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

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

[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])]

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Rubi [A]  time = 0.175996, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, 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 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]

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 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 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]

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 = 0.156441, size = 156, normalized size = 1.21 \[ -n \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )-n \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )-n \log (x) \log \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )-n \log (x) \log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )+\log (x) \log \left (d (a+x (b+c x))^n\right ) \]

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])]

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Maple [C]  time = 0.063, size = 315, normalized size = 2.4 \begin{align*} \ln \left ( x \right ) \ln \left ( \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) -\ln \left ( x \right ) \ln \left ({ \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) n-\ln \left ( x \right ) \ln \left ({ \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) n-{\it dilog} \left ({ \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) n-{\it dilog} \left ({ \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) n-{\frac{i}{2}}\ln \left ( x \right ) \pi \,{\it csgn} \left ( id \right ){\it csgn} \left ( i \left ( c{x}^{2}+bx+a \right ) ^{n} \right ){\it csgn} \left ( id \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) +{\frac{i}{2}}\ln \left ( x \right ) \pi \,{\it csgn} \left ( id \right ) \left ({\it csgn} \left ( id \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \right ) ^{2}+{\frac{i}{2}}\ln \left ( x \right ) \pi \,{\it csgn} \left ( i \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \left ({\it csgn} \left ( id \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \right ) ^{2}-{\frac{i}{2}}\ln \left ( x \right ) \pi \, \left ({\it csgn} \left ( id \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \right ) ^{3}+\ln \left ( x \right ) \ln \left ( d \right ) \end{align*}

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((b+2*c*x+
(-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((b+2*c*x+(-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)

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

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)

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

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)

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

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)

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

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)