Optimal. Leaf size=79 \[ \frac {n \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n \log \left (a+b x+c x^2\right )}{2 c}-2 n x \]
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Rubi [A] time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2523, 773, 634, 618, 206, 628} \[ \frac {n \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n \log \left (a+b x+c x^2\right )}{2 c}-2 n x \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 773
Rule 2523
Rubi steps
\begin {align*} \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=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 ) \operatorname {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*}
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Mathematica [A] time = 0.06, size = 78, normalized size = 0.99 \[ \frac {2 n \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+2 c x \left (\log \left (d (a+x (b+c x))^n\right )-2 n\right )+b n \log (a+x (b+c x))}{2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 190, normalized size = 2.41 \[ \left [-\frac {4 \, c n x - 2 \, c x \log \relax (d) - \sqrt {b^{2} - 4 \, a c} n \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (2 \, c n x + b n\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c}, -\frac {4 \, c n x - 2 \, c x \log \relax (d) - 2 \, \sqrt {-b^{2} + 4 \, a c} n \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left (2 \, c n x + b n\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 92, normalized size = 1.16 \[ n x \log \left (c x^{2} + b x + a\right ) - {\left (2 \, n - \log \relax (d)\right )} x + \frac {b n \log \left (c x^{2} + b x + a\right )}{2 \, c} - \frac {{\left (b^{2} n - 4 \, a c n\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 118, normalized size = 1.49 \[ \frac {4 a n \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {b^{2} n \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {b n \ln \left (c \,x^{2}+b x +a \right )}{2 c}-2 n x +x \ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right ) \]
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 [B] time = 0.00, size = 120, normalized size = 1.52 \[ x\,\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )-2\,n\,x-\frac {n\,\mathrm {atan}\left (\frac {b\,n\,\sqrt {4\,a\,c-b^2}}{2\,\left (\frac {b^2\,n}{2}-2\,a\,c\,n\right )}-\frac {n\,x\,\sqrt {4\,a\,c-b^2}}{2\,a\,n-\frac {b^2\,n}{2\,c}}\right )\,\sqrt {4\,a\,c-b^2}}{c}+\frac {b\,n\,\ln \left (c\,x^2+b\,x+a\right )}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 65.13, size = 275, normalized size = 3.48 \[ \begin {cases} \frac {b n \log {\left (\frac {b^{2}}{4 c} + b x + c x^{2} \right )}}{2 c} + n x \log {\left (\frac {b^{2}}{4 c} + b x + c x^{2} \right )} - 2 n x + x \log {\relax (d )} & \text {for}\: a = \frac {b^{2}}{4 c} \\\frac {a n \log {\left (a + b x \right )}}{b} + n x \log {\left (a + b x \right )} - n x + x \log {\relax (d )} & \text {for}\: c = 0 \\\frac {2 a n \log {\left (a + b x + c x^{2} \right )}}{\sqrt {- 4 a c + b^{2}}} - \frac {4 a n \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{\sqrt {- 4 a c + b^{2}}} - \frac {b^{2} n \log {\left (a + b x + c x^{2} \right )}}{2 c \sqrt {- 4 a c + b^{2}}} + \frac {b^{2} n \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{c \sqrt {- 4 a c + b^{2}}} + \frac {b n \log {\left (a + b x + c x^{2} \right )}}{2 c} + n x \log {\left (a + b x + c x^{2} \right )} - 2 n x + x \log {\relax (d )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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