Integrand size = 30, antiderivative size = 118 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^2 g^2 x}{3 d^2}-\frac {B (b c-a d) g^2 (a+b x)^2}{6 b d}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b}-\frac {B (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3} \] Output:
1/3*B*(-a*d+b*c)^2*g^2*x/d^2-1/6*B*(-a*d+b*c)*g^2*(b*x+a)^2/b/d+1/3*g^2*(b *x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/b-1/3*B*(-a*d+b*c)^3*g^2*ln(d*x+c)/b/d ^3
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^2 \left ((a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {B (-b c+a d) \left (d \left (a^2 d+4 a b d x+b^2 x (-2 c+d x)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{2 d^3}\right )}{3 b} \] Input:
Integrate[(a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
Output:
(g^2*((a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + (B*(-(b*c) + a*d) *(d*(a^2*d + 4*a*b*d*x + b^2*x*(-2*c + d*x)) + 2*(b*c - a*d)^2*Log[c + d*x ]))/(2*d^3)))/(3*b)
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2948, 27, 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.
| \(\displaystyle \int (a g+b g x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2948 |
| \(\displaystyle \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}-\frac {B (b c-a d) \int \frac {g^3 (a+b x)^2}{c+d x}dx}{3 b g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}-\frac {B g^2 (b c-a d) \int \frac {(a+b x)^2}{c+d x}dx}{3 b}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}-\frac {B g^2 (b c-a d) \int \left (\frac {(a d-b c)^2}{d^2 (c+d x)}-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}\right )dx}{3 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}-\frac {B g^2 (b c-a d) \left (\frac {(b c-a d)^2 \log (c+d x)}{d^3}-\frac {b x (b c-a d)}{d^2}+\frac {(a+b x)^2}{2 d}\right )}{3 b}\) |
Input:
Int[(a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
Output:
(g^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b) - (B*(b*c - a *d)*g^2*(-((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)
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])
Time = 0.94 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.75
| method | result | size |
| risch | \(\frac {\left (b x +a \right )^{3} g^{2} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 b}+\frac {g^{2} b^{2} A \,x^{3}}{3}+g^{2} b A a \,x^{2}+\frac {g^{2} b B a \,x^{2}}{6}-\frac {g^{2} b^{2} B c \,x^{2}}{6 d}+g^{2} A \,a^{2} x +\frac {g^{2} B \ln \left (d x +c \right ) a^{3}}{3 b}-\frac {g^{2} B \ln \left (d x +c \right ) a^{2} c}{d}+\frac {g^{2} b B \ln \left (d x +c \right ) a \,c^{2}}{d^{2}}-\frac {g^{2} b^{2} B \ln \left (d x +c \right ) c^{3}}{3 d^{3}}+\frac {2 g^{2} B \,a^{2} x}{3}-\frac {g^{2} b B a c x}{d}+\frac {g^{2} b^{2} B \,c^{2} x}{3 d^{2}}\) | \(207\) |
| parallelrisch | \(\frac {2 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c \,d^{3} g^{2}+2 A \,x^{3} a \,b^{3} c \,d^{3} g^{2}+6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c \,d^{3} g^{2}+6 A \,x^{2} a^{2} b^{2} c \,d^{3} g^{2}+B \,x^{2} a^{2} b^{2} c \,d^{3} g^{2}-B \,x^{2} a \,b^{3} c^{2} d^{2} g^{2}+6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b c \,d^{3} g^{2}+6 A x \,a^{3} b c \,d^{3} g^{2}+2 B \ln \left (b x +a \right ) a^{4} c \,d^{3} g^{2}-6 B \ln \left (b x +a \right ) a^{3} b \,c^{2} d^{2} g^{2}+6 B \ln \left (b x +a \right ) a^{2} b^{2} c^{3} d \,g^{2}-2 B \ln \left (b x +a \right ) a \,b^{3} c^{4} g^{2}+4 B x \,a^{3} b c \,d^{3} g^{2}-6 B x \,a^{2} b^{2} c^{2} d^{2} g^{2}+2 B x a \,b^{3} c^{3} d \,g^{2}+6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d^{2} g^{2}-6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c^{3} d \,g^{2}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c^{4} g^{2}}{6 a b c \,d^{3}}\) | \(421\) |
