Optimal. Leaf size=39 \[ -\frac{2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0142451, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {682} \[ -\frac{2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0205791, size = 38, normalized size = 0.97 \[ -\frac{2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 38, normalized size = 1. \begin{align*}{\frac{2\,{d}^{2} \left ( 2\,cx+b \right ) ^{3}}{12\,ac-3\,{b}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 5.55427, size = 304, normalized size = 7.79 \begin{align*} -\frac{2 \,{\left (8 \, c^{3} d^{2} x^{3} + 12 \, b c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x + b^{3} d^{2}\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19467, size = 296, normalized size = 7.59 \begin{align*} -\frac{2 \,{\left (2 \,{\left (\frac{2 \,{\left (b^{2} c^{3} d^{2} - 4 \, a c^{4} d^{2}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (b^{3} c^{2} d^{2} - 4 \, a b c^{3} d^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (b^{4} c d^{2} - 4 \, a b^{2} c^{2} d^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{5} d^{2} - 4 \, a b^{3} c d^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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