\(\int (f+g x) (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [233]

Optimal result

Integrand size = 25, antiderivative size = 109 \[ \int (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=-\frac {B (b c-a d) g x}{2 b d}-\frac {B (b f-a g)^2 \log (a+b x)}{2 b^2 g}+\frac {(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g}+\frac {B (d f-c g)^2 \log (c+d x)}{2 d^2 g} \]

[Out]

-1/2*B*(-a*d+b*c)*g*x/b/d-1/2*B*(-a*g+b*f)^2*ln(b*x+a)/b^2/g+1/2*(g*x+f)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/g+1/2*B
*(-c*g+d*f)^2*ln(d*x+c)/d^2/g

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2548, 84} \[ \int (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {(f+g x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g}-\frac {B (b f-a g)^2 \log (a+b x)}{2 b^2 g}-\frac {B g x (b c-a d)}{2 b d}+\frac {B (d f-c g)^2 \log (c+d x)}{2 d^2 g} \]

[In]

Int[(f + g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

-1/2*(B*(b*c - a*d)*g*x)/(b*d) - (B*(b*f - a*g)^2*Log[a + b*x])/(2*b^2*g) + ((f + g*x)^2*(A + B*Log[(e*(a + b*
x))/(c + d*x)]))/(2*g) + (B*(d*f - c*g)^2*Log[c + d*x])/(2*d^2*g)

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2548

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] - Dist[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])

Rubi steps \begin{align*} \text {integral}& = \frac {(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g}-\frac {(B (b c-a d)) \int \frac {(f+g x)^2}{(a+b x) (c+d x)} \, dx}{2 g} \\ & = \frac {(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g}-\frac {(B (b c-a d)) \int \left (\frac {g^2}{b d}+\frac {(b f-a g)^2}{b (b c-a d) (a+b x)}+\frac {(d f-c g)^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{2 g} \\ & = -\frac {B (b c-a d) g x}{2 b d}-\frac {B (b f-a g)^2 \log (a+b x)}{2 b^2 g}+\frac {(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 g}+\frac {B (d f-c g)^2 \log (c+d x)}{2 d^2 g} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.05 \[ \int (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {-B d^2 (b f-a g)^2 \log (a+b x)+b \left (d \left (B (-b c+a d) g^2 x+A b d (f+g x)^2\right )+b B d^2 (f+g x)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+b B (d f-c g)^2 \log (c+d x)\right )}{2 b^2 d^2 g} \]

[In]

Integrate[(f + g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(-(B*d^2*(b*f - a*g)^2*Log[a + b*x]) + b*(d*(B*(-(b*c) + a*d)*g^2*x + A*b*d*(f + g*x)^2) + b*B*d^2*(f + g*x)^2
*Log[(e*(a + b*x))/(c + d*x)] + b*B*(d*f - c*g)^2*Log[c + d*x]))/(2*b^2*d^2*g)

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {B x \left (g x +2 f \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2}+\frac {A \,x^{2} g}{2}+A f x +\frac {B \ln \left (-d x -c \right ) c^{2} g}{2 d^{2}}-\frac {B \ln \left (-d x -c \right ) c f}{d}-\frac {B \ln \left (b x +a \right ) a^{2} g}{2 b^{2}}+\frac {B \ln \left (b x +a \right ) a f}{b}+\frac {B x a g}{2 b}-\frac {B x c g}{2 d}\)	\(122\)
parallelrisch	\(\frac {B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} d^{2} g +A \,x^{2} b^{2} d^{2} g +2 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} d^{2} f +2 A \,b^{2} d^{2} f x -B \ln \left (b x +a \right ) a^{2} d^{2} g +2 B \ln \left (b x +a \right ) a b \,d^{2} f +B \ln \left (b x +a \right ) b^{2} c^{2} g -2 B \ln \left (b x +a \right ) b^{2} c d f +B x a b \,d^{2} g -B x \,b^{2} c d g -B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} c^{2} g +2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} c d f -A a b c d g -2 A a b \,d^{2} f -2 A \,b^{2} c d f -B \,a^{2} d^{2} g +B \,b^{2} c^{2} g}{2 b^{2} d^{2}}\)	\(260\)
parts	\(A \left (\frac {1}{2} g \,x^{2}+f x \right )-\frac {B \left (a d -c b \right ) e \left (d e g \left (a d -c b \right ) \left (-\frac {1}{2 e b d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}-\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{2 e^{2} b^{2} d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -2 b e \right )}{2 e^{2} b^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}\right )-d \left (c g -d f \right ) \left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right )\right )}{d^{2}}\)	\(425\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-A \,d^{2} \left (\frac {e g \left (a d -c b \right )}{2 d^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {c g -d f}{d^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )-B \,d^{2} \left (-\frac {e g \left (a d -c b \right ) \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}+\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )}{d}+\frac {\left (c g -d f \right ) \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d}\right )\right )}{d^{2}}\)	\(528\)
default	\(-\frac {e \left (a d -c b \right ) \left (-A \,d^{2} \left (\frac {e g \left (a d -c b \right )}{2 d^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {c g -d f}{d^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )-B \,d^{2} \left (-\frac {e g \left (a d -c b \right ) \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}+\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )}{d}+\frac {\left (c g -d f \right ) \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d}\right )\right )}{d^{2}}\)	\(528\)

[In]

int((g*x+f)*(A+B*ln(e*(b*x+a)/(d*x+c))),x,method=_RETURNVERBOSE)

[Out]

1/2*B*x*(g*x+2*f)*ln(e*(b*x+a)/(d*x+c))+1/2*A*x^2*g+A*f*x+1/2/d^2*B*ln(-d*x-c)*c^2*g-1/d*B*ln(-d*x-c)*c*f-1/2/
b^2*B*ln(b*x+a)*a^2*g+1/b*B*ln(b*x+a)*a*f+1/2/b*B*x*a*g-1/2/d*B*x*c*g

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.38 \[ \int (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A b^{2} d^{2} g x^{2} + {\left (2 \, A b^{2} d^{2} f - {\left (B b^{2} c d - B a b d^{2}\right )} g\right )} x + {\left (2 \, B a b d^{2} f - B a^{2} d^{2} g\right )} \log \left (b x + a\right ) - {\left (2 \, B b^{2} c d f - B b^{2} c^{2} g\right )} \log \left (d x + c\right ) + {\left (B b^{2} d^{2} g x^{2} + 2 \, B b^{2} d^{2} f x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, b^{2} d^{2}} \]

[In]

integrate((g*x+f)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

1/2*(A*b^2*d^2*g*x^2 + (2*A*b^2*d^2*f - (B*b^2*c*d - B*a*b*d^2)*g)*x + (2*B*a*b*d^2*f - B*a^2*d^2*g)*log(b*x +
a) - (2*B*b^2*c*d*f - B*b^2*c^2*g)*log(d*x + c) + (B*b^2*d^2*g*x^2 + 2*B*b^2*d^2*f*x)*log((b*e*x + a*e)/(d*x
+ c)))/(b^2*d^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (90) = 180\).

Time = 1.42 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.92 \[ \int (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A g x^{2}}{2} - \frac {B a \left (a g - 2 b f\right ) \log {\left (x + \frac {B a^{2} c d g + \frac {B a^{2} d^{2} \left (a g - 2 b f\right )}{b} + B a b c^{2} g - 4 B a b c d f - B a c d \left (a g - 2 b f\right )}{B a^{2} d^{2} g - 2 B a b d^{2} f + B b^{2} c^{2} g - 2 B b^{2} c d f} \right )}}{2 b^{2}} + \frac {B c \left (c g - 2 d f\right ) \log {\left (x + \frac {B a^{2} c d g + B a b c^{2} g - 4 B a b c d f - B a b c \left (c g - 2 d f\right ) + \frac {B b^{2} c^{2} \left (c g - 2 d f\right )}{d}}{B a^{2} d^{2} g - 2 B a b d^{2} f + B b^{2} c^{2} g - 2 B b^{2} c d f} \right )}}{2 d^{2}} + x \left (A f + \frac {B a g}{2 b} - \frac {B c g}{2 d}\right ) + \left (B f x + \frac {B g x^{2}}{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

