Optimal. Leaf size=243 \[ \frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (-\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
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Rubi [A] time = 0.23, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2418, 2394, 2393, 2391} \[ \frac {n \text {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (-\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rubi steps
\begin {align*} \int \frac {\log \left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx &=\int \left (\frac {2 f \log \left (c (a+b x)^n\right )}{\sqrt {e^2-4 d f} \left (e-\sqrt {e^2-4 d f}+2 f x\right )}-\frac {2 f \log \left (c (a+b x)^n\right )}{\sqrt {e^2-4 d f} \left (e+\sqrt {e^2-4 d f}+2 f x\right )}\right ) \, dx\\ &=\frac {(2 f) \int \frac {\log \left (c (a+b x)^n\right )}{e-\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 f) \int \frac {\log \left (c (a+b x)^n\right )}{e+\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}\\ &=\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {(b n) \int \frac {\log \left (\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{a+b x} \, dx}{\sqrt {e^2-4 d f}}+\frac {(b n) \int \frac {\log \left (\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{a+b x} \, dx}{\sqrt {e^2-4 d f}}\\ &=\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 f x}{-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 f x}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}}\\ &=\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 194, normalized size = 0.80 \[ \frac {\log \left (c (a+b x)^n\right ) \left (\log \left (\frac {b \left (\sqrt {e^2-4 d f}-e-2 f x\right )}{2 a f+b \sqrt {e^2-4 d f}+b (-e)}\right )-\log \left (\frac {b \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{b \left (\sqrt {e^2-4 d f}+e\right )-2 a f}\right )\right )+n \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f+b \left (\sqrt {e^2-4 d f}-e\right )}\right )-n \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x + a\right )}^{n} c\right )}{f x^{2} + e x + d}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (b x + a\right )}^{n} c\right )}{f x^{2} + e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 689, normalized size = 2.84 \[ \frac {b n \ln \left (\frac {2 a f -b e -2 \left (b x +a \right ) f +\sqrt {-4 b^{2} d f +b^{2} e^{2}}}{2 a f -b e +\sqrt {-4 b^{2} d f +b^{2} e^{2}}}\right ) \ln \left (b x +a \right )}{\sqrt {-4 b^{2} d f +b^{2} e^{2}}}-\frac {b n \ln \left (\frac {-2 a f +b e +2 \left (b x +a \right ) f +\sqrt {-4 b^{2} d f +b^{2} e^{2}}}{-2 a f +b e +\sqrt {-4 b^{2} d f +b^{2} e^{2}}}\right ) \ln \left (b x +a \right )}{\sqrt {-4 b^{2} d f +b^{2} e^{2}}}-\frac {i \pi \arctan \left (\frac {2 f x +e}{\sqrt {4 d f -e^{2}}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{n}\right )}{\sqrt {4 d f -e^{2}}}+\frac {i \pi \arctan \left (\frac {2 f x +e}{\sqrt {4 d f -e^{2}}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2}}{\sqrt {4 d f -e^{2}}}+\frac {i \pi \arctan \left (\frac {2 f x +e}{\sqrt {4 d f -e^{2}}}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2}}{\sqrt {4 d f -e^{2}}}-\frac {i \pi \arctan \left (\frac {2 f x +e}{\sqrt {4 d f -e^{2}}}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{n}\right )^{3}}{\sqrt {4 d f -e^{2}}}+\frac {b n \dilog \left (\frac {2 a f -b e -2 \left (b x +a \right ) f +\sqrt {-4 b^{2} d f +b^{2} e^{2}}}{2 a f -b e +\sqrt {-4 b^{2} d f +b^{2} e^{2}}}\right )}{\sqrt {-4 b^{2} d f +b^{2} e^{2}}}-\frac {b n \dilog \left (\frac {-2 a f +b e +2 \left (b x +a \right ) f +\sqrt {-4 b^{2} d f +b^{2} e^{2}}}{-2 a f +b e +\sqrt {-4 b^{2} d f +b^{2} e^{2}}}\right )}{\sqrt {-4 b^{2} d f +b^{2} e^{2}}}+\frac {2 \left (-n \ln \left (b x +a \right )+\ln \left (\left (b x +a \right )^{n}\right )\right ) b \arctan \left (\frac {-2 a f +b e +2 \left (b x +a \right ) f}{\sqrt {4 b^{2} d f -b^{2} e^{2}}}\right )}{\sqrt {4 b^{2} d f -b^{2} e^{2}}}+\frac {2 \arctan \left (\frac {2 f x +e}{\sqrt {4 d f -e^{2}}}\right ) \ln \relax (c )}{\sqrt {4 d f -e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (c\,{\left (a+b\,x\right )}^n\right )}{f\,x^2+e\,x+d} \,d x \]
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 \[ \int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}}{d + e x + f x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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