Optimal. Leaf size=36 \[ \frac {2}{5} a^2 x^{5/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{9} b^2 x^{9/2} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ \frac {2}{5} a^2 x^{5/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{9} b^2 x^{9/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x^{3/2} (a+b x)^2 \, dx &=\int \left (a^2 x^{3/2}+2 a b x^{5/2}+b^2 x^{7/2}\right ) \, dx\\ &=\frac {2}{5} a^2 x^{5/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{9} b^2 x^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.78 \[ \frac {2}{315} x^{5/2} \left (63 a^2+90 a b x+35 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 29, normalized size = 0.81 \[ \frac {2}{315} \, {\left (35 \, b^{2} x^{4} + 90 \, a b x^{3} + 63 \, a^{2} x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 24, normalized size = 0.67 \[ \frac {2}{9} \, b^{2} x^{\frac {9}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.69 \[ \frac {2 \left (35 b^{2} x^{2}+90 a b x +63 a^{2}\right ) x^{\frac {5}{2}}}{315} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 24, normalized size = 0.67 \[ \frac {2}{9} \, b^{2} x^{\frac {9}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 0.67 \[ \frac {2\,x^{5/2}\,\left (63\,a^2+90\,a\,b\,x+35\,b^2\,x^2\right )}{315} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 34, normalized size = 0.94 \[ \frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {7}{2}}}{7} + \frac {2 b^{2} x^{\frac {9}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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