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

Optimal. Leaf size=129 \[ -n \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )-n \log (x) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )-n \log (x) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )+\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right ) \]

[Out]

ln(x)*ln(d*(c*x^2+b*x+a)^n)-n*ln(x)*ln(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))-n*ln(x)*ln(1+2*c*x/(b+(-4*a*c+b^2)^(1/2
)))-n*polylog(2,-2*c*x/(b-(-4*a*c+b^2)^(1/2)))-n*polylog(2,-2*c*x/(b+(-4*a*c+b^2)^(1/2)))

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2524, 2357, 2317, 2391} \[ -n \text {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \text {PolyLog}\left (2,-\frac {2 c x}{\sqrt {b^2-4 a c}+b}\right )-n \log (x) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )-n \log (x) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )+\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(a + b*x + c*x^2)^n]/x,x]

[Out]

-(n*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])]) - n*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])] + Log
[x]*Log[d*(a + b*x + c*x^2)^n] - n*PolyLog[2, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c])] - n*PolyLog[2, (-2*c*x)/(b + S
qrt[b^2 - 4*a*c])]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x} \, dx &=\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-n \int \frac {(b+2 c x) \log (x)}{a+b x+c x^2} \, dx\\ &=\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-n \int \left (\frac {2 c \log (x)}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {2 c \log (x)}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx\\ &=\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-(2 c n) \int \frac {\log (x)}{b-\sqrt {b^2-4 a c}+2 c x} \, dx-(2 c n) \int \frac {\log (x)}{b+\sqrt {b^2-4 a c}+2 c x} \, dx\\ &=-n \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )+\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )+n \int \frac {\log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{x} \, dx+n \int \frac {\log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx\\ &=-n \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )+\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-n \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 156, normalized size = 1.21 \[ -n \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )-n \log (x) \log \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \log (x) \log \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )+\log (x) \log \left (d (a+x (b+c x))^n\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(n*Log[x]*Log[(b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c])]) - n*Log[x]*Log[(b + Sqrt[b^2 - 4*a*c]
+ 2*c*x)/(b + Sqrt[b^2 - 4*a*c])] + Log[x]*Log[d*(a + x*(b + c*x))^n] - n*PolyLog[2, (-2*c*x)/(b - Sqrt[b^2 -
4*a*c])] - n*PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])]

________________________________________________________________________________________

fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((c*x^2 + b*x + a)^n*d)/x, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((c*x^2 + b*x + a)^n*d)/x, x)

________________________________________________________________________________________

maple [C]  time = 0.34, size = 315, normalized size = 2.44 \[ -\frac {i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right ) \ln \relax (x )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2} \ln \relax (x )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{3} \ln \relax (x )}{2}-n \ln \relax (x ) \ln \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )-n \ln \relax (x ) \ln \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )-n \dilog \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )-n \dilog \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )+\ln \relax (d ) \ln \relax (x )+\ln \relax (x ) \ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)*ln((c*x^2+b*x+a)^n)-ln(x)*ln((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2)))*n-ln(x)*ln((2*c*x+b+
(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*n-dilog((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2)))*n-d
ilog((2*c*x+b+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*n-1/2*I*ln(x)*Pi*csgn(I*d)*csgn(I*(c*x^2+b*x+a)^n)*c
sgn(I*d*(c*x^2+b*x+a)^n)+1/2*I*ln(x)*Pi*csgn(I*d)*csgn(I*d*(c*x^2+b*x+a)^n)^2+1/2*I*ln(x)*Pi*csgn(I*(c*x^2+b*x
+a)^n)*csgn(I*d*(c*x^2+b*x+a)^n)^2-1/2*I*ln(x)*Pi*csgn(I*d*(c*x^2+b*x+a)^n)^3+ln(x)*ln(d)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((c*x^2 + b*x + a)^n*d)/x, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(d*(a + b*x + c*x^2)^n)/x,x)

[Out]

int(log(d*(a + b*x + c*x^2)^n)/x, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(d*(a + b*x + c*x**2)**n)/x, x)

________________________________________________________________________________________