| parts | \(\frac {A \,g^{2} \left (b x +a \right )^{3}}{3 b}-\frac {B \,g^{2} \left (d a -b c \right ) e \left (\left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right ) \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )+2 \left (-\frac {1}{2 b e d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}-\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{2 b^{2} e^{2} d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -2 b e \right )}{2 b^{2} e^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}\right ) b e \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )+\left (\frac {1}{3 b^{2} e^{2} d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}-\frac {1}{6 b e d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}+\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{3 b^{3} e^{3} d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (3 b^{2} e^{2}-3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{3}}\right ) b^{2} e^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )\right )}{d^{2}}\) | \(798\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (A \,d^{2} g^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (-\frac {b e}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {1}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {b^{2} e^{2}}{3 d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}\right )+B \,d^{2} g^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (\frac {2 \left (\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right ) b e}{d^{2}}+\frac {\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}}{d^{2}}+\frac {\left (-\frac {1}{6 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}-\frac {1}{3 b^{2} e^{2} d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{3 b^{3} e^{3} d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (3 b^{2} e^{2}-3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}\right ) b^{2} e^{2}}{d^{2}}\right )\right )}{d^{2}}\) | \(913\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (A \,d^{2} g^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (-\frac {b e}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {1}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {b^{2} e^{2}}{3 d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}\right )+B \,d^{2} g^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (\frac {2 \left (\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right ) b e}{d^{2}}+\frac {\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}}{d^{2}}+\frac {\left (-\frac {1}{6 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}-\frac {1}{3 b^{2} e^{2} d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{3 b^{3} e^{3} d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (3 b^{2} e^{2}-3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}\right ) b^{2} e^{2}}{d^{2}}\right )\right )}{d^{2}}\) | \(913\) |
Input:
int((b*g*x+a*g)^2*(A+B*ln(e*(b*x+a)/(d*x+c))),x,method=_RETURNVERBOSE)
Output:
1/3*(b*x+a)^3*g^2*B/b*ln(e*(b*x+a)/(d*x+c))+1/3*g^2*b^2*A*x^3+g^2*b*A*a*x^ 2+1/6*g^2*b*B*a*x^2-1/6*g^2*b^2/d*B*c*x^2+g^2*A*a^2*x+1/3*g^2/b*B*ln(d*x+c )*a^3-g^2/d*B*ln(d*x+c)*a^2*c+g^2*b/d^2*B*ln(d*x+c)*a*c^2-1/3*g^2*b^2/d^3* B*ln(d*x+c)*c^3+2/3*g^2*B*a^2*x-g^2*b/d*B*a*c*x+1/3*g^2*b^2/d^2*B*c^2*x
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (110) = 220\).
Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.88 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {2 \, A b^{3} d^{3} g^{2} x^{3} + 2 \, B a^{3} d^{3} g^{2} \log \left (b x + a\right ) - {\left (B b^{3} c d^{2} - {\left (6 \, A + B\right )} a b^{2} d^{3}\right )} g^{2} x^{2} + 2 \, {\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} + {\left (3 \, A + 2 \, B\right )} a^{2} b d^{3}\right )} g^{2} x - 2 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} \log \left (d x + c\right ) + 2 \, {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{6 \, b d^{3}} \] Input:
integrate((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas" )
Output:
1/6*(2*A*b^3*d^3*g^2*x^3 + 2*B*a^3*d^3*g^2*log(b*x + a) - (B*b^3*c*d^2 - ( 6*A + B)*a*b^2*d^3)*g^2*x^2 + 2*(B*b^3*c^2*d - 3*B*a*b^2*c*d^2 + (3*A + 2* B)*a^2*b*d^3)*g^2*x - 2*(B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*g^ 2*log(d*x + c) + 2*(B*b^3*d^3*g^2*x^3 + 3*B*a*b^2*d^3*g^2*x^2 + 3*B*a^2*b* d^3*g^2*x)*log((b*e*x + a*e)/(d*x + c)))/(b*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (100) = 200\).
