3.93 \(\int \frac{A+B \log (\frac{e (a+b x)}{c+d x})}{(a g+b g x)^2} \, dx\)

Optimal. Leaf size=63 \[ -\frac{(c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac{B}{b g^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.078977, antiderivative size = 102, normalized size of antiderivative = 1.62, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{b g^2 (a+b x)}-\frac{B d \log (a+b x)}{b g^2 (b c-a d)}+\frac{B d \log (c+d x)}{b g^2 (b c-a d)}-\frac{B}{b g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

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] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a g+b g x)^2} \, dx &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{b g^2 (a+b x)}+\frac{B \int \frac{b c-a d}{g (a+b x)^2 (c+d x)} \, dx}{b g}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{b g^2 (a+b x)}+\frac{(B (b c-a d)) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b g^2}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{b g^2 (a+b x)}+\frac{(B (b c-a d)) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b g^2}\\ &=-\frac{B}{b g^2 (a+b x)}-\frac{B d \log (a+b x)}{b (b c-a d) g^2}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{b g^2 (a+b x)}+\frac{B d \log (c+d x)}{b (b c-a d) g^2}\\ \end{align*}

Mathematica [A]  time = 0.0591164, size = 105, normalized size = 1.67 \[ \frac{a A d+(a B d-b B c) \log \left (\frac{e (a+b x)}{c+d x}\right )-B d (a+b x) \log (a+b x)+a B d \log (c+d x)+a B d-A b c+b B d x \log (c+d x)-b B c}{b g^2 (a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.047, size = 373, normalized size = 5.9 \begin{align*}{\frac{deAa}{ \left ( ad-bc \right ) ^{2}{g}^{2}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}-{\frac{eAbc}{ \left ( ad-bc \right ) ^{2}{g}^{2}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}+{\frac{deBa}{ \left ( ad-bc \right ) ^{2}{g}^{2}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}-{\frac{eBbc}{ \left ( ad-bc \right ) ^{2}{g}^{2}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}+{\frac{deBa}{ \left ( ad-bc \right ) ^{2}{g}^{2}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}-{\frac{eBbc}{ \left ( ad-bc \right ) ^{2}{g}^{2}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \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)))/(b*g*x+a*g)^2,x)

[Out]

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

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Maxima [B]  time = 1.15851, size = 178, normalized size = 2.83 \begin{align*} -B{\left (\frac{\log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac{1}{b^{2} g^{2} x + a b g^{2}} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac{A}{b^{2} g^{2} x + a b g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.07407, size = 177, normalized size = 2.81 \begin{align*} -\frac{{\left (A + B\right )} b c -{\left (A + B\right )} a d +{\left (B b d x + B b c\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x +{\left (a b^{2} c - a^{2} b d\right )} g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.56011, size = 231, normalized size = 3.67 \begin{align*} - \frac{B \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{a b g^{2} + b^{2} g^{2} x} - \frac{B d \log{\left (x + \frac{- \frac{B a^{2} d^{3}}{a d - b c} + \frac{2 B a b c d^{2}}{a d - b c} + B a d^{2} - \frac{B b^{2} c^{2} d}{a d - b c} + B b c d}{2 B b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac{B d \log{\left (x + \frac{\frac{B a^{2} d^{3}}{a d - b c} - \frac{2 B a b c d^{2}}{a d - b c} + B a d^{2} + \frac{B b^{2} c^{2} d}{a d - b c} + B b c d}{2 B b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} - \frac{A + B}{a b g^{2} + b^{2} g^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**2,x)

[Out]

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

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Giac [A]  time = 1.35968, size = 157, normalized size = 2.49 \begin{align*} -\frac{B d \log \left (b x + a\right )}{b^{2} c g^{2} - a b d g^{2}} + \frac{B d \log \left (d x + c\right )}{b^{2} c g^{2} - a b d g^{2}} - \frac{B \log \left (\frac{b x + a}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac{A + 2 \, B}{b^{2} g^{2} x + a b g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="giac")

[Out]

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