[In]

integrate((g*x+f)*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*g*x**2/2 - B*a*(a*g - 2*b*f)*log(x + (B*a**2*c*d*g + B*a**2*d**2*(a*g - 2*b*f)/b + B*a*b*c**2*g - 4*B*a*b*c*
d*f - B*a*c*d*(a*g - 2*b*f))/(B*a**2*d**2*g - 2*B*a*b*d**2*f + B*b**2*c**2*g - 2*B*b**2*c*d*f))/(2*b**2) + B*c
*(c*g - 2*d*f)*log(x + (B*a**2*c*d*g + B*a*b*c**2*g - 4*B*a*b*c*d*f - B*a*b*c*(c*g - 2*d*f) + B*b**2*c**2*(c*g
- 2*d*f)/d)/(B*a**2*d**2*g - 2*B*a*b*d**2*f + B*b**2*c**2*g - 2*B*b**2*c*d*f))/(2*d**2) + x*(A*f + B*a*g/(2*b
) - B*c*g/(2*d)) + (B*f*x + B*g*x**2/2)*log(e*(a + b*x)/(c + d*x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.28 \[ \int (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{2} \, A g x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B f + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B g + A f x \]

[In]

integrate((g*x+f)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/2*A*g*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*f + 1/2*(x^2*lo
g(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*g +
A*f*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1145 vs. \(2 (101) = 202\).