Time = 1.48 (sec) , antiderivative size = 491, normalized size of antiderivative = 4.16 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A b^{2} g^{2} x^{3}}{3} + \frac {B a^{3} g^{2} \log {\left (x + \frac {\frac {B a^{4} d^{3} g^{2}}{b} + 3 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 b} - \frac {B c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {4 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2} - B a c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac {B b c^{2} g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 d^{3}} + x^{2} \left (A a b g^{2} + \frac {B a b g^{2}}{6} - \frac {B b^{2} c g^{2}}{6 d}\right ) + x \left (A a^{2} g^{2} + \frac {2 B a^{2} g^{2}}{3} - \frac {B a b c g^{2}}{d} + \frac {B b^{2} c^{2} g^{2}}{3 d^{2}}\right ) + \left (B a^{2} g^{2} x + B a b g^{2} x^{2} + \frac {B b^{2} g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \] Input:
integrate((b*g*x+a*g)**2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
Output:
A*b**2*g**2*x**3/3 + B*a**3*g**2*log(x + (B*a**4*d**3*g**2/b + 3*B*a**3*c* d**2*g**2 - 3*B*a**2*b*c**2*d*g**2 + B*a*b**2*c**3*g**2)/(B*a**3*d**3*g**2 + 3*B*a**2*b*c*d**2*g**2 - 3*B*a*b**2*c**2*d*g**2 + B*b**3*c**3*g**2))/(3 *b) - B*c*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)*log(x + (4*B*a**3*c*d **2*g**2 - 3*B*a**2*b*c**2*d*g**2 + B*a*b**2*c**3*g**2 - B*a*c*g**2*(3*a** 2*d**2 - 3*a*b*c*d + b**2*c**2) + B*b*c**2*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)/d)/(B*a**3*d**3*g**2 + 3*B*a**2*b*c*d**2*g**2 - 3*B*a*b**2*c** 2*d*g**2 + B*b**3*c**3*g**2))/(3*d**3) + x**2*(A*a*b*g**2 + B*a*b*g**2/6 - B*b**2*c*g**2/(6*d)) + x*(A*a**2*g**2 + 2*B*a**2*g**2/3 - B*a*b*c*g**2/d + B*b**2*c**2*g**2/(3*d**2)) + (B*a**2*g**2*x + B*a*b*g**2*x**2 + B*b**2*g **2*x**3/3)*log(e*(a + b*x)/(c + d*x))
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (110) = 220\).
Time = 0.04 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.37 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{3} \, A b^{2} g^{2} x^{3} + A a b g^{2} x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a^{2} g^{2} + {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a b g^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} g^{2} + A a^{2} g^{2} x \] Input:
integrate((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima" )
Output:
1/3*A*b^2*g^2*x^3 + A*a*b*g^2*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*g^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*b*g^2 + 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*g^2 + A*a^2*g^2*x
Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (110) = 220\).
Time = 0.23 (sec) , antiderivative size = 1742, normalized size of antiderivative = 14.76 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:
integrate((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")
Output:
1/6*(2*(B*b^6*c^4*e^4*g^2 - 4*B*a*b^5*c^3*d*e^4*g^2 + 6*B*a^2*b^4*c^2*d^2* e^4*g^2 - 4*B*a^3*b^3*c*d^3*e^4*g^2 + B*a^4*b^2*d^4*e^4*g^2 - 3*(b*e*x + a *e)*B*b^5*c^4*d*e^3*g^2/(d*x + c) + 12*(b*e*x + a*e)*B*a*b^4*c^3*d^2*e^3*g ^2/(d*x + c) - 18*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3*g^2/(d*x + c) + 12*( b*e*x + a*e)*B*a^3*b^2*c*d^4*e^3*g^2/(d*x + c) - 3*(b*e*x + a*e)*B*a^4*b*d ^5*e^3*g^2/(d*x + c) + 3*(b*e*x + a*e)^2*B*b^4*c^4*d^2*e^2*g^2/(d*x + c)^2 - 12*(b*e*x + a*e)^2*B*a*b^3*c^3*d^3*e^2*g^2/(d*x + c)^2 + 18*(b*e*x + a* e)^2*B*a^2*b^2*c^2*d^4*e^2*g^2/(d*x + c)^2 - 12*(b*e*x + a*e)^2*B*a^3*b*c* d^5*e^2*g^2/(d*x + c)^2 + 3*(b*e*x + a*e)^2*B*a^4*d^6*e^2*g^2/(d*x + c)^2) *log((b*e*x + a*e)/(d*x + c))/(b^3*d^3*e^3 - 3*(b*e*x + a*e)*b^2*d^4*e^2/( d*x + c) + 3*(b*e*x + a*e)^2*b*d^5*e/(d*x + c)^2 - (b*e*x + a*e)^3*d^6/(d* x + c)^3) + (2*A*b^6*c^4*e^4*g^2 + 3*B*b^6*c^4*e^4*g^2 - 8*A*a*b^5*c^3*d*e ^4*g^2 - 12*B*a*b^5*c^3*d*e^4*g^2 + 12*A*a^2*b^4*c^2*d^2*e^4*g^2 + 18*B*a^ 2*b^4*c^2*d^2*e^4*g^2 - 8*A*a^3*b^3*c*d^3*e^4*g^2 - 12*B*a^3*b^3*c*d^3*e^4 *g^2 + 2*A*a^4*b^2*d^4*e^4*g^2 + 3*B*a^4*b^2*d^4*e^4*g^2 - 6*(b*e*x + a*e) *A*b^5*c^4*d*e^3*g^2/(d*x + c) - 7*(b*e*x + a*e)*B*b^5*c^4*d*e^3*g^2/(d*x + c) + 24*(b*e*x + a*e)*A*a*b^4*c^3*d^2*e^3*g^2/(d*x + c) + 28*(b*e*x + a* e)*B*a*b^4*c^3*d^2*e^3*g^2/(d*x + c) - 36*(b*e*x + a*e)*A*a^2*b^3*c^2*d^3* e^3*g^2/(d*x + c) - 42*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3*g^2/(d*x + c) + 24*(b*e*x + a*e)*A*a^3*b^2*c*d^4*e^3*g^2/(d*x + c) + 28*(b*e*x + a*e)*...
