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

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Rubi [A]  time = 0.027389, antiderivative size = 52, normalized size of antiderivative = 1., 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 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]

Rule 31

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

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 = 0.007627, size = 52, normalized size = 1. \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}-\frac{B (b c-a d) \log (c+d x)}{b d}+A x \]

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)

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Maple [B]  time = 0.149, size = 418, normalized size = 8. \begin{align*} Ax-{\frac{Ba}{b}\ln \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) }+{\frac{Bc}{d}\ln \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) }+{eBa\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}}-{\frac{eBbc}{d}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}}+{\frac{eB{a}^{2}d}{b \left ( dx+c \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}}-2\,{\frac{eBac}{dx+c}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}}+{\frac{eB{c}^{2}b}{ \left ( dx+c \right ) d}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{ade}{dx+c}}-{\frac{bec}{dx+c}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16848, size = 73, normalized size = 1.4 \begin{align*}{\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 \end{align*}

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

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Fricas [A]  time = 1.30415, size = 132, normalized size = 2.54 \begin{align*} \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} \end{align*}

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)

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Sympy [A]  time = 1.16599, size = 83, normalized size = 1.6 \begin{align*} 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 )} \end{align*}

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))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError