3.4.51 \(\int \frac {x^3 \log (x)}{a+b x+c x^2} \, dx\) [351]

Optimal. Leaf size=286 \[ \frac {b x}{c^2}-\frac {x^2}{4 c}-\frac {b x \log (x)}{c^2}+\frac {x^2 \log (x)}{2 c}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^3} \]

[Out]

b*x/c^2-1/4*x^2/c-b*x*ln(x)/c^2+1/2*x^2*ln(x)/c+1/2*ln(x)*ln(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))*(b^2-a*c-b*(-3*a*
c+b^2)/(-4*a*c+b^2)^(1/2))/c^3+1/2*polylog(2,-2*c*x/(b-(-4*a*c+b^2)^(1/2)))*(b^2-a*c-b*(-3*a*c+b^2)/(-4*a*c+b^
2)^(1/2))/c^3+1/2*ln(x)*ln(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))*(b^2-a*c+b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/c^3+1/2
*polylog(2,-2*c*x/(b+(-4*a*c+b^2)^(1/2)))*(b^2-a*c+b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/c^3

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Rubi [A]
time = 0.29, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2404, 2332, 2341, 2354, 2438} \begin {gather*} \frac {\left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \text {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \text {PolyLog}\left (2,-\frac {2 c x}{\sqrt {b^2-4 a c}+b}\right )}{2 c^3}+\frac {\log (x) \left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c^3}+\frac {\log (x) \left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c^3}+\frac {b x}{c^2}-\frac {b x \log (x)}{c^2}-\frac {x^2}{4 c}+\frac {x^2 \log (x)}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*Log[x])/(a + b*x + c*x^2),x]

[Out]

(b*x)/c^2 - x^2/(4*c) - (b*x*Log[x])/c^2 + (x^2*Log[x])/(2*c) + ((b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a
*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*c^3) + ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a
*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*c^3) + ((b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a
*c])*PolyLog[2, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*c^3) + ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c]
)*PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*c^3)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2354

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 2404

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 2438

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 {x^3 \log (x)}{a+b x+c x^2} \, dx &=\int \left (-\frac {b \log (x)}{c^2}+\frac {x \log (x)}{c}+\frac {\left (a b+\left (b^2-a c\right ) x\right ) \log (x)}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a b+\left (b^2-a c\right ) x\right ) \log (x)}{a+b x+c x^2} \, dx}{c^2}-\frac {b \int \log (x) \, dx}{c^2}+\frac {\int x \log (x) \, dx}{c}\\ &=\frac {b x}{c^2}-\frac {x^2}{4 c}-\frac {b x \log (x)}{c^2}+\frac {x^2 \log (x)}{2 c}+\frac {\int \left (\frac {\left (b^2-a c+\frac {b \left (-b^2+3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x)}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (b^2-a c-\frac {b \left (-b^2+3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x)}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{c^2}\\ &=\frac {b x}{c^2}-\frac {x^2}{4 c}-\frac {b x \log (x)}{c^2}+\frac {x^2 \log (x)}{2 c}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {\log (x)}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{c^2}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {\log (x)}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{c^2}\\ &=\frac {b x}{c^2}-\frac {x^2}{4 c}-\frac {b x \log (x)}{c^2}+\frac {x^2 \log (x)}{2 c}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^3}-\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {\log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{x} \, dx}{2 c^3}-\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {\log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx}{2 c^3}\\ &=\frac {b x}{c^2}-\frac {x^2}{4 c}-\frac {b x \log (x)}{c^2}+\frac {x^2 \log (x)}{2 c}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^3}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 464, normalized size = 1.62 \begin {gather*} \frac {4 b c x-c^2 x^2-4 b c x \log (x)+2 c^2 x^2 \log (x)+\frac {4 a b c \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+2 \left (b^2-a c\right ) \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )-\frac {4 a b c \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+2 \left (b^2-a c\right ) \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )+\frac {4 a b c \text {Li}_2\left (\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+2 \left (b^2-a c\right ) \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )-\frac {4 a b c \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+2 \left (b^2-a c\right ) \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{4 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Log[x])/(a + b*x + c*x^2),x]

[Out]

