3.234 \(\int (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\)

Optimal. Leaf size=52 \[ \frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b}-\frac {B (b c-a d) \log (c+d x)}{b d}+A x \]

[Out]

A*x+B*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/b-B*(-a*d+b*c)*ln(d*x+c)/b/d

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2486, 31} \[ \frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b}-\frac {B (b c-a d) \log (c+d x)}{b d}+A x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[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, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rubi steps

\begin {align*} \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=A x+B \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx\\ &=A x+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b}-\frac {(B (b c-a d)) \int \frac {1}{c+d x} \, dx}{b}\\ &=A x+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b}-\frac {B (b c-a d) \log (c+d x)}{b d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 52, normalized size = 1.00 \[ \frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b}-\frac {B (b c-a d) \log (c+d x)}{b d}+A x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.32, size = 56, normalized size = 1.08 \[ \frac {B b d x \log \left (\frac {b e x + a e}{d x + c}\right ) + A b d x + B a d \log \left (b x + a\right ) - B b c \log \left (d x + c\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*d*x*log((b*e*x + a*e)/(d*x + c)) + A*b*d*x + B*a*d*log(b*x + a) - B*b*c*log(d*x + c))/(b*d)

________________________________________________________________________________________

giac [B]  time = 0.52, size = 427, normalized size = 8.21 \[ -{\left ({\left (b^{2} c^{2} e^{2} - 2 \, a b c d e^{2} + a^{2} d^{2} e^{2}\right )} {\left (\frac {e^{\left (-1\right )} \log \left (\frac {{\left | b x e + a e \right |}}{{\left | d x + c \right |}}\right )}{b d} - \frac {e^{\left (-1\right )} \log \left ({\left | -b e + \frac {{\left (b x e + a e\right )} d}{d x + c} \right |}\right )}{b d}\right )} - \frac {{\left (b^{2} c^{2} e^{2} - 2 \, a b c d e^{2} + a^{2} d^{2} e^{2}\right )} \log \left (\frac {{\left (a - \frac {b {\left (\frac {a}{b c - a d} - \frac {{\left (b x e + a e\right )} c}{{\left (b c e - a d e\right )} {\left (d x + c\right )}}\right )}}{\frac {b}{b c - a d} - \frac {{\left (b x e + a e\right )} d}{{\left (b c e - a d e\right )} {\left (d x + c\right )}}}\right )} e}{c - \frac {d {\left (\frac {a}{b c - a d} - \frac {{\left (b x e + a e\right )} c}{{\left (b c e - a d e\right )} {\left (d x + c\right )}}\right )}}{\frac {b}{b c - a d} - \frac {{\left (b x e + a e\right )} d}{{\left (b c e - a d e\right )} {\left (d x + c\right )}}}}\right )}{{\left (b e - \frac {{\left (b x e + a e\right )} d}{d x + c}\right )} d}\right )} B {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} + A x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^2*c^2*e^2 - 2*a*b*c*d*e^2 + a^2*d^2*e^2)*(e^(-1)*log(abs(b*x*e + a*e)/abs(d*x + c))/(b*d) - e^(-1)*log(ab
s(-b*e + (b*x*e + a*e)*d/(d*x + c)))/(b*d)) - (b^2*c^2*e^2 - 2*a*b*c*d*e^2 + a^2*d^2*e^2)*log((a - b*(a/(b*c -
a*d) - (b*x*e + a*e)*c/((b*c*e - a*d*e)*(d*x + c)))/(b/(b*c - a*d) - (b*x*e + a*e)*d/((b*c*e - a*d*e)*(d*x +
c))))*e/(c - d*(a/(b*c - a*d) - (b*x*e + a*e)*c/((b*c*e - a*d*e)*(d*x + c)))/(b/(b*c - a*d) - (b*x*e + a*e)*d/
((b*c*e - a*d*e)*(d*x + c)))))/((b*e - (b*x*e + a*e)*d/(d*x + c))*d))*B*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a
*d/((b*c*e - a*d*e)*(b*c - a*d))) + A*x

________________________________________________________________________________________

maple [B]  time = 0.13, size = 418, normalized size = 8.04 \[ \frac {B \,a^{2} d e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) \left (d x +c \right ) b}-\frac {2 B a c e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) \left (d x +c \right )}+\frac {B b \,c^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) \left (d x +c \right ) d}+\frac {B a e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\frac {a d e}{d x +c}-\frac {b c e}{d x +c}}-\frac {B b c e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (\frac {a d e}{d x +c}-\frac {b c e}{d x +c}\right ) d}+A x -\frac {B a \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{b}+\frac {B c \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(B*ln((b*x+a)/(d*x+c)*e)+A,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.62, size = 54, normalized size = 1.04 \[ {\left (x \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + \frac {\frac {a e \log \left (b x + a\right )}{b} - \frac {c e \log \left (d x + c\right )}{d}}{e}\right )} B + A x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*log((b*x + a)*e/(d*x + c)) + (a*e*log(b*x + a)/b - c*e*log(d*x + c)/d)/e)*B + A*x

________________________________________________________________________________________

mupad [B]  time = 4.12, size = 47, normalized size = 0.90 \[ A\,x+B\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )+\frac {B\,a\,\ln \left (a+b\,x\right )}{b}-\frac {B\,c\,\ln \left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*x + B*x*log((e*(a + b*x))/(c + d*x)) + (B*a*log(a + b*x))/b - (B*c*log(c + d*x))/d

________________________________________________________________________________________

sympy [A]  time = 1.00, size = 83, normalized size = 1.60 \[ A x + \frac {B a \log {\left (x + \frac {\frac {B a^{2} d}{b} + B a c}{B a d + B b c} \right )}}{b} - \frac {B c \log {\left (x + \frac {B a c + \frac {B b c^{2}}{d}}{B a d + B b c} \right )}}{d} + B x \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________