Optimal. Leaf size=39 \[ \frac{2 \left (a+b x+c x^2\right )^{3/2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
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Rubi [A] time = 0.0140489, 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} \[ \frac{2 \left (a+b x+c x^2\right )^{3/2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^4} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{3/2}}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}\\ \end{align*}
Mathematica [A] time = 0.0184316, size = 38, normalized size = 0.97 \[ \frac{2 (a+x (b+c x))^{3/2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 38, normalized size = 1. \begin{align*} -{\frac{2}{3\, \left ( 2\,cx+b \right ) ^{3}{d}^{4} \left ( 4\,ac-{b}^{2} \right ) } \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 = 5.27809, size = 205, normalized size = 5.26 \begin{align*} \frac{2 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{3 \,{\left (8 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{3} + 12 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + 6 \,{\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{4} x +{\left (b^{5} - 4 \, a b^{3} c\right )} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28705, size = 277, normalized size = 7.1 \begin{align*} \frac{12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} c^{\frac{5}{2}} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} b c^{2} + 18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} b^{2} c^{\frac{3}{2}} + 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b^{3} c + b^{4} \sqrt{c} - 2 \, a b^{2} c^{\frac{3}{2}} + 4 \, a^{2} c^{\frac{5}{2}}}{12 \,{\left (2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b \sqrt{c} + b^{2} - 2 \, a c\right )}^{3} c^{2} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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