Optimal. Leaf size=34 \[ -\frac {2 a (a+b x)^{5/2}}{5 b^2}+\frac {2 (a+b x)^{7/2}}{7 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {2 (a+b x)^{7/2}}{7 b^2}-\frac {2 a (a+b x)^{5/2}}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x (a+b x)^{3/2} \, dx &=\int \left (-\frac {a (a+b x)^{3/2}}{b}+\frac {(a+b x)^{5/2}}{b}\right ) \, dx\\ &=-\frac {2 a (a+b x)^{5/2}}{5 b^2}+\frac {2 (a+b x)^{7/2}}{7 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.71 \begin {gather*} \frac {2 (a+b x)^{5/2} (-2 a+5 b x)}{35 b^2} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 1.99, size = 53, normalized size = 1.56 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (-2 a^3+a^2 b x+b^2 x^2 \left (8 a+5 b x\right )\right ) \sqrt {a+b x}}{35 b^2},b\text {!=}0\right \}\right \},\frac {a^{\frac {3}{2}} x^2}{2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.09, size = 26, normalized size = 0.76
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-5 b x +2 a \right )}{35 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{2}}\) | \(26\) |
trager | \(-\frac {2 \left (-5 b^{3} x^{3}-8 a \,b^{2} x^{2}-a^{2} b x +2 a^{3}\right ) \sqrt {b x +a}}{35 b^{2}}\) | \(43\) |
risch | \(-\frac {2 \left (-5 b^{3} x^{3}-8 a \,b^{2} x^{2}-a^{2} b x +2 a^{3}\right ) \sqrt {b x +a}}{35 b^{2}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}}}{7 \, b^{2}} - \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} a}{5 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 41, normalized size = 1.21 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} + a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x + a}}{35 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 80, normalized size = 2.35 \begin {gather*} \begin {cases} - \frac {4 a^{3} \sqrt {a + b x}}{35 b^{2}} + \frac {2 a^{2} x \sqrt {a + b x}}{35 b} + \frac {16 a x^{2} \sqrt {a + b x}}{35} + \frac {2 b x^{3} \sqrt {a + b x}}{7} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (26) = 52\).
time = 0.00, size = 185, normalized size = 5.44 \begin {gather*} \frac {\frac {2 b^{2} \left (\frac {1}{7} \sqrt {a+b x} \left (a+b x\right )^{3}-\frac {3}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a+\sqrt {a+b x} \left (a+b x\right ) a^{2}-\sqrt {a+b x} a^{3}\right )}{b^{3}}+\frac {4 a b \left (\frac {1}{5} \sqrt {a+b x} \left (a+b x\right )^{2}-\frac {2}{3} \sqrt {a+b x} \left (a+b x\right ) a+\sqrt {a+b x} a^{2}\right )}{b^{2}}+\frac {2 a^{2} \left (\frac {1}{3} \sqrt {a+b x} \left (a+b x\right )-a \sqrt {a+b x}\right )}{b}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 25, normalized size = 0.74 \begin {gather*} -\frac {14\,a\,{\left (a+b\,x\right )}^{5/2}-10\,{\left (a+b\,x\right )}^{7/2}}{35\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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