Optimal. Leaf size=36 \[ \frac {2}{3} a^2 x^{3/2}+\frac {4}{5} a b x^{5/2}+\frac {2}{7} b^2 x^{7/2} \]
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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} \[ \frac {2}{3} a^2 x^{3/2}+\frac {4}{5} a b x^{5/2}+\frac {2}{7} b^2 x^{7/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \sqrt {x} (a+b x)^2 \, dx &=\int \left (a^2 \sqrt {x}+2 a b x^{3/2}+b^2 x^{5/2}\right ) \, dx\\ &=\frac {2}{3} a^2 x^{3/2}+\frac {4}{5} a b x^{5/2}+\frac {2}{7} b^2 x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.78 \[ \frac {2}{105} x^{3/2} \left (35 a^2+42 a b x+15 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 27, normalized size = 0.75 \[ \frac {2}{105} \, {\left (15 \, b^{2} x^{3} + 42 \, a b x^{2} + 35 \, a^{2} x\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 24, normalized size = 0.67 \[ \frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{5} \, a b x^{\frac {5}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \]
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 \[ \frac {2 \left (15 b^{2} x^{2}+42 a b x +35 a^{2}\right ) x^{\frac {3}{2}}}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 24, normalized size = 0.67 \[ \frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{5} \, a b x^{\frac {5}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \]
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 \[ \frac {2\,x^{3/2}\,\left (35\,a^2+42\,a\,b\,x+15\,b^2\,x^2\right )}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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