Integrand size = 28, antiderivative size = 81 \[ \int (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=-\frac {B (b c-a d) i x}{2 b}-\frac {B (b c-a d)^2 i \log (a+b x)}{2 b^2 d}+\frac {i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d} \]
-1/2*B*(-a*d+b*c)*i*x/b-1/2*B*(-a*d+b*c)^2*i*ln(b*x+a)/b^2/d+1/2*i*(d*x+c) ^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/d
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {i \left (-\frac {B (b c-a d) (b d x+(b c-a d) \log (a+b x))}{b^2}+(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )}{2 d} \]
(i*(-((B*(b*c - a*d)*(b*d*x + (b*c - a*d)*Log[a + b*x]))/b^2) + (c + d*x)^ 2*(A + B*Log[(e*(a + b*x))/(c + d*x)])))/(2*d)
Time = 0.24 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.
| \(\displaystyle \int (c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2948 |
| \(\displaystyle \frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d}-\frac {B (b c-a d) \int \frac {i^2 (c+d x)}{a+b x}dx}{2 d i}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d}-\frac {B i (b c-a d) \int \frac {c+d x}{a+b x}dx}{2 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d}-\frac {B i (b c-a d) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right )dx}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d}-\frac {B i (b c-a d) \left (\frac {(b c-a d) \log (a+b x)}{b^2}+\frac {d x}{b}\right )}{2 d}\) |
-1/2*(B*(b*c - a*d)*i*((d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2))/d + (i*( c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d)
3.1.4.3.1 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])
Time = 0.56 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(\frac {i B x \left (d x +2 c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2}+\frac {i d A \,x^{2}}{2}+i A c x -\frac {B \,c^{2} i \ln \left (-d x -c \right )}{2 d}-\frac {i d B \ln \left (b x +a \right ) a^{2}}{2 b^{2}}+\frac {i B \ln \left (b x +a \right ) a c}{b}+\frac {i d B a x}{2 b}-\frac {i B c x}{2}\) | \(108\) |
| parallelrisch | \(\frac {B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} d^{2} i +A \,x^{2} b^{2} d^{2} i +2 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} c d i +2 A x \,b^{2} c d i -B \ln \left (b x +a \right ) a^{2} d^{2} i +2 B \ln \left (b x +a \right ) a b c d i -B \ln \left (b x +a \right ) b^{2} c^{2} i +B x a b \,d^{2} i -B x \,b^{2} c d i +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} c^{2} i -3 A a b c d i -2 A \,b^{2} c^{2} i -B \,a^{2} d^{2} i +B \,b^{2} c^{2} i}{2 b^{2} d}\) | \(210\) |
| parts | \(A i \left (\frac {1}{2} d \,x^{2}+x c \right )-B i \left (a d -c b \right )^{2} e^{2} \left (-\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{2 e^{2} b^{2} d}-\frac {1}{2 e b d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -2 b e \right )}{2 e^{2} b^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}\right )\) | \(259\) |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {A d e i \left (a d -c b \right )}{2 \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}-B \,d^{2} e i \left (a d -c b \right ) \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}-\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )\right )}{d^{2}}\) | \(310\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {A d e i \left (a d -c b \right )}{2 \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}-B \,d^{2} e i \left (a d -c b \right ) \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}-\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )\right )}{d^{2}}\) | \(310\) |
1/2*i*B*x*(d*x+2*c)*ln(e*(b*x+a)/(d*x+c))+1/2*i*d*A*x^2+i*A*c*x-1/2*B*c^2* i/d*ln(-d*x-c)-1/2*i/b^2*d*B*ln(b*x+a)*a^2+i/b*B*ln(b*x+a)*a*c+1/2*i/b*d*B *a*x-1/2*i*B*c*x
Time = 0.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.57 \[ \int (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A b^{2} d^{2} i x^{2} - B b^{2} c^{2} i \log \left (d x + c\right ) + {\left ({\left (2 \, A - B\right )} b^{2} c d + B a b d^{2}\right )} i x + {\left (2 \, B a b c d - B a^{2} d^{2}\right )} i \log \left (b x + a\right ) + {\left (B b^{2} d^{2} i x^{2} + 2 \, B b^{2} c d i x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, b^{2} d} \]
1/2*(A*b^2*d^2*i*x^2 - B*b^2*c^2*i*log(d*x + c) + ((2*A - B)*b^2*c*d + B*a *b*d^2)*i*x + (2*B*a*b*c*d - B*a^2*d^2)*i*log(b*x + a) + (B*b^2*d^2*i*x^2 + 2*B*b^2*c*d*i*x)*log((b*e*x + a*e)/(d*x + c)))/(b^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (68) = 136\).
Time = 0.94 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.12 \[ \int (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A d i x^{2}}{2} - \frac {B a i \left (a d - 2 b c\right ) \log {\left (x + \frac {B a^{2} c d i + \frac {B a^{2} d i \left (a d - 2 b c\right )}{b} - 3 B a b c^{2} i - B a c i \left (a d - 2 b c\right )}{B a^{2} d^{2} i - 2 B a b c d i - B b^{2} c^{2} i} \right )}}{2 b^{2}} - \frac {B c^{2} i \log {\left (x + \frac {B a^{2} c d i - 2 B a b c^{2} i - \frac {B b^{2} c^{3} i}{d}}{B a^{2} d^{2} i - 2 B a b c d i - B b^{2} c^{2} i} \right )}}{2 d} + x \left (A c i + \frac {B a d i}{2 b} - \frac {B c i}{2}\right ) + \left (B c i x + \frac {B d i x^{2}}{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]
A*d*i*x**2/2 - B*a*i*(a*d - 2*b*c)*log(x + (B*a**2*c*d*i + B*a**2*d*i*(a*d - 2*b*c)/b - 3*B*a*b*c**2*i - B*a*c*i*(a*d - 2*b*c))/(B*a**2*d**2*i - 2*B *a*b*c*d*i - B*b**2*c**2*i))/(2*b**2) - B*c**2*i*log(x + (B*a**2*c*d*i - 2 *B*a*b*c**2*i - B*b**2*c**3*i/d)/(B*a**2*d**2*i - 2*B*a*b*c*d*i - B*b**2*c **2*i))/(2*d) + x*(A*c*i + B*a*d*i/(2*b) - B*c*i/2) + (B*c*i*x + B*d*i*x** 2/2)*log(e*(a + b*x)/(c + d*x))
Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.78 \[ \int (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{2} \, A d i x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B c i + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B d i + A c i x \]
1/2*A*d*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*c*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*d* i + A*c*i*x
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (75) = 150\).