(4*b*c*x - c^2*x^2 - 4*b*c*x*Log[x] + 2*c^2*x^2*Log[x] + (4*a*b*c*Log[x]*Log[(b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(
b - Sqrt[b^2 - 4*a*c])])/Sqrt[b^2 - 4*a*c] + 2*(b^2 - a*c)*(1 - b/Sqrt[b^2 - 4*a*c])*Log[x]*Log[(b - Sqrt[b^2
- 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c])] - (4*a*b*c*Log[x]*Log[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2
- 4*a*c])])/Sqrt[b^2 - 4*a*c] + 2*(b^2 - a*c)*(1 + b/Sqrt[b^2 - 4*a*c])*Log[x]*Log[(b + Sqrt[b^2 - 4*a*c] + 2
*c*x)/(b + Sqrt[b^2 - 4*a*c])] + (4*a*b*c*PolyLog[2, (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/Sqrt[b^2 - 4*a*c] + 2*
(b^2 - a*c)*(1 - b/Sqrt[b^2 - 4*a*c])*PolyLog[2, (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] - (4*a*b*c*PolyLog[2, (-2*c
*x)/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[b^2 - 4*a*c] + 2*(b^2 - a*c)*(1 + b/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/
(b + Sqrt[b^2 - 4*a*c])])/(4*c^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs. \(2(258)=516\).
time = 0.85, size = 766, normalized size = 2.68

method	result	size
default	\(\frac {\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}}{c}-\frac {\left (\ln \left (x \right ) x -x \right ) b}{c^{2}}+\frac {-\frac {\ln \left (x \right ) \left (\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}+3 \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a b c -\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}+\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}-3 \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a b c +\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}\right )}{2 c \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}-3 \dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a b c +\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}+\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}+3 \dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a b c -\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 c \sqrt {-4 c a +b^{2}}}}{c^{2}}\)	\(766\)
risch	\(\frac {x^{2} \ln \left (x \right )}{2 c}-\frac {x^{2}}{4 c}-\frac {b x \ln \left (x \right )}{c^{2}}+\frac {b x}{c^{2}}-\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a}{2 c^{2}}+\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{3}}-\frac {3 \ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a b}{2 c^{2} \sqrt {-4 c a +b^{2}}}+\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 c^{3} \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a}{2 c^{2}}+\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{3}}+\frac {3 \ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a b}{2 c^{2} \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 c^{3} \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a}{2 c^{2}}+\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{3}}+\frac {3 \dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a b}{2 c^{2} \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 c^{3} \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a}{2 c^{2}}+\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{3}}-\frac {3 \dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a b}{2 c^{2} \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 c^{3} \sqrt {-4 c a +b^{2}}}\)	\(791\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*ln(x)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

(1/2*x^2*ln(x)-1/4*x^2)/c-(ln(x)*x-x)*b/c^2+(-1/2*ln(x)*(ln((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)
))*(-4*a*c+b^2)^(1/2)*a*c-ln((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*b^2+3*ln(
(b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*a*b*c-ln((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/
2)))*b^3+ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*a*c-ln((-2*c*x+(-4*a*c+b
^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*b^2-3*ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^
2)^(1/2)))*a*b*c+ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^3)/c/(-4*a*c+b^2)^(1/2)-1/2*(dilo
g((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*a*c-dilog((-2*c*x+(-4*a*c+b^2)^(1/
2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*b^2-3*dilog((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(
1/2)))*a*b*c+dilog((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^3+dilog((b+2*c*x+(-4*a*c+b^2)^(1/2
))/(b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)*a*c-dilog((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*(
-4*a*c+b^2)^(1/2)*b^2+3*dilog((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*a*b*c-dilog((b+2*c*x+(-4*a*
c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*b^3)/c/(-4*a*c+b^2)^(1/2))/c^2

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(x)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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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(x^3*log(x)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

integral(x^3*log(x)/(c*x^2 + b*x + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \log {\left (x \right )}}{a + b x + c x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*ln(x)/(c*x**2+b*x+a),x)

[Out]

Integral(x**3*log(x)/(a + b*x + c*x**2), x)

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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(x^3*log(x)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

integrate(x^3*log(x)/(c*x^2 + b*x + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\ln \left (x\right )}{c\,x^2+b\,x+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*log(x))/(a + b*x + c*x^2),x)

[Out]

int((x^3*log(x))/(a + b*x + c*x^2), x)

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