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

Optimal. Leaf size=519 \[ \frac{1}{12} \left (-24 \left (\text{PolyLog}\left (4,-\frac{b x}{a}\right )+\text{PolyLog}\left (4,\frac{b x}{a+b x}\right )-\text{PolyLog}\left (4,\frac{b x}{a}+1\right )\right )+12 \left (\log ^2\left (-\frac{b x}{a}\right )-2 \left (\log \left (-\frac{b x}{a+b x}\right )+\log \left (\frac{b x}{a}+1\right )\right ) \log \left (-\frac{b x}{a}\right )+2 \log (x) \left (\log \left (\frac{b x}{a}+1\right )-\log (a+b x)\right )\right ) \text{PolyLog}\left (2,-\frac{b x}{a}\right )-12 \log ^2\left (-\frac{b x}{a+b x}\right ) \text{PolyLog}\left (2,\frac{b x}{a+b x}\right )+12 \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{b x}{a+b x}\right )\right )^2 \text{PolyLog}\left (2,\frac{b x}{a}+1\right )+24 \left (\log (x)-\log \left (-\frac{b x}{a}\right )\right ) \log \left (\frac{b x}{a}+1\right ) \text{PolyLog}\left (2,\frac{b x}{a}+1\right )+24 \left (\log \left (-\frac{b x}{a+b x}\right )+\log (a+b x)\right ) \text{PolyLog}\left (3,-\frac{b x}{a}\right )+24 \log \left (-\frac{b x}{a+b x}\right ) \text{PolyLog}\left (3,\frac{b x}{a+b x}\right )+24 \left (\log \left (-\frac{b x}{a+b x}\right )-\log (x)\right ) \text{PolyLog}\left (3,\frac{b x}{a}+1\right )+\log ^4\left (-\frac{b x}{a}\right )+6 \log ^2\left (-\frac{b x}{a+b x}\right ) \log ^2\left (-\frac{b x}{a}\right )-4 \log \left (-\frac{b x}{a+b x}\right ) \left (\log \left (-\frac{b x}{a}\right )+3 \log \left (\frac{b x}{a}+1\right )\right ) \log ^2\left (-\frac{b x}{a}\right )+\log ^4\left (-\frac{b x}{a+b x}\right )-4 \left (\log \left (-\frac{b x}{a}\right )+\log \left (\frac{a}{a+b x}\right )\right ) \log ^3\left (-\frac{b x}{a+b x}\right )+6 \log ^2(x) \log ^2(a+b x)+6 \left (\log (x)-\log \left (-\frac{b x}{a}\right )\right ) \left (3 \log \left (-\frac{b x}{a}\right )+\log (x)\right ) \log ^2\left (\frac{b x}{a}+1\right )+4 \left (2 \log ^3\left (-\frac{b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (\frac{b x}{a}+1\right )\right ) \]

[Out]

(Log[-((b*x)/a)]^4 + 6*Log[-((b*x)/a)]^2*Log[-((b*x)/(a + b*x))]^2 - 4*(Log[-((b*x)/a)] + Log[a/(a + b*x)])*Lo
g[-((b*x)/(a + b*x))]^3 + Log[-((b*x)/(a + b*x))]^4 + 6*Log[x]^2*Log[a + b*x]^2 + 4*(2*Log[-((b*x)/a)]^3 - 3*L
og[x]^2*Log[a + b*x])*Log[1 + (b*x)/a] + 6*(Log[x] - Log[-((b*x)/a)])*(Log[x] + 3*Log[-((b*x)/a)])*Log[1 + (b*
x)/a]^2 - 4*Log[-((b*x)/a)]^2*Log[-((b*x)/(a + b*x))]*(Log[-((b*x)/a)] + 3*Log[1 + (b*x)/a]) + 12*(Log[-((b*x)
/a)]^2 - 2*Log[-((b*x)/a)]*(Log[-((b*x)/(a + b*x))] + Log[1 + (b*x)/a]) + 2*Log[x]*(-Log[a + b*x] + Log[1 + (b
*x)/a]))*PolyLog[2, -((b*x)/a)] - 12*Log[-((b*x)/(a + b*x))]^2*PolyLog[2, (b*x)/(a + b*x)] + 12*(Log[-((b*x)/a
)] - Log[-((b*x)/(a + b*x))])^2*PolyLog[2, 1 + (b*x)/a] + 24*(Log[x] - Log[-((b*x)/a)])*Log[1 + (b*x)/a]*PolyL
og[2, 1 + (b*x)/a] + 24*(Log[-((b*x)/(a + b*x))] + Log[a + b*x])*PolyLog[3, -((b*x)/a)] + 24*Log[-((b*x)/(a +
b*x))]*PolyLog[3, (b*x)/(a + b*x)] + 24*(-Log[x] + Log[-((b*x)/(a + b*x))])*PolyLog[3, 1 + (b*x)/a] - 24*(Poly
Log[4, -((b*x)/a)] + PolyLog[4, (b*x)/(a + b*x)] - PolyLog[4, 1 + (b*x)/a]))/12

