Optimal. Leaf size=51 \[ \frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 51, 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}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \sqrt {x} (a+b x)^3 \, dx &=\int \left (a^3 \sqrt {x}+3 a^2 b x^{3/2}+3 a b^2 x^{5/2}+b^3 x^{7/2}\right ) \, dx\\ &=\frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 39, normalized size = 0.76 \begin {gather*} \frac {2}{315} x^{3/2} \left (105 a^3+189 a^2 b x+135 a b^2 x^2+35 b^3 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 36, normalized size = 0.71
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) | \(36\) |
derivativedivides | \(\frac {2 a^{3} x^{\frac {3}{2}}}{3}+\frac {6 a^{2} b \,x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\) | \(36\) |
default | \(\frac {2 a^{3} x^{\frac {3}{2}}}{3}+\frac {6 a^{2} b \,x^{\frac {5}{2}}}{5}+\frac {6 a \,b^{2} x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\) | \(36\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) | \(36\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right )}{315}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.59, size = 38, normalized size = 0.75 \begin {gather*} \frac {2}{315} \, {\left (35 \, b^{3} x^{4} + 135 \, a b^{2} x^{3} + 189 \, a^{2} b x^{2} + 105 \, a^{3} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,a^3\,x^{3/2}}{3}+\frac {2\,b^3\,x^{9/2}}{9}+\frac {6\,a^2\,b\,x^{5/2}}{5}+\frac {6\,a\,b^2\,x^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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