Optimal. Leaf size=19 \[ \frac{2}{7} d \left (a+b x+c x^2\right )^{7/2} \]
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Rubi [A] time = 0.0066538, 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} \[ \frac{2}{7} d \left (a+b x+c x^2\right )^{7/2} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{2}{7} d \left (a+b x+c x^2\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0127724, size = 18, normalized size = 0.95 \[ \frac{2}{7} d (a+x (b+c x))^{7/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 16, normalized size = 0.8 \begin{align*}{\frac{2\,d}{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 = 1.09422, size = 20, normalized size = 1.05 \begin{align*} \frac{2}{7} \,{\left (c x^{2} + b x + a\right )}^{\frac{7}{2}} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.8411, size = 207, normalized size = 10.89 \begin{align*} \frac{2}{7} \,{\left (c^{3} d x^{6} + 3 \, b c^{2} d x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} d x^{4} + 3 \, a^{2} b d x +{\left (b^{3} + 6 \, a b c\right )} d x^{3} + a^{3} d + 3 \,{\left (a b^{2} + a^{2} c\right )} d 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 = 4.38953, size = 260, normalized size = 13.68 \begin{align*} \frac{2 a^{3} d \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a^{2} b d x \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a^{2} c d x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a b^{2} d x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{12 a b c d x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a c^{2} d x^{4} \sqrt{a + b x + c x^{2}}}{7} + \frac{2 b^{3} d x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 b^{2} c d x^{4} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 b c^{2} d x^{5} \sqrt{a + b x + c x^{2}}}{7} + \frac{2 c^{3} d 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 = 1.19344, size = 161, normalized size = 8.47 \begin{align*} \frac{2}{7} \,{\left (a^{3} d +{\left (3 \, a^{2} b d +{\left ({\left ({\left ({\left (c^{3} d x + 3 \, b c^{2} d\right )} x + \frac{3 \,{\left (b^{2} c^{7} d + a c^{8} d\right )}}{c^{6}}\right )} x + \frac{b^{3} c^{6} d + 6 \, a b c^{7} d}{c^{6}}\right )} x + \frac{3 \,{\left (a b^{2} c^{6} d + a^{2} c^{7} d\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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