Optimal. Leaf size=37 \[ \frac{\left (a+b x+c x^2\right )^2}{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^4} \]
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Rubi [A] time = 0.010164, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {682} \[ \frac{\left (a+b x+c x^2\right )^2}{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^4} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{(b d+2 c d x)^5} \, dx &=\frac{\left (a+b x+c x^2\right )^2}{2 \left (b^2-4 a c\right ) d^5 (b+2 c x)^4}\\ \end{align*}
Mathematica [A] time = 0.0148447, size = 38, normalized size = 1.03 \[ -\frac{4 c \left (a+2 c x^2\right )+b^2+8 b c x}{32 c^2 d^5 (b+2 c x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 42, normalized size = 1.1 \begin{align*}{\frac{1}{{d}^{5}} \left ( -{\frac{1}{16\,{c}^{2} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{4\,ac-{b}^{2}}{32\,{c}^{2} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22513, size = 115, normalized size = 3.11 \begin{align*} -\frac{8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{32 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.4989, size = 178, normalized size = 4.81 \begin{align*} -\frac{8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{32 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.869247, size = 90, normalized size = 2.43 \begin{align*} - \frac{4 a c + b^{2} + 8 b c x + 8 c^{2} x^{2}}{32 b^{4} c^{2} d^{5} + 256 b^{3} c^{3} d^{5} x + 768 b^{2} c^{4} d^{5} x^{2} + 1024 b c^{5} d^{5} x^{3} + 512 c^{6} d^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18428, size = 81, normalized size = 2.19 \begin{align*} \frac{\frac{b^{2}}{{\left (2 \, c d x + b d\right )}^{4} c^{2}} - \frac{4 \, a}{{\left (2 \, c d x + b d\right )}^{4} c} - \frac{2}{{\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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