Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{24 c^2 d^4 (b+2 c x)^3}-\frac{1}{8 c^2 d^4 (b+2 c x)} \]
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Rubi [A] time = 0.0317319, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {683} \[ \frac{b^2-4 a c}{24 c^2 d^4 (b+2 c x)^3}-\frac{1}{8 c^2 d^4 (b+2 c x)} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{(b d+2 c d x)^4} \, dx &=\int \left (\frac{-b^2+4 a c}{4 c d^4 (b+2 c x)^4}+\frac{1}{4 c d^4 (b+2 c x)^2}\right ) \, dx\\ &=\frac{b^2-4 a c}{24 c^2 d^4 (b+2 c x)^3}-\frac{1}{8 c^2 d^4 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.0156746, size = 38, normalized size = 0.84 \[ -\frac{2 c \left (a+3 c x^2\right )+b^2+6 b c x}{12 c^2 d^4 (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 42, normalized size = 0.9 \begin{align*}{\frac{1}{{d}^{4}} \left ( -{\frac{4\,ac-{b}^{2}}{24\,{c}^{2} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{1}{8\,{c}^{2} \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 = 1.16272, size = 96, normalized size = 2.13 \begin{align*} -\frac{6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c}{12 \,{\left (8 \, c^{5} d^{4} x^{3} + 12 \, b c^{4} d^{4} x^{2} + 6 \, b^{2} c^{3} d^{4} x + b^{3} c^{2} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35558, size = 149, normalized size = 3.31 \begin{align*} -\frac{6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c}{12 \,{\left (8 \, c^{5} d^{4} x^{3} + 12 \, b c^{4} d^{4} x^{2} + 6 \, b^{2} c^{3} d^{4} x + b^{3} c^{2} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.651829, size = 75, normalized size = 1.67 \begin{align*} - \frac{2 a c + b^{2} + 6 b c x + 6 c^{2} x^{2}}{12 b^{3} c^{2} d^{4} + 72 b^{2} c^{3} d^{4} x + 144 b c^{4} d^{4} x^{2} + 96 c^{5} d^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25477, size = 50, normalized size = 1.11 \begin{align*} -\frac{6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c}{12 \,{\left (2 \, c x + b\right )}^{3} c^{2} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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