3.89 \(\int \frac{\log ^2(\frac{c x}{a+b x})}{x (a+b x)} \, dx\)

Optimal. Leaf size=20 \[ \frac{\log ^3\left (\frac{c x}{a+b x}\right )}{3 a} \]

[Out]

Log[(c*x)/(a + b*x)]^3/(3*a)

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Rubi [A]  time = 0.0579114, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {2505} \[ \frac{\log ^3\left (\frac{c x}{a+b x}\right )}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[(c*x)/(a + b*x)]^3/(3*a)

Rule 2505

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(
b*c - a*d)), x] /; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && NeQ[s, -1]

Rubi steps

\begin{align*} \int \frac{\log ^2\left (\frac{c x}{a+b x}\right )}{x (a+b x)} \, dx &=\frac{\log ^3\left (\frac{c x}{a+b x}\right )}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0961977, size = 20, normalized size = 1. \[ \frac{\log ^3\left (\frac{c x}{a+b x}\right )}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[(c*x)/(a + b*x)]^3/(3*a)

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Maple [A]  time = 0.059, size = 29, normalized size = 1.5 \begin{align*}{\frac{1}{3\,a} \left ( \ln \left ({\frac{c}{b}}-{\frac{ac}{b \left ( bx+a \right ) }} \right ) \right ) ^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x/(b*x+a))^2/x/(b*x+a),x)

[Out]

1/3/a*ln(c/b-a*c/b/(b*x+a))^3

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Maxima [B]  time = 1.19479, size = 190, normalized size = 9.5 \begin{align*} -{\left (\frac{\log \left (b x + a\right )}{a} - \frac{\log \left (x\right )}{a}\right )} \log \left (\frac{c x}{b x + a}\right )^{2} - \frac{{\left (c \log \left (b x + a\right )^{2} - 2 \, c \log \left (b x + a\right ) \log \left (x\right ) + c \log \left (x\right )^{2}\right )} \log \left (\frac{c x}{b x + a}\right )}{a c} - \frac{c^{2} \log \left (b x + a\right )^{3} - 3 \, c^{2} \log \left (b x + a\right )^{2} \log \left (x\right ) + 3 \, c^{2} \log \left (b x + a\right ) \log \left (x\right )^{2} - c^{2} \log \left (x\right )^{3}}{3 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x/(b*x+a))^2/x/(b*x+a),x, algorithm="maxima")

[Out]

-(log(b*x + a)/a - log(x)/a)*log(c*x/(b*x + a))^2 - (c*log(b*x + a)^2 - 2*c*log(b*x + a)*log(x) + c*log(x)^2)*
log(c*x/(b*x + a))/(a*c) - 1/3*(c^2*log(b*x + a)^3 - 3*c^2*log(b*x + a)^2*log(x) + 3*c^2*log(b*x + a)*log(x)^2
- c^2*log(x)^3)/(a*c^2)

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Fricas [A]  time = 1.98203, size = 38, normalized size = 1.9 \begin{align*} \frac{\log \left (\frac{c x}{b x + a}\right )^{3}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x/(b*x+a))^2/x/(b*x+a),x, algorithm="fricas")

[Out]

1/3*log(c*x/(b*x + a))^3/a

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Sympy [A]  time = 0.390677, size = 14, normalized size = 0.7 \begin{align*} \frac{\log{\left (\frac{c x}{a + b x} \right )}^{3}}{3 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x/(b*x+a))**2/x/(b*x+a),x)

[Out]

log(c*x/(a + b*x))**3/(3*a)

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Giac [A]  time = 1.15435, size = 24, normalized size = 1.2 \begin{align*} \frac{\log \left (\frac{c x}{b x + a}\right )^{3}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x/(b*x+a))^2/x/(b*x+a),x, algorithm="giac")

[Out]

1/3*log(c*x/(b*x + a))^3/a