Optimal. Leaf size=53 \[ \frac {2 a^2 (a+b x)^{7/2}}{7 b^3}-\frac {4 a (a+b x)^{9/2}}{9 b^3}+\frac {2 (a+b x)^{11/2}}{11 b^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 53, 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 a^2 (a+b x)^{7/2}}{7 b^3}+\frac {2 (a+b x)^{11/2}}{11 b^3}-\frac {4 a (a+b x)^{9/2}}{9 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^2 (a+b x)^{5/2} \, dx &=\int \left (\frac {a^2 (a+b x)^{5/2}}{b^2}-\frac {2 a (a+b x)^{7/2}}{b^2}+\frac {(a+b x)^{9/2}}{b^2}\right ) \, dx\\ &=\frac {2 a^2 (a+b x)^{7/2}}{7 b^3}-\frac {4 a (a+b x)^{9/2}}{9 b^3}+\frac {2 (a+b x)^{11/2}}{11 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (8 a^2-28 a b x+63 b^2 x^2\right )}{693 b^3} \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 = 2.38, size = 76, normalized size = 1.43 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (8 a^5-4 a^4 b x+3 a^3 b^2 x^2+b^3 x^3 \left (113 a^2+161 a b x+63 b^2 x^2\right )\right ) \sqrt {a+b x}}{693 b^3},b\text {!=}0\right \}\right \},\frac {a^{\frac {5}{2}} x^3}{3}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.09, size = 38, normalized size = 0.72
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (63 x^{2} b^{2}-28 a b x +8 a^{2}\right )}{693 b^{3}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {4 a \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 a^{2} \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {4 a \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 a^{2} \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{3}}\) | \(38\) |
trager | \(\frac {2 \left (63 b^{5} x^{5}+161 a \,b^{4} x^{4}+113 a^{2} b^{3} x^{3}+3 a^{3} b^{2} x^{2}-4 a^{4} b x +8 a^{5}\right ) \sqrt {b x +a}}{693 b^{3}}\) | \(65\) |
risch | \(\frac {2 \left (63 b^{5} x^{5}+161 a \,b^{4} x^{4}+113 a^{2} b^{3} x^{3}+3 a^{3} b^{2} x^{2}-4 a^{4} b x +8 a^{5}\right ) \sqrt {b x +a}}{693 b^{3}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 41, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}}}{11 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {9}{2}} a}{9 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 64, normalized size = 1.21 \begin {gather*} \frac {2 \, {\left (63 \, b^{5} x^{5} + 161 \, a b^{4} x^{4} + 113 \, a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} - 4 \, a^{4} b x + 8 \, a^{5}\right )} \sqrt {b x + a}}{693 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 124, normalized size = 2.34 \begin {gather*} \begin {cases} \frac {16 a^{5} \sqrt {a + b x}}{693 b^{3}} - \frac {8 a^{4} x \sqrt {a + b x}}{693 b^{2}} + \frac {2 a^{3} x^{2} \sqrt {a + b x}}{231 b} + \frac {226 a^{2} x^{3} \sqrt {a + b x}}{693} + \frac {46 a b x^{4} \sqrt {a + b x}}{99} + \frac {2 b^{2} x^{5} \sqrt {a + b x}}{11} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{2}} x^{3}}{3} & \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. 233 vs.
\(2 (41) = 82\).
time = 0.00, size = 391, normalized size = 7.38 \begin {gather*} \frac {\frac {2 b^{3} \left (\frac {1}{11} \sqrt {a+b x} \left (a+b x\right )^{5}-\frac {5}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a+\frac {10}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{2}-2 \sqrt {a+b x} \left (a+b x\right )^{2} a^{3}+\frac {5}{3} \sqrt {a+b x} \left (a+b x\right ) a^{4}-\sqrt {a+b x} a^{5}\right )}{b^{5}}+\frac {6 a b^{2} \left (\frac {1}{9} \sqrt {a+b x} \left (a+b x\right )^{4}-\frac {4}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a+\frac {6}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{2}-\frac {4}{3} \sqrt {a+b x} \left (a+b x\right ) a^{3}+\sqrt {a+b x} a^{4}\right )}{b^{4}}+\frac {6 a^{2} b \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 {2 a^{3} \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}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {126\,{\left (a+b\,x\right )}^{11/2}-308\,a\,{\left (a+b\,x\right )}^{9/2}+198\,a^2\,{\left (a+b\,x\right )}^{7/2}}{693\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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