Time = 0.49 (sec) , antiderivative size = 1145, normalized size of antiderivative = 10.50 \[ \int (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{2} \, {\left (\frac {{\left (2 \, B b^{3} c^{2} d e^{3} f - 4 \, B a b^{2} c d^{2} e^{3} f + 2 \, B a^{2} b d^{3} e^{3} f - B b^{3} c^{3} e^{3} g + B a b^{2} c^{2} d e^{3} g + B a^{2} b c d^{2} e^{3} g - B a^{3} d^{3} e^{3} g - \frac {2 \, {\left (b e x + a e\right )} B b^{2} c^{2} d^{2} e^{2} f}{d x + c} + \frac {4 \, {\left (b e x + a e\right )} B a b c d^{3} e^{2} f}{d x + c} - \frac {2 \, {\left (b e x + a e\right )} B a^{2} d^{4} e^{2} f}{d x + c} + \frac {2 \, {\left (b e x + a e\right )} B b^{2} c^{3} d e^{2} g}{d x + c} - \frac {4 \, {\left (b e x + a e\right )} B a b c^{2} d^{2} e^{2} g}{d x + c} + \frac {2 \, {\left (b e x + a e\right )} B a^{2} c d^{3} e^{2} g}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} d^{2} e^{2} - \frac {2 \, {\left (b e x + a e\right )} b d^{3} e}{d x + c} + \frac {{\left (b e x + a e\right )}^{2} d^{4}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, A b^{4} c^{2} d e^{3} f - 4 \, A a b^{3} c d^{2} e^{3} f + 2 \, A a^{2} b^{2} d^{3} e^{3} f - A b^{4} c^{3} e^{3} g - B b^{4} c^{3} e^{3} g + A a b^{3} c^{2} d e^{3} g + 3 \, B a b^{3} c^{2} d e^{3} g + A a^{2} b^{2} c d^{2} e^{3} g - 3 \, B a^{2} b^{2} c d^{2} e^{3} g - A a^{3} b d^{3} e^{3} g + B a^{3} b d^{3} e^{3} g - \frac {2 \, {\left (b e x + a e\right )} A b^{3} c^{2} d^{2} e^{2} f}{d x + c} + \frac {4 \, {\left (b e x + a e\right )} A a b^{2} c d^{3} e^{2} f}{d x + c} - \frac {2 \, {\left (b e x + a e\right )} A a^{2} b d^{4} e^{2} f}{d x + c} + \frac {2 \, {\left (b e x + a e\right )} A b^{3} c^{3} d e^{2} g}{d x + c} + \frac {{\left (b e x + a e\right )} B b^{3} c^{3} d e^{2} g}{d x + c} - \frac {4 \, {\left (b e x + a e\right )} A a b^{2} c^{2} d^{2} e^{2} g}{d x + c} - \frac {3 \, {\left (b e x + a e\right )} B a b^{2} c^{2} d^{2} e^{2} g}{d x + c} + \frac {2 \, {\left (b e x + a e\right )} A a^{2} b c d^{3} e^{2} g}{d x + c} + \frac {3 \, {\left (b e x + a e\right )} B a^{2} b c d^{3} e^{2} g}{d x + c} - \frac {{\left (b e x + a e\right )} B a^{3} d^{4} e^{2} g}{d x + c}}{b^{3} d^{2} e^{2} - \frac {2 \, {\left (b e x + a e\right )} b^{2} d^{3} e}{d x + c} + \frac {{\left (b e x + a e\right )}^{2} b d^{4}}{{\left (d x + c\right )}^{2}}} + \frac {{\left (2 \, B b^{3} c^{2} d e f - 4 \, B a b^{2} c d^{2} e f + 2 \, B a^{2} b d^{3} e f - B b^{3} c^{3} e g + B a b^{2} c^{2} d e g + B a^{2} b c d^{2} e g - B a^{3} d^{3} e g\right )} \log \left (-b e + \frac {{\left (b e x + a e\right )} d}{d x + c}\right )}{b^{2} d^{2}} - \frac {{\left (2 \, B b^{3} c^{2} d e f - 4 \, B a b^{2} c d^{2} e f + 2 \, B a^{2} b d^{3} e f - B b^{3} c^{3} e g + B a b^{2} c^{2} d e g + B a^{2} b c d^{2} e g - B a^{3} d^{3} e g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} d^{2}}\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 )} \]

[In]

integrate((g*x+f)*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")

[Out]

