Optimal. Leaf size=66 \[ -\frac{\left (b^2-4 a c\right )^2}{96 c^3 d^4 (b+2 c x)^3}+\frac{b^2-4 a c}{16 c^3 d^4 (b+2 c x)}+\frac{x}{16 c^2 d^4} \]
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Rubi [A] time = 0.0541672, antiderivative size = 66, 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} \[ -\frac{\left (b^2-4 a c\right )^2}{96 c^3 d^4 (b+2 c x)^3}+\frac{b^2-4 a c}{16 c^3 d^4 (b+2 c x)}+\frac{x}{16 c^2 d^4} \]
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)^4} \, dx &=\int \left (\frac{1}{16 c^2 d^4}+\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^4 (b+2 c x)^4}+\frac{-b^2+4 a c}{8 c^2 d^4 (b+2 c x)^2}\right ) \, dx\\ &=\frac{x}{16 c^2 d^4}-\frac{\left (b^2-4 a c\right )^2}{96 c^3 d^4 (b+2 c x)^3}+\frac{b^2-4 a c}{16 c^3 d^4 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.0567088, size = 53, normalized size = 0.8 \[ \frac{-\frac{\left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac{6 \left (b^2-4 a c\right )}{b+2 c x}+6 c x}{96 c^3 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 67, normalized size = 1. \begin{align*}{\frac{1}{{d}^{4}} \left ({\frac{x}{16\,{c}^{2}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{96\,{c}^{3} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{4\,ac-{b}^{2}}{16\,{c}^{3} \left ( 2\,cx+b \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23916, size = 157, normalized size = 2.38 \begin{align*} \frac{5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2} + 24 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 24 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x}{96 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}} + \frac{x}{16 \, c^{2} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02515, size = 263, normalized size = 3.98 \begin{align*} \frac{48 \, c^{4} x^{4} + 72 \, b c^{3} x^{3} + 5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2} + 12 \,{\left (5 \, b^{2} c^{2} - 8 \, a c^{3}\right )} x^{2} + 6 \,{\left (5 \, b^{3} c - 16 \, a b c^{2}\right )} x}{96 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.35265, size = 117, normalized size = 1.77 \begin{align*} - \frac{16 a^{2} c^{2} + 16 a b^{2} c - 5 b^{4} + x^{2} \left (96 a c^{3} - 24 b^{2} c^{2}\right ) + x \left (96 a b c^{2} - 24 b^{3} c\right )}{96 b^{3} c^{3} d^{4} + 576 b^{2} c^{4} d^{4} x + 1152 b c^{5} d^{4} x^{2} + 768 c^{6} d^{4} x^{3}} + \frac{x}{16 c^{2} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1374, size = 111, normalized size = 1.68 \begin{align*} \frac{x}{16 \, c^{2} d^{4}} + \frac{24 \, b^{2} c^{2} x^{2} - 96 \, a c^{3} x^{2} + 24 \, b^{3} c x - 96 \, a b c^{2} x + 5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2}}{96 \,{\left (2 \, c x + b\right )}^{3} c^{3} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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