frac {(a+b x)^2}{2 d}\right )}{3 b}\right )}{4 b}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b}\) |
(g^2*i*(a + b*x)^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b) + ((b*c - a*d)*g^2*i*(((a + b*x)^3*(A - B + B*Log[(e*(a + b*x))/(c + d*x)]) )/(3*b) - (B*(b*c - a*d)*(-((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))/(3*b)))/(4*b)
3.1.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*( (A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Simp[B*n*((b*c - a*d)/(g*(m + 1))) Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] / ; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && !(EqQ[m, -2] && IntegerQ[n])
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_)), x_Symbol] :> Sim p[(f + g*x)^(m + 1)*(h + i*x)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g* (m + 2))), x] + Simp[i*((b*c - a*d)/(b*d*(m + 2))) Int[(f + g*x)^m*(A - B *n + B*Log[e*((a + b*x)^n/(c + d*x)^n)]), x], x] /; FreeQ[{a, b, c, d, e, f , g, h, i, A, B, m, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d , 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IGtQ[m, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs. \(2(170)=340\).
Time = 0.80 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.20
method | result | size |
risch | \(\frac {i \,g^{2} B x \left (3 b^{2} d \,x^{3}+8 a b d \,x^{2}+4 b^{2} c \,x^{2}+6 a^{2} d x +12 a b c x +12 a^{2} c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12}+\frac {i \,g^{2} b^{2} d A \,x^{4}}{4}+\frac {2 i \,g^{2} b d A a \,x^{3}}{3}+\frac {i \,g^{2} b^{2} A c \,x^{3}}{3}+\frac {i \,g^{2} b d B a \,x^{3}}{12}-\frac {i \,g^{2} b^{2} B c \,x^{3}}{12}+\frac {i \,g^{2} d A \,a^{2} x^{2}}{2}+i \,g^{2} b A a c \,x^{2}+\frac {5 i \,g^{2} d B \,a^{2} x^{2}}{24}-\frac {i \,g^{2} b B a c \,x^{2}}{6}-\frac {i \,g^{2} b^{2} B \,c^{2} x^{2}}{24 d}+i \,g^{2} A \,a^{2} c x -\frac {i \,g^{2} d B \ln \left (b x +a \right ) a^{4}}{12 b^{2}}+\frac {i \,g^{2} B \ln \left (b x +a \right ) a^{3} c}{3 b}-\frac {i \,g^{2} B \ln \left (-d x -c \right ) a^{2} c^{2}}{2 d}+\frac {i \,g^{2} b B \ln \left (-d x -c \right ) a \,c^{3}}{3 d^{2}}-\frac {i \,g^{2} b^{2} B \ln \left (-d x -c \right ) c^{4}}{12 d^{3}}+\frac {i \,g^{2} d B \,a^{3} x}{12 b}+\frac {i \,g^{2} B \,a^{2} c x}{6}-\frac {i \,g^{2} b B a \,c^{2} x}{3 d}+\frac {i \,g^{2} b^{2} B \,c^{3} x}{12 d^{2}}\) | \(396\) |
parallelrisch | \(\frac {16 A \,x^{3} a \,b^{3} d^{4} g^{2} i +8 A \,x^{3} b^{4} c \,d^{3} g^{2} i -36 A \,a^{3} b c \,d^{3} g^{2} i +6 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} d^{4} g^{2} i +16 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} d^{4} g^{2} i +7 B a \,b^{3} c^{3} d \,g^{2} i +8 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} g^{2} i -12 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} g^{2} i +8 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d \,g^{2} i +8 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c \,d^{3} g^{2} i +12 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} d^{4} g^{2} i +24 A \,x^{2} a \,b^{3} c \,d^{3} g^{2} i -4 B \,x^{2} a \,b^{3} c \,d^{3} g^{2} i +24 A x \,a^{2} b^{2} c \,d^{3} g^{2} i +4 B x \,a^{2} b^{2} c \,d^{3} g^{2} i -8 B x a \,b^{3} c^{2} d^{2} g^{2} i +12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c^{2} d^{2} g^{2} i -8 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c^{3} d \,g^{2} i -11 B \,a^{3} b c \,d^{3} g^{2} i +8 B \,a^{2} b^{2} c^{2} d^{2} g^{2} i -48 A \,a^{2} b^{2} c^{2} d^{2} g^{2} i +24 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c \,d^{3} g^{2} i +24 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c \,d^{3} g^{2} i -2 B \,a^{4} d^{4} g^{2} i -2 B \,b^{4} c^{4} g^{2} i +2 B \,x^{3} a \,b^{3} d^{4} g^{2} i -2 B \,x^{3} b^{4} c \,d^{3} g^{2} i +12 A \,x^{2} a^{2} b^{2} d^{4} g^{2} i +5 B \,x^{2} a^{2} b^{2} d^{4} g^{2} i -B \,x^{2} b^{4} c^{2} d^{2} g^{2} i +2 B x \,a^{3} b \,d^{4} g^{2} i +2 B x \,b^{4} c^{3} d \,g^{2} i +6 A \,x^{4} b^{4} d^{4} g^{2} i +2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c^{4} g^{2} i -2 B \ln \left (b x +a \right ) a^{4} d^{4} g^{2} i -2 B \ln \left (b x +a \right ) b^{4} c^{4} g^{2} i}{24 b^{2} d^{3}}\) | \(772\) |
parts | \(\text {Expression too large to display}\) | \(1100\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1210\) |
default | \(\text {Expression too large to display}\) | \(1210\) |
1/12*i*g^2*B*x*(3*b^2*d*x^3+8*a*b*d*x^2+4*b^2*c*x^2+6*a^2*d*x+12*a*b*c*x+1 2*a^2*c)*ln(e*(b*x+a)/(d*x+c))+1/4*i*g^2*b^2*d*A*x^4+2/3*i*g^2*b*d*A*a*x^3 +1/3*i*g^2*b^2*A*c*x^3+1/12*i*g^2*b*d*B*a*x^3-1/12*i*g^2*b^2*B*c*x^3+1/2*i *g^2*d*A*a^2*x^2+i*g^2*b*A*a*c*x^2+5/24*i*g^2*d*B*a^2*x^2-1/6*i*g^2*b*B*a* c*x^2-1/24*i*g^2*b^2/d*B*c^2*x^2+i*g^2*A*a^2*c*x-1/12*i*g^2/b^2*d*B*ln(b*x +a)*a^4+1/3*i*g^2/b*B*ln(b*x+a)*a^3*c-1/2*i*g^2/d*B*ln(-d*x-c)*a^2*c^2+1/3 *i*g^2*b/d^2*B*ln(-d*x-c)*a*c^3-1/12*i*g^2*b^2/d^3*B*ln(-d*x-c)*c^4+1/12*i *g^2/b*d*B*a^3*x+1/6*i*g^2*B*a^2*c*x-1/3*i*g^2*b/d*B*a*c^2*x+1/12*i*g^2*b^ 2/d^2*B*c^3*x
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (170) = 340\).
Time = 0.38 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.06 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} g^{2} i x^{4} + 2 \, {\left ({\left (4 \, A - B\right )} b^{4} c d^{3} + {\left (8 \, A + B\right )} a b^{3} d^{4}\right )} g^{2} i x^{3} - {\left (B b^{4} c^{2} d^{2} - 4 \, {\left (6 \, A - B\right )} a b^{3} c d^{3} - {\left (12 \, A + 5 \, B\right )} a^{2} b^{2} d^{4}\right )} g^{2} i x^{2} + 2 \, {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 2 \, {\left (6 \, A + B\right )} a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g^{2} i x + 2 \, {\left (4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} g^{2} i \log \left (b x + a\right ) - 2 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} g^{2} i \log \left (d x + c\right ) + 2 \, {\left (3 \, B b^{4} d^{4} g^{2} i x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i x + 4 \, {\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i x^{3} + 6 \, {\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{24 \, b^{2} d^{3}} \]
1/24*(6*A*b^4*d^4*g^2*i*x^4 + 2*((4*A - B)*b^4*c*d^3 + (8*A + B)*a*b^3*d^4 )*g^2*i*x^3 - (B*b^4*c^2*d^2 - 4*(6*A - B)*a*b^3*c*d^3 - (12*A + 5*B)*a^2* b^2*d^4)*g^2*i*x^2 + 2*(B*b^4*c^3*d - 4*B*a*b^3*c^2*d^2 + 2*(6*A + B)*a^2* b^2*c*d^3 + B*a^3*b*d^4)*g^2*i*x + 2*(4*B*a^3*b*c*d^3 - B*a^4*d^4)*g^2*i*l og(b*x + a) - 2*(B*b^4*c^4 - 4*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2)*g^2*i* log(d*x + c) + 2*(3*B*b^4*d^4*g^2*i*x^4 + 12*B*a^2*b^2*c*d^3*g^2*i*x + 4*( B*b^4*c*d^3 + 2*B*a*b^3*d^4)*g^2*i*x^3 + 6*(2*B*a*b^3*c*d^3 + B*a^2*b^2*d^ 4)*g^2*i*x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^2*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 850 vs. \(2 (163) = 326\).
