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 = {45}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 45
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 \begin {gather*} \frac {2}{315} x^{5/2} \left (63 a^2+90 a b x+35 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 25, normalized size = 0.69
method | result | size |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (35 x^{2} b^{2}+90 a b x +63 a^{2}\right )}{315}\) | \(25\) |
derivativedivides | \(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a b \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} x^{\frac {9}{2}}}{9}\) | \(25\) |
default | \(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a b \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} x^{\frac {9}{2}}}{9}\) | \(25\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (35 x^{2} b^{2}+90 a b x +63 a^{2}\right )}{315}\) | \(25\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (35 x^{2} b^{2}+90 a b x +63 a^{2}\right )}{315}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 0.67 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 29, normalized size = 0.81 \begin {gather*} \frac {2}{315} \, {\left (35 \, b^{2} x^{4} + 90 \, a b x^{3} + 63 \, a^{2} x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 34, normalized size = 0.94 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 24, normalized size = 0.67 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \frac {2\,x^{5/2}\,\left (63\,a^2+90\,a\,b\,x+35\,b^2\,x^2\right )}{315} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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