Optimal. Leaf size=18 \[ \frac{2}{7} \left (a+b x+c x^2\right )^{7/2} \]
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Rubi [A] time = 0.0061167, 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} \[ \frac{2}{7} \left (a+b x+c x^2\right )^{7/2} \]
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 )^{5/2} \, dx &=\frac{2}{7} \left (a+b x+c x^2\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0119581, size = 17, normalized size = 0.94 \[ \frac{2}{7} (a+x (b+c x))^{7/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{2}{7} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974065, size = 19, normalized size = 1.06 \begin{align*} \frac{2}{7} \,{\left (c x^{2} + b x + a\right )}^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68967, size = 188, normalized size = 10.44 \begin{align*} \frac{2}{7} \,{\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{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 = 2.50157, size = 243, normalized size = 13.5 \begin{align*} \frac{2 a^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a^{2} b x \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a^{2} c x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a b^{2} x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{12 a b c x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a c^{2} x^{4} \sqrt{a + b x + c x^{2}}}{7} + \frac{2 b^{3} x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 b^{2} c x^{4} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 b c^{2} x^{5} \sqrt{a + b x + c x^{2}}}{7} + \frac{2 c^{3} x^{6} \sqrt{a + b x + c x^{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.49906, size = 146, normalized size = 8.11 \begin{align*} \frac{2}{7} \,{\left (a^{3} +{\left (3 \, a^{2} b +{\left ({\left ({\left ({\left (c^{3} x + 3 \, b c^{2}\right )} x + \frac{3 \,{\left (b^{2} c^{7} + a c^{8}\right )}}{c^{6}}\right )} x + \frac{b^{3} c^{6} + 6 \, a b c^{7}}{c^{6}}\right )} x + \frac{3 \,{\left (a b^{2} c^{6} + a^{2} c^{7}\right )}}{c^{6}}\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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