Time = 0.41 (sec) , antiderivative size = 627, normalized size of antiderivative = 7.74 \[ \int (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{2} \, {\left (\frac {{\left (B b^{3} c^{3} e^{3} i - 3 \, B a b^{2} c^{2} d e^{3} i + 3 \, B a^{2} b c d^{2} e^{3} i - B a^{3} d^{3} e^{3} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} d e^{2} - \frac {2 \, {\left (b e x + a e\right )} b d^{2} e}{d x + c} + \frac {{\left (b e x + a e\right )}^{2} d^{3}}{{\left (d x + c\right )}^{2}}} + \frac {A b^{4} c^{3} e^{3} i - B b^{4} c^{3} e^{3} i - 3 \, A a b^{3} c^{2} d e^{3} i + 3 \, B a b^{3} c^{2} d e^{3} i + 3 \, A a^{2} b^{2} c d^{2} e^{3} i - 3 \, B a^{2} b^{2} c d^{2} e^{3} i - A a^{3} b d^{3} e^{3} i + B a^{3} b d^{3} e^{3} i + \frac {{\left (b e x + a e\right )} B b^{3} c^{3} d e^{2} i}{d x + c} - \frac {3 \, {\left (b e x + a e\right )} B a b^{2} c^{2} d^{2} e^{2} i}{d x + c} + \frac {3 \, {\left (b e x + a e\right )} B a^{2} b c d^{3} e^{2} i}{d x + c} - \frac {{\left (b e x + a e\right )} B a^{3} d^{4} e^{2} i}{d x + c}}{b^{3} d e^{2} - \frac {2 \, {\left (b e x + a e\right )} b^{2} d^{2} e}{d x + c} + \frac {{\left (b e x + a e\right )}^{2} b d^{3}}{{\left (d x + c\right )}^{2}}} + \frac {{\left (B b^{3} c^{3} e i - 3 \, B a b^{2} c^{2} d e i + 3 \, B a^{2} b c d^{2} e i - B a^{3} d^{3} e i\right )} \log \left (-b e + \frac {{\left (b e x + a e\right )} d}{d x + c}\right )}{b^{2} d} - \frac {{\left (B b^{3} c^{3} e i - 3 \, B a b^{2} c^{2} d e i + 3 \, B a^{2} b c d^{2} e i - B a^{3} d^{3} e i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} d}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
1/2*((B*b^3*c^3*e^3*i - 3*B*a*b^2*c^2*d*e^3*i + 3*B*a^2*b*c*d^2*e^3*i - B* a^3*d^3*e^3*i)*log((b*e*x + a*e)/(d*x + c))/(b^2*d*e^2 - 2*(b*e*x + a*e)*b *d^2*e/(d*x + c) + (b*e*x + a*e)^2*d^3/(d*x + c)^2) + (A*b^4*c^3*e^3*i - B *b^4*c^3*e^3*i - 3*A*a*b^3*c^2*d*e^3*i + 3*B*a*b^3*c^2*d*e^3*i + 3*A*a^2*b ^2*c*d^2*e^3*i - 3*B*a^2*b^2*c*d^2*e^3*i - A*a^3*b*d^3*e^3*i + B*a^3*b*d^3 *e^3*i + (b*e*x + a*e)*B*b^3*c^3*d*e^2*i/(d*x + c) - 3*(b*e*x + a*e)*B*a*b ^2*c^2*d^2*e^2*i/(d*x + c) + 3*(b*e*x + a*e)*B*a^2*b*c*d^3*e^2*i/(d*x + c) - (b*e*x + a*e)*B*a^3*d^4*e^2*i/(d*x + c))/(b^3*d*e^2 - 2*(b*e*x + a*e)*b ^2*d^2*e/(d*x + c) + (b*e*x + a*e)^2*b*d^3/(d*x + c)^2) + (B*b^3*c^3*e*i - 3*B*a*b^2*c^2*d*e*i + 3*B*a^2*b*c*d^2*e*i - B*a^3*d^3*e*i)*log(-b*e + (b* e*x + a*e)*d/(d*x + c))/(b^2*d) - (B*b^3*c^3*e*i - 3*B*a*b^2*c^2*d*e*i + 3 *B*a^2*b*c*d^2*e*i - B*a^3*d^3*e*i)*log((b*e*x + a*e)/(d*x + c))/(b^2*d))* (b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
Time = 1.07 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.56 \[ \int (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=x\,\left (\frac {i\,\left (2\,A\,a\,d+4\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{2\,b}-\frac {A\,i\,\left (2\,a\,d+2\,b\,c\right )}{2\,b}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,d\,i\,x^2}{2}+B\,c\,i\,x\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^2\,d\,i-2\,B\,a\,b\,c\,i\right )}{2\,b^2}+\frac {A\,d\,i\,x^2}{2}-\frac {B\,c^2\,i\,\ln \left (c+d\,x\right )}{2\,d} \]