Optimal. Leaf size=37 \[ -\frac{d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.0145966, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {682} \[ -\frac{d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0243607, size = 38, normalized size = 1.03 \[ -\frac{d^3 \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )}{2 (a+x (b+c x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 40, normalized size = 1.1 \begin{align*}{\frac{{d}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ( -4\,{c}^{2}{x}^{2}-4\,bcx-2\,ac-{\frac{{b}^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13308, size = 96, normalized size = 2.59 \begin{align*} -\frac{8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x +{\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03639, size = 154, normalized size = 4.16 \begin{align*} -\frac{8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x +{\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.43457, size = 80, normalized size = 2.16 \begin{align*} - \frac{4 a c d^{3} + b^{2} d^{3} + 8 b c d^{3} x + 8 c^{2} d^{3} x^{2}}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18235, size = 65, normalized size = 1.76 \begin{align*} -\frac{8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x + b^{2} d^{3} + 4 \, a c d^{3}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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