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

Optimal. Leaf size=64 \[ -\frac{A-B}{b g^2 (a+b x)}-\frac{B (c+d x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{g^2 (a+b x) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0767233, antiderivative size = 101, normalized size of antiderivative = 1.58, 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} \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{b g^2 (a+b x)}+\frac{B d \log (a+b x)}{b g^2 (b c-a d)}-\frac{B d \log (c+d x)}{b g^2 (b c-a d)}+\frac{B}{b g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

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

Mathematica [A]  time = 0.0593092, size = 86, normalized size = 1.34 \[ \frac{-(b c-a d) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A-B\right )-B d (a+b x) \log (c+d x)+B d (a+b x) \log (a+b x)}{b g^2 (a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.051, size = 520, normalized size = 8.1 \begin{align*}{\frac{A{d}^{2}a}{b \left ( ad-bc \right ) ^{2}{g}^{2}}}-{\frac{Adc}{ \left ( ad-bc \right ) ^{2}{g}^{2}}}-{\frac{A{a}^{2}{d}^{2}}{b \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }}+2\,{\frac{Aadc}{ \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }}-{\frac{bA{c}^{2}}{ \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }}+{\frac{B{d}^{2}a}{b \left ( ad-bc \right ) ^{2}{g}^{2}}\ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) }-{\frac{Bdc}{ \left ( ad-bc \right ) ^{2}{g}^{2}}\ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) }-{\frac{B{a}^{2}{d}^{2}}{b \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }\ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) }+2\,{\frac{Badc}{ \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }\ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) }-{\frac{bB{c}^{2}}{ \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }\ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) }+{\frac{B{a}^{2}{d}^{2}}{b \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }}-2\,{\frac{Badc}{ \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }}+{\frac{bB{c}^{2}}{ \left ( ad-bc \right ) ^{2}{g}^{2} \left ( bx+a \right ) }}-{\frac{B{d}^{2}a}{b \left ( ad-bc \right ) ^{2}{g}^{2}}}+{\frac{Bdc}{ \left ( ad-bc \right ) ^{2}{g}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.11624, size = 181, normalized size = 2.83 \begin{align*} -B{\left (\frac{\log \left (\frac{d e x}{b x + a} + \frac{c e}{b x + a}\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*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^2,x, algorithm="maxima")

[Out]

-B*(log(d*e*x/(b*x + a) + c*e/(b*x + a))/(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 = 1.01728, size = 177, normalized size = 2.77 \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{d e x + c e}{b x + a}\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*(d*x+c)/(b*x+a)))/(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((d*e*x + c*e)/(b*x + a)))/((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 = 1.8312, size = 231, normalized size = 3.61 \begin{align*} - \frac{B \log{\left (\frac{e \left (c + d x\right )}{a + b 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*(d*x+c)/(b*x+a)))/(b*g*x+a*g)**2,x)

[Out]

-B*log(e*(c + d*x)/(a + b*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 = 1.32796, size = 151, normalized size = 2.36 \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{d x + c}{b x + a}\right )}{b^{2} g^{2} x + a b g^{2}} - \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*(d*x+c)/(b*x+a)))/(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((d*x + c)/(b*x + a
))/(b^2*g^2*x + a*b*g^2) - A/(b^2*g^2*x + a*b*g^2)