Optimal. Leaf size=78 \[ -\frac{1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{4 c \log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )^2}+\frac{8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \]
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Rubi [A] time = 0.0382858, antiderivative size = 78, 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 = {687, 681, 31, 628} \[ -\frac{1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{4 c \log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )^2}+\frac{8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
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Rule 687
Rule 681
Rule 31
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )^2} \, dx &=-\frac{1}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}-\frac{(4 c) \int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c}\\ &=-\frac{1}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}-\frac{(4 c) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^2}+\frac{\left (16 c^2\right ) \int \frac{1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^2 d}\\ &=-\frac{1}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}+\frac{8 c \log (b+2 c x)}{\left (b^2-4 a c\right )^2 d}-\frac{4 c \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^2 d}\\ \end{align*}
Mathematica [A] time = 0.0556991, size = 59, normalized size = 0.76 \[ \frac{-\frac{b^2-4 a c}{a+x (b+c x)}-4 c \log (a+x (b+c x))+8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 119, normalized size = 1.5 \begin{align*} 4\,{\frac{ac}{d \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{{b}^{2}}{d \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-4\,{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) }{d \left ( 4\,ac-{b}^{2} \right ) ^{2}}}+8\,{\frac{c\ln \left ( 2\,cx+b \right ) }{d \left ( 4\,ac-{b}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17233, size = 163, normalized size = 2.09 \begin{align*} -\frac{4 \, c \log \left (c x^{2} + b x + a\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d} + \frac{8 \, c \log \left (2 \, c x + b\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d} - \frac{1}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94166, size = 309, normalized size = 3.96 \begin{align*} -\frac{b^{2} - 4 \, a c + 4 \,{\left (c^{2} x^{2} + b c x + a c\right )} \log \left (c x^{2} + b x + a\right ) - 8 \,{\left (c^{2} x^{2} + b c x + a c\right )} \log \left (2 \, c x + b\right )}{{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d x +{\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.2666, size = 102, normalized size = 1.31 \begin{align*} \frac{8 c \log{\left (\frac{b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )^{2}} - \frac{4 c \log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )^{2}} + \frac{1}{4 a^{2} c d - a b^{2} d + x^{2} \left (4 a c^{2} d - b^{2} c d\right ) + x \left (4 a b c d - b^{3} d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15679, size = 146, normalized size = 1.87 \begin{align*} \frac{8 \, c^{2} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{4} c d - 8 \, a b^{2} c^{2} d + 16 \, a^{2} c^{3} d} - \frac{4 \, c \log \left (c x^{2} + b x + a\right )}{b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d} - \frac{1}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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