Time = 27.52 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.46 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=x^2\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{6\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{3\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )}{3\,b\,d}-\frac {a\,g^2\,\left (3\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{d}+\frac {A\,a\,b\,c\,g^2}{d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a^2\,g^2\,x+B\,a\,b\,g^2\,x^2+\frac {B\,b^2\,g^2\,x^3}{3}\right )-\frac {\ln \left (c+d\,x\right )\,\left (3\,B\,a^2\,c\,d^2\,g^2-3\,B\,a\,b\,c^2\,d\,g^2+B\,b^2\,c^3\,g^2\right )}{3\,d^3}+\frac {A\,b^2\,g^2\,x^3}{3}+\frac {B\,a^3\,g^2\,\ln \left (a+b\,x\right )}{3\,b} \] Input:
int((a*g + b*g*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))),x)
Output:
x^2*((b*g^2*(9*A*a*d + 3*A*b*c + B*a*d - B*b*c))/(6*d) - (A*b*g^2*(3*a*d + 3*b*c))/(6*d)) - x*(((3*a*d + 3*b*c)*((b*g^2*(9*A*a*d + 3*A*b*c + B*a*d - B*b*c))/(3*d) - (A*b*g^2*(3*a*d + 3*b*c))/(3*d)))/(3*b*d) - (a*g^2*(3*A*a *d + 3*A*b*c + B*a*d - B*b*c))/d + (A*a*b*c*g^2)/d) + log((e*(a + b*x))/(c + d*x))*((B*b^2*g^2*x^3)/3 + B*a^2*g^2*x + B*a*b*g^2*x^2) - (log(c + d*x) *(B*b^2*c^3*g^2 + 3*B*a^2*c*d^2*g^2 - 3*B*a*b*c^2*d*g^2))/(3*d^3) + (A*b^2 *g^2*x^3)/3 + (B*a^3*g^2*log(a + b*x))/(3*b)
Time = 0.15 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.25 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^{2} \left (2 \,\mathrm {log}\left (d x +c \right ) a^{3} d^{3}-6 \,\mathrm {log}\left (d x +c \right ) a^{2} b c \,d^{2}+6 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c^{2} d -2 \,\mathrm {log}\left (d x +c \right ) b^{3} c^{3}+2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} d^{3}+6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b \,d^{3} x +6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} d^{3} x^{2}+2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{3} d^{3} x^{3}+6 a^{3} d^{3} x +6 a^{2} b \,d^{3} x^{2}+4 a^{2} b \,d^{3} x -6 a \,b^{2} c \,d^{2} x +2 a \,b^{2} d^{3} x^{3}+a \,b^{2} d^{3} x^{2}+2 b^{3} c^{2} d x -b^{3} c \,d^{2} x^{2}\right )}{6 d^{3}} \] Input:
int((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x)
Output:
(g**2*(2*log(c + d*x)*a**3*d**3 - 6*log(c + d*x)*a**2*b*c*d**2 + 6*log(c + d*x)*a*b**2*c**2*d - 2*log(c + d*x)*b**3*c**3 + 2*log((a*e + b*e*x)/(c + d*x))*a**3*d**3 + 6*log((a*e + b*e*x)/(c + d*x))*a**2*b*d**3*x + 6*log((a* e + b*e*x)/(c + d*x))*a*b**2*d**3*x**2 + 2*log((a*e + b*e*x)/(c + d*x))*b* *3*d**3*x**3 + 6*a**3*d**3*x + 6*a**2*b*d**3*x**2 + 4*a**2*b*d**3*x - 6*a* b**2*c*d**2*x + 2*a*b**2*d**3*x**3 + a*b**2*d**3*x**2 + 2*b**3*c**2*d*x - b**3*c*d**2*x**2))/(6*d**3)