Optimal. Leaf size=18 \[ \frac{2}{5} \left (a+b x+c x^2\right )^{5/2} \]
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Rubi [A] time = 0.0061973, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {629} \[ \frac{2}{5} \left (a+b x+c x^2\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2}{5} \left (a+b x+c x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0145059, size = 17, normalized size = 0.94 \[ \frac{2}{5} (a+x (b+c x))^{5/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{2}{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 = 0.999222, size = 19, normalized size = 1.06 \begin{align*} \frac{2}{5} \,{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94998, size = 115, normalized size = 6.39 \begin{align*} \frac{2}{5} \,{\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 )} \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 = 0.548518, size = 136, normalized size = 7.56 \begin{align*} \frac{2 a^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{4 a b x \sqrt{a + b x + c x^{2}}}{5} + \frac{4 a c x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{2 b^{2} x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{4 b c x^{3} \sqrt{a + b x + c x^{2}}}{5} + \frac{2 c^{2} 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 = 1.17879, size = 78, normalized size = 4.33 \begin{align*} \frac{2}{5} \, \sqrt{c x^{2} + b x + a}{\left (a^{2} +{\left (2 \, a b +{\left ({\left (c^{2} x + 2 \, b c\right )} x + \frac{b^{2} c^{4} + 2 \, a c^{5}}{c^{4}}\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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