Optimal. Leaf size=19 \[ \frac{2}{5} d \left (a+b x+c x^2\right )^{5/2} \]
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Rubi [A] time = 0.0066193, antiderivative size = 19, 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 = {629} \[ \frac{2}{5} d \left (a+b x+c x^2\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (b d+2 c d x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2}{5} d \left (a+b x+c x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0144772, size = 18, normalized size = 0.95 \[ \frac{2}{5} d (a+x (b+c x))^{5/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 16, normalized size = 0.8 \begin{align*}{\frac{2\,d}{5} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12318, size = 20, normalized size = 1.05 \begin{align*} \frac{2}{5} \,{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.63561, size = 128, normalized size = 6.74 \begin{align*} \frac{2}{5} \,{\left (c^{2} d x^{4} + 2 \, b c d x^{3} + 2 \, a b d x +{\left (b^{2} + 2 \, a c\right )} d x^{2} + a^{2} d\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 [B] time = 0.939766, size = 146, normalized size = 7.68 \begin{align*} \frac{2 a^{2} d \sqrt{a + b x + c x^{2}}}{5} + \frac{4 a b d x \sqrt{a + b x + c x^{2}}}{5} + \frac{4 a c d x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{2 b^{2} d x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{4 b c d x^{3} \sqrt{a + b x + c x^{2}}}{5} + \frac{2 c^{2} d x^{4} \sqrt{a + b x + c x^{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22153, size = 88, normalized size = 4.63 \begin{align*} \frac{2}{5} \,{\left (a^{2} d +{\left (2 \, a b d +{\left ({\left (c^{2} d x + 2 \, b c d\right )} x + \frac{b^{2} c^{4} d + 2 \, a c^{5} d}{c^{4}}\right )} x\right )} x\right )} \sqrt{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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