Optimal. Leaf size=70 \[ \frac{\log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^2}+\frac{1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^2} \]
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Rubi [A] time = 0.0403733, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {693, 681, 31, 628} \[ \frac{\log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^2}+\frac{1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
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Rule 693
Rule 681
Rule 31
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx &=\frac{1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac{\int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) d^2}\\ &=\frac{1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac{\int \frac{b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^4}-\frac{(4 c) \int \frac{1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^2 d^3}\\ &=\frac{1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{\left (b^2-4 a c\right )^2 d^3}+\frac{\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^2 d^3}\\ \end{align*}
Mathematica [A] time = 0.0391309, size = 65, normalized size = 0.93 \[ \frac{\frac{\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^2}+\frac{1}{\left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{\left (b^2-4 a c\right )^2}}{d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 78, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}}-2\,{\frac{\ln \left ( 2\,cx+b \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}}-{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20086, size = 174, normalized size = 2.49 \begin{align*} \frac{1}{4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{3} x +{\left (b^{4} - 4 \, a b^{2} c\right )} d^{3}} + \frac{\log \left (c x^{2} + b x + a\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3}} - \frac{2 \, \log \left (2 \, c x + b\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99394, size = 338, normalized size = 4.83 \begin{align*} \frac{b^{2} - 4 \, a c +{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (2 \, c x + b\right )}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{3} x^{2} + 4 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{3} x +{\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34491, size = 119, normalized size = 1.7 \begin{align*} - \frac{1}{4 a b^{2} c d^{3} - b^{4} d^{3} + x^{2} \left (16 a c^{3} d^{3} - 4 b^{2} c^{2} d^{3}\right ) + x \left (16 a b c^{2} d^{3} - 4 b^{3} c d^{3}\right )} - \frac{2 \log{\left (\frac{b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{2}} + \frac{\log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16075, size = 150, normalized size = 2.14 \begin{align*} -\frac{2 \, c \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{4} c d^{3} - 8 \, a b^{2} c^{2} d^{3} + 16 \, a^{2} c^{3} d^{3}} + \frac{\log \left (c x^{2} + b x + a\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} + \frac{1}{{\left (b^{2} - 4 \, a c\right )}{\left (2 \, c x + b\right )}^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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