Integrand size = 15, antiderivative size = 79 \[ \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=-2 n x+\frac {\sqrt {b^2-4 a c} n \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+\frac {b n \log \left (a+b x+c x^2\right )}{2 c}+x \log \left (d \left (a+b x+c x^2\right )^n\right ) \]
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Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2603, 787, 648, 632, 212, 642} \[ \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\frac {n \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n \log \left (a+b x+c x^2\right )}{2 c}-2 n x \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 787
Rule 2603
Rubi steps \begin{align*} \text {integral}& = x \log \left (d \left (a+b x+c x^2\right )^n\right )-n \int \frac {x (b+2 c x)}{a+b x+c x^2} \, dx \\ & = -2 n x+x \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {n \int \frac {-2 a c-b c x}{a+b x+c x^2} \, dx}{c} \\ & = -2 n x+x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {(b n) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c}-\frac {\left (\left (b^2-4 a c\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c} \\ & = -2 n x+\frac {b n \log \left (a+b x+c x^2\right )}{2 c}+x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (\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 )}{c} \\ & = -2 n x+\frac {\sqrt {b^2-4 a c} n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+\frac {b n \log \left (a+b x+c x^2\right )}{2 c}+x \log \left (d \left (a+b x+c x^2\right )^n\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99 \[ \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\frac {2 \sqrt {b^2-4 a c} n \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+b n \log (a+x (b+c x))+2 c x \left (-2 n+\log \left (d (a+x (b+c x))^n\right )\right )}{2 c} \]
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Time = 0.47 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.13
method | result | size |
default | \(x \ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )-n \left (2 x -\frac {b \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-2 a +\frac {b^{2}}{2 c}\right ) \arctan \left (\frac {2 x c +b}{\sqrt {4 c a -b^{2}}}\right )}{\sqrt {4 c a -b^{2}}}\right )\) | \(89\) |
parts | \(x \ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )-n \left (2 x -\frac {b \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-2 a +\frac {b^{2}}{2 c}\right ) \arctan \left (\frac {2 x c +b}{\sqrt {4 c a -b^{2}}}\right )}{\sqrt {4 c a -b^{2}}}\right )\) | \(89\) |
risch | \(x \ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right )+\frac {i {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{2} \operatorname {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) x \pi }{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right ) \operatorname {csgn}\left (i d \right )}{2}-\frac {i \pi x {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{3}}{2}+\frac {i \operatorname {csgn}\left (i d \right ) {\operatorname {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{2} x \pi }{2}+\frac {n \ln \left (-2 x c \sqrt {-4 c a +b^{2}}-b \sqrt {-4 c a +b^{2}}+4 c a -b^{2}\right ) \sqrt {-4 c a +b^{2}}}{2 c}+\frac {n \ln \left (-2 x c \sqrt {-4 c a +b^{2}}-b \sqrt {-4 c a +b^{2}}+4 c a -b^{2}\right ) b}{2 c}-\frac {n \ln \left (2 x c \sqrt {-4 c a +b^{2}}+b \sqrt {-4 c a +b^{2}}+4 c a -b^{2}\right ) \sqrt {-4 c a +b^{2}}}{2 c}+\frac {n \ln \left (2 x c \sqrt {-4 c a +b^{2}}+b \sqrt {-4 c a +b^{2}}+4 c a -b^{2}\right ) b}{2 c}+\ln \left (d \right ) x -2 n x\) | \(357\) |
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Time = 0.33 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.41 \[ \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\left [-\frac {4 \, c n x - 2 \, c x \log \left (d\right ) - \sqrt {b^{2} - 4 \, a c} n \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (2 \, c n x + b n\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c}, -\frac {4 \, c n x - 2 \, c x \log \left (d\right ) - 2 \, \sqrt {-b^{2} + 4 \, a c} n \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left (2 \, c n x + b n\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (75) = 150\).
Time = 34.76 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.47 \[ \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\begin {cases} \frac {a \log {\left (d \left (a + b x\right )^{n} \right )}}{b} - n x + x \log {\left (d \left (a + b x\right )^{n} \right )} & \text {for}\: c = 0 \\\frac {b \log {\left (d \left (\frac {b^{2}}{4 c} + b x + c x^{2}\right )^{n} \right )}}{2 c} - 2 n x + x \log {\left (d \left (\frac {b^{2}}{4 c} + b x + c x^{2}\right )^{n} \right )} & \text {for}\: a = \frac {b^{2}}{4 c} \\- \frac {4 a n \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{\sqrt {- 4 a c + b^{2}}} + \frac {2 a \log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{\sqrt {- 4 a c + b^{2}}} + \frac {b^{2} n \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{c \sqrt {- 4 a c + b^{2}}} - \frac {b^{2} \log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{2 c \sqrt {- 4 a c + b^{2}}} + \frac {b \log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )}}{2 c} - 2 n x + x \log {\left (d \left (a + b x + c x^{2}\right )^{n} \right )} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=\text {Exception raised: ValueError} \]
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Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.16 \[ \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=n x \log \left (c x^{2} + b x + a\right ) - {\left (2 \, n - \log \left (d\right )\right )} x + \frac {b n \log \left (c x^{2} + b x + a\right )}{2 \, c} - \frac {{\left (b^{2} n - 4 \, a c n\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} \]
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Time = 1.50 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.52 \[ \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx=x\,\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )-2\,n\,x-\frac {n\,\mathrm {atan}\left (\frac {b\,n\,\sqrt {4\,a\,c-b^2}}{2\,\left (\frac {b^2\,n}{2}-2\,a\,c\,n\right )}-\frac {n\,x\,\sqrt {4\,a\,c-b^2}}{2\,a\,n-\frac {b^2\,n}{2\,c}}\right )\,\sqrt {4\,a\,c-b^2}}{c}+\frac {b\,n\,\ln \left (c\,x^2+b\,x+a\right )}{2\,c} \]
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