Optimal. Leaf size=52 \[ -\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0233337, antiderivative size = 52, 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} \[ -\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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 )^{5/2}} \, dx &=-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac{1}{3} \left (8 c d^2\right ) \int \frac{b d+2 c d x}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0292933, size = 42, normalized size = 0.81 \[ -\frac{2 d^3 \left (4 c \left (2 a+3 c x^2\right )+b^2+12 b c x\right )}{3 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 39, normalized size = 0.8 \begin{align*} -{\frac{2\,{d}^{3} \left ( 12\,{c}^{2}{x}^{2}+12\,bcx+8\,ac+{b}^{2} \right ) }{3} \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 [A] time = 5.68305, size = 186, normalized size = 3.58 \begin{align*} -\frac{2 \,{\left (12 \, c^{2} d^{3} x^{2} + 12 \, b c d^{3} x +{\left (b^{2} + 8 \, a c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\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 = 1.99567, size = 264, normalized size = 5.08 \begin{align*} - \frac{16 a c d^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 b^{2} d^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{24 b c d^{3} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{24 c^{2} d^{3} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c 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 = 1.14325, size = 275, normalized size = 5.29 \begin{align*} -\frac{12 \,{\left (\frac{{\left (b^{4} c^{2} d^{3} - 8 \, a b^{2} c^{3} d^{3} + 16 \, a^{2} c^{4} d^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{b^{5} c d^{3} - 8 \, a b^{3} c^{2} d^{3} + 16 \, a^{2} b c^{3} d^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{6} d^{3} - 48 \, a^{2} b^{2} c^{2} d^{3} + 128 \, a^{3} c^{3} d^{3}}{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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