Optimal. Leaf size=48 \[ 16 c d^3 \sqrt{a+b x+c x^2}-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.0229013, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {686, 629} \[ 16 c d^3 \sqrt{a+b x+c x^2}-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 686
Rule 629
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}}+\left (8 c d^2\right ) \int \frac{b d+2 c d x}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}}+16 c d^3 \sqrt{a+b x+c x^2}\\ \end{align*}
Mathematica [A] time = 0.0287474, size = 40, normalized size = 0.83 \[ \frac{d^3 \left (8 c \left (2 a+c x^2\right )-2 b^2+8 b c x\right )}{\sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 41, normalized size = 0.9 \begin{align*} 2\,{\frac{{d}^{3} \left ( 4\,{c}^{2}{x}^{2}+4\,bcx+8\,ac-{b}^{2} \right ) }{\sqrt{c{x}^{2}+bx+a}}} \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 [A] time = 3.55734, size = 101, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x -{\left (b^{2} - 8 \, a c\right )} d^{3}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.36838, size = 92, normalized size = 1.92 \begin{align*} \frac{16 a c d^{3}}{\sqrt{a + b x + c x^{2}}} - \frac{2 b^{2} d^{3}}{\sqrt{a + b x + c x^{2}}} + \frac{8 b c d^{3} x}{\sqrt{a + b x + c x^{2}}} + \frac{8 c^{2} d^{3} x^{2}}{\sqrt{a + b x + c x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17413, size = 188, normalized size = 3.92 \begin{align*} \frac{2 \,{\left (4 \,{\left (\frac{{\left (b^{2} c^{3} d^{3} - 4 \, a c^{4} d^{3}\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{b^{3} c^{2} d^{3} - 4 \, a b c^{3} d^{3}}{b^{2} c - 4 \, a c^{2}}\right )} x - \frac{b^{4} c d^{3} - 12 \, a b^{2} c^{2} d^{3} + 32 \, a^{2} c^{3} d^{3}}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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