Optimal. Leaf size=129 \[ -n \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )+\log (x) \log \left (d \left (a+b x+c x^2\right )^n\right )-n \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right ) \]
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Rubi [A]
time = 0.12, 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 = {2604, 2404,
2354, 2438} \begin {gather*} -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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2354
Rule 2404
Rule 2438
Rule 2604
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*}
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Mathematica [A]
time = 0.12, size = 156, normalized size = 1.21 \begin {gather*} -n \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )-n \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )+\log (x) \log \left (d (a+x (b+c x))^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 {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.04, size = 315, normalized size = 2.44
method | result | size |
risch | \(\ln \left (x \right ) \ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right )-\ln \left (\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right ) \ln \left (x \right ) n -\ln \left (\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right ) \ln \left (x \right ) n -\dilog \left (\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right ) n -\dilog \left (\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right ) n -\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}{2}+\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2}}{2}+\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2}}{2}-\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{3}}{2}+\ln \left (x \right ) \ln \left (d \right )\) | \(315\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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