Integrand size = 19, antiderivative size = 121 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^3} \, dx=-\frac {b n}{2 a x}-\frac {b \sqrt {b^2-4 a c} n \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 a^2}-\frac {\left (b^2-2 a c\right ) n \log (x)}{2 a^2}+\frac {\left (b^2-2 a c\right ) n \log \left (a+b x+c x^2\right )}{4 a^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2} \]
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Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2605, 814, 648, 632, 212, 642} \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^3} \, dx=-\frac {b n \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 a^2}+\frac {n \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{4 a^2}-\frac {n \log (x) \left (b^2-2 a c\right )}{2 a^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2}-\frac {b n}{2 a x} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 814
Rule 2605
Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2}+\frac {1}{2} n \int \frac {b+2 c x}{x^2 \left (a+b x+c x^2\right )} \, dx \\ & = -\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2}+\frac {1}{2} n \int \left (\frac {b}{a x^2}+\frac {-b^2+2 a c}{a^2 x}+\frac {b \left (b^2-3 a c\right )+c \left (b^2-2 a c\right ) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx \\ & = -\frac {b n}{2 a x}-\frac {\left (b^2-2 a c\right ) n \log (x)}{2 a^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2}+\frac {n \int \frac {b \left (b^2-3 a c\right )+c \left (b^2-2 a c\right ) x}{a+b x+c x^2} \, dx}{2 a^2} \\ & = -\frac {b n}{2 a x}-\frac {\left (b^2-2 a c\right ) n \log (x)}{2 a^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2}+\frac {\left (b \left (b^2-4 a c\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{4 a^2}+\frac {\left (\left (b^2-2 a c\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{4 a^2} \\ & = -\frac {b n}{2 a x}-\frac {\left (b^2-2 a c\right ) n \log (x)}{2 a^2}+\frac {\left (b^2-2 a c\right ) n \log \left (a+b x+c x^2\right )}{4 a^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2}-\frac {\left (b \left (b^2-4 a c\right ) n\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{2 a^2} \\ & = -\frac {b n}{2 a x}-\frac {b \sqrt {b^2-4 a c} n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 a^2}-\frac {\left (b^2-2 a c\right ) n \log (x)}{2 a^2}+\frac {\left (b^2-2 a c\right ) n \log \left (a+b x+c x^2\right )}{4 a^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^3} \, dx=-\frac {\frac {n x \left (2 a b+2 b \sqrt {b^2-4 a c} x \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+2 \left (b^2-2 a c\right ) x \log (x)-\left (b^2-2 a c\right ) x \log (a+x (b+c x))\right )}{a^2}+2 \log \left (d (a+x (b+c x))^n\right )}{4 x^2} \]
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Time = 0.57 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.20
method | result | size |
parts | \(-\frac {\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )}{2 x^{2}}+\frac {n \left (-\frac {b}{a x}+\frac {\left (2 c a -b^{2}\right ) \ln \left (x \right )}{a^{2}}+\frac {\frac {\left (-2 a \,c^{2}+b^{2} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-3 a b c +b^{3}-\frac {\left (-2 a \,c^{2}+b^{2} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 x c +b}{\sqrt {4 c a -b^{2}}}\right )}{\sqrt {4 c a -b^{2}}}}{a^{2}}\right )}{2}\) | \(145\) |
risch | \(\text {Expression too large to display}\) | \(1178\) |
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Time = 0.36 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.16 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^3} \, dx=\left [\frac {\sqrt {b^{2} - 4 \, a c} b n x^{2} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 2 \, {\left (b^{2} - 2 \, a c\right )} n x^{2} \log \left (x\right ) - 2 \, a b n x - 2 \, a^{2} \log \left (d\right ) + {\left ({\left (b^{2} - 2 \, a c\right )} n x^{2} - 2 \, a^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{4 \, a^{2} x^{2}}, -\frac {2 \, \sqrt {-b^{2} + 4 \, a c} b n x^{2} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (b^{2} - 2 \, a c\right )} n x^{2} \log \left (x\right ) + 2 \, a b n x + 2 \, a^{2} \log \left (d\right ) - {\left ({\left (b^{2} - 2 \, a c\right )} n x^{2} - 2 \, a^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{4 \, a^{2} x^{2}}\right ] \]
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Timed out. \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^3} \, dx=\frac {{\left (b^{2} n - 2 \, a c n\right )} \log \left (c x^{2} + b x + a\right )}{4 \, a^{2}} - \frac {n \log \left (c x^{2} + b x + a\right )}{2 \, x^{2}} - \frac {{\left (b^{2} n - 2 \, a c n\right )} \log \left (x\right )}{2 \, a^{2}} + \frac {{\left (b^{3} n - 4 \, a b c n\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{2}} - \frac {b n x + a \log \left (d\right )}{2 \, a x^{2}} \]
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Time = 1.89 (sec) , antiderivative size = 474, normalized size of antiderivative = 3.92 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^3} \, dx=\frac {\ln \left (\frac {b^3\,c^2\,n^2-2\,a\,b\,c^3\,n^2}{4\,a^2}+\frac {\left (b^2\,n-2\,a\,c\,n+b\,n\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {\left (\frac {x\,\left (24\,a^3\,c^2-8\,a^2\,b^2\,c\right )}{4\,a^2}-a\,b\,c\right )\,\left (b^2\,n-2\,a\,c\,n+b\,n\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^2}-\frac {2\,a\,b^3\,c\,n-6\,a^2\,b\,c^2\,n}{4\,a^2}+\frac {x\,\left (12\,a^2\,c^3\,n-4\,a\,b^2\,c^2\,n\right )}{4\,a^2}\right )}{4\,a^2}+\frac {b^2\,c^3\,n^2\,x}{4\,a^2}\right )\,\left (b^2\,n-2\,a\,c\,n+b\,n\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^2}-\frac {\ln \left (x\right )\,\left (b^2\,n-2\,a\,c\,n\right )}{2\,a^2}-\frac {\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{2\,x^2}-\frac {\ln \left (\frac {b^3\,c^2\,n^2-2\,a\,b\,c^3\,n^2}{4\,a^2}+\frac {\left (2\,a\,c\,n-b^2\,n+b\,n\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {2\,a\,b^3\,c\,n-6\,a^2\,b\,c^2\,n}{4\,a^2}+\frac {\left (\frac {x\,\left (24\,a^3\,c^2-8\,a^2\,b^2\,c\right )}{4\,a^2}-a\,b\,c\right )\,\left (2\,a\,c\,n-b^2\,n+b\,n\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^2}-\frac {x\,\left (12\,a^2\,c^3\,n-4\,a\,b^2\,c^2\,n\right )}{4\,a^2}\right )}{4\,a^2}+\frac {b^2\,c^3\,n^2\,x}{4\,a^2}\right )\,\left (2\,a\,c\,n-b^2\,n+b\,n\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^2}-\frac {b\,n}{2\,a\,x} \]
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