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

Optimal. Leaf size=54 \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b}-\frac{2 B (b c-a d) \log (c+d x)}{b d}+A x \]

[Out]

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

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Rubi [A]  time = 0.0271851, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2486, 31} \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b}-\frac{2 B (b c-a d) \log (c+d x)}{b d}+A x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0232233, size = 54, normalized size = 1. \[ \frac{B (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b}-\frac{2 B (b c-a d) \log (c+d x)}{b d}+A x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.224, size = 233, normalized size = 4.3 \begin{align*} Ax+B\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) x+{\frac{Bc}{d}\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) }-2\,{\frac{B\ln \left ( \left ( dx+c \right ) ^{-1} \right ) a}{b}}+2\,{\frac{B\ln \left ( \left ( dx+c \right ) ^{-1} \right ) c}{d}}+2\,{\frac{Bd{a}^{2}}{b \left ( ad-bc \right ) }\ln \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) }-4\,{\frac{Bac}{ad-bc}\ln \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) }+2\,{\frac{B{c}^{2}b}{d \left ( ad-bc \right ) }\ln \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*x+B*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*x+B/d*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*c-2*B/b*ln(1/(
d*x+c))*a+2*B/d*ln(1/(d*x+c))*c+2*B*d/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^2-4*B/(a*d-b*c)*ln(1/(d*x+
c)*a*d-b*c/(d*x+c)+b)*a*c+2*B/d/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^2*b

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Maxima [A]  time = 1.23, size = 77, normalized size = 1.43 \begin{align*}{\left (x \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + \frac{2 \,{\left (\frac{a e \log \left (b x + a\right )}{b} - \frac{c e \log \left (d x + c\right )}{d}\right )}}{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)^2/(d*x+c)^2),x, algorithm="maxima")

[Out]

(x*log((b*x + a)^2*e/(d*x + c)^2) + 2*(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 = 1.01892, size = 184, normalized size = 3.41 \begin{align*} \frac{B b d x \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + A b d x + 2 \, B a d \log \left (b x + a\right ) - 2 \, 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)^2/(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.14493, size = 104, normalized size = 1.93 \begin{align*} A x + \frac{2 B a \log{\left (x + \frac{\frac{2 B a^{2} d}{b} + 2 B a c}{2 B a d + 2 B b c} \right )}}{b} - \frac{2 B c \log{\left (x + \frac{2 B a c + \frac{2 B b c^{2}}{d}}{2 B a d + 2 B b c} \right )}}{d} + B x \log{\left (\frac{e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^2/(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError