3.4.74 \(\int \frac {\log (x) \log ^2(a+b x)}{x} \, dx\) [374]

Optimal. Leaf size=519 \[ \frac {1}{12} \left (\log ^4\left (-\frac {b x}{a}\right )+6 \log ^2\left (-\frac {b x}{a}\right ) \log ^2\left (-\frac {b x}{a+b x}\right )-4 \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {a}{a+b x}\right )\right ) \log ^3\left (-\frac {b x}{a+b x}\right )+\log ^4\left (-\frac {b x}{a+b x}\right )+6 \log ^2(x) \log ^2(a+b x)+4 \left (2 \log ^3\left (-\frac {b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (1+\frac {b x}{a}\right )+6 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \left (\log (x)+3 \log \left (-\frac {b x}{a}\right )\right ) \log ^2\left (1+\frac {b x}{a}\right )-4 \log ^2\left (-\frac {b x}{a}\right ) \log \left (-\frac {b x}{a+b x}\right ) \left (\log \left (-\frac {b x}{a}\right )+3 \log \left (1+\frac {b x}{a}\right )\right )+12 \left (\log ^2\left (-\frac {b x}{a}\right )-2 \log \left (-\frac {b x}{a}\right ) \left (\log \left (-\frac {b x}{a+b x}\right )+\log \left (1+\frac {b x}{a}\right )\right )+2 \log (x) \left (-\log (a+b x)+\log \left (1+\frac {b x}{a}\right )\right )\right ) \text {Li}_2\left (-\frac {b x}{a}\right )-12 \log ^2\left (-\frac {b x}{a+b x}\right ) \text {Li}_2\left (\frac {b x}{a+b x}\right )+12 \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {b x}{a+b x}\right )\right )^2 \text {Li}_2\left (1+\frac {b x}{a}\right )+24 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \log \left (1+\frac {b x}{a}\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )+24 \left (\log \left (-\frac {b x}{a+b x}\right )+\log (a+b x)\right ) \text {Li}_3\left (-\frac {b x}{a}\right )+24 \log \left (-\frac {b x}{a+b x}\right ) \text {Li}_3\left (\frac {b x}{a+b x}\right )+24 \left (-\log (x)+\log \left (-\frac {b x}{a+b x}\right )\right ) \text {Li}_3\left (1+\frac {b x}{a}\right )-24 \left (\text {Li}_4\left (-\frac {b x}{a}\right )+\text {Li}_4\left (\frac {b x}{a+b x}\right )-\text {Li}_4\left (1+\frac {b x}{a}\right )\right )\right ) \]

[Out]

1/12*ln(-b*x/a)^4+1/2*ln(-b*x/a)^2*ln(-b*x/(b*x+a))^2-1/3*(ln(-b*x/a)+ln(a/(b*x+a)))*ln(-b*x/(b*x+a))^3+1/12*l
n(-b*x/(b*x+a))^4+1/2*ln(x)^2*ln(b*x+a)^2+1/3*(2*ln(-b*x/a)^3-3*ln(x)^2*ln(b*x+a))*ln(1+b*x/a)+1/2*(ln(x)-ln(-
b*x/a))*(ln(x)+3*ln(-b*x/a))*ln(1+b*x/a)^2-1/3*ln(-b*x/a)^2*ln(-b*x/(b*x+a))*(ln(-b*x/a)+3*ln(1+b*x/a))+(ln(-b
*x/a)^2-2*ln(-b*x/a)*(ln(-b*x/(b*x+a))+ln(1+b*x/a))+2*ln(x)*(-ln(b*x+a)+ln(1+b*x/a)))*polylog(2,-b*x/a)-ln(-b*
x/(b*x+a))^2*polylog(2,b*x/(b*x+a))+(ln(-b*x/a)-ln(-b*x/(b*x+a)))^2*polylog(2,1+b*x/a)+2*(ln(x)-ln(-b*x/a))*ln
(1+b*x/a)*polylog(2,1+b*x/a)+2*(ln(-b*x/(b*x+a))+ln(b*x+a))*polylog(3,-b*x/a)+2*ln(-b*x/(b*x+a))*polylog(3,b*x
/(b*x+a))+2*(-ln(x)+ln(-b*x/(b*x+a)))*polylog(3,1+b*x/a)-2*polylog(4,-b*x/a)-2*polylog(4,b*x/(b*x+a))+2*polylo
g(4,1+b*x/a)

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Rubi [F]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\log (x) \log ^2(a+b x)}{x} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.07, size = 519, normalized size = 1.00 \begin {gather*} \frac {1}{12} \left (\log ^4\left (-\frac {b x}{a}\right )+6 \log ^2\left (-\frac {b x}{a}\right ) \log ^2\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 )+\log ^4\left (-\frac {b x}{a+b x}\right )+6 \log ^2(x) \log ^2(a+b x)+4 \left (2 \log ^3\left (-\frac {b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (1+\frac {b x}{a}\right )+6 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \left (\log (x)+3 \log \left (-\frac {b x}{a}\right )\right ) \log ^2\left (1+\frac {b x}{a}\right )-4 \log ^2\left (-\frac {b x}{a}\right ) \log \left (-\frac {b x}{a+b x}\right ) \left (\log \left (-\frac {b x}{a}\right )+3 \log \left (1+\frac {b x}{a}\right )\right )+12 \left (\log ^2\left (-\frac {b x}{a}\right )-2 \log \left (-\frac {b x}{a}\right ) \left (\log \left (-\frac {b x}{a+b x}\right )+\log \left (1+\frac {b x}{a}\right )\right )+2 \log (x) \left (-\log (a+b x)+\log \left (1+\frac {b x}{a}\right )\right )\right ) \text {Li}_2\left (-\frac {b x}{a}\right )-12 \log ^2\left (-\frac {b x}{a+b x}\right ) \text {Li}_2\left (\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 {Li}_2\left (1+\frac {b x}{a}\right )+24 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \log \left (1+\frac {b x}{a}\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )+24 \left (\log \left (-\frac {b x}{a+b x}\right )+\log (a+b x)\right ) \text {Li}_3\left (-\frac {b x}{a}\right )+24 \log \left (-\frac {b x}{a+b x}\right ) \text {Li}_3\left (\frac {b x}{a+b x}\right )+24 \left (-\log (x)+\log \left (-\frac {b x}{a+b x}\right )\right ) \text {Li}_3\left (1+\frac {b x}{a}\right )-24 \left (\text {Li}_4\left (-\frac {b x}{a}\right )+\text {Li}_4\left (\frac {b x}{a+b x}\right )-\text {Li}_4\left (1+\frac {b x}{a}\right )\right )\right ) \end {gather*}

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 = 0.31, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (x \right ) \ln \left (b x +a \right )^{2}}{x}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - 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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\ln \left (a+b\,x\right )}^2\,\ln \left (x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(a + b*x)^2*log(x))/x,x)

[Out]

int((log(a + b*x)^2*log(x))/x, x)

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