Optimal. Leaf size=79 \[ -\frac{\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3}+\frac{b x}{16 c^2 d^3}+\frac{x^2}{16 c d^3} \]
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Rubi [A] time = 0.0635056, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ -\frac{\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3}+\frac{b x}{16 c^2 d^3}+\frac{x^2}{16 c d^3} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx &=\int \left (\frac{b}{16 c^2 d^3}+\frac{x}{8 c d^3}+\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^3 (b+2 c x)^3}+\frac{-b^2+4 a c}{8 c^2 d^3 (b+2 c x)}\right ) \, dx\\ &=\frac{b x}{16 c^2 d^3}+\frac{x^2}{16 c d^3}-\frac{\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0406992, size = 61, normalized size = 0.77 \[ \frac{-\frac{\left (b^2-4 a c\right )^2}{(b+2 c x)^2}-4 \left (b^2-4 a c\right ) \log (b+2 c x)+4 b c x+4 c^2 x^2}{64 c^3 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 115, normalized size = 1.5 \begin{align*}{\frac{{x}^{2}}{16\,c{d}^{3}}}+{\frac{bx}{16\,{c}^{2}{d}^{3}}}-{\frac{{a}^{2}}{4\,c{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{{b}^{2}a}{8\,{c}^{2}{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{{b}^{4}}{64\,{c}^{3}{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{\ln \left ( 2\,cx+b \right ) a}{4\,{c}^{2}{d}^{3}}}-{\frac{\ln \left ( 2\,cx+b \right ){b}^{2}}{16\,{c}^{3}{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13658, size = 130, normalized size = 1.65 \begin{align*} -\frac{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{64 \,{\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}} + \frac{c x^{2} + b x}{16 \, c^{2} d^{3}} - \frac{{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{16 \, c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02038, size = 312, normalized size = 3.95 \begin{align*} \frac{16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 20 \, b^{2} c^{2} x^{2} + 4 \, b^{3} c x - b^{4} + 8 \, a b^{2} c - 16 \, a^{2} c^{2} - 4 \,{\left (b^{4} - 4 \, a b^{2} c + 4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{64 \,{\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.08025, size = 102, normalized size = 1.29 \begin{align*} \frac{b x}{16 c^{2} d^{3}} - \frac{16 a^{2} c^{2} - 8 a b^{2} c + b^{4}}{64 b^{2} c^{3} d^{3} + 256 b c^{4} d^{3} x + 256 c^{5} d^{3} x^{2}} + \frac{x^{2}}{16 c d^{3}} + \frac{\left (4 a c - b^{2}\right ) \log{\left (b + 2 c x \right )}}{16 c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13045, size = 119, normalized size = 1.51 \begin{align*} -\frac{{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{16 \, c^{3} d^{3}} - \frac{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{64 \,{\left (2 \, c x + b\right )}^{2} c^{3} d^{3}} + \frac{c^{5} d^{3} x^{2} + b c^{4} d^{3} x}{16 \, c^{6} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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