Optimal. Leaf size=47 \[ \frac {a^3 x^3}{3}+\frac {6}{7} a^2 b x^{7/2}+\frac {3}{4} a b^2 x^4+\frac {2}{9} b^3 x^{9/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} \frac {a^3 x^3}{3}+\frac {6}{7} a^2 b x^{7/2}+\frac {3}{4} a b^2 x^4+\frac {2}{9} b^3 x^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx &=2 \text {Subst}\left (\int x^5 (a+b x)^3 \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (a^3 x^5+3 a^2 b x^6+3 a b^2 x^7+b^3 x^8\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^3 x^3}{3}+\frac {6}{7} a^2 b x^{7/2}+\frac {3}{4} a b^2 x^4+\frac {2}{9} b^3 x^{9/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 0.91 \begin {gather*} \frac {1}{252} \left (84 a^3 x^3+216 a^2 b x^{7/2}+189 a b^2 x^4+56 b^3 x^{9/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 36, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {a^{3} x^{3}}{3}+\frac {6 a^{2} b \,x^{\frac {7}{2}}}{7}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\) | \(36\) |
default | \(\frac {a^{3} x^{3}}{3}+\frac {6 a^{2} b \,x^{\frac {7}{2}}}{7}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\) | \(36\) |
trager | \(\frac {a \left (9 b^{2} x^{3}+4 a^{2} x^{2}+9 b^{2} x^{2}+4 a^{2} x +9 b^{2} x +4 a^{2}+9 b^{2}\right ) \left (x -1\right )}{12}+\frac {2 b \,x^{\frac {7}{2}} \left (7 b^{2} x +27 a^{2}\right )}{63}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (35) = 70\).
time = 0.30, size = 98, normalized size = 2.09 \begin {gather*} \frac {2 \, {\left (b \sqrt {x} + a\right )}^{9}}{9 \, b^{6}} - \frac {5 \, {\left (b \sqrt {x} + a\right )}^{8} a}{4 \, b^{6}} + \frac {20 \, {\left (b \sqrt {x} + a\right )}^{7} a^{2}}{7 \, b^{6}} - \frac {10 \, {\left (b \sqrt {x} + a\right )}^{6} a^{3}}{3 \, b^{6}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{5} a^{4}}{b^{6}} - \frac {{\left (b \sqrt {x} + a\right )}^{4} a^{5}}{2 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 41, normalized size = 0.87 \begin {gather*} \frac {3}{4} \, a b^{2} x^{4} + \frac {1}{3} \, a^{3} x^{3} + \frac {2}{63} \, {\left (7 \, b^{3} x^{4} + 27 \, a^{2} b x^{3}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.81, size = 44, normalized size = 0.94 \begin {gather*} \frac {a^{3} x^{3}}{3} + \frac {6 a^{2} b x^{\frac {7}{2}}}{7} + \frac {3 a b^{2} x^{4}}{4} + \frac {2 b^{3} x^{\frac {9}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 35, normalized size = 0.74 \begin {gather*} \frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {3}{4} \, a b^{2} x^{4} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {1}{3} \, a^{3} x^{3} \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 = 0.74 \begin {gather*} \frac {a^3\,x^3}{3}+\frac {2\,b^3\,x^{9/2}}{9}+\frac {3\,a\,b^2\,x^4}{4}+\frac {6\,a^2\,b\,x^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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