1/2*((2*B*b^3*c^2*d*e^3*f - 4*B*a*b^2*c*d^2*e^3*f + 2*B*a^2*b*d^3*e^3*f - B*b^3*c^3*e^3*g + B*a*b^2*c^2*d*e^3*
g + B*a^2*b*c*d^2*e^3*g - B*a^3*d^3*e^3*g - 2*(b*e*x + a*e)*B*b^2*c^2*d^2*e^2*f/(d*x + c) + 4*(b*e*x + a*e)*B*
a*b*c*d^3*e^2*f/(d*x + c) - 2*(b*e*x + a*e)*B*a^2*d^4*e^2*f/(d*x + c) + 2*(b*e*x + a*e)*B*b^2*c^3*d*e^2*g/(d*x
+ c) - 4*(b*e*x + a*e)*B*a*b*c^2*d^2*e^2*g/(d*x + c) + 2*(b*e*x + a*e)*B*a^2*c*d^3*e^2*g/(d*x + c))*log((b*e*
x + a*e)/(d*x + c))/(b^2*d^2*e^2 - 2*(b*e*x + a*e)*b*d^3*e/(d*x + c) + (b*e*x + a*e)^2*d^4/(d*x + c)^2) + (2*A
*b^4*c^2*d*e^3*f - 4*A*a*b^3*c*d^2*e^3*f + 2*A*a^2*b^2*d^3*e^3*f - A*b^4*c^3*e^3*g - B*b^4*c^3*e^3*g + A*a*b^3
*c^2*d*e^3*g + 3*B*a*b^3*c^2*d*e^3*g + A*a^2*b^2*c*d^2*e^3*g - 3*B*a^2*b^2*c*d^2*e^3*g - A*a^3*b*d^3*e^3*g + B
*a^3*b*d^3*e^3*g - 2*(b*e*x + a*e)*A*b^3*c^2*d^2*e^2*f/(d*x + c) + 4*(b*e*x + a*e)*A*a*b^2*c*d^3*e^2*f/(d*x +
c) - 2*(b*e*x + a*e)*A*a^2*b*d^4*e^2*f/(d*x + c) + 2*(b*e*x + a*e)*A*b^3*c^3*d*e^2*g/(d*x + c) + (b*e*x + a*e)
*B*b^3*c^3*d*e^2*g/(d*x + c) - 4*(b*e*x + a*e)*A*a*b^2*c^2*d^2*e^2*g/(d*x + c) - 3*(b*e*x + a*e)*B*a*b^2*c^2*d
^2*e^2*g/(d*x + c) + 2*(b*e*x + a*e)*A*a^2*b*c*d^3*e^2*g/(d*x + c) + 3*(b*e*x + a*e)*B*a^2*b*c*d^3*e^2*g/(d*x
+ c) - (b*e*x + a*e)*B*a^3*d^4*e^2*g/(d*x + c))/(b^3*d^2*e^2 - 2*(b*e*x + a*e)*b^2*d^3*e/(d*x + c) + (b*e*x +
a*e)^2*b*d^4/(d*x + c)^2) + (2*B*b^3*c^2*d*e*f - 4*B*a*b^2*c*d^2*e*f + 2*B*a^2*b*d^3*e*f - B*b^3*c^3*e*g + B*a
*b^2*c^2*d*e*g + B*a^2*b*c*d^2*e*g - B*a^3*d^3*e*g)*log(-b*e + (b*e*x + a*e)*d/(d*x + c))/(b^2*d^2) - (2*B*b^3
*c^2*d*e*f - 4*B*a*b^2*c*d^2*e*f + 2*B*a^2*b*d^3*e*f - B*b^3*c^3*e*g + B*a*b^2*c^2*d*e*g + B*a^2*b*c*d^2*e*g -
B*a^3*d^3*e*g)*log((b*e*x + a*e)/(d*x + c))/(b^2*d^2))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d
*e)*(b*c - a*d)))

Mupad [B] (verification not implemented)

Time = 1.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.32 \[ \int (f+g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,g\,x^2}{2}+B\,f\,x\right )+x\,\left (\frac {2\,A\,a\,d\,g+2\,A\,b\,c\,g+2\,A\,b\,d\,f+B\,a\,d\,g-B\,b\,c\,g}{2\,b\,d}-\frac {A\,g\,\left (2\,a\,d+2\,b\,c\right )}{2\,b\,d}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^2\,g-2\,B\,a\,b\,f\right )}{2\,b^2}+\frac {\ln \left (c+d\,x\right )\,\left (B\,c^2\,g-2\,B\,c\,d\,f\right )}{2\,d^2}+\frac {A\,g\,x^2}{2} \]

[In]

int((f + g*x)*(A + B*log((e*(a + b*x))/(c + d*x))),x)

[Out]

log((e*(a + b*x))/(c + d*x))*(B*f*x + (B*g*x^2)/2) + x*((2*A*a*d*g + 2*A*b*c*g + 2*A*b*d*f + B*a*d*g - B*b*c*g
)/(2*b*d) - (A*g*(2*a*d + 2*b*c))/(2*b*d)) - (log(a + b*x)*(B*a^2*g - 2*B*a*b*f))/(2*b^2) + (log(c + d*x)*(B*c
^2*g - 2*B*c*d*f))/(2*d^2) + (A*g*x^2)/2