Time = 2.48 (sec) , antiderivative size = 850, normalized size of antiderivative = 4.72 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A b^{2} d g^{2} i x^{4}}{4} - \frac {B a^{3} g^{2} i \left (a d - 4 b c\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{2} i + \frac {B a^{4} d^{3} g^{2} i \left (a d - 4 b c\right )}{b} - 10 B a^{3} b c^{2} d^{2} g^{2} i - B a^{3} c d^{2} g^{2} i \left (a d - 4 b c\right ) + 4 B a^{2} b^{2} c^{3} d g^{2} i - B a b^{3} c^{4} g^{2} i}{B a^{4} d^{4} g^{2} i - 4 B a^{3} b c d^{3} g^{2} i - 6 B a^{2} b^{2} c^{2} d^{2} g^{2} i + 4 B a b^{3} c^{3} d g^{2} i - B b^{4} c^{4} g^{2} i} \right )}}{12 b^{2}} - \frac {B c^{2} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{2} i - 10 B a^{3} b c^{2} d^{2} g^{2} i + 4 B a^{2} b^{2} c^{3} d g^{2} i - B a b^{3} c^{4} g^{2} i + B a b c^{2} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) - \frac {B b^{2} c^{3} g^{2} i \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{2} i - 4 B a^{3} b c d^{3} g^{2} i - 6 B a^{2} b^{2} c^{2} d^{2} g^{2} i + 4 B a b^{3} c^{3} d g^{2} i - B b^{4} c^{4} g^{2} i} \right )}}{12 d^{3}} + x^{3} \cdot \left (\frac {2 A a b d g^{2} i}{3} + \frac {A b^{2} c g^{2} i}{3} + \frac {B a b d g^{2} i}{12} - \frac {B b^{2} c g^{2} i}{12}\right ) + x^{2} \left (\frac {A a^{2} d g^{2} i}{2} + A a b c g^{2} i + \frac {5 B a^{2} d g^{2} i}{24} - \frac {B a b c g^{2} i}{6} - \frac {B b^{2} c^{2} g^{2} i}{24 d}\right ) + x \left (A a^{2} c g^{2} i + \frac {B a^{3} d g^{2} i}{12 b} + \frac {B a^{2} c g^{2} i}{6} - \frac {B a b c^{2} g^{2} i}{3 d} + \frac {B b^{2} c^{3} g^{2} i}{12 d^{2}}\right ) + \left (B a^{2} c g^{2} i x + \frac {B a^{2} d g^{2} i x^{2}}{2} + B a b c g^{2} i x^{2} + \frac {2 B a b d g^{2} i x^{3}}{3} + \frac {B b^{2} c g^{2} i x^{3}}{3} + \frac {B b^{2} d g^{2} i x^{4}}{4}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]
A*b**2*d*g**2*i*x**4/4 - B*a**3*g**2*i*(a*d - 4*b*c)*log(x + (B*a**4*c*d** 3*g**2*i + B*a**4*d**3*g**2*i*(a*d - 4*b*c)/b - 10*B*a**3*b*c**2*d**2*g**2 *i - B*a**3*c*d**2*g**2*i*(a*d - 4*b*c) + 4*B*a**2*b**2*c**3*d*g**2*i - B* a*b**3*c**4*g**2*i)/(B*a**4*d**4*g**2*i - 4*B*a**3*b*c*d**3*g**2*i - 6*B*a **2*b**2*c**2*d**2*g**2*i + 4*B*a*b**3*c**3*d*g**2*i - B*b**4*c**4*g**2*i) )/(12*b**2) - B*c**2*g**2*i*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)*log(x + (B*a**4*c*d**3*g**2*i - 10*B*a**3*b*c**2*d**2*g**2*i + 4*B*a**2*b**2*c**3* d*g**2*i - B*a*b**3*c**4*g**2*i + B*a*b*c**2*g**2*i*(6*a**2*d**2 - 4*a*b*c *d + b**2*c**2) - B*b**2*c**3*g**2*i*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2) /d)/(B*a**4*d**4*g**2*i - 4*B*a**3*b*c*d**3*g**2*i - 6*B*a**2*b**2*c**2*d* *2*g**2*i + 4*B*a*b**3*c**3*d*g**2*i - B*b**4*c**4*g**2*i))/(12*d**3) + x* *3*(2*A*a*b*d*g**2*i/3 + A*b**2*c*g**2*i/3 + B*a*b*d*g**2*i/12 - B*b**2*c* g**2*i/12) + x**2*(A*a**2*d*g**2*i/2 + A*a*b*c*g**2*i + 5*B*a**2*d*g**2*i/ 24 - B*a*b*c*g**2*i/6 - B*b**2*c**2*g**2*i/(24*d)) + x*(A*a**2*c*g**2*i + B*a**3*d*g**2*i/(12*b) + B*a**2*c*g**2*i/6 - B*a*b*c**2*g**2*i/(3*d) + B*b **2*c**3*g**2*i/(12*d**2)) + (B*a**2*c*g**2*i*x + B*a**2*d*g**2*i*x**2/2 + B*a*b*c*g**2*i*x**2 + 2*B*a*b*d*g**2*i*x**3/3 + B*b**2*c*g**2*i*x**3/3 + B*b**2*d*g**2*i*x**4/4)*log(e*(a + b*x)/(c + d*x))
Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (170) = 340\).
