Optimal. Leaf size=18 \[ \frac {2}{5} \left (a+b x+c x^2\right )^{5/2} \]
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Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{5} \left (a+b x+c x^2\right )^{5/2} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} \frac {2}{5} (a+x (b+c x))^{5/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {2}{5} \left (a+b x+c x^2\right )^{5/2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 49, normalized size = 2.72 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 15, normalized size = 0.83 \begin {gather*} \frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.92, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.61, size = 136, normalized size = 7.56 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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