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Rubi [F]  time = 0.048933, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log (x) \log ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Log[x]*Log[a + b*x]^2)/x,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{\log (x) \log ^2(a+b x)}{x} \, dx &=\frac{1}{2} \log ^2(x) \log ^2(a+b x)-b \int \frac{\log ^2(x) \log (a+b x)}{a+b x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0948574, size = 519, normalized size = 1. \[ \frac{1}{12} \left (-24 \left (\text{PolyLog}\left (4,-\frac{b x}{a}\right )+\text{PolyLog}\left (4,\frac{b x}{a+b x}\right )-\text{PolyLog}\left (4,\frac{b x}{a}+1\right )\right )+12 \left (\log ^2\left (-\frac{b x}{a}\right )-2 \left (\log \left (-\frac{b x}{a+b x}\right )+\log \left (\frac{b x}{a}+1\right )\right ) \log \left (-\frac{b x}{a}\right )+2 \log (x) \left (\log \left (\frac{b x}{a}+1\right )-\log (a+b x)\right )\right ) \text{PolyLog}\left (2,-\frac{b x}{a}\right )-12 \log ^2\left (-\frac{b x}{a+b x}\right ) \text{PolyLog}\left (2,\frac{b x}{a+b x}\right )+12 \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{b x}{a+b x}\right )\right )^2 \text{PolyLog}\left (2,\frac{b x}{a}+1\right )+24 \left (\log (x)-\log \left (-\frac{b x}{a}\right )\right ) \log \left (\frac{b x}{a}+1\right ) \text{PolyLog}\left (2,\frac{b x}{a}+1\right )+24 \left (\log \left (-\frac{b x}{a+b x}\right )+\log (a+b x)\right ) \text{PolyLog}\left (3,-\frac{b x}{a}\right )+24 \log \left (-\frac{b x}{a+b x}\right ) \text{PolyLog}\left (3,\frac{b x}{a+b x}\right )+24 \left (\log \left (-\frac{b x}{a+b x}\right )-\log (x)\right ) \text{PolyLog}\left (3,\frac{b x}{a}+1\right )+\log ^4\left (-\frac{b x}{a}\right )+6 \log ^2\left (-\frac{b x}{a+b x}\right ) \log ^2\left (-\frac{b x}{a}\right )-4 \log \left (-\frac{b x}{a+b x}\right ) \left (\log \left (-\frac{b x}{a}\right )+3 \log \left (\frac{b x}{a}+1\right )\right ) \log ^2\left (-\frac{b x}{a}\right )+\log ^4\left (-\frac{b x}{a+b x}\right )-4 \left (\log \left (-\frac{b x}{a}\right )+\log \left (\frac{a}{a+b x}\right )\right ) \log ^3\left (-\frac{b x}{a+b x}\right )+6 \log ^2(x) \log ^2(a+b x)+6 \left (\log (x)-\log \left (-\frac{b x}{a}\right )\right ) \left (3 \log \left (-\frac{b x}{a}\right )+\log (x)\right ) \log ^2\left (\frac{b x}{a}+1\right )+4 \left (2 \log ^3\left (-\frac{b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (\frac{b x}{a}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[-((b*x)/a)]^4 + 6*Log[-((b*x)/a)]^2*Log[-((b*x)/(a + b*x))]^2 - 4*(Log[-((b*x)/a)] + Log[a/(a + b*x)])*Lo
g[-((b*x)/(a + b*x))]^3 + Log[-((b*x)/(a + b*x))]^4 + 6*Log[x]^2*Log[a + b*x]^2 + 4*(2*Log[-((b*x)/a)]^3 - 3*L
og[x]^2*Log[a + b*x])*Log[1 + (b*x)/a] + 6*(Log[x] - Log[-((b*x)/a)])*(Log[x] + 3*Log[-((b*x)/a)])*Log[1 + (b*
x)/a]^2 - 4*Log[-((b*x)/a)]^2*Log[-((b*x)/(a + b*x))]*(Log[-((b*x)/a)] + 3*Log[1 + (b*x)/a]) + 12*(Log[-((b*x)
/a)]^2 - 2*Log[-((b*x)/a)]*(Log[-((b*x)/(a + b*x))] + Log[1 + (b*x)/a]) + 2*Log[x]*(-Log[a + b*x] + Log[1 + (b
*x)/a]))*PolyLog[2, -((b*x)/a)] - 12*Log[-((b*x)/(a + b*x))]^2*PolyLog[2, (b*x)/(a + b*x)] + 12*(Log[-((b*x)/a
)] - Log[-((b*x)/(a + b*x))])^2*PolyLog[2, 1 + (b*x)/a] + 24*(Log[x] - Log[-((b*x)/a)])*Log[1 + (b*x)/a]*PolyL
og[2, 1 + (b*x)/a] + 24*(Log[-((b*x)/(a + b*x))] + Log[a + b*x])*PolyLog[3, -((b*x)/a)] + 24*Log[-((b*x)/(a +
b*x))]*PolyLog[3, (b*x)/(a + b*x)] + 24*(-Log[x] + Log[-((b*x)/(a + b*x))])*PolyLog[3, 1 + (b*x)/a] - 24*(Poly
Log[4, -((b*x)/a)] + PolyLog[4, (b*x)/(a + b*x)] - PolyLog[4, 1 + (b*x)/a]))/12

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( x \right ) \left ( \ln \left ( bx+a \right ) \right ) ^{2}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, \log \left (b x + a\right )^{2} \log \left (x\right )^{2} - b \int \frac{\log \left (b x + a\right ) \log \left (x\right )^{2}}{b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(b*x + a)^2*log(x)^2 - b*integrate(log(b*x + a)*log(x)^2/(b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (b x + a\right )^{2} \log \left (x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(b*x + a)^2*log(x)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - b \int \frac{\log{\left (x \right )}^{2} \log{\left (a + b x \right )}}{a + b x}\, dx + \frac{\log{\left (x \right )}^{2} \log{\left (a + b x \right )}^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*Integral(log(x)**2*log(a + b*x)/(a + b*x), x) + log(x)**2*log(a + b*x)**2/2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (b x + a\right )^{2} \log \left (x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(b*x + a)^2*log(x)/x, x)