Time = 0.22 (sec) , antiderivative size = 671, normalized size of antiderivative = 3.73 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{4} \, A b^{2} d g^{2} i x^{4} + \frac {1}{3} \, A b^{2} c g^{2} i x^{3} + \frac {2}{3} \, A a b d g^{2} i x^{3} + A a b c g^{2} i x^{2} + \frac {1}{2} \, A a^{2} d g^{2} i x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a^{2} c g^{2} i + {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a b c g^{2} i + \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} c g^{2} i + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a^{2} d g^{2} i + \frac {1}{3} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a b d g^{2} i + \frac {1}{24} \, {\left (6 \, x^{4} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b^{2} d g^{2} i + A a^{2} c g^{2} i x \]
1/4*A*b^2*d*g^2*i*x^4 + 1/3*A*b^2*c*g^2*i*x^3 + 2/3*A*a*b*d*g^2*i*x^3 + A* a*b*c*g^2*i*x^2 + 1/2*A*a^2*d*g^2*i*x^2 + (x*log(b*e*x/(d*x + c) + a*e/(d* x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B*a^2*c*g^2*i + (x^2*log(b* e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d ^2 - (b*c - a*d)*x/(b*d))*B*a*b*c*g^2*i + 1/6*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2* c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*b^2*c*g^2*i + 1 /2*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2* log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a^2*d*g^2*i + 1/3*(2*x^3*log(b*e *x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c )/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a *b*d*g^2*i + 1/24*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log( b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3 *(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b^2*d *g^2*i + A*a^2*c*g^2*i*x
Leaf count of result is larger than twice the leaf count of optimal. 2418 vs. \(2 (170) = 340\).
Time = 0.46 (sec) , antiderivative size = 2418, normalized size of antiderivative = 13.43 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]
1/24*(2*(B*b^7*c^5*e^5*g^2*i - 5*B*a*b^6*c^4*d*e^5*g^2*i + 10*B*a^2*b^5*c^ 3*d^2*e^5*g^2*i - 10*B*a^3*b^4*c^2*d^3*e^5*g^2*i + 5*B*a^4*b^3*c*d^4*e^5*g ^2*i - B*a^5*b^2*d^5*e^5*g^2*i - 4*(b*e*x + a*e)*B*b^6*c^5*d*e^4*g^2*i/(d* x + c) + 20*(b*e*x + a*e)*B*a*b^5*c^4*d^2*e^4*g^2*i/(d*x + c) - 40*(b*e*x + a*e)*B*a^2*b^4*c^3*d^3*e^4*g^2*i/(d*x + c) + 40*(b*e*x + a*e)*B*a^3*b^3* c^2*d^4*e^4*g^2*i/(d*x + c) - 20*(b*e*x + a*e)*B*a^4*b^2*c*d^5*e^4*g^2*i/( d*x + c) + 4*(b*e*x + a*e)*B*a^5*b*d^6*e^4*g^2*i/(d*x + c) + 6*(b*e*x + a* e)^2*B*b^5*c^5*d^2*e^3*g^2*i/(d*x + c)^2 - 30*(b*e*x + a*e)^2*B*a*b^4*c^4* d^3*e^3*g^2*i/(d*x + c)^2 + 60*(b*e*x + a*e)^2*B*a^2*b^3*c^3*d^4*e^3*g^2*i /(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a^3*b^2*c^2*d^5*e^3*g^2*i/(d*x + c)^2 + 30*(b*e*x + a*e)^2*B*a^4*b*c*d^6*e^3*g^2*i/(d*x + c)^2 - 6*(b*e*x + a*e) ^2*B*a^5*d^7*e^3*g^2*i/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/(b^4*d^3* e^4 - 4*(b*e*x + a*e)*b^3*d^4*e^3/(d*x + c) + 6*(b*e*x + a*e)^2*b^2*d^5*e^ 2/(d*x + c)^2 - 4*(b*e*x + a*e)^3*b*d^6*e/(d*x + c)^3 + (b*e*x + a*e)^4*d^ 7/(d*x + c)^4) + (2*A*b^8*c^5*e^5*g^2*i + B*b^8*c^5*e^5*g^2*i - 10*A*a*b^7 *c^4*d*e^5*g^2*i - 5*B*a*b^7*c^4*d*e^5*g^2*i + 20*A*a^2*b^6*c^3*d^2*e^5*g^ 2*i + 10*B*a^2*b^6*c^3*d^2*e^5*g^2*i - 20*A*a^3*b^5*c^2*d^3*e^5*g^2*i - 10 *B*a^3*b^5*c^2*d^3*e^5*g^2*i + 10*A*a^4*b^4*c*d^4*e^5*g^2*i + 5*B*a^4*b^4* c*d^4*e^5*g^2*i - 2*A*a^5*b^3*d^5*e^5*g^2*i - B*a^5*b^3*d^5*e^5*g^2*i - 8* (b*e*x + a*e)*A*b^7*c^5*d*e^4*g^2*i/(d*x + c) - 2*(b*e*x + a*e)*B*b^7*c...
Time = 1.59 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.54 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=x^3\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{12}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{36}\right )-x^2\,\left (\frac {\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{24\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2-B\,b^2\,c^2+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\right )}{6\,d}+\frac {A\,a\,b\,c\,g^2\,i}{2}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a^2\,c\,g^2\,i\,x+\frac {B\,a\,g^2\,i\,x^2\,\left (a\,d+2\,b\,c\right )}{2}+\frac {B\,b\,g^2\,i\,x^3\,\left (2\,a\,d+b\,c\right )}{3}+\frac {B\,b^2\,d\,g^2\,i\,x^4}{4}\right )+x\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{12\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2-B\,b^2\,c^2+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\right )}{3\,d}+A\,a\,b\,c\,g^2\,i\right )}{12\,b\,d}-\frac {a\,c\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{b\,d}+\frac {a\,g^2\,i\,\left (2\,A\,a^2\,d^2+6\,A\,b^2\,c^2+B\,a^2\,d^2-2\,B\,b^2\,c^2+12\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\right )}{2\,b\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (6\,B\,i\,a^2\,c^2\,d^2\,g^2-4\,B\,i\,a\,b\,c^3\,d\,g^2+B\,i\,b^2\,c^4\,g^2\right )}{12\,d^3}-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4\,d\,g^2\,i-4\,B\,a^3\,b\,c\,g^2\,i\right )}{12\,b^2}+\frac {A\,b^2\,d\,g^2\,i\,x^4}{4} \]
x^3*((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/12 - (A*b*g^2*i*(12*a* d + 12*b*c))/36) - x^2*((((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12)*(12*a*d + 12*b*c))/(24*b*d) - (g^2*i* (9*A*a^2*d^2 + 3*A*b^2*c^2 + 2*B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d - B*a* b*c*d))/(6*d) + (A*a*b*c*g^2*i)/2) + log((e*(a + b*x))/(c + d*x))*(B*a^2*c *g^2*i*x + (B*a*g^2*i*x^2*(a*d + 2*b*c))/2 + (B*b*g^2*i*x^3*(2*a*d + b*c)) /3 + (B*b^2*d*g^2*i*x^4)/4) + x*(((12*a*d + 12*b*c)*((((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12)*(12*a*d + 12*b*c))/(12*b*d) - (g^2*i*(9*A*a^2*d^2 + 3*A*b^2*c^2 + 2*B*a^2*d^2 - B* b^2*c^2 + 18*A*a*b*c*d - B*a*b*c*d))/(3*d) + A*a*b*c*g^2*i))/(12*b*d) - (a *c*((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d - B*b*c))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12))/(b*d) + (a*g^2*i*(2*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2 - 2*B*b^2*c^2 + 12*A*a*b*c*d + B*a*b*c*d))/(2*b*d)) - (log(c + d*x)*(B*b^2*c ^4*g^2*i + 6*B*a^2*c^2*d^2*g^2*i - 4*B*a*b*c^3*d*g^2*i))/(12*d^3) - (log(a + b*x)*(B*a^4*d*g^2*i - 4*B*a^3*b*c*g^2*i))/(12*b^2) + (A*b^2*d*g^2*i*x^4 )/4